A Moly

Martel F T Costecalde

Annual BCI AWARDS 2020: 2 nd place

Je tiens ensuite à remercier de tout mon coeur le Docteur Tetiana AKSENOVA pour tous ses conseils qui ont permis de mener à terme les recherches de cette thèse et l'écriture du manuscrit. Je n'aurais pas pu réaliser tout ce travail sans sa patience, sa gentillesse et sa disponibilité durant les trois dernières années. Merci de m'avoir expliqué plusieurs dizaines de fois les mêmes notions mathématiques sans perdre votre calme. Merci également de votre efficacité pour la correction du manuscrit de thèse. Cette dernière année aura été un challenge que nous avons bien géré.

Merci à Guillaume CHARVET pour toutes ses astuces et stratégies pour améliorer mes présentations et toujours voir les plus petits détails qui échappent à tout le monde. Grâce à tes conseils je maitrîse toutes les arcanes de Power Point. Merci également à l'ensemble des personnes travaillant à CLINATEC avec qui j'aurai passé trois superbes années et qui m'auront soutenu tout au long de la thèse.

Je tiens bien évidemment à remercier toute ma famille, ma mère, mon père, ma soeur, mon beaufrère, et mes grands parents. A ma mère et mon père, merci de m'avoir toujours aidé dans tous mes problèmes et pour votre réconfort moral qui a été permanent durant ces trois longues années. Je n'aurais jamais pu faire cette thèse sans vous. A ma soeur Agathe, son futur mari Arnaud, merci pour votre bonne humeur permanante. Vous retrouvez été toujours un immense moment de bonheur et j'espère avoir plus de temps pour venir vous voir (je n'oublie pas Odin que je remercie aussi). Malgré tous les problèmes de cette fin d'année, je vous souhaite tous mes voeux de bonheur et il me tarde de vous retrouver au plus vite pour célébrer votre union.

A partir d'ici commence la longue liste des personnes que je dois remercier pour m'avoir soutenu d'une façon ou d'une autre durant cette longue aventure. Merci Julie et Félix, j'ai passé beaucoup de temps avec vous, nous avons beaucoup ri durant ces trois années et je me souviendrai toujours de toutes nos soirées jeux et de nos lundis films.

Antoine et Maud, merci pour votre bonne humeur et les nombreuses discussions que nous avons eu (merci aussi pour l'aide au déménagement, il me tarde de venir voir votre belle maison).

Vincent, nous avons passé beaucoup de temps à discuter, que ce soit pour le travail ou dans tout autre domaine, nos conversations ont toujours été passionantes et ton humour m'a toujours remonté le moral (nous continuerons de parler manga, jeux vidéo, etc.).

Tamara, merci pour tous ces bons moments, ma deuxième année aurait été bien différente sans toi (il me tarde de pouvoir râler et critiquer encore avec toi). Courage pour la thèse, je sais que tu feras du très bon travail comme toujours.

Lucas, merci d'avoir été là durant cette deuxième année de thèse, nous aurons bien ri et parlé de mangas. Nous nous ressemblons beaucoup, il est très pratique d'avoir quelqu'un qui soit toujours du même avis que soi. Il me tarde de te revoir au plus vite.

Marion, que dire Marion, d'une simple rencontre à CLINATEC s'est créée une vrai amitié, merci pour tous tes conseils et ta gentillesse constante. Je ne pense pas connaître de personne plus douce IV Acknowledgements et gentille. Tous mes voeux de bonheur et il me tarde d'être en 2021 pour fêter cet heureux évènement.

Coline, nous aurons bien bavardé durant ces deux premières années dans notre carré magique avec toi et Clémentine. C'est toi qui m'a accueilli dans les locaux et qui m'a intégré à l'équipe. Un membre fondateur des « jeunes de CLINATEC », il me tarde de te revoir et que tu me racontes toutes tes histoires d'école vétérinaire.

Maxime et Guillaume, vous étiez là pendant 6 mois pour vos stages de fin d'étude, merci pour votre bonne humeur communicative qui me motivait tous les jours pour aller travailler.

A tous mes amis de l'escalade, sans vous, j'aurai sûrement perdu la raison. Partiquer le sport avec vous m'a permis d'évacuer bien du stress. Merci donc à Manon, Meije, Gaspard, Anto, Grand Guillaume et bien d'autres.

Alexia, Tamara, Marie, Camille et Benjamin, cela fait maintenant plus de huit ans que l'on se connait et que l'on fait les quatre cent coups ensemble. Je tiens à vous remercier car ma vie avant et pendant le doctorat aurait été bien différente sans vous. Il me tarde de vous revoir et de partager de bons moments avec vous.

Le groupe des « Grenoblois », (Anaïs, Agathe, Alice, Estelle, Alessia, Damien, Paul, Jeremy, Jordan, Vincent, Loïs, etc.), j'imagine qu'ils se reconnaitront tous et toutes, merci de m'avoir si bien intégré dans votre groupe depuis un an, j'ai passé des moments inoubliables avec vous et il me tarde déjà d'être à nos prochains repas/ week-end/ soirées tous ensemble.

Enfin, les meilleurs pour la fin…

Merci Gaël pour beaucoup trop de choses pour être mentionnées ici. Merci pour ta gentillesse, ta rigueur, ton attention, ta bonne humeur permanente, etc. Tu m'as toujours poussé à faire mieux et tu as toujours été présent durant la thèse pour me faire à la fois décompresser et mieux réfléchir à mon sujet. J'espère pouvoir encore partager beaucoup de bons moments avec toi. Nous allons bien rire durant les prochaines années ! Merci Clémentine, pour ces trois années passées dos à dos, ces trois années à se plaindre, ces trois années à se supporter jour après jour afin de réussir à boucler nos thèses. Il me tarde déjà d'assister à ta soutenance qui, je le sais, sera incroyable. Tu m'as aidé tellement de fois dans ma vie personnelle et professionnelle que j'ai arrêté de tenir les comptes. Il me tarde de passer encore de nombreuses années à discuter et me plaindre de tout avec toi. Que tu le veuilles ou non, malheuresement, tu devras faire avec moi en tant qu'ami durant tes années futures Enfin, Gwen, merci pour cette dernière année. L'année 2020 aura été bien étrange sur de nombreux aspects, mais tu es restée à mes cotés durant cette période difficile. Malgré mon mauvais caractère et mon envie permanente de dormir et faire la sieste, nous avons beaucoup ri, nous nous sommes plaints sur de nombreux sujets mais cette dernière année restera inoubliable grâce à toi. Merci du fond du coeur, j'ai hâte de voir ce que l'avenir nous réserve. Plus de chats ?

VII

Résumé

Chapitre 1 : Introduction Les Interfaces Cerveau-Machine (ICM), plus connues sous le nom anglais de « Brain Computer Interface », ou BCI, sont des systèmes permettant de traduire l'activité cérébrale d'un individu grâce à un ordinateur afin d'effectuer des tâches nécessitant normalement une action des nerfs périphériques et/ou musculaires. L'essor des ICMs a permis de créer de nombreuses applications dans divers domaines comme la communication, la neuro-réhabilitation ou neuro-rééducation, le domaine du jeux vidéo, etc. Cependant, le domaine des ICMs a particulièrement été développé dans le but d'aider des patients souffrant de handicaps moteurs sévères. Les disfonctionnements moteurs peuvent être une des conséquences d'accidents comme une lésion de la moelle épinière, un accident vasculaire cérébral (AVC), ou encore la conséquence de maladies neurodégénératives ou neuromusculaires comme une sclérose latérale amyotrophique, ou une Myopathie de Duchenne, etc. Les ICMs dont le but est de palier, remplacer ou aider à la rééducation fonctionnelle de patients souffrant d'un handicap moteur sont appelés ICM motrices. Avec le développement de l'intelligence artificielle, l'augmentation de la puissance de calcul des ordinateurs et considérant les bénéfices hypothétiques pour les patients handicapés, la recherche sur les ICMs est devenue un domaine suscitant beaucoup d'espoir. Ce champ de recherche, et les applications qui en découlent, proposent de nombreux challenges à relever dans de nombreux domaines (en électronique, biologie, mathématiques, mécanique, etc.).

Les éléments composant une ICMs.

Une ICM permet l'interaction d'un patient avec son environnement grâce au contrôle d'un effecteur dont les actions sont régies par l'activité cérébrale du sujet. Une ICM est composée de quatre éléments principaux.

• Système d'acquisition.

Premièrement, une ICM comprend un système d'acquisition dont le but est d'enregistrer l'activité cérébrale du sujet. De nombreux systèmes d'enregistrement ont été développés au fil du temps avec pour objectif premier d'améliorer le ratio entre la qualité du signal et l'invasivité du système. Un large spectre de système d'acquisition existe allant du système très invasif enregistrant l'activité cérébrale proche de la source des signaux (les neurones) au système non-invasif recueillant l'activité globale du cerveau.

Les matrices de microélectrodes ou « Microelectrode array » (MEA) en anglais regroupent les systèmes les plus invasifs et les plus précis utilisés à ce jour pour l'acquisition de signaux neuronaux. Les MEAs enregistrent directement le milieu extracellulaire proche des neurones dans une zone restreinte du cerveau. A titre d'exemple, les MEAs les plus communes dans le domaine des ICMs sont les matrices VIII Résumé Utah qui sont composées d'une centaine d'électrodes réparties sur quelques mm 2 . Les électrodes font entre 0.5mm et 1.5mm de long, 400µm de large et sont espacées d'environ 0.4mm. Bien que les MEAs permettent l'acquisition des signaux cérébraux avec une grande qualité et un très bon rapport signal-sur-bruit, une dégradation de la qualité des enregistrements au cours du temps a été signalée dans de nombreux articles. En ne considérant pas les quelques exemples de recherches utilisant des MEAs plus de 1000 jours après implantation, les MEAs génèrent une réaction immunitaire dégradant les signaux. A ce jour, les MEAs sont limitées à des enregistrements filaires. Cette limitation est problématique dans le cas d'enregistrements chroniques et dans le cas d'une utilisation quotidienne de par l'augmentation du risque d'infection en comparaison avec des systèmes sans fil. Les expériences d'ICMs cliniques sur une longue période de temps utilisant des MEAs sont donc peu nombreuses et souvent limitées aux contrôles d'effecteurs avec un faible nombre de degrés de liberté.

L'électroencéphalographie (EEG) est une méthode d'acquisition des signaux cérébraux à l'autre bout du spectre des possibles par rapport aux MEAs. L'EEG est un système d'acquisition non-invasif utilisant des macro-électrodes (entre 64 et 256) positionnées à la surface du crâne. Un signal EEG enregistre l'activité de millions de neurones distribués sur une grande région du cerveau (10 cm 2 ou plus) à la surface du crâne. De par sa simplicité d'utilisation, son cout et son risque zéro pour le patient, l'EEG est le système d'acquisition le plus commun dans le domaine des ICMs. Cependant, ce système d'acquisition comporte de nombreuses limites en comparaison aux systèmes d'acquisition plus invasifs. La résolution spatiale est limitée tandis que les bandes de fréquences exploitables sont cantonnées entre 0Hz et 100Hz. De plus, les signaux EEGs sont bruités et très sensibles aux artefacts provenant de plusieurs sources (mouvements des yeux, activités musculaires, rythme cardiaque, lignes électriques à 50Hz,etc.). Enfin, la mise en place du casque EEG est une tâche complexe, technique et couteuse en temps qui requiert une personne qualifiée afin d'obtenir des signaux exploitables. De plus, il est très difficile (voire impossible) de replacer exactement au même endroit les électrodes entre deux expériences, ce qui augmente encore la variabilité des signaux cérébraux enregistrés entre deux expériences.

L'électrocorticographie (ECoG) est un système d'acquisition se situant entre l'EEG et les MEA. L'ECoG est un système composé de macro-électrodes (avec un diamètre de l'ordre du mm) réparties sur une grille implantée au-dessus (on parlera de ECoG épidurale) ou en dessous (on parlera de ECoG subdurale) de la Dure-mère. Tout comme l'EEG, les ECoG enregistrent la somme des signaux cérébraux de milliers de neurones. Cependant les ECoGs ont démontré une bien meilleure qualité de signaux, une plus grande bande fréquence exploitable et moins d'artéfacts que les enregistrements EEG. Bien que les systèmes ECoGs n'enregistrent pas directement l'activité neuronale comme les MEAs, ils sont beaucoup moins invasifs et ont montré une stabilité d'enregistrement dans le temps plus importante.

Résumé IX

• Le décodage des signaux cérébraux.

Après avoir été enregistrés, les signaux cérébraux doivent être traduits en commandes pour contrôler un effecteur. Cette étape de traduction des signaux cérébraux se déroule en quatre phases, le pré-processing, l'extraction des caractéristiques, le décodage et le post-processing. Durant la phase de pré-processing, les signaux cérébraux sont filtrés et « nettoyés » afin d'améliorer le rapport signal-sur-bruit des signaux en enlevant les éventuels artéfacts par exemple. L'extraction des caractéristiques a pour but de calculer, à partir des signaux cérébraux bruts, les informations utiles au décodage. L'étape de décodage, comme son nom l'indique, analyse les caractéristiques précédemment calculées afin de générer une commande pour l'effecteur. Les décodeurs peuvent générer des commandes discrètes (un état mental, une action binaire, etc.) ou continues (déplacer un curseur dans l'espace, bouger un bras robotique, etc.). De nombreux algorithmes ont été développés pour obtenir le décodage des signaux cérébraux le plus précis possible, mais à ce jour, aucun consensus sur la meilleure approche n'a été trouvé. Enfin, le post-processing traite la commande générée à l'étape de décodage afin d'y appliquer des contraintes supplémentaires ou d'améliorer la prédiction. L'ajout d'un seuil d'activation, de transition entre différents états ou l'application d'une moyenne glissante sur les prédictions du décodeur sont des exemples courants de post-processing.

• Le contrôle d'effecteurs.

Les commandes générées sont réalisées par un effecteur. De nombreux effecteurs de tous types ont été utilisés dans le domaine des ICMs. Ils peuvent être classés dans deux catégories : les effecteurs virtuels comme les curseurs d'ordinateurs, les avatars virtuels, les jeux vidéos, les systèmes de communications, etc. et les effecteurs réels regroupant tous les systèmes impliquant une interaction directe entre l'environnement et le sujet comme les bras robotiques, les orthèses, les prothèses, les fauteuils roulants, les exosquelettes, etc.

• Le retour sensoriel (feedback).

Enfin, une étape essentielle des ICMs est le retour sensoriel (feedback) suscité par les actions de l'effecteur sur le patient. Dans la plupart des ICMs controlées par des patients handicapés, le retour sensoriel généré par leurs actions est visuel. Cette étape est très importante car le retour sensoriel influe directement sur l'activité cérébrale du sujet. On parle de système en boucle fermé (closed-loop). Quelques exemples d'expériences ICMs intégrant des retours haptiques ont été menées avec des patients handicapés.

Stratégie de contrôle d'une ICM.

Afin de contrôler une ICM, plusieurs stratégies ont été mises en place. Les stratégies de contrôle peuvent se diviser en deux catégories : le contrôle exogène et endogène. Les systèmes exogènes utilisent des stimuli extérieurs (auditifs, visuels, etc.) afin de générer une modulation des signaux cérébraux du patient facilement reconnaissable et interprétable. A l'inverse, les systèmes endogènes permettent le contrôle d'effecteurs via X Résumé les variations des signaux cérébraux directement induites par le patient sans stimuli extérieurs. Ce manuscrit se concentre plus particulièrement sur les ICMs endogènes. Les stratégies endogènes se décomposent en deux approches : le « remapping » somatotopique et le décodage neural direct.

Le « remapping » somatotopique, ou imagerie motrice (IM) associe une commande spécifique de l'effecteur (bouger un curseur vers le haut, la gauche, activer la marche d'un robot, etc.) a une action réelle ou imaginaire arbitrairement choisie comme bouger la langue, le coude, l'épaule, les jambes, les doigts, etc. Chaque action réelle ou imaginaire génère une activité cérébrale différente qui peut être liée artificiellement à une commande spécifique de l'effecteur.

Le décodage neural direct ou biomimétique utilise l'activité cérébrale enregistrée lors de la réalisation d'un mouvement (ou de l'imagination de celui-ci) pour générer un ordre de l'effecteur directement lié aux signaux cérébraux. Comme son nom l'indique, le décodage neural direct traduit directement les intentions du patient en action.

A titre d'exemple, considérons une expérience où un patient a pour tâche de contrôler un bras robotique dans le but d'atteindre une cible située en hauteur. Une stratégie de type IM non-direct associe cette tâche à une action du patient (imaginée ou non) sans lien avec la tâche à accomplir (bouger son épaule, son coude, sa langue, etc.). Dans le cas d'un décodage neural direct, afin de bouger le bras robotique vers le haut, le patient devra simplement imaginer bouger son bras vers le haut.

Bien que la stratégie de contrôle neural direct soit bien plus complexe à décoder, elle présente de nombreux avantages pour le patient. La stratégie de décodage neural direct est plus naturelle et demande une charge mentale moins importante pour le patient afin de réaliser des tâches complexes. De plus, la stratégie IM est limitée dans le nombre possible d'actions réalisables dans la même expérience : après avoir associé les jambes, les poignets, les coudes, etc. à une tâche spécifique, il devient difficile d'ajouter d'autres degrés de liberté. Cependant, à ce jour, le décodage en temps réel des mouvements d'un effecteur durant des expériences cliniques utilisant la stratégie de décodage neural direct n'a été réalisé qu'avec le système d'acquisition MEA.

ICM pour une utilisation dans la vie quotidienne.

Le but final des ICMs est de créer des systèmes utilisables par des personnes handicapées dans la vie quotidienne afin d'améliorer leur niveau de vie et leur autonomie. Dans cette optique, les ICMs doivent répondre à de nombreuses exigences.

Pour une application quotidienne, le système d'enregistrement doit permettre un enregistrement chronique et stable des signaux cérébraux. Bien que quelques études aient démontré la stabilité des implants de type MEA sur plusieurs centaines de jours, l'enregistrement direct de l'activité extracellulaire a montré une grande variabilité dans la stabilité des implants (dégradation au cours du temps) et une grande variabilité dans Résumé XI l'activité des neurones. De plus, les MEAs sont pour l'instant limitées à des enregistrements filaires. Les enregistrements ECoG ont montré de meilleures performances avec de nombreuses études démontrant la qualité et la stabilité des enregistrements plusieurs années après implantation. Bien que les systèmes de type EEG soientt non-invasif, la vulnérabilité aux artefacts, la grande variabilité des enregistrements et la nécessité de replacer au même endroit les électrodes à chaque utilisation ne font pas des EEGs de bons candidats pour une utilisation quotidienne.

Il est nécessaire que l'ICM créée soit suffisamment précise et offre assez de liberté à l'utilisateur. A ce jour, les ICMs ayant démontrées un grand nombre de degrés de liberté sont celles basées sur les systèmes d'acquisition MEAs. Un patient tétraplégique a pu contrôler, grâce à une MEA implantée dans le cortex moteur, un bras robotique avec 10 degrés de liberté (déplacement du bras dans l'espace 3D, rotation 3D du bras ainsi que 4 positions de main différentes). Pour les systèmes moins invasifs, plusieurs études ont démontré le contrôle 3D de curseurs et bras robotiques à partir d'ICM utilisant des enregistrements ECoGs, tandis que le décodage de signaux EEG se limite généralement à du contrôle 1D ou 2D.

Les actions de la vie courante nécessitent souvent l'utilisation simultanée ou séquentielle de plusieurs membres comme la réalisation de mouvements bimanuels. Le décodage multi-membres est un domaine très peu étudié dans le domaine des ICMs bien que la possibilité de réaliser des mouvements simultanés ou alternatifs de plusieurs membres puisse être un grand avantage pour les personnes handicapées. La plupart des expériences d'ICM se concentrent sur le décodage d'un membre en particulier ou une action spécifique d'un effecteur. Le peu d'expériences multi-membres réalisées se limite à des études précliniques sur singe ou des expériences cliniques de décodage des mouvements des doigts.

L'un des critères principaux du développement des ICMs est la capacité à proposer aux utilisateurs un contrôle asynchrone sur l'effecteur. Les systèmes synchrones donnent au sujet le contrôle d'un effecteur dans des fenêtres de temps limitées périodiquement proposées par un opérateur afin d'activer ou désactiver le contrôle de l'ICMs. A l'inverse, un ICM asynchrone est disponible en permanence. Il est capable de décoder un état mental de repos nommé « idle state » (IS) dans ce manuscrit. Lorsque l'état de repos est détecté, aucune action n'est réalisée par l'effecteur.

La plupart des expériences de l'état de l'art sont des ICMs synchrones. Ceci est d'autant plus vrai dans les expériences dont le but est le contrôle de nombreux degrés de liberté comme par exemple, l'étude citée précédemment où un patient utilisait un bras robotique avec 10 degrés de liberté. Un contrôle asynchrone est obligatoire dans le cas d'ICM pour la vie courante et d'autant plus dans le cas de contrôle d'effecteurs multimembres.

XII

Résumé

Un ICM utilisable quotidiennement doit décoder les signaux en temps réel afin de limiter la latence entre l'action désirée par le sujet et la réalisation de celle-ci par l'effecteur (~10-20Hz). La majorité des études d'ICMs sont pratiquées sur des jeux de données en offline. Dans ces conditions, le temps et la puissance de calcul nécessaires pour décoder les signaux cérébraux ne sont pas des facteurs limitants. Pour répondre aux besoins d'une utilisation en temps réel, les algorithmes de décodage doivent être suffisamment simples et optimisés pour nécessiter un temps de calcul très faible.

En plus d'être utilisable en temps réel, l'algorithme de décodage doit garder des performances stables au cours du temps. Les signaux cérébraux sont non-stationnaires, ils changent au cours du temps et ne génèrent pas les mêmes variations d'amplitudes au cours du temps pour la même action. Cette évolution des signaux cérébraux au cours du temps engendre une baisse des performances des modèles de décodage et requiert donc de recalibrer les modèles. Cependant une étape de recalibration quotidienne des modèles de décodage (voir plus) n'est pas envisageable dans le cas d'utilisation d'ICM dans la vie courante. Ce sont des évènements complexes qui peuvent s'avérer fatiguants pour le patient et qui doivent rester exceptionnels. Des stratégies pour mettre à jour les modèles facilement doivent être considérées pour une utilisation quotidienne d'un ICM.

Enfin, comme mentionné précédemment, le retour visuel (feedback) lié à l'utilisation d'un effecteur modifie l'activité cérébrale de l'utilisateur. Plusieurs études ont démontré que l'intégration dans la phase de calibration des signaux cérébraux générés par le retour visuel peut améliorer grandement les performances de décodage durant des expériences en boucle fermée (closed-loop). Plusieurs stratégies ont été développées et sont regroupées sous le nom d'adaptation du décodeur en boucle fermée : (« Closed-Loop Decoder Adaptation », CLDA, en anglais). Ces stratégies mettent à jour le modèle en prenant en compte les feedbacks du patient grâce à des procédures de calibration offlines ou en temps réel. Les algorithmes intégrant une procédure de CLDA ont montré de meilleures performances et une meilleure stabilité au cours du temps lors d'expériences d'ICM en temps réel. L'intégration de procédures similaires semble nécessaire pour garantir la stabilité et les performances de tout ICM dédiée à la vie quotidienne.

Recherche de doctorat.

Les recherches de thèses présentées dans ce manuscrit ont été menées dans le but de créer de nouveaux algorithmes de décodage répondant aux critères des ICMs précédemment mentionnés. Les algorithmes de cette thèse auront donc pour but d'être utilisables en temps réel, d'être asynchrones, d'être utilisables pour le contrôle multimembres et d'être capables d'être facilement mis à jour.

Ce doctorat est mené dans le cadre du projet « BCI and Tétraplégie» de CLINATEC. CLINATEC est un centre de recherche biomédical situé sur le site du CEA-Grenoble en collaboration avec l'Université Grenoble Alpes (UGA) et le Centre Hospitalier Résumé XIII Universitaire de Grenoble Alpes (CHUGA). Le but de l'essai clinique est de permettre à des patients tétraplégiques de contrôler des effecteurs complexes tel qu'un exosquelette quatre membres via un décodage neural direct de signaux cérébraux chroniques enregistrés grâce à deux implants sans fil ECoG épiduraux nommés WIMAGINE. L'ensemble des études et expériences menées dans le cadre de la thèse reposeront sur des enregistrements ECoG uniques, et viseront à répondre aux besoins de l'essai clinique et à permettre au patient de contrôler les différents membres de l'exosquelette.

Chapitre 2 : L'essai clinique « BCI et Tétraplégie » L'essai clinique de CLINATEC « BCI et Tétraplégie » a pour but de faire la preuve de concept qu'un patient tétraplégique implanté avec des électrodes ECoG épidurales peut contrôler un effecteur complexe tel qu'un exosquelette en utilisant un décodage neural direct. L'essai clinique toujours en cours a été référencé dans le registre ClinicalTrials.gov sous l'identifiant NCT02550522 le 11/09/2015. Cet essai clinique a été approuvé par l'Agence Nationale de Sécurité du Médicament et des produits de santé (ANSM) sous l'identifiant 2015-A00650-49 et le Comité de Protection des Personnes (CPP) sous le nom 15-CHUG-19.

Entre le début de l'essai clinique et juin 2020, trois patients ont été inclus dans le protocole clinique. Le premier patient a été implanté en mai 2016. Au fait d'un problème technique lié aux implants, le patient a été explanté et retiré de l'essai clinique. Le deuxième patient a été implanté en Juin 2017. Un troisième patient a été inclu dans l'essai clinique fin 2019. Les résultats décrits dans cette thèse ne se concentrent que sur les analyses, les études et les expériences menées avec le deuxième patient. Ce patient tétraplégique a subi une lésion complète de la moelle épinière au niveau des vertèbres C4-C5. Son score ASIA (American Spinal Injury Association Impairment) a été évalué à 4 et 5 pour le côté droit et gauche du corps tandis que la contraction des extenseurs du poignet gauche (droit) a reçu la note de 3 (0). Tous les autres muscles ont été évalués à un score ASIA de 0.

Durant l'essai clinique, le deuxième patient a réalisé plusieurs expériences avec différents effecteurs réels et virtuels (exosquelette, fauteuil roulant, avatar virtuel, jeux vidéos, etc.). Les résultats obtenus dans cette thèse se concentrent uniquement sur les performances du patient à contrôler l'exosquelette et l'avatar virtuel. L'avatar virtuel est une reproduction virtuelle de l'exosquelette permettant au patient de simuler le contrôle de l'exosquelette à domicile.

Le patient a été entrainé à contrôler graduellement des effecteurs de plus en plus complexes. L'exosquelette et l'avatar virtuel proposent 12 degrés de liberté : le contrôle continu 3D des mouvements de la main gauche et droite, le contrôle continu 1D de la rotation des poignets du bras gauche et droit, un système de préhension 1D discret de la main gauche et droite, la marche et l'état de repos. Le patient a d'abord appris à contrôler XIV Résumé individuellement chaque degré de liberté avant d'essayer des expériences plus complexes où les paradigmes expérimentaux ont été combinés. Dans chaque expérience, il est demandé au patient d'atteindre des cibles dépendantes du paradigme expérimental utilisé. Par exemple, dans le cas du mouvement 3D de la main gauche, il est demandé au patient d'atteindre une cible avec la main gauche. Lors d'une tâche de rotation du poignet, le patient doit faire tourner le poignet de l'exosquelette jusqu'à une certaine limite.

Une expérience d'ICM en temps réel se déroule en deux phases. Premièrement une phase optionnelle de calibration du modèle dans le cas où le modèle est initialisé à zéro ou si le modèle doit être mis à jour. Durant la phase de calibration, les signaux cérébraux sont labélisés afin de mettre à jour le modèle en temps réel. Durant la seconde phase, le modèle est fixé et des tâches similaires aux tâches proposées durant la phase de calibration sont effectuées afin d'évaluer les performances du modèle.

Chapitre 3 : Décodeur de Signaux Cérébraux

Les types de décodeurs

De nombreux algorithmes de décodage ont été développés et utilisés dans la communauté des ICMs dans le but de décoder le plus précisément possible les signaux cérébraux. Les algorithmes de décodage peuvent se diviser en deux catégories, les algorithmes dont le but est de classer l'activité cérébrale dans des états discrets (classifieurs) et ceux dont le but est de traduire l'activité cérébrale par une variable continue (modèle de régression ou filtre bayésiens). La plupart des études évaluant les performances de classifieur sont réalisées via des enregistrements de type EEG tandis que le décodage de variables continues est le domaine de prédilection des MEAs. A ce jour, il n'existe aucun consensus sur les meilleurs algorithmes à utiliser pour le décodage des signaux cérébraux car leurs performances sont très variables selon la tâche à effectuer, le type d'enregistrement etc.

En général, les modèles de régression et les filtres bayésiens échouent à fournir des estimations neutres (associées à une prédiction en vitesse nulle) qui ne génèrent aucun mouvement de l'effecteur. Cette caractéristique des modèles de prédictions continues est très problématique dans le cadre de la création d'un ICM asynchrone qui doit être capable de fournir un état de repos fiable à l'utilisateur.

Pour résoudre ce problème, deux stratégies ont émergé, à savoir : la détection de l'état de repos par post-traitement, et l'intégration directe de la détection de l'état de repos dans l'algorithme de décodage. Dans le deuxième cas, la détection de l'état de repos et souvent synonyme d'introduction de non-linéarités dans le décodeur. Pour ce faire, des algorithmes dit hybrides utilisant un algorithme de classification en parallèle d'un ou de plusieurs modèles de régressions ont été testés. Le but des algorithmes hybrides est Résumé XV d'utiliser la variable latente discrète décodée par le modèle de classification afin d'introduire un comportement non-linéaire dans le décodeur de variables continues. Dans le cas de la détection de l'état de repos, le classifieur est en mesure d'inhiber le décodeur de variables continues afin de générer une commande de contrôle neutre (ex. vitesse de prédiction nulle).

La mise à jour des modèles

Afin de prendre en compte les variations des signaux cérébraux au cours du temps et les signaux cérébraux liés aux retours sensitifs du patient lors de l'utilisation d'un effecteur, il est nécessaire d'utiliser des algorithmes intégrant une procédure CLDA. Les procédures CLDA existantes peuvent être classées selon la fréquence de mise à jour du modèle.

Le modèle peut être mis à jour à la fin de chaque session, à chaque essai, toutes les 15 minutes etc. Généralement, dans le cas où le temps entre deux mises à jour est assez long, la procédure se déroule en offline. A l'inverse, la mise à jour peut être opérée dans des intervalles de temps beaucoup plus courts de l'ordre de la minute, de quelques secondes voir moins. Dans ce cas, la procédure est généralement réalisée en temps réel durant l'expérience. Les procédures CLDA en temps réel utilisent souvent des algorithmes adaptatifs dits incrémentaux. Un algorithme incrémental modifie le modèle de décodage en se basant sur les nouvelles données enregistrées en temps réel au cours de l'expérience et le modèle précédemment calibré.

Les algorithmes adaptatifs incrémentaux ont été utilisés dans plusieurs études cliniques et précliniques dans le but de mettre à jour en temps réel les modèles de décodage. Les algorithmes adaptatifs incrémentaux ont montré des résultats prometteurs pour contrer la variabilité naturelle des signaux cérébraux et pour intégrer directement le patient dans le processus de calibration du modèle. Le patient apprend du modèle tout autant que le modèle apprend du patient.

La plupart des classifieurs adaptatifs incrémentaux ont été testés dans des expériences utilisant des enregistrements de type EEGs. Ils sont basés sur des algorithmes bien connus dans le domaine du machine learning mais adaptés afin d'intégrer une mise à jour incrémentale. Parmi eux, nous pouvons citer l'« incremental adaptive linear discriminant analysis (LDA)», l'« incremental adaptive Kalman-LDA», le « adaptive information matrix (ADIM) » ou encore le « incremental Support Vector Machine (ISVM) »etc.

A l'inverse, les algorithmes incrémentaux de décodage de variables continues ont surtout été étudiés en se basant sur l'analyse de signaux cérébraux enregistrés via MEAs. Les algorithmes adaptatifs de ce type les plus connus font partie de la famille des filtres de Kalman. Un seul algorithme de décodage de variables continues a été créé dans le cadre de décodage de signaux ECoG. Cet algorithme incrémental se nomme « Recursive Exponentially Weighted N-way Partial Least Square (REW-NPLS) ». A notre XVI Résumé connaissance, aucun algorithme incrémental adaptatif hybride n'a été appliqué dans le domaine des ICMs.

Chapitre 4 : Décodeur REW-MSLM

Le but de cette thèse est de proposer un algorithme incrémental adaptatif permettant le contrôle asynchrone en temps réel d'un effecteur multi-membres. Cette thèse propose donc un nouvel algorithme nommé « Recursive Eponentially Weighted Markov Switching multi-Linear Model (REW-MSLM)» inspiré des algorithmes « Recursive Exponentially Weighted N-way Partial Least Square (REW-NPLS) » et « Markov Switching Linear Model (MSLM) ». L'algorithme REW-MSLM s'inspire de la structure et de l'algorithme de classification dynamique utilisé dans MSLM.

REW-MSLM est un modèle se basant sur une architecture de « Mixtures d'Experts (ME)». Le principe des ME est d'estimer les prédictions de plusieurs décodeurs de variables continues (régression) que l'on nomme « experts ». La prédiction de chaque expert est alors pondérée (inhibée ou stimulée) par un poids reflétant la probabilité que chaque expert a d'être actif. Le modèle estimant le poids de chaque expert est un classifieur discret appelé « gate ». La valeur de cette variable discrète latente (le poids de chaque expert) est directement déduite des signaux cérébraux et de leurs distributions. Tout comme le MSLM, le REW-MSLM réalise une estimation dynamique de la probabilité d'activation des experts (la valeur de la variable latente discrète). Dans l'algorithme REW-MSLM, la séquence de probabilité des états latents est générée par une chaîne de Markov d'ordre 1. Plus précisément, les probabilités sont estimées par un Modèle de Markov Caché nommé « Hidden Markov Model (HMM) » en anglais. Intégrer un classifieur dynamique de type HMM dans un modèle de type ME a pour but de réduire le nombre de fausses détections des états actifs et de l'état de repos.

REW-MSLM regroupe donc plusieurs sous-modèles : les experts et la gate. Chaque sousmodèle doit être calibré afin de traduire les signaux cérébraux en commande pour l'effecteur. La calibration de chaque expert et de la gate est réalisée individuellement en utilisant une procédure d'apprentissage supervisé. L'apprentissage supervisé de chaque sous-modèle est effectué grâce à l'algorithme incrémental adaptatif REW-NPLS. REW-NPLS est une évolution de l'algorithme « Partial Least Square (PLS) » connu pour sa stabilité dans le traitement de données à grandes dimensions. REW-NPLS traite les données sous forme tensorielle et effectue une estimation des paramètres d'un modèle multilinéaires en temps réel de façon incrémentale. En plus de l'évaluation du modèle en temps réel, une procédure nommée Validation Récursive ou « Recursive Validation (RV) » est intégrée dans l'algorithme afin de déterminer en temps réel l'hyperparamètre de l'algorithme REW-NPLS. L'hyperparamètre de REW-NPLS noté 𝑓 est la dimension de l'espace des variables latentes dans lequel sont projetées les données afin d'estimer le modèle de décodage. Résumé XVII La combinaison de l'algorithme MSLM et REW-NPLS permet la création d'un modèle de type ME applicable en temps réel avec des sous-modèles recalibrés de manière incrémentale en temps réel. REW-MSLM présente de nombreux avantages pour contrôler un effecteur asynchrone multi-membres en temps réel.

Premièrement, chaque expert peut être associé à un membre ou groupe de degrés de liberté. Chaque expert est alors calibré individuellement pour réaliser une action spécifique : aucun expert ne doit apprendre tous les degrés de liberté. C'est une création de modèle « par morceaux » où chaque expert à son domaine de prédiction. Dans le cas d'application de ce manuscrit, chaque expert est associé à un mouvement spécifique de l'effecteur comme bouger le bras gauche, tourner le poignet droit, etc. La gate a pour rôle de choisir quel expert sélectionner, et donc quel mouvement doit être réalisé par l'effecteur. De surcroît, la gate est calibrée pour détecter l'état de repos (idle state : IS) afin de proposer un contrôle asynchrone de l'effecteur au patient.

Chapitre 5 : Décodeur PREW-NPLS et APREW-NPLS

Le décodage de signaux cérébraux, bien que reposant sur certains signaux neurologiques généralisables à tous les individus, est extrêmement dépendant du sujet. Afin de créer une ICM pouvant être calibrée et utilisée par le plus grand nombre, de nombreuses caractéristiques (features) sont extraites des signaux cérébraux. La phase de calibration a pour rôle de déterminer, parmi toutes les caractéristiques calculées pour un sujet donné les caractéristiques les plus pertinentes pour décoder les signaux cérébraux.

Afin de s'assurer que les caractéristiques les plus pertinentes soient calculées pour chaque sujet, l'espace des caractéristiques est souvent de grande dimension. Cependant, le décodage des signaux cérébraux à partir d'un grand nombre de caractéristiques peut engendrer plusieurs problèmes.

Du point de vue de l'apprentissage des modèles, si les caractéristiques sont corrélées ou si la base de données d'entraînement est trop petite, un grand nombre de caractéristiques peut mener à un problème de surapprentissage ou fléau de la dimension (« overfitting ou curse of dimentionality »). De plus, un grand nombre de caractéristiques demande une puissance de calcul plus importante pour calibrer et appliquer le modèle, ce qui peut être problématique dans le cas d'application d'une ICM portable (avec une puissance de calcul limitée). Enfin un modèle créé à partir d'un nombre de caractéristiques important est plus compliqué à interpréter.

Afin de résoudre les difficultés citées précédemment, les algorithmes Penalized REW-NPSL (PREW-NPLS) et Automatic PREW-NPLS (APREW-NPLS) ont été développés et testés. Ces algorithmes ont pour but d'être intégrés dans REW-MSLM à la place de REW-NPLS. XVIII Résumé PREW-NPLS est une version pénalisée de REW-NPLS. Plus précisément, PREW-NPLS intègre une pénalisation de la norme L0, L0.5 ou L1 au modèle REW-NPLS. Ces trois types de régularisation ont pour but de réduire le nombre de caractéristiques impliquées dans la prédiction des signaux cérébraux en imposant un poids de 0 aux caractéristiques les moins utiles. La pénalisation de REW-NPLS se déroule durant l'algorithme de décomposition tensorielle (PARAFAC).

PREW-NPLS est, tout comme REW-NPLS, un algorithme adaptatif incrémental qui peut être calibré et appliqué en temps réel. Cependant PREW-NPLS ajoute un nouvel hyperparamètre nommé coefficient de pénalisation noté 𝜆. Le coefficient de pénalisation est une variable qui détermine le « degré » de pénalisation (la pénalité) imposé aux coefficients du modèle. Plus 𝜆 est important, plus le modèle est pénalisé et donc plus le nombre de paramètres du modèle fixés avec un poids de 0 augmente. Cet hyperparamètre a une grande influence sur le modèle estimé et la performance de celuici. Malheureusement, comme tout hyperparamètre, 𝜆 doit être choisi avant la phase de calibration du modèle. Il est nécessaire de mener des études préliminaires afin de déterminer la valeur 𝜆 la plus optimale avant d'utiliser PREW-NPLS en temps réel.

Cette procédure préliminaire d'optimisation de 𝜆, généralement réalisée en offline, est contre-intuitive pour un algorithme incrémental adaptatif en temps réel. De plus, aucune étude ne permet d'affirmer que l'hyperparamètre de pénalisation optimal 𝜆 déterminé durant des analyses offline soit le même que le coefficient de pénalisation optimal pour un application en temps réel, et que celui-ci ne change pas au cours du temps (et de l'expérience).

L'algorithme APREW-NPLS a été développé afin de déterminer en temps réel durant la période de calibration le meilleur hyperparamètre de pénalisation 𝜆 parmi un groupe de valeurs disponibles. APREW-NPLS considère la sélection du coefficient de pénalisation comme le problème du bandit manchot («Multi-armed bandit problem »). Le problème est résolu en utilisant une stratégie d'apprentissage par renforcement (reinforcement learning). Les différents modèles PREW-NPLS (avec différents hyperparamètres de pénalisation) sont mis en compétition durant la phase de calibration incrémentale. A chaque incrément de calibration, les modèles obtenant les meilleures performances dans l'algorithme de Validation Récursive (RV) sont sélectionnés pour être recalibrés à la prochaine mise à jour tandis que les modèles les moins performants ne le sont pas. Cette stratégie permet de choisir un coefficient de pénalisation optimal en temps réel parmi plusieurs coefficients 𝜆 disponibles.

Résumé XIX Chapitre 6 : Classifieur H2M2 pour la gate de REW-MSLM Dans le but d'améliorer les performances de prédiction de la gate de REW-MSLM, un nouvel algorithme (H2M2) inspiré du Modèle de Markov Caché Hiérarchique ou « Hierarchical Hidden Markov Model (HHMM) » est proposé dans cette thèse.

A l'inverse du HMM qui considère tous les états discrets de façon équivalente, HHMM structure les états en couches hiérarchisées. Les états d'une couche inférieure sont dépendants des états de la couche supérieure. Ce type de modèle s'avère intéressant dans le cas de données naturellement structurées.

Dans notre cas d'application, les états permettant le contrôle des membres d'un exosquelette peuvent être hiérarchisés. Considérons le cas de contrôle asynchrone du mouvement 3D de la main gauche et de la main droite ainsi que la rotation 1D du poignet gauche et droit (5 états discrets avec l'état de repos). Dans ce cas, le nouvel algorithme H2M2 est composé de trois sous-modèles de type HMM. Le premier HMM de la couche supérieure se concentre sur l'activation de l'état de repos (IS) ou l'état « membres du côté gauche » (regroupant les états de mouvement de la main et de rotation du poignet du bras gauche) ou l'activation de l'état « membres du côté droit ». Ce premier HMM classifie trois états différents. Les deux autres HMMs font partie de la couche inférieure et se concentrent sur la classification binaire entre l'état de mouvement ou de rotation d'une des mains (un HMM pour la main gauche et l'autre pour la main droite).

Cette architecture a pour but d'améliorer les performances de décodage et réduire le temps de latence de détection des états.

Chapitre 7: Données expérimentales

Tous les algorithmes ont été testés durant une première phase de test offline en utilisant une procédure dite pseudo-online. Le but des études pseudo-onlines est de simuler en offline les conditions d'entraînement et de calibration réalisées durant les expériences en temps réel. Bien que les résultats obtenus durant les études pseudo-onlines ne soient pas généralisables au cas de l'application en temps réel, ils permettent de donner une tendance sur les résultats pouvant être obtenus durant les expériences onlines. Tous les jeux de données utilisés pour les études pseudo-online sont tirés d'expériences réalisées en temps réel par le patient.

Jeux de données d'évaluation de l'algorithme REW-MSLM.

Les tests pseudo-onlines ont été réalisés sur des jeux de données provenant d'expériences en temps réel où le patient contrôlait alternativement le mouvement des deux bras dans l'espace 3D (expériences 6D). Durant ces expériences en temps réel, l'algorithmes REW-NPLS a été utilisé pour prédire le mouvement des deux bras. Afin d'évaluer au mieux les performances de REW-MSLM, l'algorithme a été testé en pseudoonline selon trois paradigmes différents. Durant la première série d'expériences, chaque XX Résumé expérience est considérée individuellement, les modèles sont initialisés à zéro, avant d'être calibrés et testés durant la même session. Durant la deuxième série d'expériences, les modèles sont calibrés et testés durant chaque session mais les modèles ne sont pas réinitialisés à zéro entre deux expériences (la calibration est cumulative au cours des sessions). Enfin, durant la troisième série, le modèle calibré à la dernière session de la série précédente est testé sur de nouvelles données sans réaliser de recalibration au début de chaque session dans le but d'évaluer la stabilité du modèle.

Après avoir évalué les performances du modèle avec des études pseudo-onlines, REW-MSLM a été intégré dans la plateforme de décodage en temps réel de CLINATEC nommé « Adaptive Brain Signal Decoder (ABSD) » afin de réaliser des expériences 8D en temps réel. Durant ces expériences le patient devait contrôler 8 degrés de liberté continue regroupés en 5 états : le contrôle discret de l'état de repos, le contrôle continu des mouvements de la main gauche et droite dans l'espace 3D et le contrôle 1D de la rotation du poignet gauche et droit. Durant ces expériences, le patient contrôlait l'exosquelette ou l'avatar virtuel selon si l'expérience se déroulait à CLINATEC ou au domicile du patient. Un modèle différent a été calibré pour le contrôle de chaque effecteur. Chaque modèle a été calibré durant 6 expériences. Le modèle calibré pour le contrôle de l'exosquelette a été testé durant 15 expériences réparties entre 0 et 167 jours après la fin de la calibration du modèle. Le modèle calibré pour le contrôle de l'avatar virtuel a quant à lui été testé durant 37 expériences réparties entre 5 et 203 jours après la fin de la calibration du modèle.

Jeux de données d'évaluation de l'algorithme PREW-NPLS et APREW-NPLS.

Les expériences 8D réalisées avec l'avatar virtuel et l'algorithme REW-MSLM en temps réel, présentées précédemment, ont été utilisées pour mener des études pseudo-onlines des performances des algorithmes PREW-NPLS et APREW-NPLS. Plus particulièrement, les analyses pseudo-onlines se sont concentrées sur les données de contrôle 3D des mouvements de la main gauche et de la main droite.

Jeux de données d'évaluation de l'algorithme H2M2.

De nouvelles expériences de contrôle de l'avatar virtuel ont été réalisées avec le patient afin d'évaluer les performances pseudo-onlines de H2M2. Durant ces expériences, l'algorithme REW-MSLM a été utilisé pour décoder les signaux cérébraux en temps réel. Le patient devait contrôler 5D continus regroupées en 5 états : le contrôle discret de l'état de repos, le contrôle continu des mouvements de la main gauche et droite dans l'espace 1D et le contrôle 1D de la rotation du poignet gauche et droit. Le but des expériences était de réaliser le plus de transitions possibles entre les 5 états (l'état de repos, l'état de contrôle du mouvement de la main gauche, l'état de contrôle de rotation de la main gauche, l'état de contrôle du mouvement de la main droite, l'état de contrôle de rotation de la main droite) afin d'étudier la capacité de H2M2 à transiter de façon efficace entre les états.

XXVIII Abstract

The REW-MSLM is based on a mixture of experts (ME) architecture composed of several continuous "expert" models decoding continuous movements from the neural signals and a dynamic "gating" model activating or inhibiting the expert continuous outputs. In the REW-MSLM, the continuous linear expert models were evaluated using the Recursive Exponentially Weighted N-way Partial Least Squares (REW-NPLS) algorithm whereas the gating model is a Hidden Markov Model (HMM). The switching of several linear models (experts) may explain complex non-linear behaviors with a moderate computational load compatible with real-time neural signal decoding applications.

The PREW-NPLS and APREW-NPLS are two adaptive group-wise sparse decoders designed to reduce the feature space dimension, to improve the decoding performance and be potentially integrated in the REW-MSLM algorithms as sparse gating and/or expert models. Finally, H2M2 dynamic classifier is a dynamic hierarchical model designed to enhance the gating model responsiveness.

Before their integration into online closed-loop BCI experiments, all the proposed algorithms were evaluated offline. The REW-MSLM offline study induced the benefits of using a gating model for the detection of the idle state and the different active states related to each limb movements. Moreover, the interest of cross-session training in order to obtain decoder more robust to brain and experimental condition variability was stressed. The PREW-NPLS algorithms highlighted that significantly higher decoding performances could be obtained with group-wise sparse models whereas APREW-NPLS preliminary study induced the possibility to set in an online procedure the hyperparameters that were commonly tuned during offline studies in other state of the art research. APREW-NPLS hyperparameter tuning is based on a reinforcement learning strategies in order to confront the decoding performance of several models during the incremental calibration procedure. Finally, the H2M2 offline preliminary study highlighted the interest of using a hierarchical classifier structure in the case of complex classification tasks to improve the decoder responsiveness.

The REW-MSLM decoder highlighted promising results in multiple offline pseudoonline studies. Therefore, REW-MSLM was integrated into the online BCI platform Adaptive Brain Signal Decoder (ABSD) to perform asynchronous multi-limb online closed-loop BCI experiments. Using the REW-MSLM algorithm, a tetraplegic patient performed the online high-dimensional control of an exoskeleton and a virtual avatar. The patient achieved asynchronous 8D continuous control including alternative 3D hand reaching tasks and 1D wrist rotation for each hand distributed into 5 discrete states: idle state, left and right hands translation and left and right wrists rotation. The discrete and continuous decoding performance highlighted stable results over 6 months of clinical experiments after the last model recalibration for both effectors. As an example, for the exoskeleton experiments carried out from 0 to 37 days after the last model calibration experiments, the decoding performance highlighted a hit score of 71 ± 12% and 99 ± 2% for the 3D hand translation and 1D wrist rotation tasks whereas the Abstarct XXIX dynamic classifier showed a five-state classification F-score performance of 77 ± 14% . For the experiments achieved from 0 to 167 days, the decoding performance highlighted hit scores of 67 ± 21% and 93 ± 12% for the hand translation and wrist rotation tasks whereas the dynamic classifier demonstrated a five-state classification F-score performance of 75 ± 12% .

All the pseudo offline and online closed-loop BCI experiments confirmed in a long term study that direct neural decoding is not limited to individual neuron action potential driven (MEA-based) BCIs and can be achieved from population neuron recordings, particularly from epidural ECoG neural signals. These results challenge the empirical evidence that population neuron recordings are limited to the control of fewer dimensions due to lower spatio-temporal resolution and the restricted number of possible somatotopic remapping combination.

The nearest perspectives of the presented study is to integrate the developed REW-MSLM, AREW-MSLM and H2M2 algorithms into the online BCI platform in order to evaluate the benefits of these algorithms during online closed-loop experiments.

The doctoral manuscript is organized in eleven chapters describing the BCI state of the art research, the designed algorithms and the obtained offline and online results. Specifically, the Chapter 1 introduces the principle requirements of a BCI system dedicated to daily life applications. The Chapter 2 presents in detail the "BCI and Tetraplegia" clinical trial from the paradigm of control to the training timeline. Chapter 3 reports the state-of-the-art BCI transducer (preprocessing, feature extraction and decoder). The REW-MSLM, PREW-NPLS, APREW-NPLS and H2M2 algorithms are detailed in the Chapter 4, 5 and 6 whereas experiments description, integration of the decoder into the CLINATEC online BCI platform and decoder performance evaluation are presented in the Chapter 7, 8 and 9 respectively. Chapter 10 clusters the results of each decoders. Finally, Chapter 11 regroups the results discussion, the benefit of this study, the implications of the presented results in the BCI fields and the research perspective. The manuscript is completed with an abstract of the PhD research, a

Definition of the norm

The tensor Frobenius norm of 𝐗 is defined as

‖𝐗‖ = √∑ … 𝐼 1 𝑖 1 =1 ∑ 𝑥 𝑖 1 ,… ,𝑖 𝑁 2 𝐼 𝑁 𝑖 𝑁 =1
, which is analogue of standard definitions for matrices (Frobenius norm) and vectors (Euclidian norm). In the manuscript, ‖•‖, always referred to L2 norm (Frobenius, Euclidian norm depending on the variable dimensions).

Tensor unfolding transformation

Transformation of a tensor into a matrix is named unfolding, flattening or matricization. This process flattens a tensor 𝑿 ∈ ℝ 𝐼 1 ×…×𝐼 𝑁 into a matrix along a specific dimension (or mode) 𝑛 and will be noted as 𝐗 (𝑛) ∈ ℝ 𝐼 𝑛 ×𝐼 1 𝐼 2 …𝐼 𝑛-1 𝐼 𝑛+1 …𝐼 𝑁 [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF] [ [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]. An example of the three possible unfolding transformation of a third order tensor 𝑿 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 is presented in the Figure 12. As mentioned, unfolding can be Mathematical operators and notations XXXIX performed following each tensor dimension leading in this three order tensor example to the three matrices 𝑿 (1) = ℝ 𝐼 1 ×𝐼 2 𝐼 3 , 𝑿 (2) = ℝ 𝐼 2 ×𝐼 1 𝐼 3 and 𝑿 (3) = ℝ 𝐼 3 ×𝐼 1 𝐼 2 .

Figure 1-2: Example of third-order tensor 𝑿 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 unfolding along the 3 possible modes leading to the three possible matrices 𝑿 (1) = ℝ 𝐼 1 ×𝐼 2 𝐼 3 , 𝑿 (2) = ℝ 𝐼 2 ×𝐼 1 𝐼 3 and 𝑿 (3) = ℝ 𝐼 3 ×𝐼 1 𝐼 2 .

Outer Product

Let 𝑿 ∈ ℝ 𝐼 1 ×…×𝐼 𝑁 be a N-order tensor and 𝐰 𝑛 ∈ ℝ 𝐼 𝑛 (𝑛 = 1 … 𝑁) 𝑁 vectors, the outer product (noted " ∘ ") of 𝐰 𝑛 is defined as:

𝑿 = 𝐰 1 ∘ 𝐰 2 ∘ … ∘ 𝐰 𝑁 𝑤𝑖𝑡ℎ 𝑥 𝑖 1 ,… ,𝑖 𝑁 = 𝑤 𝑖1 1 𝑤 𝑖2 2 … 𝑤 𝑖𝑁 𝑁 .
𝑋 is a rank one tensor as it can be expressed as the outer product of 𝑁 vectors.

N-mode product

The n-mode product (noted " × 𝑛 ") between a tensor 𝑿 ∈ ℝ 𝐼 1 ×…×𝐼 𝑁 and a matrix 𝐘 ∈ ℝ 𝐾 𝑛 ×𝐼 𝑛 is noted 𝐂 = 𝑿 × 𝑛 𝐘 with 𝐂 ∈ ℝ 𝐼 1 ×…×𝐼 𝑁-1 ×𝐾 𝑛 ×𝐼 𝑁+1 ×…×𝐼 𝑁 and 𝑪 𝑖 1 …𝑖 𝑛-1 𝑘 𝑛 𝑖 𝑛+1 …𝑖 𝑁 = ∑ 𝑥 𝑖 1 …𝑖 𝑛-1 𝑖 𝑛 𝑖 𝑛+1 …𝑖 𝑁 𝑦 𝑘 𝑛 𝑖 𝑛 𝐼 𝑛 𝑖 𝑛 =1

.

The n-mode product 𝐂 = 𝑿 × 𝑛 𝐘 can be expressed using the unfolded tensor expression:

𝐂 (𝑛) = 𝐘𝐗 (𝑛) .

Kroenecker product:

Let 𝐗 ∈ ℝ 𝐼 1 ×𝐼 2 and 𝐘 ∈ ℝ 𝐽 1 ×𝐽 2 be two matrices, the Kronecker product of these two matrices noted 𝐂 = 𝐗⨂𝐘 with 𝐂 ∈ ℝ (𝐼 1 𝐽 1 )×(𝐼

Khatri-Rao product:

The Khatri-Rao product is the column wise Kronecker product. Let 𝐗 ∈ ℝ 𝐼 1 ×𝐾 and 𝐘 ∈ ℝ 𝐽 1 ×𝐾 be two matrices, The Khatri-Rao product is denoted 𝐂 = 𝐗⨀𝐘 with 𝐂 ∈ ℝ (𝐼 1 𝐽 1 )×𝐾 and is defined by

𝐂 = 𝐗⨀𝐘 = [𝐱 1 ⨂𝐲 1 𝐱 2 ⨂𝐲 2 … 𝐱 𝐾 ⨂𝐲 𝐾 ].

Khatri-Rao product property

These unfolded matrices and tensors products have many properties [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]. For the next chapters, the following property is mentioned:

(𝐗⨀𝐘) † = ((𝐗 T 𝐗) * (𝐘 T 𝐘)) † (𝐗⨀𝐘) T , (0. 1)

where 𝐗 ∈ ℝ 𝐼 1 ×𝐾 , 𝐘 ∈ ℝ 𝐽 1 ×𝐾 and 𝐗 † is the Moore-Penrose pseudoinverse of 𝐗 [START_REF] Kolda | Tensor Decompositions and Applications[END_REF] The elementwise matrix (Hadamard) product

Let 𝐗 and 𝐘 ∈ ℝ 𝐼 1 ×𝐼 2 two matrices of the same dimensions, the elementwise matrix product noted 𝐂 = 𝐗 * 𝐘 is defined by ]. Brain-Computer Interfaces (BCIs) or brain machine interfaces (BMIs) are systems allowing the control of external devices thanks to the brain neural signals without using the natural neuromuscular activation. BCIs create a new communication pathway between the brain and an effector [START_REF] Vidal | Toward Direct Brain-Computer Communication[END_REF]] [Lebedev and Nicolelis, 2006[START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF]. BCIs have been investigated for many applications such as communication, neurorehabilitation, drowsiness monitoring, computer gaming [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Lee | Clinical neuroprosthetics: Today and tomorrow[END_REF][START_REF] Ramadan | Basics of Brain Computer Interface[END_REF], etc. Nevertheless, BCIs were particularly developed for patient suffering from severe motor disabilities. In this domain, BCIs are referred as motor BCI. Motor dysfunctions may be the consequences of neurological/neuromuscular disorders as spinal-cord injury [W. Wang et al., 2013], hemiplegia, brain stroke [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF], amyotrophic lateral sclerosis (ALS), cerebral palsy [Lebedev andNicolelis, 2006] [Mak andWolpaw, 2009] or brain, muscular diseases as the Duchenne muscular dystrophy (DMD) [START_REF] Utsumi | Operation of a P300-based brain-computer interface in patients with Duchenne muscular dystrophy[END_REF]. BCIs have been used for functional rehabilitations or robotic assistance of individuals suffering from muscular activity deterioration to help patients and brain plasticity to recover [START_REF] Carvalho | Brain-machine interface of upper limb recovery in stroke patients rehabilitation: A systematic review[END_REF][START_REF] Donati | Long-Term Training with a Brain-Machine Interface-Based Gait Protocol Induces Partial Neurological Recovery in Paraplegic Patients[END_REF][START_REF] López-Larraz | Brain-machine interfaces for rehabilitation in stroke: A review[END_REF][START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF] or complete functional compensation for patients enable to perform any muscular activation [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF], 2006[START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF] with numerous effectors such as drones [START_REF] Lafleur | Quadcopter control in threedimensional space using a noninvasive motor imagery-based brain-computer interface[END_REF], wheelchairs [START_REF] Huang | Electroencephalography (EEG)-Based Brain-Computer Interface (BCI): A 2-D Virtual Wheelchair Control Based on Event-Related Desynchronization/Synchronization and State Control[END_REF][START_REF] Leeb | Self-Paced (Asynchronous) BCI Control of a Wheelchair in Virtual Environments: A Case Study with a Tetraplegic[END_REF]] [Li et al., 2013], robotic limbs [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF], or exoskeleton [Eliseyev et al., 2014[START_REF] Morinière | EMY: a dual arm exoskeleton dedicated to the evaluation of Brain Machine Interface in clinical trials[END_REF]. Spinal cord injury affects 17 000 people per year in United States with an estimated prevalence of 280 000 injured persons [START_REF] Eckert | Trauma: Spinal Cord Injury[END_REF] whereas stroke is one of the main causes of long-term motor disability worldwide, and generally drives to functional deficits in motor control [START_REF] Langhorne | Stroke rehabilitation[END_REF][START_REF] López-Larraz | Brain-machine interfaces for rehabilitation in stroke: A review[END_REF]. Considering the possible benefits for the disabled patient, the development of new technologies related to the implantable recording devices, the artificial intelligence and the exponential increase of computational power, BCIs translated from niche area of research to a broad and complicated field of research to analyze (Figure 1-1).

XLI

Table of contents

Figure 1-1 : Evolution of the BCI research field across years and domain of application. Statistics were extracted from Scopus website with the key words "BCI", "brain computer interface" and "Brain machine interface" on August 2020. The "Others" category clusters fields with less than 2% of the articles Chapitre 1 : Introduction

In the next section, basic neuroscience notions are provided to understand the brain activity modulations and their possible exploitation to control various systems using BCI and motor BCI.

Basic Neuroscience for BCI

The nervous system, particularly the brain, is composed of nerve cells named neurons and supporting cells called neuroglia. Neurons are specialized in electrical signaling to interact with each other and transfer/process information whereas supporting cells do not produce electrical signals but assist the nerve cells [START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF]. Many diverse neurons exist to handle different functions, nevertheless, they all generally have the same basic components which are the dendrites, the soma and the axon. The dendrites retrieve information from the chemical components released in the synapses by other neurons. The information conveyed in the synapses up to the neuronal dendrites is integrated and processed at the origin of the axon [START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF]. The axon hillock, the base of the axon in the soma, generates (or not) the fundamental unit of electrical information called the action potential that carries signals at high speed across the axon to the axon termination named telodendria. In the axon termination, synaptic contacts are made with other neurons. The information is transmitted to other neurons thanks to chemical components released in the synapse. The concentration of the released chemical component is dependent on the received electrical information [START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF] (Figure 12). An action potential is a brief (1ms) all-or-nothing change of the neuron transmembrane potential from negative to positive which is triggered if an input stimulus recorded at the dendrites is above an activation threshold [START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF].

The action potential amplitude, shape and maximal firing rate are fixed properties dependent on the neurons type (excitatory, inhibitory etc).

The human brain (encephalon) is part of the central nervous system (CNS) with the spinal cord and is estimated to contain 100 billion neurons with complex interactions called neural networks (or neural circuits) and several times as many supporting cells [START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF]. Neural circuits process input information of the brain and provide the foundation of sensation, perception and behavior. Encephalon is composed of the brainstem (midbrain, the pons, the medulla oblongata), the diencephalon and the telencephalon. Depending on the chemical components transmitted as well as the postsynaptic cell receptor type, the probability of generating an action potential in the axon hillock is increased or decreased [START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF]. If an action potential is generated, it is sent through the axon to the synaptic terminals to transmit the information to the next neurons.

The brainstem ensures numerous major functions. It is the link between the spinal cord and the rest of the central nervous system, regroups the nuclei of the cranial nerves and is involved in cardiovascular, respiratory and consciousness control. The diencephalon clusters the basal ganglia, the thalamus, hypothalamus and plays a critical role in sensory information transmission, auditory, visual, sensory, motor and emotional information [Kandel andTollet, 2016] [Purves et al., 2004]. The telencephalon is composed of the two cerebral hemispheres connected through white matter (as the corpus callosum). Telencephalon anatomy formed of crests (gyri) and grooves (sulci) can be separated into the occipital, temporal, parietal and frontal lobes with the central sulcus splitting the brain in two between the parietal and frontal lobes (Figure 1-3) [Kandel andTollet, 2016] [Purves et al., 2004]. All the lobes have specific functions, occipital lobe functions are oriented on the processing of visual stimuli, the temporal lobe is engaged in the auditory functions, the parietal lobe involves proprioceptive and sensory information whereas the frontal lobe is dedicated to the motor and cognitive functions. The motor cortex (following the central sulcus) is dedicated to voluntary movements. The firing rate of the motor cortex neurons is related to low-and high-level information of attempted and realized movements. et al., 2004].

The neural system and, particularly the brain have a functional topographic organization (somatotopic). The spatial distribution of neurons in the cortical regions of the brain is dependent on their functions. For the motor cortex, studies highlighted the contralateral somatotopic organization of human motor control. Moreover, it was proven that neurons action potential firing rate of premotor and motor cortex are directly tuned depending on arm intended direction and action [START_REF] Georgopoulos | Neuronal population coding of movement direction[END_REF] • Gamma (ɣ) for signal frequencies above 30 to 35 Hz.

In the case of motor control pathways degradation, the intended movement may not be transmitted to the muscles (due to spinal cord damage, or motor neuron degradation, etc.). However, the action potentials of neurons in the motor cortex are still firing/activated. Motor BCIs aim to record these neural activities and translate it into order for an effector (an actioner) that will substitute the non-realized movement by an action of the effector. To perform, BCI systems are composed of basic elements described in the next section.

BCI components

A Brain computer interface is built on three fundamental components that are common to every system, namely the acquisition system, the transducer and the effector. A BCI is a system allowing the interaction of a patient with the environment through the control of an effector using an acquisition system to record the brain neural signals and a signal-processing block to translate the neural activity variations into commands to the effector. Finally, the information (success, failure, etc.) about the actions carried out by the effector are sent to the patient through a feedback (visual, auditory, proprioceptive, etc.) (Figure 1-4) [Lebedev andNicolelis, 2017] [Mak andWolpaw, 2009]. Sensory feedback is a key element of the BCIs dedicated to daily life applications which must be considered.

Acquisition systems

The acquisition system aims to record the brain activity from neurophysiologic signals or using indirect measures. Acquisition systems attempt to optimize the ratio between invasiveness and resolution. Therefore, numerous devices more or less invasive have been designed with various spatio-temporal resolutions (Figure 1-5) [START_REF] Kim | A study on a robot arm driven by three-dimensional trajectories predicted from non-invasive neural signals[END_REF][START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. Invasive systems record information closer to the sources providing simpler signals to analyze compared to less invasive devices which record neurons activity integrated/filtered and spread into a large area due to multiple layers protecting the brain (Pia, Arachnoid, dura matter, skull and scalp). This section is reviewing the different recording systems from the more to the less invasive techniques.

Figure 1-5: Invasiveness and resolution of BCI recording systems [START_REF] Jorfi | Progress Towards Biocompatible Intracortical Microelectrodes for Neural Interfacing Applications[END_REF]] [Kim et al., 2015].

Possible recording systems cover a wide range of invasiveness going from highly invasive systems to recording systems without direct interaction with the patient. A trade-off between invasiveness and resolution is to consider to select the more adapted recording system.

Microelectrode arrays (MEA) are intracortical recording systems for BCI applications which are implanted to directly sample the activity of the neurons from small local brain area. As an example, the Utah array is composed of hundreds of shanks distributed in few mm 2 with an electrode length between 0.5 and 1.5mm, 400µm pitch and spaced from each other of around 0.4mm [START_REF] Gunasekera | Intracortical Recording Interfaces: Current Challenges to Chronic Recording Function[END_REF][START_REF] Jorfi | Progress Towards Biocompatible Intracortical Microelectrodes for Neural Interfacing Applications[END_REF][START_REF] Stieglitz | Brain-computer interfaces: an overview of the hardware to record neural signals from the cortex[END_REF].

MEA recording system provides high spatial and temporal signal resolution and allows recording Single-Unit Activity (SUA) and Multi-Unit Activity (MUA) which provide a direct representation of the neuronal activity near the electrodes. Depending on the signal processing of MEA neural signals, the Local Field Potentials (LFP) can also be extracted. The LFPs are the signals recorded from the summation of synchronous action potentials from thousands of neuron populations close to the electrode tip [Lebedev andNicolelis, 2017] [Schaeffer and[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. Disabled patients controlled in real-time complex effector up to 10 Degrees of freedom (DoF) based on MEA neural signal decoding [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. However, this control required frequent recalibration (each session or daily recalibration) due to highly intra and inter-day signal instabilities [START_REF] Perge | Intra-day signal instabilities affect decoding performance in an intracortical neural interface system[END_REF]. Few Long-term studies were reported with long-term implantation of MEA up to 1000 days [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF][START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF][START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF]] [Simeral et al., 2011], nevertheless, degradation of the recorded signals related to the loss of electrodes or recorded neurons is recurrent and is an important limitation for clinical and daily life application. The implantation of a MEA device leads to acute damage, which generates body immune system reaction to protect brain cells and repair the damaged tissues. This reaction generates a glial encapsulation of the intracortical microelectrodes which may deteriorate the electrodes and significantly reduce the amplitude of the recorded spikes [START_REF] Gunasekera | Intracortical Recording Interfaces: Current Challenges to Chronic Recording Function[END_REF][START_REF] Jorfi | Progress Towards Biocompatible Intracortical Microelectrodes for Neural Interfacing Applications[END_REF][START_REF] Kozai | Brain Tissue Responses to Neural Implants Impact Signal Sensitivity and Intervention Strategies[END_REF][START_REF] Marin | Biocompatibility of intracortical microelectrodes: current status and future prospects[END_REF][START_REF] Moran | Evolution of brain-computer interface: action potentials, local field potentials and electrocorticograms[END_REF][START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]] [Ward et al., 2009]. This biocompatibility issue and degradation of the electrodes imply difficulties for chronic/long term SUA and MUA recordings and is one the main research topics in MEA recording system domain [START_REF] Jorfi | Progress Towards Biocompatible Intracortical Microelectrodes for Neural Interfacing Applications[END_REF]. LFP is less sensitive to the spike amplitude reduction by the nature of the signal which integrates the behavior of many more neurons [START_REF] Moran | Evolution of brain-computer interface: action potentials, local field potentials and electrocorticograms[END_REF]. Finally, to this day, MEA recording systems applied to BCI and motor BCI are limited to wired systems due to the massive stream of data recorded at a high sampling rate. Wired-systems are not suited to daily life applications and enhance the possible risk of infection.

Electrocorticographic arrays (ECoG) are grids of plane macro-electrodes (diameter in mm). The grid is in the order of the cm 2 and the electrodes are distanced by few mm. ECoG are implanted under the skull either above the Dura Matter (epidural ECoG) or below (subdural ECoG). ECoG can be considered as "semi-invasive" recordings as the patient underwent a craniotomy but the brain integrity is not affected by the operation [Lebedev andNicolelis, 2017] [Rak et al., 2012]. Due to the dimensions of the electrodes and the distance between the electrodes and the neurons, ECoG is limited to the neural population recording of the superficial layers of the cortex. ECoG recorded signals represent the sum of the synchronous extracellular potential of the neurons [START_REF] Buzsáki | The origin of extracellular fields and currents -EEG, ECoG, LFP and spikes[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. ECoG recording systems is a good trade-off between invasiveness and signal to noise ratio, it contains broader bandwidth, higher amplitude and fewer artifacts than non-invasive recording systems whereas clinical risk is reduced compared to MEA even though ECoG has lower spatial and temporal resolution [START_REF] Leuthardt | Electrocorticography-based brain computer Interface-the seattle experience[END_REF], 2004[START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF]. Moreover, ECoG recording systems highlighted real-time motor BCI experiments with up to 3 DoF control by tetraplegic patients [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] [W. Wang et al., 2013] as well as long-term stability properties [START_REF] Leuthardt | A brain-computer interface using electrocorticographic signals in humans[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] in preclinical [START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF]] [Costecalde et al., 2017] [Sauter-Starace et al., 2019] and clinical studies [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF][START_REF] Nurse | Consistency of Long-Term Subdural Electrocorticography in Humans[END_REF]. Nevertheless, clinical BCI research based on ECoG array implantation was seldom realized for BCI purposes. In most of the studies, ECoG arrays were implanted to detect the epileptic sources of patient before surgery which limits the relevance of the study for disabled patients and the duration of the reported state of the Electroencephalography (EEG) is a non-invasive recording device based on a helmet/headset with a large number of electrodes (64 to 256) placed on the surface of the scalp. EEG signals integrate the extracellular currents of a large neural population over a large region (10 cm 2 or more) [START_REF] Buzsáki | The origin of extracellular fields and currents -EEG, ECoG, LFP and spikes[END_REF][START_REF] Rak | Brain-computer interface as measurement and control system The review paper[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF].

Similarly to ECoG recordings, EEG is limited to the recording of the low-pass filtered synchronous extracellular current activity of neurons at the surface of the motor cortex. EEG is the most widely used brain activity recording system and presents many advantages compared to the previously mentioned recording systems. The noninvasiveness, ease of use and low cost of EEG recording systems tend to apply it for research with a humongous number of studies in the BCI and the motor BCI field in the past years [Lebedev andNicolelis, 2017] [Lotte et al., 2018]. Additionally, EEG recording was used for epileptic, sleep or brain disorder detection [START_REF] Rak | Brain-computer interface as measurement and control system The review paper[END_REF]. Nevertheless, EEG-based BCIs present several limitations compared to more invasive neural recording systems. EEG recording devices have limited spatial resolution, lower than more invasive systems and restricted frequency resolution with a bandwidth from 0 to around 100 Hz with the major information between 0 and 40 Hz due to the low pass-filtering of the multiple layers protecting the brain (Pia, Arachnoid, Dura Matter, skull, skin) [START_REF] Lebedev | Brain-machine interfaces: past, present and future[END_REF][START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. Recorded signals are small (from 10 μV to 100 μV), noisy and highly artifact sensitive from different sources (eye movements, muscle activity, heart pulses, power line at 50 Hz) [Lebedev andNicolelis, 2017] [Rak et al., 2012]. Moreover, EEG recorded signals are not stable in time and showed high inter and intra subject variability [Clerc et al., 2016a].

Finally, EEG headset commonly required a skilled person for device preparation and electrodes positioning which is a time-consuming and complicated procedure.

In summary, MEA provides the best spatial resolution allowing to directly record the action potentials of the neurons whereas ECoG and EEG are limited to population neurons recordings equal to the sum of the individual neuron action potentials. Nevertheless, MEA recordings are highly invasive recording systems with a higher risk of infection, important immune response and low temporal stability due to the constant modification of the neurons activity. EEG recording system, while the spatial and timefrequency resolution is bad compared to the other presented recording systems, is noninvasive and is widely used in various BCI experiments. ECoG recording system is a trade-off between MEA and EEG. ECoG has better recording properties for BCI application than EEG and is less invasive than MEA, particularly the epidural ECoG recording systems. EEG, ECoG and MEA are the recording systems generally used in BCI applications. Each of them has strengths and weaknesses [Nicolas-Alonso and Gomez-Gil, 2012] that are summarized in Figure 1-6. However, other types of recording systems have been developed or adapted to BCI applications. Magnetoencephalography (MEG) is a non-invasive recording technique which measures at distance, out of contact with the patient, the brain magnetic field intensity and its variation at few centimeters above the skin. It uses arrays of superconducting quantum inference devices (SQUIDs) or spin-exchange-relaxation-free magnetometer (SERF) to record the small magnetic fields (around 10 fT to 1 pT) generated by the intracellular currents flowing through the cortical pyramidal neurons dendrites [START_REF] Buzsáki | The origin of extracellular fields and currents -EEG, ECoG, LFP and spikes[END_REF][START_REF] Hämäläinen | Magnetoencephalography---theory, instrumentation, and applications to noninvasive studies of the working human brain[END_REF][START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF] [Lopes da [START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. MEG recordings present several advantages compared to the other recording systems. Firstly, as MEG is a measurement of the magnetic field at distance from the patient, there is no contact between the patient and the recording system. MEG signals are less dependent on the extracellular space conductivity (skin, muscles, Dura Matter, etc.) and have a better signal-to-noise ratio than EEG specifically in the brain high frequency bandwidth (above 30 Hz) [START_REF] Buzsáki | The origin of extracellular fields and currents -EEG, ECoG, LFP and spikes[END_REF][START_REF] Yelisyeyev | Brain-Computer Interface with cortical electrical activity recording[END_REF]. MEG system has a better spatial and frequency (above 1ms) resolution than EEG [START_REF] Buzsáki | The origin of extracellular fields and currents -EEG, ECoG, LFP and spikes[END_REF][START_REF] Hämäläinen | Magnetoencephalography---theory, instrumentation, and applications to noninvasive studies of the working human brain[END_REF]. Nevertheless, MEG recording system presents numerous practical drawbacks. MEG instrumentation is very expensive, is cumbersome requiring a lot of spaces to install the system (cooling device, magnetic recording device, etc.), required long patient preparation and the patient need to stay immobile during experiments [START_REF] Hämäläinen | Magnetoencephalography---theory, instrumentation, and applications to noninvasive studies of the working human brain[END_REF][START_REF] Rak | Brain-computer interface as measurement and control system The review paper[END_REF]. It is good to notice that MEG recording system can be used to realized pre-surgical studies and evaluate the potential performance of ECoG recording systems or localize the more optimal localization of invasive/semi-invasive recording systems [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF][START_REF] Fukuma | Real-Time Control of a Neuroprosthetic Hand by Magnetoencephalographic Signals from Paralysed Patients[END_REF]. MEG neural signal processing was reported during BCI and motor BCI experiments [START_REF] Jerbi | Inferring hand movement kinematics from MEG, EEG and intracranial EEG: From brain-machine interfaces to motor rehabilitation. IRBM[END_REF][START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Waldert | Hand Movement Direction Decoded from MEG and EEG[END_REF].

Functional Near-Infrared Spectroscopy (fNIRS) is an indirect brain activity measurement system. Brain neural activity leads to variation in brain blood oxygenation (hemodynamic responses) with increases in the brain oxygen concentration in the blood (oxyhemoglobin HbO) or decreases of the oxygen concentration (deoxyhemoglobin HbR) [START_REF] Sirpal | fNIRS improves seizure detection in multimodal EEG-fNIRS recordings[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. The fNIRS recording systems are based on a helmet composed of optodes and receptors. The optodes apply light in the near infrared spectrum (600 to 1000 nm) through the skull and the cortex [Lebedev andNicolelis, 2017] [Sirpal et al., 2019]. The unabsorbed light by the brain tissue is retrieved by the receptors. The brain near infrared light absorption is related to the brain oxygenation which allows to record the brain oxygen and thus brain activity. fNIRS recordings yield higher spatial resolution than EEG (around 1 cm) [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Roy | Chapter 98 -EEG and FNIRS Connectivity Features for Mental Workload Assessment: A Preliminary Study[END_REF][START_REF] Sirpal | fNIRS improves seizure detection in multimodal EEG-fNIRS recordings[END_REF] whereas, due to the slow dynamics of hemodynamic response, fNIRS systems have a lower temporal resolution (around 100 ms) [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Roy | Chapter 98 -EEG and FNIRS Connectivity Features for Mental Workload Assessment: A Preliminary Study[END_REF][START_REF] Sirpal | fNIRS improves seizure detection in multimodal EEG-fNIRS recordings[END_REF]. The fNIRS systems are not expensive and ease to use. fNIRS recording systems were reported in motor BCI [START_REF] Khan | fNIRS-based Neurorobotic Interface for gait rehabilitation[END_REF] and BCI [Naseer et al., 2016a] experiments. Moreover, multiple studies mixed EEG and fNIRS recordings to bring the best of both worlds and improve BCI performance [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Roy | Chapter 98 -EEG and FNIRS Connectivity Features for Mental Workload Assessment: A Preliminary Study[END_REF][START_REF] Sirpal | fNIRS improves seizure detection in multimodal EEG-fNIRS recordings[END_REF].

Functional Magnetic Resonance Imaging (fMRI)

. is an indirect brain activity recording system using an MRI scanner. As fNIRS, it is a measurement of blood oxygen concentration [START_REF] Rak | Brain-computer interface as measurement and control system The review paper[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF], specifically from blood oxygen level dependent (BOLD) activity [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF]. The major advantage of fMRI is its good spatial resolution for the entire brain (not only the cortex surface) allowing 3D representation of the brain activity (generally around 3 to 4 mm 3 voxels) [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF]. Nevertheless, fMRI clusters numerous drawbacks such as a low temporal resolution (around 1 to 2 seconds) and a significant delay between the brain activity and the BOLD response (around 3 to 6 seconds) [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF]. Moreover, fMRI instrumentation presents the same disadvantages as MEG recordings systems with very expensive and bulky platform requirements [START_REF] Rak | Brain-computer interface as measurement and control system The review paper[END_REF].

Neural signal processing

The digitized neural data are processed in a signal translation block also called transducer [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. Transducer block groups all the signalprocessing steps which allow the patient to generate a command for the effector from his recorded neural signals. To do so, BCI signal processing generally consists of several steps referred to as brain signal pre-processing, neural feature extraction, decoding and post-processing (Figure 1234567). Each of these blocks is highly dependent on numerous criteria such as the recording system, the mental strategy applied during the experiments by the patient, the goal of the BCI (motor BCI, rehabilitation BCI, communication BCI, etc.) and the effector to control (wheelchair, drone, speller, computer browsing, etc.). Nevertheless, the principle of each block can be generally defined. Pre-processing block aims to enhance the signal quality and improve the signal-to-noise ratio for future steps [START_REF] Anitha | Brain-Computer Interface for Persons with Motor Disabilities -A Review[END_REF]] [Bashashati et al., 2007a[START_REF] Ramadan | Basics of Brain Computer Interface[END_REF].

Different operations can be performed to enhance the brain signals such as temporal and/or spatial filters or down-sampling operations as anti-aliasing filters [START_REF] Hassan | Design and Implementation of Pre-processing Chip for Brain Computer Interface Machine[END_REF][START_REF] Mcfarland | Spatial filter selection for EEG-based communication[END_REF][START_REF] Syan | Comparison of Pre-Processing and Classification Techniques for Single-Trial and Multi-Trial P300-Based Brain Computer Interfaces[END_REF]. Artifacts related to eye blinking, muscle activity or power line at 50 Hz can be removed/reduced through filters or decoders dedicated to the recognition of such biological patterns [START_REF] Anitha | Brain-Computer Interface for Persons with Motor Disabilities -A Review[END_REF]] [Fatourechi et al., 2007]. This step is crucial in non-invasive recording systems such as EEG which is highly sensitive to artifacts and other noise sources.

Neural feature extraction is a crucial step which extracts from the pre-processed neural signals the features that will be used by the decoding algorithm to generate orders to the effector. Consequently, this step aims to extract the neural features that are the most discriminative to the subject desired mental tasks [START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF]Wolpaw, 2009] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. Feature extraction can be defined by a step of feature generation and an optional step of feature selection or feature space dimension reduction [Bashashati et al., 2007a]. Feature extraction is dependent on the analyzed neural signals. Spike-count strategies are specific to MEA-based BCIs to evaluate the firing rate variation of the individual neurons [START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF]Wolpaw, 2009] [Schaeffer and[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. The features extracted from the neural population recording systems are more focused on the Chapitre 1 : Introduction modulation of the neural signal amplitude, power and phase across time or timefrequency domains [START_REF] Anitha | Brain-Computer Interface for Persons with Motor Disabilities -A Review[END_REF][START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. Time domain analysis computes interesting features to track slow dynamics responses. Time-frequency domain analysis is, for example, useful to track amplitude spectrogram modulations during movements as their related neural signals are commonly characterized by strong amplitude modulation in the low and high frequency bands [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF].

Numerous features can be extracted from several electrodes, at several frequency bands and different time segments leading generally to a high dimensional feature space [Eliseyev et al., 2017] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. High dimensional feature space may lead to numerous issues such as important computing power requirements, high computational load, irrelevant or redundant information and "curse of dimensionality" problem in the decoder training step [START_REF] Bellman | Adaptive Control Processes: A Guided Tour[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF]. Dimensional reduction and feature selection algorithms aim to reduce the feature space dimension to avoid the "curse of dimensionality" and improve the decoding performances. Additionally, reduction of the feature space dimension may also drastically lower the required computing time by allocating less computing resources to feature extraction step (do not compute the irrelevant features), avoid overfitting, reduce the training time of the decoder, remove correlated features (numerical stabilization), denoise the signals and lead to an easier interpretation of the results [START_REF] Haufe | Dimensionality reduction for the analysis of brain oscillations[END_REF]. In the case of high dimensional neural signal processing, all these aspects are relevant [START_REF] Haufe | Dimensionality reduction for the analysis of brain oscillations[END_REF][START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF][START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF]] and more specifically for realtime BCI application with real-time data flow processing and decoding. Feature selection family regroups filter-based, wrapper-based and embedded techniques [START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF]Dhanalakshmi, 2019] [Remeseiro and[START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF]. Another approach called dimension reduction or projection aims to project the feature space into a subspace of lower dimension by a linear or non-linear combination of the initial feature space components to create few highly informative features [START_REF] Haufe | Dimensionality reduction for the analysis of brain oscillations[END_REF].

Decoding step purpose is to use a linear or non-linear model to map/translate the neural feature space to the space of possible commands/orders to send to the effector. Decoder changes independent variables (signal features) into estimates of the user movement intention dependent variables (effector control commands) [START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF][START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. Nevertheless, the model parameters are generally patient-specific and data-driven. Therefore, various BCI decoders were tested or designed to improve the neural signal decoding. The decoding algorithms applied in the BCI field can be clustered into specific nested families.

Decoders can be regrouped depending on the type of expected user's intention. Discrete decoders also called classifiers, cluster the neural features into a limited number of defined states or classes. Classifiers can be binary decoders (classification between 2 states) or multi-class decoders (classification between 𝑁 states). Classification models (discrete decoders) have been created to detect discrete mental states allowing an accurate classification of patient's intentions (e.g., open or close the hand, walk or stand, etc.) [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. Continuous, regression decoders predict continuous variables to be realized by the effector (e.g. position or displacement of the hands in the space, etc.). For example, continuous decoders can predict the trajectories of a cursor or limb based on its position, velocity, acceleration or a combination of these components [Bishop, 2006] [Marathe and[START_REF] Marathe | Decoding position, velocity, or goal: Does it matter for brain-machine interfaces?[END_REF]. Another approach consists in mixing both continuous and discrete decoders to create so-called hybrid decoding models. These models mostly rely on switching between (multiple) continuous models. The selection of the continuous decoder is handled by the discrete output of a classifier: the more likely continuous decoder is selected [Schaeffer and Aksenova, 2016a].

The majority of BCI decoders (discrete, continuous and hybrid) are defined as static decoders. They rely on traditional statistical algorithms which assume that successive input and output variables are temporally independent. Nevertheless, in biological systems such as the neural signals (and BCI in general), this assumption is an important restriction which leads to a loss of information for the decoders. To take into consideration the natural temporal dependencies of the brain neural signals, several decoders referred to as dynamic or sequential decoders were implemented in BCI applications [Lotte et al., 2007] [Schaeffer and[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF].

Discrete, continuous and hybrid decoders were exploited in motor BCI applications to control various effectors [START_REF] Han | Brain-Switches for Asynchronous Brain-Computer Interfaces: A Systematic Review[END_REF][START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF], 2007[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF]. To this date, there is no consensus on the best decoder as the reported decoding performance were highly dependent on the patient, the recording system, the experimental paradigm, etc.

The model parameters can be automatically estimated based on artificial intelligence strategies such as machine learning and deep learning. These techniques use a finite "training" dataset representative of the relation between the neural feature space and the patient's intentions to estimate automatically the model parameters. Three major training categories have been conceptualized referred as: supervised, unsupervised and reinforcement learning depending on which information (independent and dependent variables) are provided in the "training" dataset to estimate the model parameters [START_REF] Ayodele | Types of Machine Learning Algorithms[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Supervised learning algorithms create a model from examples (training dataset) where each of the input/independent variables (e.g. neural features) are associated with the desired decoder output/dependent variables (e.g. labels or movements) [START_REF] Ayodele | Types of Machine Learning Algorithms[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Unsupervised learning strategies group the algorithms which extract a model from the input variables without any information on the corresponding desired output variables. These algorithms aim to find groups in which input data can be clustered because of their input neural features similarities [START_REF] Ayodele | Types of Machine Learning Algorithms[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Reinforcement learning strategies are learning the most Chapitre 1 : Introduction suitable actions to perform depending on the input variables to maximize a reward signal (representative of the output variables). The optimal output variables are unknown and must be discovered by trial and error processes [START_REF] Ayodele | Types of Machine Learning Algorithms[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF].

Commonly, initial model parameters estimation for BCI application was performed based on supervised learning strategies. Nowadays, BCI experiments were realized in a defined environment where output variables were easily accessible. While few BCI fully unsupervised training procedures were tested during offline and online EEG-based [START_REF] Hüebner | Unsupervised Learning for Brain-Computer Interfaces Based on Event-Related Potentials: Review and Online Comparison [Research Frontier[END_REF][START_REF] Kindermans | True Zero-Training Brain-Computer Interfacing -An Online Study[END_REF] and online MEA [START_REF] Paraskevopoulou | Hierarchical Adaptive Means (HAM) clustering for hardware-efficient, unsupervised and real-time spike sorting[END_REF] experiments, unsupervised and reinforcement learning strategies were poorly explored for BCI applications and were preferred when output variables were not recorded.

Finally, all the presented algorithms groups are separable between the offline, online and online incremental/adaptive algorithms. The offline algorithm category gathers the algorithms limited to applications in post-treatments after the recording of the data. This limitation is generally related to the required computation time, cross-validation optimization requirements or the need to have the entire dataset (from the start to the end) to process the input data. Online algorithms denote the algorithms that can be applied during closed-loop online experiments with a fixed model trained previously offline. The training of the algorithms is heavy and cannot be realized during the experiments, nevertheless, the application of the model is sufficiently optimized to process in real-time the neural feature data-flow. Incremental/adaptive algorithms encompass the algorithms which can be applied and evaluated in real-time. Incremental adaptive decoders incrementally update their parameters, optimizing in real-time the model parameters to the user brain signals variations.

Contrary to offline and initial model parameter estimations which are generally based on supervised training strategy, adaptive algorithms were reported using both supervised and unsupervised re-updating strategies. For example, several adaptive algorithms using supervised and non-supervised adaptation during BCI experiments using EEG recordings were reviewed in [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. However, the majority of the online BCI algorithms were tested during offline dataset analysis. Tests during online BCI experiments should be a gold standard and deeper investigation during online BCI experiments must be achieved to evaluate the benefits of the online non-adaptive and adaptive algorithms. Online non-adaptive or online adaptive algorithms are mandatory for daily life BCI application, nevertheless, the online property of the algorithm brings new requirements and specifications to the algorithms such as computational load, complexity, [START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF], etc.

Post-processing methods are generally used to smooth and/or reduce unlikely predictions and errors of the decoder to improve prediction performance. Post-processing can also be used to integrate a priori knowledge on the effector possible operations, limitations or restrictions (e.g. maximum velocity limitation, physical barriers to not cross, do not allow walking and sitting state transition too fast, etc.) [Bashashati et al., 2007a] [Schaeffer and[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF].

Effectors

Numerous effectors have been designed and integrated into BCI systems in various domains such as the entertainment industry [START_REF] Lafleur | Quadcopter control in threedimensional space using a noninvasive motor imagery-based brain-computer interface[END_REF][START_REF] Mudgal | Brain computer interface advancement in neurosciences: Applications and issues[END_REF] [Nicolas-Alonso and Gomez-Gil, 2012], the healthcare domain with diverse prevention and detection applications [START_REF] Lee | Clinical neuroprosthetics: Today and tomorrow[END_REF][START_REF] Mudgal | Brain computer interface advancement in neurosciences: Applications and issues[END_REF][START_REF] Ramadan | Basics of Brain Computer Interface[END_REF][START_REF] Roy | Chapter 98 -EEG and FNIRS Connectivity Features for Mental Workload Assessment: A Preliminary Study[END_REF], etc.

BCI committed to healthcare applications and more particularly BCI controlled by patients suffering from severe motor disabilities required to remain safe and harmless in any condition. Numerous effectors were controlled using BCI by disabled patients for various objectives and applications.

The BCIs dedicated to communication such as spellers were designed and controlled using BCI [START_REF] Kim | Elastic net ensemble classifier for event-related potential based automatic spelling[END_REF][START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF] but do not present direct risk and threat to the user even though false activations remain problematic. On the opposite, motor BCIs interact directly with the environment and the user. False activations may be problematic in out-of-lab applications.

Numerous studies using real (not virtual) effectors have been designed. In particular, several BCIs and motor BCIs created for the complete functional compensation of disabled patients were reported in [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF], 2006[START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF] with numerous effectors. In clinical experiments, robotic assistance of individuals unable to perform any muscular activation was performed using neuro-orthoses or prostheses for various tasks such as grasping, upper [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Edelman | Noninvasive neuroimaging enhances continuous neural tracking for robotic device control[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF][START_REF] Morinière | EMY: a dual arm exoskeleton dedicated to the evaluation of Brain Machine Interface in clinical trials[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF] or lower [START_REF] He | Brain-machine interfaces for controlling lower-limb powered robotic systems[END_REF][START_REF] Kwak | A lower limb exoskeleton control system based on steady state visual evoked potentials[END_REF][START_REF] López-Larraz | Control of an Ambulatory Exoskeleton with a Brain-Machine Interface for Spinal Cord Injury Gait Rehabilitation[END_REF]] [Zhang et al., 2018] limb movements or all the tasks together using an exoskeleton [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]] [Eliseyev et al., 2014].

A 6 DoF commercialized robotic arm JACO (from Kinova Robotics company) was used in EEG-based BCI studies for 2D and 3D movements control [START_REF] Baxter | Noninvasive control of a robotic arm in multiple dimensions using scalp electroencephalogram[END_REF][START_REF] Bhattacharyya | Interval type-2 fuzzy logic based multiclass ANFIS algorithm for real-time EEG based movement control of a robot arm[END_REF], 2017a[START_REF] Edelman | Noninvasive neuroimaging enhances continuous neural tracking for robotic device control[END_REF]] [Huang et al., 2019[START_REF] Meng | Noninvasive Electroencephalogram Based Control of a Robotic Arm for Reach and Grasp Tasks[END_REF][START_REF] Postelnicu | Towards Hybrid Multimodal Brain Computer Interface for Robotic Arm Command[END_REF]. The DLR Light-Weight Robot III combined with the Five-Finger Hand was controlled in an end-point velocity space (7 DoF) by a tetraplegic patient using MEA neural signal decoder [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF]. The same experiments were performed using the DEKA robotic arm for 6 DoF control in [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF]. The DLR prosthetic system provides potentially a 7 DoF arm and 15 DoF hand (shoulder Chapitre 1 : Introduction abduction, shoulder flexion, humeral rotation and elbow flexion, wrist flexion, wrist rotation) and 4 DoF in the hand [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF].

John Hopkins University designed the modular prosthetic limb, an anthropomorphic prosthesis which enables in end-point-control mode to command independently 16 degrees of freedom (3D translation and 3D orientation of the hand, as well as 1D flexion/extension of each finger, ab/adduction of the index finger, combined ab/adduction of the little and ring fingers, and 4D control of the thumb) [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Johannes | An overview of the developmental process for the modular prosthetic limb[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. This prosthesis was used to performed 10 DoF control by a tetraplegic patient using a MEA recording system [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. Additionally, full-body exoskeleton [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]] [Eliseyev et al., 2014] have been designed and controlled via neural signals decoding. A whole-body exoskeleton with 14 actuated DoF was used for BCI experiments using epidural ECoG recording systems [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF].

Numerous studies on lower or upper limb control of an exoskeleton using EEG acquisition system were reported in [START_REF] Al-Quraishi | EEG-Based Control for Upper and Lower Limb Exoskeletons and Prostheses: A Systematic Review[END_REF]. However, these EEG experiments were carried out with a lower number of DoF than experiments with more invasive recording systems and were generally performed with healthy subjects.

A commonly controlled effector providing high mobility to disable patients is the wheelchair. Wheelchairs were adapted to BCI control giving back mobility to patients using EEG recordings [START_REF] Huang | Electroencephalography (EEG)-Based Brain-Computer Interface (BCI): A 2-D Virtual Wheelchair Control Based on Event-Related Desynchronization/Synchronization and State Control[END_REF][START_REF] Leeb | Self-Paced (Asynchronous) BCI Control of a Wheelchair in Virtual Environments: A Case Study with a Tetraplegic[END_REF]] [Li et al., 2013[START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF].

BCIs were also used for functional rehabilitation of individuals suffering from muscular activity deterioration to help patients and brain plasticity to recover [START_REF] David | Contralesional Brain-Computer Interface Control of a Powered Exoskeleton for Motor Recovery in Chronic Stroke Survivors[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Carvalho | Brain-machine interface of upper limb recovery in stroke patients rehabilitation: A systematic review[END_REF][START_REF] Donati | Long-Term Training with a Brain-Machine Interface-Based Gait Protocol Induces Partial Neurological Recovery in Paraplegic Patients[END_REF][START_REF] Frolov | Post-stroke Rehabilitation Training with a Motor-Imagery-Based Brain-Computer Interface (BCI)-Controlled Hand Exoskeleton: A Randomized Controlled Multicenter Trial[END_REF][START_REF] López-Larraz | Brain-machine interfaces for rehabilitation in stroke: A review[END_REF][START_REF] Mak | Clinical Applications of Brain-Computer Interfaces: Current State and Future Prospects[END_REF][START_REF] Mattia | Brain Computer Interface for Hand Motor Function Restoration and Rehabilitation[END_REF][START_REF] Mudgal | Brain computer interface advancement in neurosciences: Applications and issues[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Pfurtscheller | Rehabilitation with Brain-Computer Interface Systems[END_REF][START_REF] Qin | eConHand: A Wearable Brain-Computer Interface System for Stroke Rehabilitation[END_REF][START_REF] Van Dokkum | Brain computer interfaces for neurorehabilitation -its current status as a rehabilitation strategy post-stroke[END_REF][START_REF] Webb | Towards a portable assistive arm exoskeleton for stroke patient rehabilitation controlled through a brain computer interface[END_REF]. Experiments oriented for functional rehabilitation applications were reported using real effectors dedicated to grasping or wrist rotation [START_REF] David | Contralesional Brain-Computer Interface Control of a Powered Exoskeleton for Motor Recovery in Chronic Stroke Survivors[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Carvalho | Brain-machine interface of upper limb recovery in stroke patients rehabilitation: A systematic review[END_REF][START_REF] Frolov | Post-stroke Rehabilitation Training with a Motor-Imagery-Based Brain-Computer Interface (BCI)-Controlled Hand Exoskeleton: A Randomized Controlled Multicenter Trial[END_REF][START_REF] Qin | eConHand: A Wearable Brain-Computer Interface System for Stroke Rehabilitation[END_REF][START_REF] Van Dokkum | Brain computer interfaces for neurorehabilitation -its current status as a rehabilitation strategy post-stroke[END_REF], upper [START_REF] Carvalho | Brain-machine interface of upper limb recovery in stroke patients rehabilitation: A systematic review[END_REF][START_REF] Van Dokkum | Brain computer interfaces for neurorehabilitation -its current status as a rehabilitation strategy post-stroke[END_REF][START_REF] Webb | Towards a portable assistive arm exoskeleton for stroke patient rehabilitation controlled through a brain computer interface[END_REF] and lower [START_REF] Donati | Long-Term Training with a Brain-Machine Interface-Based Gait Protocol Induces Partial Neurological Recovery in Paraplegic Patients[END_REF] limb movements, functional electrical stimulation (FES) [START_REF] Carvalho | Brain-machine interface of upper limb recovery in stroke patients rehabilitation: A systematic review[END_REF][START_REF] Mattia | Brain Computer Interface for Hand Motor Function Restoration and Rehabilitation[END_REF][START_REF] Van Dokkum | Brain computer interfaces for neurorehabilitation -its current status as a rehabilitation strategy post-stroke[END_REF] etc. The systems used for functional rehabilitation generally present low DoF (1 or 2) as the possible actions provided to the patient are restricted to specific movements.

Nevertheless, the most widespread category of applications belongs to the virtual effector family due to its set up simplicity and accessibility for clinical and preclinical studies. Many experiments based on 1D, 2D or 3D cursors [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF][START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Dangi | Continuous Closed-Loop Decoder Adaptation with a Recursive Maximum Likelihood Algorithm Allows for Rapid Performance Acquisition in Brain-Machine Interfaces[END_REF]] [Kao et al., 2017[START_REF] Leuthardt | A brain-computer interface using electrocorticographic signals in humans[END_REF][START_REF] Marathe | The impact of command signal power distribution, processing delays, and speed scaling on neurally-controlled devices[END_REF]] [Orsborn et al., 2014[START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF] virtual avatar, arms or environments [START_REF] Huang | Electroencephalography (EEG)-Based Brain-Computer Interface (BCI): A 2-D Virtual Wheelchair Control Based on Event-Related Desynchronization/Synchronization and State Control[END_REF][START_REF] Ifft | A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys[END_REF]] [Lebedev and Nicolelis, 2011[START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] were reported.

In all the cases, the effectors must be designed to provide safe use to the patient and be as transparent as possible (ease of use and a high number of possible actions). Numerous safety restrictions are generally used for effectors that are in direct contact with the patients. As an example, virtual boundaries were fixed in [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF] during the post-processing step to avoid collisions between the robotics arms, the experimental setup and the patient.

Feedback

The sensory feedbacks are the information about the task evolution, success or failure provided to the patient which allows him to react and adapt to the current situation. Feedback is crucial in BCI applications. Feedback highly influences the model convergence, parameter estimation [START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF] and performance. As an example, several studies highlighted that higher decoding frequency improved effector control [START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF] whereas visual feedback delay significantly affected the decoding performance [START_REF] Marathe | The impact of command signal power distribution, processing delays, and speed scaling on neurally-controlled devices[END_REF].

In the majority of the motor BCI studies dedicated to functional compassion, the feedback provided to the patient is restricted to visual feedback. Indeed, besides being easier to integrate within the BCI experiments (using a screen as visual feedback for 2D or 3D cursors control), it is generally the only feedback that can be provided to paraplegic and tetraplegic patients who lose their other sensory feedbacks. Few studies tried to add other feedbacks to the visual feedback such as kinaesthetic (sense of body movement) for the upper limb decoding, the haptic feedback (sense of touch) for grasping control or vibrotactile feedback for cursor control [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF].

Multiple sensory feedback types are much more common in motor BCI experiments designed for functional rehabilitation. Most of the patients did not lose their entire sensory system and used multiple feedbacks to improve their rehabilitation. Regardless of the effector, numerous reported experiments combined visual, kinesthetic and proprioceptive (sense of body positioning) and/or haptic feedbacks [START_REF] Carvalho | Brain-machine interface of upper limb recovery in stroke patients rehabilitation: A systematic review[END_REF][START_REF] Mattia | Brain Computer Interface for Hand Motor Function Restoration and Rehabilitation[END_REF][START_REF] Van Dokkum | Brain computer interfaces for neurorehabilitation -its current status as a rehabilitation strategy post-stroke[END_REF]

BCI control strategies: somatotopic remapping versus direct neural decoding

Numerous BCI control strategies were experimented to obtain the most accurate neural signal predictions. BCIs and control strategies can be clustered into exogenous and endogenous BCIs [Chan et al., 2015] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]] according to the nature of the neural signals used to control the effector. On the one hand, exogenous BCI relies on patient's neural signal variations related to an external stimulus named evoked potentials (EP) and more specifically event-related potentials (ERP). This category regroups visual (VEP), auditory, P300 evoked potentials, error-related potentials (ErrP), etc. On the other hand, endogenous BCIs use the natural variations of the sensorimotor neural signal rhythms to decode the patient's intentions. They do not rely on external systems. Endogenous BCIs encompass two BCI control strategies: the direct neural decoding strategy and the somatotopic mapping/mental task strategy [Chan et al., 2015] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. Another evoked potential commonly used in BCI application, especially for communication BCI, is the P300 evoked potential. P300 evoked potentials is characterized by a positive variation of the neural signals elicited around 300 ms after scarce auditory, visual, or somatosensory stimuli [START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. P300 evoked potentials do not require patient's training, nevertheless the bit rate information is lower than VEP and the P300 amplitude modulation may be reduced due to patient's habituation [START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF].

Exogenous BCIs

Several BCI were designed based on evoked potentials decoding. Generally, they relied on EEG recording systems. Several evoked potentials were used for various BCI applications such as 2D cursor control or spellers [START_REF] Dornhege | Boosting bit rates in noninvasive EEG single-trial classifications by feature combination and multiclass paradigms[END_REF][START_REF] Huang | Electroencephalography (EEG)-Based Brain-Computer Interface (BCI): A 2-D Virtual Wheelchair Control Based on Event-Related Desynchronization/Synchronization and State Control[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF]. However, P300 and SSVEP were the most common patterns decoded in the BCI field. SSVEP was used to perform online experiments for 2D cursor control [START_REF] Trejo | Brain-computer interfaces for 1-D and 2-D cursor control: designs using volitional control of the EEG spectrum or steady-state visual evoked potentials[END_REF], wheelchair control [START_REF] Li | A Hybrid BCI System Combining P300 and SSVEP and Its Application to Wheelchair Control[END_REF]] [Müller et al., 2015], grasping control of a hand orthosis [START_REF] Ortner | An SSVEP BCI to Control a Hand Orthosis for Persons With Tetraplegia[END_REF] or lower limb exoskeleton control [START_REF] Kwak | A lower limb exoskeleton control system based on steady state visual evoked potentials[END_REF] from EEG neural signals. EEG neural signal decoders based on P300 evoked potentials were tested during online motor BCI experiments to control a wheelchair [START_REF] Annese | Wireless Brain-Computer Interface for Wheelchair Control by Using Fast Machine Learning and Real-Time Hyper-Dimensional Classification[END_REF][START_REF] Iturrate | Synchronous EEG brain-actuated wheelchair with automated navigation[END_REF]] [Li et al., 2013]. However, they were more commonly reported in online experiments [START_REF] Lin | A novel P300 BCI speller based on the Triple RSVP paradigm[END_REF][START_REF] Long | Semi-supervised joint spatio-temporal feature selection for P300-based BCI speller[END_REF][START_REF] Utsumi | Operation of a P300-based brain-computer interface in patients with Duchenne muscular dystrophy[END_REF] and offline studies [START_REF] Kim | Elastic net ensemble classifier for event-related potential based automatic spelling[END_REF]

Endogenous BCIs

Endogenous BCIs provide commands to an effector directly from brain signals variations decoding without any external stimuli. Endogenous BCIs are less restricted systems than exogenous ones which rely on few state classification task based on various external stimulation. However, endogenous BCI models are more complicated to train and are more prone to errors. Endogenous BCI can be clustered into two control strategies: direct neural decoding (also named kinematic decoding) [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF] and somatotopic remapping approach also named somatotopic mapping/mental task strategy, arbitrary-mapping paradigms [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF].

The Somatotopic remapping approach [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF], is also referred as somatotopic mapping/mental task strategy [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF] or arbitrary-mapping paradigms [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] in the literature. As mentioned in 1.1, the brain motor cortex has a somatotopic organization: specific parts of the body are associated with distinct locations and specific neural signals patterns (the left hand and left foot movement do not activate the same motor cortex area). The somatotopic mapping approach aims to associate specific BCI commands to arbitrary selected attempted (realized or imagined) movements which have distinct neural pattern activations between each other [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. As an example, Figure 1-8A, shows the somatotopic mapping strategy used by five epileptic patients to control a 2D cursor with attempted real or imagined movements in [START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF]. The Figure 1-8B highlights the distinct motor cortex activations related to actual and imagined movements of the tongue and the hand. The mental imagination of a motor task without its execution (without movements) is referred to as Motor Imagery (MI) in the literature. The somatotopic mapping strategies to control BCI systems are numerous and varied, tongue, jaw, hands, shoulders, elbows, fingers, legs, feet [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] and are not restricted to strategies based on motor cortex activations [START_REF] Müller-Putz | Chapter 18 -Electroencephalography[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. [START_REF] Scherer | Individually Adapted Imagery Improves Brain-Computer Interface Performance in End-Users with Disability[END_REF] highlighted that the use of "brain-teaser" such as mental subtraction and mental word association combined with more classical motor imagery strategies increase the classification performance compared to motor imagery strategy alone.

Mental tasks strategy was generally performed with a neural population recording system distributed in a sufficiently large area of the brain to performed distinct motor imagery (ECoG, EEG, MEG, etc.) [START_REF] Müller-Putz | Chapter 18 -Electroencephalography[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. EEG studies reported accurate 2D continuous pursuit task (cursor tracking) [START_REF] Edelman | Noninvasive neuroimaging enhances continuous neural tracking for robotic device control[END_REF] and 3D sequential reach and grasp task (2D movements then automatized 1D grasp movements) [START_REF] Meng | Noninvasive Electroencephalogram Based Control of a Robotic Arm for Reach and Grasp Tasks[END_REF] using left and right hand motor imagery for left or right movements while rest state or both hands motor imagery were associated to up and down movements. EEG based 3D virtual cursor control was reported by [START_REF] Mcfarland | ELECTROENCEPHALOGRAPHIC (EEG) CONTROL OF THREE-DIMENSIONAL MOVEMENT[END_REF] using MI ("initially employed" to refer to the article) with intensive subjects training. The four subjects underwent between 24-96min, 4 to 5 hours and 8 to 17 hours of training to control a cursor in the 1D, 2D and 3D space respectively. Virtual drones were controlled in the 3D space using hands MI in [START_REF] Lafleur | Quadcopter control in threedimensional space using a noninvasive motor imagery-based brain-computer interface[END_REF][START_REF] Royer | EEG Control of a Virtual Helicopter in 3-Dimensional Space Using Intelligent Control Strategies[END_REF].

Somatotopic remapping strategy in ECoG-based experiments is the most common control strategy performed. Early BCI studies using ECoG neural signal recording systems were carried out using somatotopic remapping strategy [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] and ECoG-driven state of the art BCI continued to exploit it. Wang et al., in two online BCI experiments [Degenhart et al., 2018] [W. Wang et al., 2013], highlighted 2D and 3D cursor control by three disabled patients using motor imagery strategy and ECoG recording system. These experiments were conducted with a tetraplegic patient caused by complete C4 level spinal cord injury seven years before the study, a patient diagnosed with amyotrophic lateral sclerosis nine years prior to the study and a subject with left brachial plexus injury three years before his enrolment in the study [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF]. These patients underwent a craniotomy to implant a high-density ECoG grid of 32 (for the two first patients) and 64 (for the third patient) electrodes embedded in a 2cm × 4cm (for the two first patients) or 4cm × 4cm grid. The proposed task was to firstly control a 2D cursor to perform a center out task with 8 targets for the first and third subjects and 4 targets for the second subject. In a next step, 3D cursor center out task was performed with 8 targets for all the subjects. Motor imagery strategies were different for each subject depending on neural signal modulation in the gamma band and spatially distinct patterns. The MI strategies performed during the experiments are represented in Figure 123456789. For each patient, the optimal motor imagery strategy was determined during prior motor screening task analysis to identify the attempted movements which generated the strongest cortical modulations [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF]. [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] to performed 3D center out task control with 8 targets. M1 and M2 were used to control the displacement of the target in the X-Y axis whereas M3 was associated to Z-axis (depth axis). "+" represents attempted movements whereas ø" shows relaxation.

Based on somatotopic remapping strategies, patients highlighted a 85 ± 6 % and 75 ± 10 % success rates (targets hits) during online 2D center out experiments and 3D center out experiments respectively. To maintain the decoding performance, the decoder was regularly recalibrated "as needed" to improve the cursor control and reduce the sudden drop off decoding performance. Re-calibration sessions were performed before and between test trials with a total of 8, 19 and 5 re-calibration sessions for the first, second and third patients, respectively.

Direct neural decoding relies on the decoding of single neurons or population neural signals directly related to movement control [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. In the mid-1980's, Georgeopoulos et al. highlighted in preclinical experiments that the action potential firing rates of the premotor and motor cortex neurons were correlated to specific movement directions. The firing rate of a premotor and motor cortex neuron was directionally tuned by the movement direction. The activity of several neurons (named population vector) of the premotor and motor cortex appeared to provide the direction of visually guided movements. [START_REF] Georgopoulos | Neuronal population coding of movement direction[END_REF][START_REF] Georgopoulos | Coding of movements in the motor cortex[END_REF][START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF] and other characteristics such as speed, velocity [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF], etc. A milestone was reached in 2006, Hochberg et al. [START_REF] Hochberg | Neuronal ensemble control of prosthetic devices by a human with tetraplegia[END_REF]] demonstrated that similar firing rate directional tuning of the motor cortex neurons could be estimated through the recording of the motor cortex neurons of a tetraplegic patient imagining hand movements using an implanted 96-microelectrode array. Moreover, Hochberg et al proved that the neural signals modulations of a tetraplegic patient imagining hand movements could be exploited to control 2D cursor position and perform center-out tasks. These results highlighted that intended (and not only realized) movements were correlated to neural population firing rate recorded through invasive intracortical electrodes even for a 3 years old spinal cord injured patient.

BCI studies based on direct neural decoder were generally performed with invasive MEA recordings. As mentioned in 1.2.1, MEA systems directly record single or multiunits neuronal activities and are by definition suited to direct neural recording. High dimensional and accurate control was performed using MEA and direct neural decoding strategy. [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF], based on Kalman filter, allowed a tetraplegic patient to perform online 3D hand translation and grasp state control of a robotic arm. [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF] highlighted the online 7 dimensional control of a robotic arm using indirect optimal linear estimation (OLE) with ridge regression whereas Wodlinger et al. [START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]] performed a 10 Dimensional control of a robotic arm by tetraplegic patient using indirect OLE. While it is less Chapitre 1 : Introduction common, kinematic decoding of lower limb bipedal walking was also highlighted in rhesus macaque using direct neural decoding strategy [START_REF] Fitzsimmons | Extracting Kinematic Parameters for Monkey Bipedal Walking from Cortical Neuronal Ensemble Activity[END_REF][START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF].

Although most studies were performed based on SUA and MUA decoding using MEA systems, correlation between neural signals amplitude variation and center-out movements from population recording systems was reported [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF].

During center out movements, similar neural signal modulation from LFP, ECoG, EEG and MEG recordings were reviewed in [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF] [START_REF] Waldert | Hand Movement Direction Decoded from MEG and EEG[END_REF] stressed significant directional tuning of the neural signals and real movements of the hand controlling a cursor in the 2D space with a joystick (four targets) (67% accuracy with MEG recordings). Moreover, Schwarz et al. [START_REF] Schwarz | Decoding hand movements from human EEG to control a robotic arm in a simulation environment[END_REF] highlighted online direct neural classification of three grasping types (palmar and lateral grasps, and wrist supinations) with 48% corrects trials performed by fifteen healthy subjects with EEG recording system. Direct neural decoding based on ECoG recording system was reported with performance highly dependent on the experimental paradigms, patients' status (healthy or disabled), analysis and electrodes localizations [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF]. Schalk et al. [START_REF] Schalk | Decoding two-dimensional movement trajectories using electrocorticographic signals in humans[END_REF] reported in 2007 an offline study where five epileptic patients implanted with subdural ECoG performed, using a joystick, a 2D cursor tracking task of a target moving in a counterclockwise circular trajectory. The average correlation between neural signals and velocity was 0.48 ± 0.09%. [START_REF] Ball | Differential representation of arm movement direction in relation to cortical anatomy and function[END_REF] highlighted the cosine tuning of four epileptic patient's between the subdural ECoG signals and their arm movements during 2D center-out tasks with four to eight targets (squared correlation coefficient of 0.67 for 34Hz-128Hz band). [START_REF] Anderson | Electrocorticographic (ECoG) correlates of human arm movements[END_REF] tested the correlation between subdural ECoG signals direction, velocity and speed from seven epileptic patients. Patients performed 2D center-out and tracing tasks with a force feedback joystick. The results underlined a higher modulation of direction, velocity and speed in the motor cortical areas (depth of modulation around 0.17, 0.38, 0.23 for the direction, velocity and speed respectively) [START_REF] Anderson | Electrocorticographic (ECoG) correlates of human arm movements[END_REF]. Finally, 3D center-out task movement reconstruction was achieved by Bundy et al [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF] from five epileptic patients implanted with subdural ECoG performing 3D real hands movements. Correlation coefficients between recorded and predicted position, velocity and speed were 0.3656 ± 0.1384 for the position, 0.3461 ± 0.1119 for the velocity and 0.6208 ± 0.1893 for the speed.

Other ECoG-based motor BCI experiments using direct neural decoding strategy were designed to decode finger movements [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF]. Based on five epileptic patients implanted with subdural ECoG who were instructed to move specific individual fingers in response to visual cues, finger movements were extracted from neural signals using various algorithms [START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF][START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF][START_REF] Xie | Decoding of finger trajectory from ECoG using deep learning[END_REF]. Flint et al [START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF] reported continuous grasp and finger joint movements decoding from 5 epileptic patients with epidural and/or subdural ECoG recording systems. They reported similar results than previous experiments, highlighting that low-frequency modulation (7)(8)(9)(10)[START_REF]Offline evaluation of the developed algorithms[END_REF](12)(13)(14)(15)[START_REF] Suminski | Online adaptive decoding of intended movements with a hybrid kinetic and kinematic brain machine interface[END_REF](17)(18)(19)(20) encodes movement onset (as grasp aperture) whereas high frequency variations (above 70 Hz) are correlated with finer movements (fingers, grip force, etc.). A study based on two epileptic patients implanted with subdural ECoG over the interhemispheric M1 area was performed to evaluate the direct neural decoding performance of ECoG recording system for lower limb control [START_REF] Mccrimmon | Electrocorticographic Encoding of Human Gait in the Leg Primary Motor Cortex[END_REF].

The results of this study suggested that high frequency 𝛾 band (40-200 Hz) provides information on the lower limb high-level motor control (walking duration and speed) and do not encode muscle activations or muscle trajectories. Nevertheless, lower limb kinematic decoding is a poorly developed BCI field and requires more studies to have a clear representation of the information which can be extracted from ECoG recordings.

Although motor imagery approaches lead to interesting results, this strategy may be limited in many aspects. Firstly, 10 to 30% of users are unable to control MI-BCIs [START_REF] Jeunet | Chapter 1 -Advances in user-training for mental-imagery-based BCI control: Psychological and cognitive factors and their neural correlates[END_REF]. Moreover, the control of complex effectors with high dimensional control may be difficult to handle for patients as motor imagery complexity highly increase with task complexity (highlighted by the density of articles using MI to control 2D cursor Chapitre 1 : Introduction compared to 3D ones). More complex effector control required more MI strategies, which are by definition limited (two hands, elbows, shoulders, etc.). Finally, for rehabilitation applications, patients must perform natural movements in the hope of improving the affected limb movements. However, the MI-based BCIs which are not using the natural somatotopic mapping of the brain might be useless compare to direct neural BCI for the specific case of rehabilitation applications.

Motor BCI requirements for daily life applications

The Motor Brain computer interfaces estimate a command from the neural activity and send it to an effector which performs the movement imagined by the patient [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF]. Motor BCIs are particularly useful for disabled patients who lost entirely or partially the natural neuromuscular activation path. Motor BCIs can be an interesting approach for both robotic assistance and neurorehabilitation therapy of individuals suffering from severe motor disabilities [START_REF] Donati | Long-Term Training with a Brain-Machine Interface-Based Gait Protocol Induces Partial Neurological Recovery in Paraplegic Patients[END_REF][START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] López-Larraz | Brain-machine interfaces for rehabilitation in stroke: A review[END_REF]. Spectacular BCI milestones have been reached over the years in the motor BCI field [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF] [W. Wang et al., 2013[START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. These milestones have sustained the aim of translating BCI-driven systems from laboratories directly into patients' home for daily life applications. In order to develop Motor BCI for future daily life applications, many challenging aspects and restrictions need to be addressed.

Acquisition system requirements

The primary challenge of motor BCIs for clinical and daily-life applications is safe, chronic and stable neural recordings over time. Biocompatibility as well as stability over time are mandatory for recording devices designed for long-term BCI clinical use. Brain signal recordings need to remain accurate in conditions less favorable than laboratories.

MEA recordings, post-surgery, are a safe recording system, nevertheless, they have biocompatibility issues with signal degradation over time (decreasing signal-to-noise ratio), loss of electrodes [START_REF] Gunasekera | Intracortical Recording Interfaces: Current Challenges to Chronic Recording Function[END_REF][START_REF] Jorfi | Progress Towards Biocompatible Intracortical Microelectrodes for Neural Interfacing Applications[END_REF][START_REF] Marin | Biocompatibility of intracortical microelectrodes: current status and future prospects[END_REF][START_REF] Moran | Evolution of brain-computer interface: action potentials, local field potentials and electrocorticograms[END_REF][START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF][START_REF] Rousche | Chronic intracortical microstimulation (ICMS) of cat sensory cortex using the Utah intracortical electrode array[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF]] [Ward et al., 2009] and have high across-day variation in the neural signals [START_REF] Perge | Intra-day signal instabilities affect decoding performance in an intracortical neural interface system[END_REF][START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF]. However, several studies highlighted significant results with MEA implanted since hundreds days [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF]] [Simeral et al., 2011[START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. Wodlinger et al [START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF] performed 7 dimensional and 10 dimensional robotic arm control by a tetraplegic patient from 32 to 280 days postsurgery. Simeral et al [START_REF] Simeral | Neural control of cursor trajectory and click by a human with tetraplegia 1000 days after implant of an intracortical microelectrode array[END_REF] demonstrated 2D point and click control using SUA recordings by a tetraplegic patient 1000 days after the surgery despite only 57 electrodes over 96 electrodes were obtained (41 electrodes utilized). Milekovic et al [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF] based on LFP recordings enable an ALS patient to control a BCI for communication 550 days after implantation. However, a chronic preclinical study evaluating the MEA recording robustness over time highlighted that 56% of the recording systems failed within a year of implantation [START_REF] Barrese | Failure mode analysis of silicon-based intracortical microelectrode arrays in non-human primates[END_REF].

MEA recording is restricted to wired connection to a computer due to the high sampling rate of the system. Such wired platform is limiting for daily life application and enhanced the infection risks.

Electrocorticography (ECoG) provides a fair compromise between invasiveness, signal resolution and quality [START_REF] Leuthardt | Electrocorticography-based brain computer Interface-the seattle experience[END_REF], 2004[START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF]. ECoG recordings have fewer biocompatibility troubles than MEA. However, as previously mentioned, ECoG arrays are generally implanted to detect the epileptic sources of patients before surgery limiting the ECoG clinical trial from several days to 1 or 2 weeks (less than 28 days) of research (with an implantation from 3 to 35 days) [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Leuthardt | A brain-computer interface using electrocorticographic signals in humans[END_REF]] [Nakanishi et al., 2013[START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF][START_REF] Schalk | Decoding two-dimensional movement trajectories using electrocorticographic signals in humans[END_REF], 2008[START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] [W. Wang et al., 2013[START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF]. Bundy et al [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF] reconstructed 3D hand movements performed by five patients with intractable epilepsy who underwent subdural ECoG arrays implantation for 5 to 14 days to localize their epileptic foci and map cortices for presurgical planning. Schalk et al's study [START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF] on 2D cursor control is based on five epileptic patients who had subdural ECoG arrays implanted for 7-14 days in preparation for surgery. Nevertheless, some pre-clinical [START_REF] Costecalde | A Long-Term BCI Study With ECoG Recordings in Freely Moving Rats[END_REF]] [Sauter-Starace et al., 2019] and clinical [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF][START_REF] Nurse | Consistency of Long-Term Subdural Electrocorticography in Humans[END_REF][START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF] studies showed good signal-to-noise ratio stability in ECoG signals over months or years, encouraging the use of BCIs in long-term applications.

Nurse et al [START_REF] Nurse | Consistency of Long-Term Subdural Electrocorticography in Humans[END_REF] highlighted that subdural ECoG arrays (two ECoG grids for a total of 16 electrodes) can robustly record high frequency neural signal activities on 15 epileptic patients who underwent ECoG monitoring for 184 to 766 days. Amyotrophic Lateral Sclerosis (ALS) patients performed stable control of a subdural ECoG based-BCI system for communication over 36 months in [START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF].

Benabid et al [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF] reported epidural ECoG stability over 24 months after implantation for the BCI control of an exoskeleton by a tetraplegic patient. ECoG recording is a reliable solution for chronic BCI system. However, as for MEA-based BCI the infection risks associated to the use of tethered cables is significant [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] but the breakthrough has been made toward the development and the test of wireless fully-implantable technologies [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF][START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF] [START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF][START_REF] Shimoda | Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques[END_REF]. While decoding performance of epidural ECoG are lower than subdural one, epidural ECoG still presents good decoding performance and is one of the safest invasive recording methods. Additionally, epidural ECoG decoding of unimodal and bimanual upper limb movements was reported during offline preclinical experiments [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]. Therefore, epidural ECoG recording systems seem to be a good trade-off between invasiveness, safety, neural signal stability, wireless recordings and decoding performance.

control system requirements

1.4.2.1. Degree of freedom and accuracy BCI system for daily life application requires providing sufficient freedom to the patients in order to enhance their independence and simplify daily life tasks. The control provided to the patients must be sufficient to reflect the user's intentions and proposes sufficient controllable degrees of freedom (DoF) or dimension to not be restricted to specific actions.

Despite that EEG-based 2D and 3D control experiments were reported in [START_REF] Lafleur | Quadcopter control in threedimensional space using a noninvasive motor imagery-based brain-computer interface[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Mcfarland | ELECTROENCEPHALOGRAPHIC (EEG) CONTROL OF THREE-DIMENSIONAL MOVEMENT[END_REF][START_REF] Royer | EEG Control of a Virtual Helicopter in 3-Dimensional Space Using Intelligent Control Strategies[END_REF][START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF][START_REF] Schwarz | Decoding hand movements from human EEG to control a robotic arm in a simulation environment[END_REF][START_REF] Vilela | Chapter 8 -Applications of brain-computer interfaces to the control of robotic and prosthetic arms[END_REF][START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF], they generally required specific motor imagery strategies [Lotte et al., 2018] [Schalk and[START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF] and high subjects training. As an example between 8 and 17 training hours were required to control a 3D cursor in [START_REF] Mcfarland | ELECTROENCEPHALOGRAPHIC (EEG) CONTROL OF THREE-DIMENSIONAL MOVEMENT[END_REF]. So far, the complexity of control carried out using EEG remains largely inferior to the needs of medical motor BCIs and less efficient than other (more invasive) brain neuronal activity acquisition systems [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF].

MEA-driven BCIs demonstrated better performances compared to less invasive clinical BCI systems. Using two 96-channels MEAs implanted in the left motor cortex Wodlinger and colleagues demonstrated that a tetraplegic patient was able to control 10 DoFs of a robotic arm (including 3D translation movements, 3D rotations and four hands shaping) [START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF].

Numerous pre-clinical and clinical studies have demonstrated the interest in ECoGbased BCIs to control effectors [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]] [Bundy et al., 2016[START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF]] [Eliseyev et al., 2017] [Nakanishi et al., 2013] [Schaeffer and Aksenova, 2016a[START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF][START_REF] Shimoda | Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques[END_REF] [W. Wang et al., 2013[START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF]. Wang et al reported 3D robotic arm and cursor control by tetraplegic (SCI) and upper limb paralysis (ALS) subjects [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] [W. Wang et al., 2013] whereas Bundy et al showed offline 3D hand movement prediction [Bundy et al., 2016] based on ECoG recording strategy of epileptic subjects.

Improving the decoding accuracy and enhancing the possible interactions between the patient and various environments are the major goals of part of the BCI research community. Nevertheless, the number of controllable DoF and the decoding accuracy are not the only BCI requirements for daily life applications as some long term BCI experiments stressed the improvement in the quality of life of disabled patients using communication BCI system with only 1 DoF available [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF][START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF][START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF]. In [START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF], an amyotrophic lateral sclerosis patient used a 1DoF communication BCI system controlled through chronic (over 36 months) subdural ECoG decoding. The article stressed that the patient reported high satisfaction with the BCI system with the exception of the wired recording system which was qualified as "unsatisfied".

Multi-limb control

Daily life actions commonly required bimanual or multi-limb movements. Multi-limb decoding is a poorly explored area of the BCI field compare to single-limb movement decoding. Alternative decoding of multiple-limb could improve greatly motor-impaired patients with simultaneous or alternative multi-limb movements.

The majority of BCI studies were focused on the control of a single limb or a single effector (generally one hand or lower limb effector) [START_REF] Lebedev | Chapter 3 -Toward a whole-body neuroprosthetic[END_REF].

Bimanual movements engage multiple brain areas which are different from unimanual movements creating new neural signals modulations to decode [START_REF] Donchin | Primary motor cortex is involved in bimanual coordination[END_REF]] [Lebedev and Nicolelis, 2011[START_REF] Oliveira | Neural interactions between motor cortical hemispheres during bimanual and unimanual arm movements[END_REF][START_REF] Steinberg | Neuronal populations in primary motor cortex encode bimanual arm movements[END_REF]. Only a few bimanual experiments were reported. Monkeys bimanual movements (2D for each arm) were decoded from MEA neural signals using an Unscented Kalman filter decoder to control a virtual avatar during online experiments [START_REF] Ifft | A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys[END_REF]. Offline preclinical ECoG-based movement detection studies using hierarchical partial least squares algorithm were reported in [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]. Multi-limb BCI systems were mainly Hybrid models were often employed for these multi-limb/ multi-finger experiments using a classifier to detect the activated finger and continuous decoders (or multiple classifiers) to predict their respective movements [START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF]] [Hotson et al., 2016a[START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF].

The limited number of studies reporting bimanual or multi-limb experiments may be related to several technical limitations. Firstly, this shortage may be explained by a lack of experiments with bilateral implantation of invasive recording devices. During intended movements, the motor cortex activity modulation is stronger on the contralateral side. Therefore, in the case of multi-limb BCI applications with direct neural decoding, bilateral implantation is a mandatory criterion. Poor resolution of noninvasive recording systems likely impedes the related study in EEG-driven BCIs.

Secondly, multi-limb effectors must be available which may be complicated to design. Finally, bimanual or multi-limb decoding requires more complex algorithms and a longer calibration procedure. Deeper investigations on such experiments should be carried out.

Asynchronous BCI with idle state support

In the scope of daily life application, BCI must be a stand-alone system which can be freely used at any time by the patient without external help or cue. This feature implies discriminating the intentional movements and the idle period from the patient's neural signals. Moreover, false activation of the BCI system must be rare events.

BCI can be clustered into the cue-based triggered (synchronous) and the self-paced (asynchronous) systems (Figure 1-12). Synchronous BCIs analyze the neural signals in predefined time windows. After visual or auditory stimuli (cue) generated by the BCI or a researcher, the patient performs the mental task during a time interval which produces a command (any neural signal produced outside the time windows are ignored) [Müller-Putz et al., 2006] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. On the other hand, Asynchronous control systems continuously analyze the ongoing brain activity without any temporal restriction [Bashashati et al., 2007a[START_REF] Müller-Putz | Brain-computer interfaces for control of neuroprostheses: from synchronous to asynchronous mode of operation / Brain-Computer Interfaces zur Steuerung von Neuroprothesen: von der synchronen zur asynchronen Funktionsweise[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]] [Williams et al., 2013]. Synchronous BCIs are easier to design, train and evaluate. They only require patient's concentration in specific time windows in which the patient must focus and not create any artifact (eye blinking, muscle activation, etc.). Moreover, a decoder is only calibrated to decode a specific known mental state associated with a specific visual or auditory cue. Nevertheless, such systems are limited and do not rely on a natural control paradigm [Müller-Putz et al., 2006] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. On the opposite, Asynchronous BCI systems act as stand-alone devices which switch between intentional control and no-control phases determined by the patient's neural signals (and not an external cue). Asynchronous BCIs are able to perform a reliable rest state decoding during intentional no-control command. Consequently, asynchronous BCI provides a more natural control paradigm which does not require any external cue/stimuli. However, asynchronous BCIs are much more complicated to design, train and evaluate [Müller-Putz et al., 2006] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. They generally highlighted lower performance than synchronous BCI (low true-positive rate and high false-positive rate) [START_REF] Han | Brain-Switches for Asynchronous Brain-Computer Interfaces: A Systematic Review[END_REF].

The majority of the state-of-art BCI decoders are synchronous [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. They do not decode an idle state. During common center-out experiments, it is assumed that subject is intentionally controlling the device at all times, nevertheless, in practice, between each trial, the cursor is replaced to the center of the screen. It is likely that during not intended control of the user, such BCI may lead to unwanted activations [START_REF] Williams | Differentiating closed-loop cortical intention from rest: building an asynchronous electrocorticographic BCI[END_REF] and is therefore not adapted to "real-life" BCI applications. During daily life application, it is mandatory that false activations remain exceptionally rare events using real effector (exoskeleton, wheelchair, etc.) due to its direct contact with the patient, and the mechanical/technical limitations of the effector. Asynchronous control for BCI application is mandatory for more realistic experiments than center our tasks.

The majority of asynchronous BCI studies have been performed based on non-invasive recording systems (generally EEG) [START_REF] Chae | Toward brain-actuated humanoid robots: Asynchronous direct control using an EEG-Based BCI[END_REF][START_REF] Han | Brain-Switches for Asynchronous Brain-Computer Interfaces: A Systematic Review[END_REF][START_REF] Kalunga | Online SSVEPbased BCI using Riemannian geometry[END_REF]] [Li et al., 2013[START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF][START_REF] Müller-Putz | Brain-computer interfaces for control of neuroprostheses: from synchronous to asynchronous mode of operation / Brain-Computer Interfaces zur Steuerung von Neuroprothesen: von der synchronen zur asynchronen Funktionsweise[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF][START_REF] Ortner | An SSVEP BCI to Control a Hand Orthosis for Persons With Tetraplegia[END_REF][START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF][START_REF] Yousefi | Development of a robust asynchronous brain-switch using ErrPbased error correction[END_REF] using various decoders and control strategies. Several ECoG-based brain switch decoders were tested to perform idle state (IS) classification during offline pre-clinical [START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Schaeffer and Aksenova, 2016b] and clinical [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF] [P. T. Wang et al., 2013] studies. Finally, MEAbased single/multi-units asynchronous BCI for pre-clinical online [START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF] experiments and offline decoding [START_REF] Achtman | Free-paced highperformance brain-computer interfaces[END_REF][START_REF] Ludwig | Use of a Bayesian maximumlikelihood classifier to generate training data for brain-machine interfaces[END_REF][START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] studies were reported.

Diverse classifiers such as Hidden Markov Model (HMM), logistic regression, linear Bayesian classifier, support-vector machine (SVM), linear discriminant analysis (LDA) have been coupled with continuous decoders such as partial least squares (PLS), Kalman Filters (KF), population vector algorithm, Laplace Gaussian filter etc. to decode offline and online asynchronous EEG-, ECoG-, MEA-driven BCIs [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Leeb | Self-Paced (Asynchronous) BCI Control of a Wheelchair in Virtual Environments: A Case Study with a Tetraplegic[END_REF][START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF][START_REF] Müller-Putz | Brain-computer interfaces for control of neuroprostheses: from synchronous to asynchronous mode of operation / Brain-Computer Interfaces zur Steuerung von Neuroprothesen: von der synchronen zur asynchronen Funktionsweise[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF][START_REF] Saa | Asynchronous decoding of finger movements from ECoG signals using long-range dependencies conditional random fields[END_REF][START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF]] [Schaeffer and Aksenova, 2016b[START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF][START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] [P. T. Wang et al., 2013[START_REF] Wu | Modeling and Decoding Motor Cortical Activity Using a Switching Kalman Filter[END_REF][START_REF] Yousefi | Development of a robust asynchronous brain-switch using ErrPbased error correction[END_REF].

Asynchronous BCI is an important field of research and numerous articles already reported relevant results for idle state detection. However, with the exception of few online MEA-based preclinical experiments e.g. [START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF][START_REF] Wu | Modeling and Decoding Motor Cortical Activity Using a Switching Kalman Filter[END_REF], most of the asynchronous BCI experiments relying on invasive neural signal recordings were achieved during offline neural signal analysis [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF]] [Schaeffer and Aksenova, 2016a[START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] [P. T. Wang et al., 2013]. In order to design daily life BCI applications, further investigation on the decoding performance of asynchronous algorithms during online experiments based on invasive recording systems must be achieved.

Decoder requirements

BCI systems dedicated to daily-life applications have several constraints to be respected in order to help patients suffering from severe motor disabilities. Particularly, several criteria specifically related to the neural signal decoder must be met.

Real-time neural signal decoding

Obviously, to use a BCI system in daily life application, the BCI system must be sufficiently efficient to perform real-time/online/closed-loop processing of the incoming neural signals. This requirement brings new theoretical and technical demands.

From a technical point of view, processing of high dimensional data flow in real-time with minimal latency [START_REF] Marathe | The impact of command signal power distribution, processing delays, and speed scaling on neurally-controlled devices[END_REF], and fast control rate (~10-20Hz) [START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF] are mandatory requirements of motor BCI to control robotic devices. However, the processing of high dimensional feature space and/or tensor data structure may lead to high computational burden and time-consuming neural signal decoding which are incompatible with real-time processing. Therefore, decoders for online applications are generally restricted to linear optimized and efficient algorithms [START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF].

State-of-the art decoder for online clinical BCI application generally relies on indirect optimal linear estimation (OLE) algorithm or decoder from the Kalman filter family. OLE was reported in MEA [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF][START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF] and ECoG [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] based experiments providing to disabled patients up to 10 DoF control based on MEA recording systems [START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]] and 3D control with ECoG neural signals decoding [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF]. Kalman filter algorithm allowed a tetraplegic patient to control a robotic arm to perform 3D reach and grasp movements through MEA neural signal decoding in [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF].

Additionally, other well-known decoders were used in online BCI applications with a lower number of DoF to decode. Long-term 1D MEA-based control (up to 138 days), reported by Milekovic et al [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF], was performed using a regularized LDA decoder. Nick F. Ramsey's team reported long-term communication BCI decoder based on smoothing and threshold optimization from ECoG neural signals [START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF]. Hotson et al highlighted the online control of the five individual fingers of a prosthetic hand by an epileptic subject performing real finger movements using a subdural ECoG recording system and a hierarchical LDA decoder [Hotson et al., 2016a].

The control of complex effectors through neural signal decoding generally requires algorithms with a high computational burden. This requirement is in contradiction with the needs of real-time data flow processing with minimal latency. Therefore, a trade-off between complexity and accuracy must be found for real-time neural signal decoding.

Robust and stable decoding over time

To perform efficient, useful, and convenient use of BCI in real-life applications, decoding models must remain efficient over time and not require daily recalibration to provide accurate transcription of the patient's wills. Robust and stable BCI decoding over a long period without any model recalibration is one of the major challenges of the current BCI field.

Brain neural signals are a highly variable non-stationary environment where firing potential patterns of the neurons continuously change over hours, days and months. The non-stationarity of decoding patterns is related to inter (subject to subject) and/or intra (session to session or trials to trials) variability [Clerc et al., 2016b]. Intra subject variability is provoked by the brain plasticity and patient's factor (inattention, habituation, mental workload, etc.) [Clerc et al., 2016b]. In addition to the brain neural signals natural variability, it is mandatory that decoders remain stable in noisy environments less restricted and constrained than the laboratory settings (e.g. at home, in the street, etc.).

Chapitre 1 : Introduction MEA-based BCI are sensitive to any brain neural signal variations as it records direct neuron activities and consequently require frequent (order of the day) recalibration and skilled engineer supports [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF]] [Simeral et al., 2011[START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. Milekovic et al provided to two patients suffering from locked-in syndrome and ALS a long-term robust and durable communication BCI based on LFP recordings using brain switch decoders (1 DoF) for a period of 76 and 138 days, respectively, without recalibration and without significant loss of performance [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF]. Schwemmer et al reported offline stable accurate four-movement classification (index flexion and extension and wrist flexion and extension) by a tetraplegic patient using 96-channel MEA in the primary motor cortex for 375 days after the end of the training period [START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF]. Communication BCI based on ECoG recordings was used for 36 months by an ALS patient using decoder without recalibration during hundreds of days and without significant loss of performance [START_REF] Pels | Stability of a chronic implanted braincomputer interface in late-stage amyotrophic lateral sclerosis[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF].

However, the presented BCI systems were designed to control a low number of dimensions. BCIs dedicated to more complicated tasks require generally constant recalibration. For example, the OLE decoder used to perform 10 DoF effector control from MEA neural signals was recalibrated every day [START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF] whereas the OLE model dedicated to the 3 DoF effector control from ECoG neural signals was frequently recalibrated after several days of experiments [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] [W. Wang et al., 2013]. Nowadays, long-term robust decoding of complex effector remains a major challenge of daily life application.

Online closed-loop adaptive model calibration

By definition, daily-life BCI applications are closed-loop experiments. The subject's neural signals are decoded to command an effector which interacts with the environment. This interaction provides sensory feedbacks to the subject influencing the generated neural signals and future predictions, etc. In order to perform accurate and stable decoding across time, it may be preferable to take into account the neural signals patterns induced by the sensory feedback. This "human-in-loop" (closed-loop) strategy is opposed to the open-loop experiments usually performed in BCI experiments.

Open-loop sessions are experiments where the patient is passive and try to perform a mental task without feedback on his neural signals and the success or failure of the task to be carried out. However, a drop in the decoding performance was repeatedly reported applying decoders adjusted offline using open-loop (without feedback) experiments' training dataset during closed-loop experiments [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF], 2006[START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]] [Orsborn et al., 2014].

Experiments taking into account patient's sensory feedback (closed-loop) during the model identification highlighted drastic different parameter choices compared to protocols with passive subjects (open loop) during the calibration phase [START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]. The changes may be explained by the modification of the neural activity patterns between open-loop motor imagery tasks and closed-loop experiments which include new neural signals related to motor imagery and effector control feedbacks [Clerc et al., 2016b[START_REF] Schlögl | Adaptive Methods in BCI Research -An Introductory Tutorial[END_REF]. Many researches underlined that closed-loop decoder identification can lead to performance rises over time [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]] [Orsborn et al., 2014[START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]. Online incremental adaptive decoders are a group of the CLDA algorithms which update their parameters in an incremental manner with new incoming data, optimizing the model parameters in real-time and providing adaptation of the decoder to the patient and vice versa. Several studies stressed the benefits of algorithms integrating online adaptive calibration such as easier and faster use than offline computed models, more convenient for disabled patients who may struggle to remain alert and engaged during long sessions of calibration, etc. [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF][START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]] [Orsborn et al., 2014[START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF].

Strategies

Adaptive real-time decoder identification is an important request of real life BCI application for easy use, faster integration of the feedback related neural signals during the model calibration and decoding performance improvements. In order to integrate decoding model identification into the closed-loop BCI session, several adaptive decoder identification procedures have been designed.

Several conventional classifiers (linear and non-linear) LDA, QDA, SVM, fuzzy inference system were adapted to closed loop decoder requirements and tested during closed loop BCI experiments [START_REF] Bamdadian | Improving session-to-session transfer performance of motor imagery-based BCI using adaptive extreme learning machine[END_REF][START_REF] Hazrati | An online EEG-based brain-computer interface for controlling hand grasp using an adaptive probabilistic neural network[END_REF][START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF]] [Rong et al., 2018a[START_REF] Schlögl | Adaptive Methods in BCI Research -An Introductory Tutorial[END_REF]] [Spüler et al., 2012b[START_REF] Vidaurre | Study of On-Line Adaptive Discriminant Analysis for EEG-Based Brain Computer Interfaces[END_REF], 2011] [START_REF] Wen | Online motor imagery BCI based on adaptive and incremental linear discriminant analysis algorithm[END_REF]. Early studies on adaptive continuous algorithms were based on Chapitre 1 : Introduction the adaptation of mu and beta rhythms amplitude linear combination during EEG recorded experiments [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF]. Nowadays, conventional adaptive continuous decoders are often belonging to KF family [START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF] such as Adaptive KF, ReFIT KF or gaussian process discriminative KF (GP-DKF) [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF][START_REF] Dangi | Continuous Closed-Loop Decoder Adaptation with a Recursive Maximum Likelihood Algorithm Allows for Rapid Performance Acquisition in Brain-Machine Interfaces[END_REF], 2011[START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF][START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF]] [Li et al., 2011[START_REF] Orsborn | Closed-Loop Decoder Adaptation on Intermediate Time-Scales Facilitates Rapid BMI Performance Improvements Independent of Decoder Initialization Conditions[END_REF], and are majorly applied in MEA based experiments. Other less conventional (MEA based BCI) strategies using Wiener filter with feedback error learning [START_REF] Suminski | Online adaptive decoding of intended movements with a hybrid kinetic and kinematic brain machine interface[END_REF], recurrent neural network [START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF], or adaptive Point Process Filters (PPF) [START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF][START_REF] Shanechi | Robust Brain-Machine Interface Design Using Optimal Feedback Control Modeling and Adaptive Point Process Filtering[END_REF] were reported.

Adaptive models with "Human-in-loop" update strategy seems to be a promising approach for accurate and robust BCI applications without daily decoder recalibration. While several adaptive linear and nonlinear regression and classification decoders have been designed for MEA [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF][START_REF] Dangi | Continuous Closed-Loop Decoder Adaptation with a Recursive Maximum Likelihood Algorithm Allows for Rapid Performance Acquisition in Brain-Machine Interfaces[END_REF][START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF]] [Li et al., 2011[START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF][START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF] and electroencephalography (EEG) [START_REF] Hazrati | An online EEG-based brain-computer interface for controlling hand grasp using an adaptive probabilistic neural network[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF]] [Rong et al., 2018a[START_REF] Vidaurre | Toward Unsupervised Adaptation of LDA for Brain-Computer Interfaces[END_REF] recording systems, only few adaptive decoders were developed for ECoG recording systems [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. Additionally, most reported adaptive algorithms designed were restricted to linear decoders, which may be limiting for complex effector control with a high number of DoFs. Therefore, deeper investigation on the development of closed-loop decoder adaptation must be achieved. Moreover, more online clinical experiments based on adaptive decoder must be carried out for BCI application, as this is the only solution to evaluate the impact of patients and decoder mutual adaptation.

BCI project at CLINATEC

The experiments and results presented in this manuscript were part of CEA-Grenoble\Leti\CLINATEC clinical trial: "BCI and tetraplegia". CLINATEC is a laboratory of the CEA-Grenoble in partnership with the University Grenoble Alpes (UGA) and University Hospital of Grenoble (CHUGA). This ongoing clinical trial was approved by the French competent authorities and is referred under the identifier NCT02550522 in ClinicalTrials.gov ["ClinicalTrial NCT02550522," n.d.]. The clinical protocol which started in end 2015 plans to include five patients in five years.

Concept and context

"BCI and Tetraplegia" clinical trial aims to bring the proof of concept that a tetraplegic patient can control a complex effector in real-time using ECOG recording system and direct neural decoding strategy. To succeed, innovative chronic epidural electrocorticographic recording arrays and a complex motorized multi-limb exoskeleton effector were designed. After preclinical experiments on monkeys [Eliseyev et al., 2014],

the clinical trial started and to this date, three tetraplegic patients were included in the protocol and underwent bilateral implantation of chronic wireless epidural electrocorticographic arrays. Two of them are still training using our homemade BCI platform in order to control multiple effectors (3D Avatar, exoskeleton and wheelchair, etc.).

Principles / Approaches

The principle of the presented BCI platform is shown in Figure 1-13. The patient neural activity is monitored through two wireless, safe and chronic "semi-invasive" epidural ECoG implants located above the dura matter of the motor cortex of both encephalon which digitize and pre-filter the neural signals. These digitized neural signals are recorded and processed using a signal processing software platform (signal translation block) which sends commands to a complex effector (four-limbs exoskeleton) to perform the attempted action of the patient and provides visual feedback to him (closed-loop BCI). Firstly, the transducer must apply to real-time uses which restricted the possible preprocessing, feature extraction, decoding and post-processing steps to low computational burden with sufficiently simple and straightforward algorithms in order to apply the transducer block (decision rate) in about a hundred millisecond time scale.

Secondly, as mentioned in 1.4.2.3, the asynchronous attribute is a major characteristic for BCI applications which control real/physical effectors. The BCI project of CLINATEC aims to control a complex multi-limb exoskeleton. Such an effector must be active and responsive to any generated command and additionally, remain static if the patient decides not to attempt any movement. Synchronous BCI systems can only be used in specific prefixed time periods which is incompatible with this application.

To control an exoskeleton with several limbs, it is mandatory to allow the patient to command independently each limb with strong discrimination between the activation of each limb. Indeed, if the patient is focusing on precise manual tasks, the lower limb must stay inactive without any unwanted activation.

The decoder used to translate the ECoG neural signals of the patient into commands should be able to control a complex effector with a high number of dimensions. The control of the four limbs of the exoskeleton represents numerous DoF to decode such as 3D space for the left hand, 3D space for the right hand, both wrists rotations, grasping, etc. The decoder must be sufficiently efficient to control each DoF using a direct neural decoding strategy.

Finally, BCI transducer must be as stable as possible to perform robust neural signal decoding without recurrent calibration sessions. Decoder with stable performance over days, weeks, months, etc. are easier to use and less frustrating for the user.

PhD goals

The

Real-time application

To imagine BCI daily life application, the BCI platform must be sufficiently efficient (low computational time) to provide to the patient a transparent control of the effector. Consequently, the online application of the decoder may be limited to simple and/or efficient algorithms [START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]. Nevertheless, these decoders are generally linear models, which might be limited for complex multi-limb control objectives proposed in the "BCI and Tetraplegia" clinical trial. Online decoding restriction is one of the major requirements for the clinical trial and every algorithms proposed in this Chapitre 1 : Introduction manuscript were especially focused on real-time application purpose and related restrictions (e.g. decision rate below 300ms).

Asynchronous multi-limbs decoder

As mentioned in 1.4.2.3, the majority of the state of the art high dimensional control experiments were performed based on synchronous limited center-out tasks and only few online continuous decoders integrated asynchronous control. However, during daily life applications, numerous situations require to sit back and not perform any movement (queue up, take the subway, rest, etc.). Moreover, in the case of real effector control (prosthetics, FES, wheelchair, exoskeleton, etc.) which has direct interaction with the user, false activations may lead to particularly disturbing and stressful situations. False activations must remain exceptionally rare events. As an example, in the case of gait cycle initiated with a false positive activation, the following false-positive activations during the gait cycle will not be taken into account (as the patient is already walking). In this situation, rare long false activations are less disturbing than highfrequency false activations.

Additionally, daily life actions commonly require bimanual (or generally multi-limb) movements. Multi-limb decoding is a poorly explored area of the BCI field compare to single limb movement decoding. Providing tetraplegic patients with simultaneous or alternative bimanual and walk control will greatly enhance patient's mobility, independence and improve their quality of life. Therefore, if a patient is attempting a high precision task with the left arm of the exoskeleton, the multi-limb decoder must be able to compute zero-velocity outputs for the other limbs. In the scope of the CLINATEC clinical trial, the new online decoders presented in the manuscript must integrate asynchronous control and highly efficient limb discrimination to enable a tetraplegic patient to perform a stable idle state (IS) and alternative multi-limb tasks at his earliest convenience.

Online incremental adaptive decoder

Finally, one of the BCI project requirements is the stability and robustness of the model across time and experiments even though it is well known that patient neural signals are non-stationary signals. The proposed solution is to design an adaptive decoder which re-estimates the model parameters across experiments. The reported adaptive decoder brought several interesting properties in the BCI field.

As mentioned in section 1.4.3.3, the adaptation of the decoder during closed-loop experiments lead to different model parameter convergence, better performance compared to decoders trained offline during open-loop experiments [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF], 2006[START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]] [Orsborn et al., 2014], easier/faster training procedure [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF] and adaptation to neural signal variations across time.

The present Ph.D. study is specifically focused on the previously presented decoder characteristics which also respond to the clinical trial requirements. In order to perform online closed-loop experiments, all the decoders designed in the Ph.D. researches are oriented towards online adaptive closed-loop asynchronous multi-limb BCI applications.

Manuscript organization

The doctoral manuscript presents the new incremental adaptive asynchronous multilimb decoders implemented in online closed-loop experiments with a tetraplegic patient or tested offline in pseudo-online decoding performance comparative studies. Chapter 2 presents in detail the "BCI and Tetraplegia" clinical trial from the paradigm of control to the training timeline. Chapter 3 reports state-of-the-art BCI transducers (preprocessing, feature extraction and decoder). The new decoders designed are detailed in Chapter 4, 5 and 6 whereas experiments description, integration of the decoder into the CLINATEC online BCI platform and decoder performance evaluation are presented in Chapter 7, 8 and 9 respectively. Chapter 10 clusters the results of each decoder. Finally, Chapter 11 regroups the discussion, the added value of this study, the implications of the presented results in the BCI field and the research perspective.

As mentioned is the first chapter, "BCI and Tetraplegia" clinical trial was created with the purpose to provide the proof of concept that a tetraplegic patient implanted with epidural ECoG can control a complex multi-limb effector through direct neural decoding strategy This chapter presents all the information related to the clinical trial and the research environment of the Ph.D. study. Particularly, this chapter is focused on the patients' condition, the BCI recording system, the experimental environment, the global software chain in which the decoding algorithms proposed in this manuscript were integrated and the controlled effectors (Figure 2-1).

Figure 2-1: CLINATEC "BCI and Tetralegia" specific BCI platform.

Inclusion criteria of the clinical trial

According to the inclusion criteria of the clinical trial defined by the Principal investigator and formulated in the World Health Organization (WHO) International

Chapter 2 : Scientific context Clinical Trials Registry Platform (ICTRP) ["ClinicalTrial NCT02550522," n.d.] ["ICTRP clinical trial NCT02550522," n.d.], the participants must be a French fluent male or female patient between 18 and 45 years old with stable neurological deficits. Moreover, the patients should claim a need for additional mobility, be registered in the French social security system, have stable ambulatory or hospitalized monitoring, and signed informed consent [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]

Participants of the clinical trial

Between the start of the clinical trial and mid-2020, three patients were included in the clinical protocol.

The first tetraplegic patient was successfully implanted in May 2016 with two WIMAGINE implants. After the surgery, the implants of the first patient stopped communicating with the base-station. Further investigation highlighted technical issues in the firmware of the microcontroller unit. The recording systems were explanted and the patient was excluded from the clinical trial. The issues were corrected and after ANSM and ethics committee revision, the clinical trial was authorized to restart in February 2017. Further details are given in the supplementary data of [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF].

The second subject, recruited in mid-2017, was a 29-year-old right-handed male with traumatic sensorimotor tetraplegia caused by a complete C4-C5 spinal cord injury 2 years prior to the study. The patient can perform neck, shoulder and small upper limb movements by contraction of the biceps at the elbow and extensors of the wrists. American Spinal Injury Association Impairment (ASIA) scores [START_REF] Roberts | Classifications In Brief: American Spinal Injury Association (ASIA) Impairment Scale[END_REF] evaluation was achieved under the supervision of Professor Benabid and Professor Chabardès. The contraction of the biceps close to the elbow was scored at 4 and 5 for the right and left body side, whereas extensors contractions were rated at 0 and 3 for the right and left wrists respectively. With the exception of the cited muscles, all other muscles below were scored 0 on the ASIA scale. Moreover, the sensory-motor deficit was complete (Figure 2-2) [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]. A third subject was included to the study in late 2019. Nevertheless, at the end of the Ph.D. experimental period, the third patient was in an early training phase and was

consequently not yet able to perform the complex task related to the Ph.D. study. Therefore, the proposed Ph.D. study is only focused on the experiments performed and the results obtained with the second patient.

Implantation

The patients underwent bilateral implantation of two long-term safe and chronic wireless implants for epidural ECoG signal recordings under general anesthesia using Image Guided Functional NeuroSurgery on May 2016, June 21 th 2017 and November 19 th 2019, respectively. The epidural ECoG wireless implants named WIMAGINE were implanted into the skull in contact with the dura mater within a 25 mm radius craniotomy placed in front of the sensory motor cortex (SMC) area. The subjects' SMC were localized clearly using functional imaging (fMRI and MEG) as they imagined upper and lower limb virtual movements or performed real motor tasks when possible. 100 and 80 trials were performed with MEG and fMRI respectively to optimize the implants positioning before the surgery. Details of the protocol are provided in the previous study [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]. The second patient implanted in June 21, 2017 started the training procedure in early July and since has been training for more than 36 months to control several effectors with various complexity.

Experimental platform 2.4.1. Recording system

The primary challenge of motor BCIs for clinical and daily-life applications is the development of safe, chronic and stable neural recording systems over time. In order to fulfill these requirements for chronic brain neural signal processing, CLINATEC designed an innovative wireless epidural ECoG recording system named Wireless Implantable Multi-channel Acquisition system for Generic Interface with NEurons (WIMAGINE) [START_REF] Mestais | WIMAGINE: Wireless 64-Channel ECoG Recording Implant for Long Term Clinical Applications[END_REF].

WIMAGINE is an active implantable medical device designed to be implanted into the skull above the dura matter to record epidural ECoG brain signals. The recorded epidural ECoG signals are transmitted wirelessly to a custom designed base station connected to a computer [START_REF] Mestais | WIMAGINE: Wireless 64-Channel ECoG Recording Implant for Long Term Clinical Applications[END_REF]. WIMAGINE implants are composed of two main structures. The recording part is a 50-mm diameter silicone-coated titanium cylinder with a flat internal surface with 64 electrodes for ECoG recording. The 64 plane platinum iridium 90/10 electrodes have a 2.3 mm diameter and have a inter-electrodes distance of 4 and 4.5 mm on the lateral and antero-posterior directions, respectively [START_REF] Sauter-Starace | Long-Term Sheep Implantation of WIMAGINE®, a Wireless 64-Channel Electrocorticogram Recorder[END_REF]. The digitized ECoG signals are low and high pass filtered in a bandwidth from 0.5Hz to 300Hz, amplified, cleaned thanks to an anti-aliasing filter and clustered into buffers [START_REF] Robinet | A Low-Power 0.7 $\mu \rm V_rms$ 32-Channel Mixed-Signal Circuit for ECoG Recordings[END_REF]. All these operations are performed using an ASIC CINESIC32 integrated into the implants (Figure 234). This integrated circuit presents a low noise amplifier (0.7 µVRMS) which requires low power alimentation (32µA per channel for a global consumption of 3mA with 32 active channels). The second part of the WIMAGINE implant is a silicon film containing high frequency (HF) antenna (13,56Mhz) for transcutaneous remote power supply and an ultra-high frequency (UHF) antenna (402-405 Mhz) for wireless data transfer. Remote power supply up to 100mW is provided to the implants through an inductive link with an external antenna integrated into a helmet worn by the patient (Figure 2345).

Limited data rates, caused by restricted radio link (≤250 kb/s) narrowed the real-time transmission of the neural signals to a maximum of 32 electrodes by implant simultaneously recorded at a 586 Hz sampling frequency.

All the required European directive 2007/47/EC and ISO standards regulation for clinical trials applications have been obtained concerning biocompatibility (ISO10993), mechanical and electrical safety (ISO45502-1, ISO60601-1, ISO14708-1), software reliability, risk management of medical device (IEC62304, ISO14971), manufacturing process (ISO13485) and electrical safety and electromagnetic compatibility of the external unit (NF EN 60601-1).

Software chain

The software chain is composed of four main components. The digitized neural signals from the WIMAGINE implants are sent to the Wireless Implant Software Control Interface (WISCI) which receives the ECoG neural signals, synchronizes the channels and formats the neural signals to real-time batch processing. In a second step, the formatted ECoG signals are treated by the Adaptive Brain Signal Decoder (ABSD) which processes the ECoG signals using an online adaptive model to estimate the patient's intended movements and control an effector. ABSD regroups the steps described as the classical signal processing block in the general architecture of BCI systems (preprocessing, feature extraction, etc.). ABSD was designed to support adaptive calibration procedures. ABSD updates an adaptive decoder using incremental batch learning in the background while decoding continues. However, ABSD requires a computationally light and optimized adaptive decoder to work. The orders generated by ABSD are sent to the EMY Motion Manager (EMM) and EMY Motion Controller (EMC) to generate appropriate motor commands for the exoskeleton. In the case of virtual effectors, an adequate portal software is used to generate commands and control the virtual effector (Figure 23456).

From the neural signals recorded at 586 Hz, the software chain extracts an order to send to the effector at a 10Hz decision rate. Moreover, the commands performed by the effector (limb activated, current hands positions, wrists angle, etc.) are recorded at 10 Hz and could be used for adaptive model calibration.

Every analysis and online experiment, including training and decoding, was performed with Matlab2017b® and Visual Studio 2015 using an Intel Xeon E5-2620v3 computer with 64 GB RAM.

Effectors

Several effectors were designed to be controlled by the patient during his training. The effectors can be clustered into the virtual and real effector categories.

Real effector

The purpose of the "BCI and Tetraplegia" clinical trial is to prove that a tetraplegic patient can control a complex multi-limb effector. The Enhancing MobilitY (EMY) exoskeleton adapted by the CEA/LIST is a wearable fully motorized four-limb robotic neuroprosthesis weighting 65 kg designed to be driven by the decoded ECoG brain signals [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF] during the clinical trial (ISO60601) [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF].

Upper limb control of both arms presents high movement amplitudes with 65, 105, 105, 100 degrees for shoulder rotation, shoulder and elbow flexion/extension and pronosupination respectively [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF][START_REF] Morinière | EMY: a dual arm exoskeleton dedicated to the evaluation of Brain Machine Interface in clinical trials[END_REF]. Each arm can be controlled using angular or Cartesian end-point coordinates. Lower limbs of the EMY exoskeleton are controlled through a configurable walking cycle. Additional grasping/prehension systems were integrated to the EMY exoskeleton for object manipulation experiments.

Independent control of the 3D Cartesian endpoint trajectory of the arm, wrist rotation, open/close hand shape for both arms, walking, sitting, and rest state represent 13 DoF.

A battery and a computer station receiving ECoG radio-emitted signals are embedded in the back of the exoskeleton. The neural signals samples are decoded and translated into incremental velocity endpoint-control commands through the decoding software. The control commands are converted into joints movements by the exoskeleton control system activating the limbs and producing the appropriate movements.

To this date, EMY exoskeleton does not manage the balance of the prosthetic and it requires ceiling-mounted support to enable the patient to practice standing experiments.

To mention, the second patient controlled other real effectors such as the Kinova JACO assistive robotic arm and a wheelchair, nevertheless, these performances will not be precisely detailed in the manuscript.

Virtual effectors

The virtual effectors used during the clinical trial were designed to train the patient to control from 1 to 13 DoF of the exoskeleton. Several video games such as walking simulators, car racing games, 1D and 2D cursor control were created to train the patients. Moreover, a virtual avatar reproducing the exoskeleton dimensions was integrated into the BCI platform. The virtual avatar is a replica of the exoskeleton and can perform every movement similarly to the exoskeleton. All the virtual environments were coded using UNITY® software.

In the entire Ph.D. manuscript, the experiments using virtual effectors were performed using the virtual avatar. Real and virtual effectors available in the BCI and tetraplegia project. Real effectors cluster the exoskeleton, the wheelchair, etc. whereas the virtual effectors regroup the virtual avatar the car racing video game, etc.

Progressive patient Training and Timeline 55

Progressive patient Training and Timeline

The patient's training began in July 2017 and is ongoing to this date. During this period, the trainings in the laboratory and at the patient's home were performed in parallel. Patient's training at home was performed every week (two days per week). Home experiments were focused on virtual effector control. Experiments inside the laboratory occurred, on average, once a month (three days in a row) and focused on real effector control. During laboratory experiments, various tasks were proposed to the patient in order to control the different DoF of the exoskeleton such as walking, moving the arms in sitting position etc.

One of CLINATEC purposes is to control all the EMY exoskeleton DoF. The training strategy followed was to increase gradually the patient's control on the effectors, unlocking new DoFs when the patient achieved good control of the effector. The progress of the patient was investigated in terms of the number of DoFs controlled over time [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]. To this point, numerous experimental paradigms were created to control the limbs of the exoskeleton (Table 1). With the exception of the "Walking" command, all the experimental paradigms can be executed with the exoskeleton in a standing or sitting position. Once a DoF is controlled independently, it is added to the pool of controlled DoF and the patient is trained to control all the paradigm in the same experiment. As an example, considering that the patient was able to control the 3D Cartesian left and right hand translation paradigm in different experiments (3D control experiments), a new paradigm with alternative control of both arm in the same experiment was created and tested (6D control), etc. This procedure was similar for both home and laboratory patient's training.

Experimental session procedure

Experiments were divided into several steps. Firstly, the patient was settled in the exoskeleton or in front of the television where the virtual effectors were displayed. The recording helmet (including the HF antennas) was placed on the patient and signal quality was checked. Then, the BCI session started. A clinician used ABSD software environment to select an experiment with specific controllable DoFs. The BCI session was divided into two main phases. The first optional phase was the model calibration/training period. Calibration period aims to create a mathematical model which evaluates the correct command to send to the effector depending on the neural signals modulation of the patient. This phase was performed during the online control of the effectors by the patient and was optional as model created during previous experiments can be applied. The experimenter, depending on the performance of the model stopped the calibration phase to start the second step of the experiment: the test phase. The test phase was used to evaluate the control performance of the patient with a fixed model. The tasks to complete were similar than the one proposed in the training phase. The only difference was that the decoding model was fixed and not updated anymore. To mention, contrarily to most of the state-of-the-art experiments, the patient could talk and interact with the experimenter without any restrictions during both phases.

A BCI session was composed of multiple tasks such as idle, left or right hand translation movements, left or right wrist rotations etc. The task to complete was selected by the experimenter. Each task was composed of several trials defined as a specific action to perform (e.g. reach the target on the left corner, rotate the wrist to a specific angle etc.).

During virtual effector based experiments, the targets to reach by the patient selected by the clinician were represented thanks to virtual spheres, gauges or icons depending on the task to complete. Using the exoskeleton, targets were represented thanks to lightened LED distributed on a panel. As previously mentioned, numerous transducer strategies were designed for BCI applications. Given that the transducer is generally dependent on the inter-subject variability, the neuronal patterns, the recording system and the effector to control, abounding variations of transducer strategies were proposed through the past twenty years. Nevertheless, several classical strategies emerged as relevant techniques to extract brain neural signal information. The following chapter provides an overview of the most popular transducers pre-processing, feature extraction and model identification blocks used in the BCI field. Moreover, in regard to the CLINATEC clinical trial and thesis scope, special emphasis is placed on online processing methods and online adaptive transducer identification procedures.

Pre-processing

Pre-processing is generally the first signal processing step which follows the neural signal recording block. Pre-processing aims to enhance the signal quality and improve the signal-to-noise ratio to extract the most relevant features without undesired nonrelevant artifacts [Bashashati et al., 2007a]. The neural signal may be amplified, downsampled to reduce the sampling frequency, band-pass filtered in neurologically interesting frequency band [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Mutasim | Computational Intelligence for Pattern Recognition in EEG Signals[END_REF], etc. Subsequent filters may be applied to remove artifacts from several sources such as non-voluntary muscular activation, power line noise, etc. particularly for EEG recordings. Power line noise around 50-60 Hz and its harmonics were generally removed through notch, sharp edge, band pass or Butterworth filters in ECoG [START_REF] Anderson | Electrocorticographic (ECoG) correlates of human arm movements[END_REF][START_REF] Branco | Optimization of sampling rate and smoothing improves classification of high frequency power in electrocorticographic brain signals[END_REF][START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF] [ [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Salari | Classification of Articulator Movements and Movement Direction from Sensorimotor Cortex Activity[END_REF] and EEG [START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Chowdhury | Online Covariate Shift Detection based Adaptive Brain-Computer Interface to Trigger Hand Exoskeleton Feedback for Neuro-Rehabilitation[END_REF][START_REF] Chu | Incremental Linear Discriminant Analysis: A Fast Algorithm and Comparisons[END_REF][START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF][START_REF] Delisle-Rodriguez | Adaptive Spatial Filter Based on Similarity Indices to Preserve the Neural Information on EEG Signals during On-Line Processing[END_REF][START_REF] Faller | Autocalibration and Recurrent Adaptation: Towards a Plug and Play Online ERD-BCI[END_REF][START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF][START_REF] Mobaien | ACSP: Adaptive CSP filter for BCI applications[END_REF]] [Nguyen et al., 2019[START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF][START_REF] Zhao | Incremental Common Spatial Pattern algorithm for BCI[END_REF]] [Rong et al., 2018b[START_REF] Song | Improving brain-computer interface classification using adaptive common spatial patterns[END_REF][START_REF] Wen | Online motor imagery BCI based on adaptive and incremental linear discriminant analysis algorithm[END_REF]] [Zhao et al., 2008] BCI experiments. For electrical muscle activities such as hand movements or eye blinking, algorithms were designed to detect these artifacts in an unsupervised manner, using for example independent component analysis (ICA) [START_REF] Moro | Towards adaptive brain-computer interfaces: Improving accuracy of detection of event-related potentials[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF], or algorithms dedicated to the processing of other types of biological signals such as electromyography (EMG) or electrooculography (EOG) [START_REF] Hazrati | An online EEG-based brain-computer interface for controlling hand grasp using an adaptive probabilistic neural network[END_REF][START_REF] Li | A self-adaptive frequency selection common spatial pattern and least squares twin support vector machine for motor imagery electroencephalography recognition[END_REF]] [Nguyen et al., 2019], etc.

Common average reference (CAR) is a spatial filter which estimates the average electrical activity measured across all electrodes. Re-referencing is achieved by creating an average of all electrodes and subtracting the resulting signal from each channel. CAR is a classic spatial filter frequently used in neural population recording systems such as LFP [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF][START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF] ECoG [Branco et Chapter 3 : Transducers in BCI al., 2017, 2018] [Chen et al., 2013[START_REF] Gunduz | Differential roles of high gamma and local motor potentials for movement preparation and execution[END_REF]] [Hotson et al., 2016a[START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF]] [Miller et al., 2009] [Nakanishi et al., 2013[START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF][START_REF] Rembado | Independent Component Decomposition of Human Somatosensory Evoked Potentials Recorded by Micro-Electrocorticography[END_REF] [P. T. Wang et al., 2016] [Wang et al., 2017] [Zhao et al., 2013a] and EEG [START_REF] Chu | Incremental Linear Discriminant Analysis: A Fast Algorithm and Comparisons[END_REF][START_REF] Cincotti | High-resolution EEG techniques for brain-computer interface applications[END_REF][START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF]] [Kim et al., 2018[START_REF] Roijendijk | Classifying Regularized Sensor Covariance Matrices: An Alternative to CSP[END_REF][START_REF] Wen | Online motor imagery BCI based on adaptive and incremental linear discriminant analysis algorithm[END_REF] BCIs.

All the transducer steps are highly dependent on the type of recorded neural signals.

Pre-processing algorithms specific to MUA and SUA neural signals were reported such as the spike sorting algorithm family which extracts the neuron action potentials from the recorded brain activity. Numerous spike sorting algorithms were designed for BCI experiments. For example, spike sorting algorithm based on amplitude-thresholding with a threshold defined as a multiple of the standard deviation of the neural signals was reported in [START_REF] Achtman | Free-paced highperformance brain-computer interfaces[END_REF][START_REF] Bibliography Fan | Intention estimation in brain-machine interfaces[END_REF][START_REF] Kim | Neural control of computer cursor velocity by decoding motor cortical spiking activity in humans with tetraplegia[END_REF]] [Li et al., 2011] [Orsborn et al., 2014] [Suminski et al., 2013].

Other algorithms were used to perform in the same time the artifact removal and the neuronal feature extraction steps. Principal component analysis (PCA) as well as independent component analysis (ICA) are two neural feature extraction algorithms which were reported for artifact signals rejection [Nicolas-Alonso and Gomez-Gil, 2012].

Neural feature extraction

Neural feature extraction block extracts the neural signal informative characteristics and discards irrelevant components to decode the mental task performed by the patient. Feature extraction can be divided into a feature generation step which extracts the relevant characteristics from the pre-processed neural signals and an optional feature dimension reduction step which selects among the computed features the best/most informative features [Bashashati et al., 2007a] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF].

Feature generation

The feature generation step is highly dependent on the recording system. Differentiation can be made between algorithms dedicated to the SUA/MUA neural signal processing and neural signal population processing.

MEA single/multi neuron recording feature extraction

Spike counts is the general feature generation strategy for BCI system based on SUA and MUA recordings. After the spike sorting preprocessing step, spike count on small time windows (from 20ms to 100ms) is performed using a threshold generally defined as a multiple of the signal standard deviation to evaluate the threshold crossing rate [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Dangi | Continuous Closed-Loop Decoder Adaptation with a Recursive Maximum Likelihood Algorithm Allows for Rapid Performance Acquisition in Brain-Machine Interfaces[END_REF][START_REF] Fitzsimmons | Extracting Kinematic Parameters for Monkey Bipedal Walking from Cortical Neuronal Ensemble Activity[END_REF][START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF], 2006[START_REF] Ifft | A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]] [Orsborn et al., 2014[START_REF] Perge | Intra-day signal instabilities affect decoding performance in an intracortical neural interface system[END_REF]] [Simeral et al., 2011] [Willett et al., 2018[START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF][START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF].

Nevertheless, other features can be evaluated to complete or replace the classic spike counts features. Young and Willet in [START_REF] Willett | A Comparison of Intention Estimation Methods for Decoder Calibration in Intracortical Brain-Computer Interfaces[END_REF][START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF] added to spike count feature the high-frequency spike power (HFSP) by taking the root mean square of the filtered spike band voltages from 250 to 5000 Hz using a 8 th order non-causal Butterworth filter. Another strategy was proposed by Shanechi [START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF][START_REF] Shanechi | Robust Brain-Machine Interface Design Using Optimal Feedback Control Modeling and Adaptive Point Process Filtering[END_REF] who directly performed BCI decoding on binary spike activity. The spikes were binned in small intervals containing at most one spike in each interval. [START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF][START_REF] Gunduz | Differential roles of high gamma and local motor potentials for movement preparation and execution[END_REF][START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF][START_REF] Mehring | Comparing information about arm movement direction in single channels of local and epicortical field potentials from monkey and human motor cortex[END_REF][START_REF] Schalk | Decoding two-dimensional movement trajectories using electrocorticographic signals in humans[END_REF]] [Wang et al., 2011]. Times features were generally computed by averaging the signal amplitude in a defined time window in a specific frequency band to isolate precise patterns [Hotson et al., 2016a] [Wang et al., 2012]. Frequency bands were extracted based on finite or infinite impulse response (IIR or FIR) low or band pass filters such as Chebyshev [START_REF] Ang | Filter Bank Common Spatial Pattern (FBCSP) algorithm using online adaptive and semi-supervised learning[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF][START_REF] Vo | Subject-Independent ERP-Based Brain-Computer Interfaces[END_REF], Butterworth [START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF][START_REF] Gunduz | Differential roles of high gamma and local motor potentials for movement preparation and execution[END_REF], Savitsky-Golay [START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF]] [Bundy et al., 2016[START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Pistohl | Prediction of arm movement trajectories from ECoG-recordings in humans[END_REF] or Gaussian filters [Chen et al., 2013[START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF]] [Hotson et al., 2016a], etc. Time features can be extracted from multiple frequency bands using a so-called filter bank strategy which evaluates in parallel sensorimotor rhythms from different frequency bands [START_REF] Chen | Prediction of Hand Trajectory from Electrocorticography Signals in Primary Motor Cortex[END_REF]] [Nakanishi et al., 2013].

Frequency and time-frequency features are based on the variation of the neural signal power across one or several frequency bands [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. As previously mentioned, a mental task induces variations and specific patterns in low and high frequency bands of the neural signals acquired with population recording systems (LFP, ECoG, EEG, etc.) [START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. Therefore, neural signals frequency and especially time-frequency features were commonly extracted in the BCI field. In the time-frequency domain, both neural signal amplitude and phase features were reported in BCI and motor-BCI applications.

Neural signals decoding relying on the extraction of amplitude-based features such as instantaneous power or magnitude were extensively studied [START_REF] Zeng | The Advantage of Low-Delta Electroencephalogram Phase Feature for Reconstructing the Center-Out Reaching Hand Movements[END_REF] and is the most represented strategy for BCI applications. To extract relevant information from the neural signal in a specific frequency, the signal can be band pass filtered using single or multiple filters or transformed into the time-frequency domain using non-parametric transformation such as Short Time Fourier Transform (STFT), Wavelet transforms, Hilbert transform or parametric ones (e.g. autoregressive model) [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2018] [Volkova et al., 2019]. All these technics aim to balance temporal and frequency resolution [START_REF] Polikar | THE WAVELET TUTORIAL SECOND EDITION[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF].

Filter banks use a set of band-pass filters (Butterworth, Gabors, Savitsky-Golay, etc ) to obtain an amplitude representation of the neural signals on several frequency bands with a good trade-off between frequency distortion and temporal delay [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF].

STFT algorithm computes the Fourier transform of the neural signal into small segments where the signal is assumed to be stationary. The segment of signals is evaluated via the convolution of the neural signals and a predetermined window function [START_REF] Polikar | THE WAVELET TUTORIAL SECOND EDITION[END_REF]. [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF], ECoG [START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF]] [Miller et al., 2009] [P. T. Wang et al., 2016[START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF], EEG [START_REF] Mend | Human computer interface with online brute force feature selection[END_REF]Kullmann, 2012] [Roijendijk et al., 2016] and MEG [START_REF] Fukuma | Real-Time Control of a Neuroprosthetic Hand by Magnetoencephalographic Signals from Paralysed Patients[END_REF][START_REF] Fukuma | Closed-Loop Control of a Neuroprosthetic Hand by Magnetoencephalographic Signals[END_REF]] neural signal recordings.

Fourier
Wavelet transforms (WT) analyzes the signal at different frequencies with different resolutions. WT was designed to overcome the shortcomings of the STFT. Instead of sinus decomposition of the signals in fixed time window, WT uses a wavelet basic function named Mother Wavelet translated and scaled to obtained variable time resolution depending on the analyzed frequency. High frequencies have a better temporal resolution than low frequencies whereas low frequencies have a better frequency resolution than high frequencies [START_REF] Polikar | THE WAVELET TUTORIAL SECOND EDITION[END_REF]. Continuous wavelet transform was frequently used in BCI studies to extract instantaneous power of MEA [START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF], ECoG [Branco et al., 2017, 2018[START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF]] [Eliseyev et al., 2017] [Eliseyev and Aksenova, 2014, 2016[START_REF] Jubien | Decoding of finger activation from ECoG data: a comparative study[END_REF][START_REF] Motrenko | Multi-way feature selection for ECoG-based Brain-Computer Interface[END_REF][START_REF] Salari | Classification of Articulator Movements and Movement Direction from Sensorimotor Cortex Activity[END_REF]] [Schaeffer and Aksenova, 2016a[START_REF] Shimoda | Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques[END_REF]] [Zhao et al., 2013a] and EEG [START_REF] Ebrahimpour | EEG-based motor imagery classification using wavelet coefficients and ensemble classifiers[END_REF][START_REF] López-Larraz | Continuous decoding of movement intention of upper limb self-initiated analytic movements from pre-movement EEG correlates[END_REF][START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF][START_REF] Robinson | Multi-class EEG classification of voluntary hand movement directions[END_REF][START_REF] Sreeja | Weighted sparse representation for classification of motor imagery EEG signals[END_REF] neural signals. A comparative study based on ECoG neural signal processing highlighted that WT may provide better frequency resolution than STFT or autoregressive analysis [START_REF] Motrenko | Multi-way feature selection for ECoG-based Brain-Computer Interface[END_REF].

The Hilbert transform computes from the neural signals a so-called analytic signal defined in the complex domain. A spectro-temporal representation of the signal may be obtained by decomposing it into neighboring frequency components (through bandpass filters) and by computing the so-called analytic signal of each component via the Hilbert transform [START_REF] Bruns | Fourier-, Hilbert-and wavelet-based signal analysis: are they really different approaches[END_REF]. Hilbert transform was applied in several ECoG [Hotson et al., 2016a] [START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF][START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF][START_REF] Vansteensel | Fully Implanted Brain-Computer Interface in a Locked-In Patient with ALS[END_REF]] [Wang et al., 2011] and EEG [START_REF] Cincotti | High-resolution EEG techniques for brain-computer interface applications[END_REF][START_REF] Hettiarachchi | Multivariate Adaptive Autoregressive Modeling and Kalman Filtering for Motor Imagery BCI[END_REF][START_REF] Iqbal | A QR decomposition based RLS algorithm with forgetting factor for adaptation of AR EEG features[END_REF]] [Spüler et al., 2012b] [Vidaurre et al., 2006b, 2007[START_REF] Wen | Online motor imagery BCI based on adaptive and incremental linear discriminant analysis algorithm[END_REF] [ [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF]] BCI experiments.

BCIs based on phase features are less common than amplitude ones. However, the phase features were exploited in few offline and online BCI studies [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2018] [Volkova et al., 2019]. Online and offline decoders based on phase features extracted with Hilbert transform were reported in [START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF][START_REF] Gysels | Phase synchronization for the recognition of mental tasks in a brain-computer interface[END_REF][START_REF] Hamner | Phase-based features for motor imagery braincomputer interfaces[END_REF][START_REF] Pourbakhtiar | Neuro-fuzzy classification of brain computer interface data using phase based feature[END_REF][START_REF] Wang | Phase Synchrony Measurement in Motor Cortex for Classifying Single-trial EEG during Motor Imagery[END_REF][START_REF] Wei | Amplitude and phase coupling measures for feature extraction in an EEG-based brain-computer interface[END_REF]. Phase features were directly used as BCI decoder input variables, but other feature generation procedures were proposed. For example, the Phase Locking Value (PLV) was used to investigate the task-induced coupling in long-range synchronization of the neural activity between two electrodes from EEG neural signals [START_REF] Gysels | Phase synchronization for the recognition of mental tasks in a brain-computer interface[END_REF][START_REF] Hamner | Phase-based features for motor imagery braincomputer interfaces[END_REF][START_REF] Pourbakhtiar | Neuro-fuzzy classification of brain computer interface data using phase based feature[END_REF][START_REF] Wang | Phase Synchrony Measurement in Motor Cortex for Classifying Single-trial EEG during Motor Imagery[END_REF][START_REF] Wei | Amplitude and phase coupling measures for feature extraction in an EEG-based brain-computer interface[END_REF]. Some EEG comparative studies highlighted better performance using phase features than amplitude ones [START_REF] Sburlea | Advantages of EEG phase patterns for the detection of gait intention in healthy and stroke subjects[END_REF][START_REF] Zeng | The Advantage of Low-Delta Electroencephalogram Phase Feature for Reconstructing the Center-Out Reaching Hand Movements[END_REF]. In [START_REF] Hammer | The role of ECoG magnitude and phase in decoding position, velocity, and acceleration during continuous motor behavior[END_REF], offline ECoG-based BCI results suggested that low frequency phases may be more informative for continuous motor decoding than magnitude-based features. Additionally, amplitude coupled with phase features showed better decoding performance than phase and 66 Chapter 3 : Transducers in BCI amplitude features separately in [START_REF] Hammer | The role of ECoG magnitude and phase in decoding position, velocity, and acceleration during continuous motor behavior[END_REF][START_REF] Sburlea | Advantages of EEG phase patterns for the detection of gait intention in healthy and stroke subjects[END_REF]. However, phase features were understudied compared to amplitude features and more investigation must be carried out to evaluate the potential of BCI decoding relying on phase or amplitude-phase features.

Commonly, the neural signals recorded from multiple electrodes are considered independently and concatenated to obtain a time-frequency description for each channel creating time-feature-space features. However, some strategies were designed to improve signal-to-noise ratio before computing time and time-frequency features.

Spatial filters combine the original recorded sources to create virtual ones with a higher signal-to-noise ratio than that of individual electrodes [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF] CSP, PCA and ICA algorithms were also used for feature dimension reduction procedure described in the next subsection.

Multi-way neural signal features. Since many years, time-frequency analysis highlighted attractive results to decode brain neural signals. Additionally, with the development of more sophisticated recording systems, electrodes/channels numbers highly increased to improve the recording spatial resolution. Therefore, modern BCIs rely generally on features with two (e.g. frequency-space) or three (e.g. time-frequencyspace) multi-way array (also referred to as tensor) to decode brain signals [START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF]. The order of a tensor is the number of its dimensions. Vector and matrices of the neural signal features are specific cases of N=1 and N=2 way tensor [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF].

Generally, BCI relied on neural signals described with a N=3 tensor of features (e.g. timefrequency-space features). Two main strategies were designed to process multi-way time-frequency-space features to decode the neural signals. The most usual procedure is named unfolding and considered the feature space as a vector or matrix feature space by concatenating supplementary dimensions. Unfolding procedure has the benefit to reduce the N-way tensor of neural signals features into a well-known domain and allows the application of generic BCI algorithms in clinical and preclinical EEG and ECoG classification and regression experiments [START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]] [Schaeffer and Aksenova, 2016a[START_REF] Shimoda | Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques[END_REF]. However, unfolding strategy exponentially increases the dimension of the feature space and limits the analysis of the features to standard pair-wise interactions which inevitably creates independent features and loses potentially existing interactions between/among the folded modes, such as time, frequency and space modes [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF][START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF]. Consequently, several approaches were designed to directly extract neural signal decoding information from high dimensional tensors using tensor factorization procedures or high dimensional tensor projectors to make the best use of the neural signal information. These procedures were reported in EEG and ECoG neural signal decoding studies [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF][START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF]] [Eliseyev et al., 2017[START_REF] Eliseyev | Recursive N-way partial least squares for brain-computer interface[END_REF], 2014, 2016] [ [START_REF] Krauss | A statistical method for analyzing and comparing spatiotemporal cortical activation patterns[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Onishi | Tensor classification for P300-based brain computer interface[END_REF] [ [START_REF] Washizawa | Tensor Based Simultaneous Feature Extraction and Sample Weighting for EEG Classification[END_REF][START_REF] Zhang | Frequency recognition in ssvep-based bci using multiset canonical correlation analysis[END_REF][START_REF] Zhang | Optimizing spatial patterns with sparse filter bands for motor-imagery based brain-computer interface[END_REF][START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]] [Zhao et al., 2013a, 2013b].

Other features were used in the context of BCI to decode neural signals [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. For example, features based on entropy measures or temporal sequence, etc. were evaluated during EEG experiments [START_REF] Boostani | A comparison approach toward finding the best feature and classifier in cue-based BCI[END_REF][START_REF] Garcia | Support vector EEG classification in the Fourier and timefrequency correlation domains[END_REF][START_REF] Hsu | EEG-based motor imagery classification using enhanced active segment selection and adaptive classifier[END_REF][START_REF] Obermaier | Hidden Markov models for online classification of single trial EEG data[END_REF][START_REF] Thulasidas | Robust classification of EEG signal for brain-computer interface[END_REF][START_REF] Vidaurre | Time Domain Parameters as a feature for EEGbased Brain-Computer Interfaces[END_REF]] [Zhang et al., 2008]. Additionally, a decoder using covariance matrices as input features were studied since the last few years. These decoders relied on the computation of symmetric positivedefinite matrices and Riemannian geometry [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. Riemannian geometry using manifold of EEG neural signal covariance matrices highlighted promising results in the last few years.

End-to-end decoding. Deep learning is a specific method of the machine-learning field whose popularity is gradually increasing for BCI applications. Deep learning is usually implemented using a neural network architecture composed of several layers. These end-to-end trained decoders do not rely on a fixed feature extraction block to decode the brain neural signals as classic machine-learning algorithms. Instead, the feature extraction step is directly integrated into the model training to automatically extract features useful for decoding rather than hand-engineering them [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF].

Popular deep neural networks approaches reported in BCI studies clustered convolutional neural networks, recurrent neural networks, restricted Boltzmann machines [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF], etc.

In conclusion, motor BCI relying in population recording system commonly used amplitude-based features. Phase or phase-amplitude-based neural decoders highlighted better decoding performance than amplitude-based decoders in several studies with ECoG [START_REF] Hammer | The role of ECoG magnitude and phase in decoding position, velocity, and acceleration during continuous motor behavior[END_REF] and EEG neural signal recording systems [START_REF] Djemal | Three-Class EEG-Based Motor Imagery Classification Using Phase-Space Reconstruction Technique[END_REF][START_REF] Kumar | EEG Based Motor Imagery Classification Using Instantaneous Phase Difference Sequence[END_REF][START_REF] Sburlea | Advantages of EEG phase patterns for the detection of gait intention in healthy and stroke subjects[END_REF][START_REF] Sun | Investigation of the phase feature of lowfrequency electroencephalography signals for decoding hand movement parameters[END_REF][START_REF] Zeng | The Advantage of Low-Delta Electroencephalogram Phase Feature for Reconstructing the Center-Out Reaching Hand Movements[END_REF]. Nevertheless, the benefits of phase-related features remain unclear due to the small number of reported offline studies and especially online experiments. In [START_REF] Krusienski | Value of amplitude, phase, and coherence features for a sensorimotor rhythm-based brain-computer interface[END_REF], EEG phase features did not lead to better classification accuracy. Phase features showed lower inter-session performance variability than amplitude-based decoder in [START_REF] Sburlea | Advantages of EEG phase patterns for the detection of gait intention in healthy and stroke subjects[END_REF]. However, decoders combining both phase and amplitude features showed better inter-session performance than amplitude and phase decoders. Due to the lack of knowledge and unclear benefits of phase features, decoders based on amplitude features remain more widespread. Deeper investigations on phase features must be carried out.

While extensive studies on BCI using population recording system were conducted, there is no consensus on the best time-frequency features to select for BCI application.

The efficiency of time-frequency and time-scale features for neural signal decoding depends on the analyzed datasets [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]]. On the one hand, parametric strategies generally outperformed non-parametric methods when the neural signals were well fitted by the selected parametric model. On the other hand, they were irrelevant if the signal was badly fitted [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. STFT requires finding a trade-off between time and frequency resolution. AR spectral estimation is preferred to Fourier Transform but it performs poorly when the signal is not stationary (which is problematic with non-stationary neural signals) and is also sensitive to the artifact [START_REF] Kevric | Comparison of signal decomposition methods in classification of EEG signals for motor-imagery BCI system[END_REF]Subasi, 2017] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]. Some comparative studies highlighted better performance with parametric models [START_REF] Herman | Comparative Analysis of Spectral Approaches to Feature Extraction for EEG-Based Motor Imagery Classification[END_REF] whereas it was outpaced by wavelet decomposition in others

Feature dimension reduction

As previously mentioned, with the multiplication of the studied frequency bands and the number of recorded channels, the feature space may have a high dimension. High dimensional feature space may be problematic for the neural signal decoders which require larger training dataset for the calibration procedure and require managing noninformative or correlated features. The application of dimensional reduction algorithms highlighted several benefits for neural signal processing. Feature space dimension reduction allows avoiding the curse of dimensionality [START_REF] Bellman | Adaptive Control Processes: A Guided Tour[END_REF], improving decoding performances, and reducing the required computing time by allocating less computing resources to the feature extraction step (do not compute the irrelevant features). In the case of high-dimensional data flow processing and real-time decoding, all the mentioned aspects are relevant [START_REF] Haufe | Dimensionality reduction for the analysis of brain oscillations[END_REF]. Dimensionality reduction algorithms are dissociated into the projection methods and the feature selection algorithms.

Projection algorithms aim to project the original feature space into a lower dimensional subspace by linear or non-linear combinations of the initial feature space components. This family clusters the principal and independent component analysis (PCA and ICA), common spatial pattern (CSP) or partial least square (PLS). They were commonly used in BCI applications [START_REF] Bousseta | EEG Based Brain Computer Interface for Controlling a Robot Arm Movement Through Thought[END_REF]] [Bundy et al., 2016[START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Eliseyev et al., 2017] [Haufe et al., 2014] [Hsu et al., 2016[START_REF] Jafarifarmand | Real-time multiclass motor imagery brain-computer interface by modified common spatial patterns and adaptive neuro-fuzzy classifier[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF] [S. P. Kim et al., 2006[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Marathe | Decoding continuous limb movements from high-density epidural electrode arrays using custom spatial filters[END_REF][START_REF] Palmer | Independent Component Analysis (ICA) Features for Electro-corticographic (ECoG) Brain-Machine Interfaces (BMIs). 臨床神経生理[END_REF][START_REF] Sannelli | Ensembles of adaptive spatial filters increase BCI performance: an online evaluation[END_REF]] [Schaeffer and Aksenova, 2016b[START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Sreenath | Classification of denoising techniques for EEG signals: A review[END_REF].

Principal component analysis (PCA) is a statistical unsupervised procedure used for dimensional reduction and feature extraction in the BCI field. PCA projectors are estimated to maximize the variance of the projected data [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF] [Nicolas-Alonso and Gomez-Gil, 2012]. In the vector space, it can be easily demonstrated that the variance is maximum if the projector is set to the eigenvector of the observation variable having the largest eigenvalue. For higher dimensional space, projectors are incrementally added by selecting the eigenvectors with the highest eigenvalues in descending order [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Eigenvalues are representative of the information provided by each eigenvector to describe the dataset distribution. Therefore, the most informative PCA features can be used whereas less informative ones can be discarded to reduce the new feature space dimension with minimal loss of information (generally 5% to 30%). PCA feature projection algorithm was applied for the dimensional reduction of MEA [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF] [ Velliste et al., 2014] [Wu and[START_REF] Wu | Direct feature extraction from multi-electrode recordings for spike sorting[END_REF], ECoG [START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF][START_REF] Wang | Human Motor Cortical Activity Recorded with Micro-ECoG Electrodes During Individual Finger Movements[END_REF] and EEG [START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]] neural feature space. The application of PCA does not always lead to performance improvements as the discriminative feature may not be relying on the selected principal components, nevertheless, PCA is a decent noise reduction method [Nicolas-Alonso and Gomez-Gil, 2012] [ [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF].

Independent component analysis (ICA) is a statistical unsupervised procedure which identifies a statistically independent basis composed of so-called "source" signals estimated from the combination (commonly linear) of multiple electrodes signals [Nicolas-Alonso and Gomez-Gil, 2012] [START_REF] Stone | Independent component analysis: an introduction[END_REF]. The number of created sources is a hyperparameter fixed before the ICA computation. Generally, the number of sources is inferior to the number of original electrodes to perform the feature dimension reduction procedure. ICA was commonly used with low spatial resolution recording system such as EEG-based BCI to extract features from the neural signals in parallel with artifact rejection [START_REF] Farooq | Motor Imagery based Multivariate EEG Signal Classification for Brain Controlled Interface Applications[END_REF] Only few BCI studies based on ECoG neural signal analysis employing ICA were reported [START_REF] Bouchard | Sparse coding of ECoG signals identifies interpretable components for speech control in human sensorimotor cortex[END_REF][START_REF] Estrin | Improving Classification Accuracy in Cortical Surface Recordings Using ICA-Based Features[END_REF][START_REF] Palmer | Independent Component Analysis (ICA) Features for Electro-corticographic (ECoG) Brain-Machine Interfaces (BMIs). 臨床神経生理[END_REF] and the benefits of this method were less obvious than its application to EEG neural signal [START_REF] Hill | Classifying Event-Related Desynchronization in EEG, ECoG and MEG Signals[END_REF][START_REF] Rembado | Independent Component Decomposition of Human Somatosensory Evoked Potentials Recorded by Micro-Electrocorticography[END_REF]. Additionally, in [START_REF] Wu | To Explore the Potentials of Independent Component Analysis in Brain-Computer Interface of Motor Imagery[END_REF], ICA algorithms highlighted worse decoding performance compared to other classical spatial filter algorithms such as CSP algorithms during online motor BCI experiments.

Partial least square (PLS) algorithm is the supervised counterpart of PCA. The input neural signals and output variables to predict are projected into a new low dimension subspace which maximizes the covariance between input and output variables projected into this lower subspace [START_REF] Bro | Multi-way analysis in the food industry: models, algorithms, and applications[END_REF][START_REF] Bro | Comparative study of band-power extraction techniques for Motor Imagery classification[END_REF]. PLS was widely used for continuous decoding in the case of high dimensional features and small training dataset [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF]] [Chen et al., 2013[START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Eliseyev | Penalized Multi-Way Partial Least Squares for Smooth Trajectory Decoding from Electrocorticographic (ECoG) Recording[END_REF][START_REF] Jubien | Decoding of finger activation from ECoG data: a comparative study[END_REF][START_REF] Shimoda | Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques[END_REF]. Nevertheless, PLS was also reported as a projection algorithm and was coupled to other classification or regression decoder [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF]Erfanian, 2018] [Schaeffer andAksenova, 2016a]. Other projector algorithms were designed for BCI application such as Spatio-Spectral Decomposition (SSD) [START_REF] Dähne | SPoC: A novel framework for relating the amplitude of neuronal oscillations to behaviorally relevant parameters[END_REF], Neighborhood Component Analysis (NCA) [START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF], or dimensionality reduction algorithms based on Riemannian geometry [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. Projection methods create a linear combination of the existing neural signal features to create new, more informative, features. In the opposite, wrapper-based techniques rely on supervised learning algorithms to evaluate the possible interactions between the features and the decoder (Figure 3-1). [START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF], sequential backward floating selection (SBFS) [START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF], dimensionality reduction mechanism (called DimReM) [START_REF] Tan | Dimensionality reduction in evolutionary algorithms-based feature selection for motor imagery brain-computer interface[END_REF], genetic algorithms [START_REF] Corralejo | Feature selection using a genetic algorithm in a motor imagerybased Brain Computer Interface[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF]] [Moro et al., 2017[START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF][START_REF] Schroder | Automated EEG feature selection for brain computer interfaces[END_REF] and others strategies [START_REF] Motrenko | Multi-way feature selection for ECoG-based Brain-Computer Interface[END_REF] were used to optimize the feature selection step in BCI applications.

Embedded techniques regroups the strategies where the feature selection step is directly integrated into the model calibration procedure (Figure 3 In summary, Dimensional reduction algorithms highlighted benefits for neural signal decoding. They can remove correlated features, improve the signal-to-noise ratio, limit the risks of overfitting, speed up the decoder calibration and/or the neural signal decoding, reduce the computational loading, etc. Many strategies were tested to select the best features [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. However, there is no consensus on the best method to be followed as it is influenced by the analyzed signals properties (highly correlated features, low or high signal to noise ratio, signal stationarity across time, etc.), the problem to solve (binary or multi-class classification problem), the length of the training dataset, the restriction related to computational complexity (offline or online applications), etc. Projector methods were commonly applied for dimensional reduction of the neural signal feature dimension. Nevertheless, such procedures lead to less interpretable features, are computationally expensive and may be not optimal in the case of non-stationary signals. Feature selection algorithms find the best subset of features among all the computed ones. These methods can be dependent or not from the decoder training and present various advantages. Filter based-methods require less computing power than embedded methods which itself is less consuming than wrapper techniques.

Nevertheless, generally, wrapper and embedded methods outperformed filter-based strategies [START_REF] Rouhi | Feature Selection in High-Dimensional Data[END_REF].

Effector control features

From the extracted neural signal features, the decoder evaluates a discrete or continuous output that will be converted into commands to control the effector.

Discrete output variables are commonly estimated to evaluate the discrete mental state of the subject. In motor BCI applications, the discrete states/classes were used to control various movement states such as the opening/closure of the hand [START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF], the waking/gait cycle activation [START_REF] López-Larraz | Control of an Ambulatory Exoskeleton with a Brain-Machine Interface for Spinal Cord Injury Gait Rehabilitation[END_REF], the movements of one specific finger [Schaeffer and Aksenova, 2016a] or limb [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF], etc. Additionally, idle state classification was often integrated into asynchronous BCI systems [Kao et al., 2017] [Schaeffer andAksenova, 2016a]. Discrete output variables were also reported to control the continuous movements of an effector. The continuous movements were discretized into a finite number of directions which are selected using a classifier [START_REF] Huang | EEG-based online two-dimensional cursor control[END_REF][START_REF] Leeb | Self-Paced (Asynchronous) BCI Control of a Wheelchair in Virtual Environments: A Case Study with a Tetraplegic[END_REF][START_REF] Trejo | Brain-computer interfaces for 1-D and 2-D cursor control: designs using volitional control of the EEG spectrum or steady-state visual evoked potentials[END_REF]] [Vidaurre et al., 2016].

Continuous dependent variables were commonly the end-point kinematic variables of the controlled effector such as the position, the velocity, the speed, the acceleration or several of these variables [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Hammer | The role of ECoG magnitude and phase in decoding position, velocity, and acceleration during continuous motor behavior[END_REF] [Z. Li et al., 2009].

However, other less conventional output variables [START_REF] Marathe | Decoding position, velocity, or goal: Does it matter for brain-machine interfaces?[END_REF] such as joint velocity [START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF], angle/orientation [START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF], force [START_REF] Carmena | Learning to Control a Brain-Machine Interface for Reaching and Grasping by Primates[END_REF], muscle activation [START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF] were reported.

Finally, PCA or other dimensionality reduction algoritms were applied to reduce the dependent variable dimension, "decorrelate" [START_REF] Acharya | Electrocorticographic amplitude predicts finger positions during slow grasping motions of the hand[END_REF]] [Hotson et al., 2014] the output variables and represent the output variable space as a linear combination of the initial dependent variables. In this case, the model decoded the PCA coordinates from the neural signals before evaluating the original coordinates through inverse PCA transformation [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF]. Dependent variable dimension reduction via the PCA algorithm was reported in MEA preclinical [START_REF] Mollazadeh | Principal components of hand kinematics and neurophysiological signals in motor cortex during reach to grasp movements[END_REF][START_REF] Schaffelhofer | Decoding a Wide Range of Hand Configurations from Macaque Motor, Premotor, and Parietal Cortices[END_REF] and ECoG clinical [START_REF] Acharya | Electrocorticographic amplitude predicts finger positions during slow grasping motions of the hand[END_REF]] [Hotson et al., 2014[START_REF] Vinjamuri | Toward Synergy-Based Brain-Machine Interfaces[END_REF] experiments.

There is no consensus on the best continuous output variable to decode. Nevertheless, position and velocity are the most widespread output variables in the BCI field. In [START_REF] Marathe | Decoding position, velocity, or goal: Does it matter for brain-machine interfaces?[END_REF], the position-based decoder underperformed compared to the velocity-based decoder particularly in the case of large and numerous decoding errors.

Decoder/Model identification

After extracting the relevant information from the recorded neural signals, it is required to evaluate a function which transforms the input feature variables into discrete or continuous output variables. In the most general case, the assumption is made that an unknown linear or non-linear function is mapping the input variable space into a continuous or discrete variable space.

Generally, in the BCI field, the input and output variables at time 𝑡 are vector or matrix variables [START_REF] Cong | Tensor decomposition of EEG signals: A brief review[END_REF]. Let 𝐱 𝑡 ∈ ℝ 𝐼 1 , 𝐲 𝑡 ∈ ℝ 𝐽 1 and 𝑧 𝑡 ∈ ℕ denote the input neural signals variable, the continuous output variable and the discrete output variable respectively where 𝐼 1 is the input neural signal feature dimension and 𝐼 2 is the continuous dependent variable dimension. It is assumed that a linear or non-linear function ℎ exists such as 𝐲 𝑡 = ℎ(𝐱 𝑡 ) + 𝛆 𝑡 or z 𝑡 = ℎ(𝐱 𝑡 ) + ε 𝑡 in the cases of continuous and discrete variables respectively where 𝛆 𝑡 and ε 𝑡 are "noise" random variables generally, supposed to be independent and identically distributed.

The model identification, calibration or training consists in finding an estimation ℎ ̂ of the unknown function ℎ which minimizes the differences between the output variables 𝐲 𝑡 or z 𝑡 and an estimated output variable 𝐲 ̂𝑡 = ℎ ̂(𝐱 𝑡 ) or z ̂𝑡 = ℎ ̂(𝐱 𝑡 ) following specific criteria (e.g. minimum least square, maximum likelihood, etc.) and using machine-learning methods [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. Numerous models were tested to control various effectors and perform various tasks. Model parameter weights may be estimated using supervised, unsupervised and reinforcement learning strategies. This review is particularly focused on the supervised decoding of the neural signals in the case of BCI applications.

To translate clinical BCI experiments into daily life BCI applications, BCI system must be applied in real-time. Nevertheless, most BCI studies were carried out offline using a database with a finite number of samples. Translation from offline studies to real-time experiments is not trivial due to the computational complexity of the decoding algorithms used to estimate the output variables from input neural signal features. Numerous models applied in offline studies are not suitable for real-time decoding and data-flow processing. As an example, in [START_REF] Cunningham | Methods for estimating neural firing rates, and their application to brain-machine interfaces[END_REF], Cunningham argued that the computation time for spike rate decoding of a one-second spike train for the simplest methods was performed on a millisecond time scale whereas more complex strategies were limited to seconds or even minutes. In the case of real-time BCI application, algorithms with a computation time longer than the analyzed neural signal window can not be integrated into the online BCI system. Therefore, the computational complexity and the computation time for neural signal decoding are key characteristics to consider. Due to the temporal limitations of the complex algorithms and the optimization procedures, simple linear models are generally promoted for online neural signal decoding [START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF].

Additionally, for online closed-loop BCI applications, several studies highlighted parameter weights and decoding performance differences between models trained using open-loop and closed-loop calibration procedures [START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]] [Orsborn et al., 2014]. Closed-loop decoder adaptation (CLDA) can be achieved using an offline or online calibration procedure with different update frequencies (e.g. after each sample, second, trial, session, day, etc.). However, several benefits of online incremental calibration procedures were stressed in [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF] such as shorter calibration sessions, rapid feedback to the patient, etc. For example, a cross-validation procedure for model selection or hyperparameter optimization is a widespread calibration method applied in the BCI field which is computationally too heavy to be implemented for realtime model calibration. Therefore, the computational burden and frequency rate of the calibration procedure are important characteristics to consider.

The following section firstly introduces the most common strategies practiced for offline model calibration and their application in offline studies or online BCI experiments. Finally, a specific focus is granted to CLDA and incremental adaptive model calibration during real-time BCI experiments.

Offline decoder calibration

Discrete output variables decoding: Classifiers

A classification problem makes the assumption that observed neural signal 𝐱 𝑡 ∈ ℝ 𝐼 1 can be clustered into a finite number of class/label states 𝑧 𝑡 ∈ ℕ. These states are related to specific mental or motor imagery tasks which create specific neural signal patterns identified by the classifier. Offline classification experiments are the most common studies in the motor-BCI fields and numerous decoders were designed to enhance classification performance.

Linear Discriminant Analysis (LDA) decoder is a classic linear classifier for binary and multi-class problems. LDA is based on a multivariate Gaussian distribution estimation for each state with the assumption of equal covariance for each class. In the case of binary classification, LDA is looking for the hyperplane which maximizes the distance between the two projected classes while minimizing the interclass variance [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF].

LDA is one of the most popular types of classifiers in the BCI field [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF] and has been used in numerous experiments. LDA classifiers were employed for offline EEG neural signal analysis and performance comparison to other algorithms [START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF]] [Kim et al., 2018[START_REF] López-Larraz | Continuous decoding of movement intention of upper limb self-initiated analytic movements from pre-movement EEG correlates[END_REF][START_REF] Lotte | Regularizing Common Spatial Patterns to Improve BCI Designs: Unified Theory and New Algorithms[END_REF][START_REF] Scherer | Individually Adapted Imagery Improves Brain-Computer Interface Performance in End-Users with Disability[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Shin | Noise robustness analysis of sparse representation based classification method for non-stationary EEG signal classification[END_REF]. During the offline studies, stepwise LDA (SWLDA) and LDA were applied for preparation versus execution tasks and cursor direction classification from ECoG neural signals [START_REF] Gunduz | Differential roles of high gamma and local motor potentials for movement preparation and execution[END_REF]. Additionally, the hand flexion and extension were classified during offline ECoG [START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF]and EEG [START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF] studies.

LDA classifiers were embedded in several online BCI systems based on both invasive and non-invasive recording systems. A BCI communication system controlled by an amyotrophic lateral sclerosis (ALS) patient and a locked-in syndrome patient was reported using LFP signals (MEA) and a LDA classifier [START_REF] Milekovic | Stable long-term BCI-enabled communication in ALS and locked-in syndrome using LFP signals[END_REF]. Additionally, the real-time click detection of another BCI communication system was controlled by two tetraplegic and two ALS patients using SUA/MUA signals in [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF]. ECoG-based real-time decoding was performed to detect idle versus active state from epileptic patients using a LDA decoder [START_REF] Kapeller | Online control of a humanoid robot through hand movement imagination using CSP and ECoG based features[END_REF] whereas a hierarchical LDA classifier was integrated into online asynchronous BCI experiments to detect the individual finger and idle states [Hotson et al., 2016a].

A tetraplegic patient [START_REF] Pfurtscheller | Brain oscillations control hand orthosis in a tetraplegic[END_REF] [START_REF] Donati | Long-Term Training with a Brain-Machine Interface-Based Gait Protocol Induces Partial Neurological Recovery in Paraplegic Patients[END_REF]. A LDA decoder was selected in [START_REF] Khan | fNIRS-based Neurorobotic Interface for gait rehabilitation[END_REF] for real-time detection of the walking gait cycle from a fNIRS recording system because LDA provided a good trade-off between the time of execution and classification accuracy. Finally, online three-states classification experiments were reported based on the EEG neural signal decoding of 6 healthy subjects using LDA classifier [START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF][START_REF] Chae | Toward brain-actuated humanoid robots: Asynchronous direct control using an EEG-Based BCI[END_REF] and 15 healthy subject using the shrinkage LDA algorithm [START_REF] Schwarz | Decoding hand movements from human EEG to control a robotic arm in a simulation environment[END_REF].

Quadratic Discriminant Analysis (QDA) is a non-linear variant of LDA in which an individual covariance matrix is estimated for every class of observations. QDA is particularly useful if there is the prior knowledge that individual classes exhibit distinct covariance matrices. Due to higher computational complexity without obvious performance improvements, its simpler version, the LDA was often preferred for BCI application. Nevertheless, QDA was regularly used in offline studies for classifier performance comparison. QDA was compared to other algorithms based on ECoG [START_REF] Jubien | Decoding of finger activation from ECoG data: a comparative study[END_REF], EEG [Bhattacharyya et al., 2017b] [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. In numerous BCI experiments, SVM outperformed other algorithms and highlighted robust classification with better generalization ability [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. Therefore, SVM was frequently applied to compare algorithm performance during offline studies using ECoG [START_REF] Jubien | Decoding of finger activation from ECoG data: a comparative study[END_REF], EEG [START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF][START_REF] Eva | Amplitude Modulation Index as Feature in a Brain Computer Interface[END_REF][START_REF] Faradji | Plausibility assessment of a 2-state self-paced mental task-based BCI using the no-control performance analysis[END_REF][START_REF] Hettiarachchi | Multivariate Adaptive Autoregressive Modeling and Kalman Filtering for Motor Imagery BCI[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF][START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF][START_REF] Schlögl | Characterization of four-class motor imagery EEG data for the BCI-competition 2005[END_REF][START_REF] Shin | Noise robustness analysis of sparse representation based classification method for non-stationary EEG signal classification[END_REF][START_REF] Tan | Dimensionality reduction in evolutionary algorithms-based feature selection for motor imagery brain-computer interface[END_REF] and fNIRS [START_REF] Khan | fNIRS-based Neurorobotic Interface for gait rehabilitation[END_REF]] [Naseer et al., 2016b] recording systems. Additionally, online experiments to control real and virtual effectors [START_REF] Al-Quraishi | EEG-Based Control for Upper and Lower Limb Exoskeletons and Prostheses: A Systematic Review[END_REF] were performed using SVM algorithm for open/closure hand movement in MEG [START_REF] Fukuma | Real-Time Control of a Neuroprosthetic Hand by Magnetoencephalographic Signals from Paralysed Patients[END_REF] as well as for P300 speller control [START_REF] Thulasidas | Robust classification of EEG signal for brain-computer interface[END_REF][START_REF] Woehrle | An Adaptive Spatial Filter for User-Independent Single Trial Detection of Event-Related Potentials[END_REF] and online 4 state classification for 2D cursor control [START_REF] Huang | EEG-based online two-dimensional cursor control[END_REF] or robot arm / prosthetic control [START_REF] Hortal | SVM-based Brain-Machine Interface for controlling a robot arm through four mental tasks[END_REF][START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF]. Multiple SVM classifiers were trained in [Bhattacharyya et al., 2017a] to perform online 3D sequential control of a hardwired Jaco robot arm. et al., 2015]. Due to the high computational burden of ANN algorithms, only few online BCI experiments were reported using ANN. A tetraplegic patient controlled in real-time a four-class functional electrical stimulation (FES) system to perform four hand shapes. The employed deep neural network was made up of Long Short Term Memory (LSTM), convolutional and fully connected layers [START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF]. Online recognition of two mental states was performed using a fNIRS recording system and Deep ANN during driving simulation experiments [START_REF] Huve | Online Recognition of the Mental States of Drivers with an fNIRS-Based Brain-Computer Interface Using Deep Neural Network[END_REF]. Nowadays, ANN is not as prevalent in the real-time BCI application as in other domain due to the lack of interpretability of the ANN models (e.g. the ANN model parameters cannot be related to fundamental neurophysiological insights) [START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] and the high computational resources required to performed neural signal decoding.

K-nearest neighbor (kNN) is a non-parametric non-linear classifier which determines the unseen samples label depending on the label of its k-nearest neighbors evaluated within the training dataset. The nearest neighbors are found according to a distance metric (e.g. Euclidian distance). The unseen sample is clustered in the same class as its closest neighbors according to the distance metrics [START_REF] Khan | fNIRS-based Neurorobotic Interface for gait rehabilitation[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. kNN algorithm was often tested during offline comparative studies of multiple algorithms as it is an easy non-linear decoder to implement. kNN experiments were predominantly reported for EEG recordings analysis. Binary classification performance comparison between kNN and other algorithms were reported based on EEG dataset [START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF][START_REF] Hettiarachchi | Multivariate Adaptive Autoregressive Modeling and Kalman Filtering for Motor Imagery BCI[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF][START_REF] Tan | Dimensionality reduction in evolutionary algorithms-based feature selection for motor imagery brain-computer interface[END_REF] as well as 3state [START_REF] Eva | Amplitude Modulation Index as Feature in a Brain Computer Interface[END_REF]] and 4-state [START_REF] Schlögl | Characterization of four-class motor imagery EEG data for the BCI-competition 2005[END_REF] classification performance comparison. Additionally, online control of a lower limb exoskeleton was reported from EEG neural signal decoding using a kNN classifier [START_REF] Kwak | A lower limb exoskeleton control system based on steady state visual evoked potentials[END_REF]. The offline upper limb movement classification (3 classes) of two epileptic subjects was reported based on ECoG recordings using kNN classifier [START_REF] Chin | Identification of arm movements using correlation of electrocorticographic spectral components and kinematic recordings[END_REF]. Finally, kNN was tested during offline binary classification comparative studies using a fNIRS recording system [START_REF] Khan | fNIRS-based Neurorobotic Interface for gait rehabilitation[END_REF]] [Naseer et al., 2016b]. This algorithm is efficient with low dimensional feature vectors. However, Euclidian distance-based kNN algorithms are highly sensitive to high dimensional feature space [Nicolas-Alonso and Gomez-Gil, 2012]. Due to its heavy computational resource requirements in the case of high dimensional data and its nonsignificant enhancement of the performance compared to other simpler methods, kNN was scarcely used for online BCI experiments.

Other discrete algorithms were applied in BCI experiments. Bayesian classifiers were reported during offline EEG and fNIRS studies [START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF], 2018] [Naseer et al., 2016b] as well as during online EEG experiments using LDA and Bayes rules [START_REF] He | Brain-machine interfaces for controlling lower-limb powered robotic systems[END_REF][START_REF] King | Performance Assessment of a Brain-Computer Interface Driven Hand Orthosis[END_REF]. A logistic regression (LR) classifier was tested during EEG [START_REF] Javed | Recognition of finger movements using EEG signals for control of upper limb prosthesis using logistic regression[END_REF] and ECoG [START_REF] Jubien | Decoding of finger activation from ECoG data: a comparative study[END_REF] offline comparative studies as well as during EEG online comparative study [Bhattacharyya et al., 2017b], online binary classification experiments [START_REF] Lehtonen | Online Classification of Single EEG Trials During Finger Movements[END_REF] and online walking detection rehabilitation experiments [START_REF] García-Cossio | Decoding Sensorimotor Rhythms during Robotic-Assisted Treadmill Walking for Brain Computer Interface (BCI) Applications[END_REF]. Additionally, supervised Gaussian Mixture Models (GMM) were integrated into a BCI system for real-time lower-body effector control from EEG neural signals [START_REF] Kilicarslan | High accuracy decoding of user intentions using EEG to control a lower-body exoskeleton[END_REF]. Riemannian geometry-based classifiers were recently developed in the BCI field [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Yger | Riemannian Approaches in Brain-Computer Interfaces: A Review[END_REF].

Few EEG-based offline [START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF][START_REF] Roijendijk | Classifying Regularized Sensor Covariance Matrices: An Alternative to CSP[END_REF] and online [START_REF] Kalunga | Online SSVEPbased BCI using Riemannian geometry[END_REF] experiments highlighted the benefits of these new classifiers. Finally, random forest (RF) [START_REF] Al-Quraishi | EEG-Based Control for Upper and Lower Limb Exoskeletons and Prostheses: A Systematic Review[END_REF] and Mahalanobis distance-based classifiers [START_REF] Eva | Amplitude Modulation Index as Feature in a Brain Computer Interface[END_REF][START_REF] Faradji | Plausibility assessment of a 2-state self-paced mental task-based BCI using the no-control performance analysis[END_REF][START_REF] Huang | EEG-based online two-dimensional cursor control[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Schlögl | Characterization of four-class motor imagery EEG data for the BCI-competition 2005[END_REF] were applied to EEG BCI experiments to detect discrete mental states.

Previously presented decoders are static decoders. These decoders assume that the observed variables are independent in time. The possible temporal dependencies between successive observed variables are not considered. However, generally, this assumption is violated in BCI and particularly in motor BCI applications [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. Therefore, several BCI studies were led to investigate the classification enhancement related to the integration of temporal information into the BCI transducer.

One explored solution was to consider the temporal dependencies in the feature extraction or post-processing steps. Several BCI experiments were reported with neural features estimated using temporal/recurrent sliding windows [START_REF] Dietterich | Machine Learning for Sequential Data: A Review[END_REF]] [Eliseyev et al., 2017[START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF] whereas the decoder output variables were post-processed using a moving average filter [Hotson et al., 2016a]. In [START_REF] Fifer | Simultaneous Neural Control of Simple Reaching and Grasping With the Modular Prosthetic Limb Using Intracranial EEG[END_REF], LDA was coupled to a manually adjusted transition probability matrix to control in real-time the reaching and grasping movement (binary classification) of a robotic HMM classifiers were reported in offline [Antelis et al., 2017] [Dobiáš and[START_REF] Dobiáš | Movement EEG classification using parallel Hidden Markov Models[END_REF] and online [START_REF] Obermaier | Hidden Markov models for online classification of single trial EEG data[END_REF]] EEG-based BCI experiments. Moreover, Markov models were integrated into more complex classifiers and tested in an offline study [START_REF] Williams | Markov Model-Based Method to Analyse Time-Varying Networks in EEG Task-Related Data[END_REF] and online experiment [START_REF] Lisi | Markov Switching Model for Quick Detection of Event Related Desynchronization in EEG[END_REF] using EEG neural signal recordings. Dynamic models were also reported with invasive recording systems. ECoG neural signal classification with Markovian process or HMM were tested offline [Onaran et al., 2011b[START_REF] Pfeiffer | Hidden Markov model based continuous decoding of finger movements with prior knowledge incorporation using bi-gram models[END_REF]] [Wang et al., 2011[START_REF] Wissel | Hidden Markov Model and Support Vector Machine based decoding of finger movements using Electrocorticography[END_REF] and online [Hotson et al., 2016a[START_REF] Moses | Real-time classification of auditory sentences using evoked cortical activity in humans[END_REF] clinical experiments whereas preclinical online [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF] and offline [START_REF] Darmanjian | Bimodal brain-machine interface for motor control of robotic prosthetic[END_REF] MEA-based BCI studies applied HMM to detect idle versus active movement state activation.

Variants of HMM such as Hierarchical HMM (HHMM) [START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF] [Suk and [START_REF] Suk | Two-Layer Hidden Markov Models for Multi-class Motor Imagery Classification[END_REF], Input-Output HMM (IOHMM) [START_REF] Chiappa | Analysis and Classification of EEG Signals using Probabilistic Models for Brain Computer Interfaces[END_REF] or Kernel-HMM [START_REF] Xu | Kernel based hidden Markov model with applications to eeg signal classification[END_REF] were tested offline using BCI EEG datasets. The HHMM generalized the HMM to a structured multi-level stochastic process where each hidden state of the HMM is composed of sub-states which are themselves modeled by a HMM and so on [START_REF] Fine | The Hierarchical Hidden Markov Model: Analysis and Applications[END_REF]. IOHMM is a discriminative decoder which models both the hidden state and observation variable succession. The Kernel-HMM combines the HMM and the maximum margin principle projected into a kernel space of a support vector machine (SVM) to enhance the classification performance [START_REF] Xu | Kernel based hidden Markov model with applications to eeg signal classification[END_REF]. Input-Output HMM (IOHMM) [START_REF] Chiappa | Analysis and Classification of EEG Signals using Probabilistic Models for Brain Computer Interfaces[END_REF] and Kernel-HMM [START_REF] Xu | Kernel based hidden Markov model with applications to eeg signal classification[END_REF] algorithms outperformed HMM in EEG-based BCI classification offline studies. The hidden semi-Markov model (HSMM), another extension of HMM, was used for offline unsupervised fMRI mapping [START_REF] Faisan | Unsupervised learning and mapping of active brain functional MRI signals based on hidden semi-Markov event sequence models[END_REF] and EEG mental state classification [START_REF] Oliver | Asynchronous Brain Computer Interface using Hidden Semi-Markov Models[END_REF].

Conditional random fields (CRF) algorithm was reported in offline motor BCI experiments based on EEG [Delgado [START_REF] Delgado Saa | Hidden conditional random fields for classification of imaginary motor tasks from EEG data[END_REF]] [Hasan and Gan, 2011a, 2011b[START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF], 2012] and clinical ECoG [START_REF] Saa | Asynchronous decoding of finger movements from ECoG signals using long-range dependencies conditional random fields[END_REF] recordings. CRF is a discriminative model where the interaction of the latent variables with each other is determined by several past latent variables (𝑧 𝑡-1 , … 𝑧 𝑡-Δ𝑡 ) and the sequence of observed variables (𝑥 𝑡-1 , … 𝑥 𝑡-Δ𝑡 ). The HMM can be seen as the generative version of a linear-chain CRF with a particular choice of feature function [Dietterich, 2002] [Sutton and[START_REF] Sutton | An Introduction to Conditional Random Fields[END_REF]. Variants of CRF such as latent dynamic CRF [START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF] or Hidden CRF [Delgado Saa and[START_REF] Delgado Saa | Hidden conditional random fields for classification of imaginary motor tasks from EEG data[END_REF][START_REF] Saa | A latent discriminative model-based approach for classification of imaginary motor tasks from EEG data[END_REF] were tested in EEG offline studies.

Neural Networks (ANN) taking into account temporal modeling of the observed variables were tested in offline BCI classification studies relying on EEG [Bashashati et al., 2017] [Bashashati and Ward, 2017] [J.-M. Cano-Izquierdo et al., 2012[START_REF] Haselsteiner | Using time-dependent neural networks for EEG classification[END_REF] or ECoG [START_REF] Du | Decoding ECoG Signal with Deep Learning Model Based on LSTM[END_REF][START_REF] Xie | Decoding of finger trajectory from ECoG using deep learning[END_REF] neural signals. Additionally, online BCI experiments using EEG [START_REF] Millan | Noninvasive brain-actuated control of a mobile robot by human EEG[END_REF] and MEA [START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF] recording system were reported. In the cited BCI experiments, various dynamic ANN algorithm such as Long-Short Term Memory (LSTM) networks [START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF] [ [START_REF] Xie | Decoding of finger trajectory from ECoG using deep learning[END_REF], Neural Networks CRF [Bashashati et al., 2017] [Bashashati andWard, 2017], time-dependent multi-layer perceptron algorithms [START_REF] Haselsteiner | Using time-dependent neural networks for EEG classification[END_REF]Pfurtscheller, 2000] [Millan et al., 2004], etc. were tested.

Whereas CRF and NN outperformed HMM in numerous articles [Bashashati et al., 2017] [Bashashati andWard, 2017] et al., 2015], the decoders based on these two strategies are computationally expensive [Dietterich, 2002] [Sutton and[START_REF] Sutton | An Introduction to Conditional Random Fields[END_REF] and may not be adapted to specific applications such as online decoding. HMM remains the most common dynamic model even though dynamic modeling remains underused for BCI state classification [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF].

In summary, various algorithms were applied to BCI mental state classification problems and few comparative studies intended to evaluate the best strategy to select [START_REF] Boostani | A comparison approach toward finding the best feature and classifier in cue-based BCI[END_REF][START_REF] Cincotti | Comparison of different feature classifiers for brain computer interfaces[END_REF][START_REF] Kanoga | A COMPARATIVE STUDY OF FEATURES AND CLASSIFIERS IN SINGLE-CHANNEL EEG-BASED MOTOR IMAGERY BCI[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Oganesyan | Comparison of Results Obtained Using Brain-Computer Interface Classifiers in a Motor Imagery Recognition Task[END_REF][START_REF] Saa | Asynchronous decoding of finger movements from ECoG signals using long-range dependencies conditional random fields[END_REF][START_REF] Wissel | Hidden Markov Model and Support Vector Machine based decoding of finger movements using Electrocorticography[END_REF]. No conclusion on the best classifier can be drawn from the comparative studies as the reported classifier performances seem to be highly dependent on, the studied dataset, the preprocessing, the feature extraction procedure, etc.

Among the static decoders, SVM, LDA and NN-based classifiers are likely the decoders with the most stable and robust performance across the studies [START_REF] Boostani | A comparison approach toward finding the best feature and classifier in cue-based BCI[END_REF] [ [START_REF] Kanoga | A COMPARATIVE STUDY OF FEATURES AND CLASSIFIERS IN SINGLE-CHANNEL EEG-BASED MOTOR IMAGERY BCI[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Oganesyan | Comparison of Results Obtained Using Brain-Computer Interface Classifiers in a Motor Imagery Recognition Task[END_REF]. Other algorithms such as kNN are not recommended in the case of high-dimensional space which is commonly the case in the neural signal processing field [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. Only few studies compared the performance of static versus dynamic decoders. With the exception of [START_REF] Cincotti | Comparison of different feature classifiers for brain computer interfaces[END_REF], dynamic decoders outperformed or at least performed as well as static decoders [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Saa | Asynchronous decoding of finger movements from ECoG signals using long-range dependencies conditional random fields[END_REF][START_REF] Wissel | Hidden Markov Model and Support Vector Machine based decoding of finger movements using Electrocorticography[END_REF]. The discrete decoding experiments reviewed in [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF] highlighted that the HMM provided better performance than other algorithms during synchronous BCI experiments but performed similarly than static decoders in asynchronous BCI experiments. However, asynchronous studies are lacking published experiments and results to bring out conclusions on the best classifier to employ.

Continuous output variable decoding

Regression is a statistical approach often used in BCI and particularly motor BCI field which decodes from the observed neural signal 𝐱 𝑡 ∈ ℝ 𝐼 1 a continuous output variable 𝐲 𝑡 ∈ ℝ 𝐽 1 . In the motor BCI field, the continuous dependent variables used to achieve endpoint effector control (cursor, prosthetic hand, etc.) were often kinematic variables such as position, velocity, acceleration or speed. With the exception of some EEG experiments [START_REF] Edelman | Noninvasive neuroimaging enhances continuous neural tracking for robotic device control[END_REF][START_REF] Mcfarland | ELECTROENCEPHALOGRAPHIC (EEG) CONTROL OF THREE-DIMENSIONAL MOVEMENT[END_REF][START_REF] Meng | Noninvasive Electroencephalogram Based Control of a Robotic Arm for Reach and Grasp Tasks[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF], continuous decoding was mostly performed with invasive recording systems (MEA and ECoG) [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. Numerous algorithms were designed to estimate linear or non-linear decoding models. A brief review of the most common strategies is described in this section.

Population Vector Algorithm (PVA) is a MEA-specific algorithm relying on the cosine tuning of the motor cortex neurons [START_REF] Georgopoulos | Neuronal population coding of movement direction[END_REF]. PVA estimates the cosine tuning of individual neuronal responses characterized by a single preferred direction in which the unit fires maximally. The vector direction is defined as the sum of the preferred directions of the recorded neuron population, weighted by the instantaneous firing rates of each cell [START_REF] Chase | Bias, optimal linear estimation, and the differences between open-loop simulation and closed-loop performance of spiking-based brain-computer interface algorithms[END_REF][START_REF] Schwartz | Extraction algorithms for cortical control of arm prosthetics[END_REF][START_REF] Velliste | Cortical control of a prosthetic arm for self-feeding[END_REF]. PVA assumes a uniform distribution of preferred directions. This algorithm was used for closed-loop BCI decoding of a prosthetic arm during monkey self-feeding experiments [START_REF] Velliste | Cortical control of a prosthetic arm for self-feeding[END_REF] and center-out experiments [START_REF] Koyama | Comparison of braincomputer interface decoding algorithms in open-loop and closed-loop control[END_REF][START_REF] Taylor | Direct Cortical Control of 3D Neuroprosthetic Devices[END_REF] [ [START_REF] Wahnoun | Selection and parameterization of cortical neurons for neuroprosthetic control[END_REF].

Optimal linear estimation (OLE) is a variant of PVA, without the assumption that the preferred directions are uniformly distributed, in order to avoid the bias introduced by PVA when the uniform distribution assumption is violated [START_REF] Chase | Bias, optimal linear estimation, and the differences between open-loop simulation and closed-loop performance of spiking-based brain-computer interface algorithms[END_REF] [START_REF] Salinas | Vector reconstruction from firing rates[END_REF]Abbott, 1994] [Wang et al., 2007] adapted to the ECoG neural signal processing, Wang and Degenhart highlighted real-time 3D cursor control by a tetraplegic, an ALS and a brachial plexus injured patients using motor imagery strategy [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] [W. Wang et al., 2013].

In the BCI field, linear regression models were frequently trained using Ordinary Least Squares (OLS) to estimate the maximum likelihood with the assumption of Gaussian noise. Nevertheless, OLS is highly unstable in high-dimensional feature space or in the case of correlated features [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2018] [Shanechi et al., 2013]. To overcome this issue, regularized/penalized linear regression were designed among which the pace regression [START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF][START_REF] Wang | Decoding Finger Flexion from Electrocorticographic Signals Using a Sparse Gaussian Process[END_REF], the sparse regression [START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF]] [Rouse et al., 2016] [Williams et al., 2013] and the ridge regression [START_REF] Kim | A study on a robot arm driven by three-dimensional trajectories predicted from non-invasive neural signals[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Shanechi | A Real-Time Brain-Machine Interface Combining Motor Target and Trajectory Intent Using an Optimal Feedback Control Design[END_REF]] [Suminski et al., 2013, 2010] [Willett et al., 2013]. These regularized OLS algorithms were exploited in all the BCI field with EEG [START_REF] Kim | A study on a robot arm driven by three-dimensional trajectories predicted from non-invasive neural signals[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF], ECoG [START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Nakanishi | Prediction of Three-Dimensional Arm Trajectories Based on ECoG Signals Recorded from Human Sensorimotor Cortex[END_REF] [ [START_REF] Rouse | Spatial co-adaptation of cortical control columns in a micro-ECoG brain-computer interface[END_REF]] [Williams et al., 2013] and MEA [START_REF] Shanechi | A Real-Time Brain-Machine Interface Combining Motor Target and Trajectory Intent Using an Optimal Feedback Control Design[END_REF]] [Suminski et al., 2013, 2010] [Willett et al., 2013] neural signal decoding, for offline [START_REF] Kim | A study on a robot arm driven by three-dimensional trajectories predicted from non-invasive neural signals[END_REF][START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Suminski | Incorporating Feedback from Multiple Sensory Modalities Enhances Brain-Machine Interface Control[END_REF] and online [START_REF] Rouse | Spatial co-adaptation of cortical control columns in a micro-ECoG brain-computer interface[END_REF][START_REF] Shanechi | A Real-Time Brain-Machine Interface Combining Motor Target and Trajectory Intent Using an Optimal Feedback Control Design[END_REF]] [Suminski et al., 2013] [Willett et al., 2013] [Williams et al., 2013] clinical and preclinical experiments.

Partial Least Square (PLS) regression family algorithms were widely used in the BCI field due to their robustness to high dimensional feature space and multi-collinearity problems. Additionally, PLS algorithms provide reliable model estimation in the case of small training dataset [Cramer, 1993] [Geladi and[START_REF] Geladi | Partial least-squares regression: a tutorial[END_REF]. Several articles reported offline 3D movements decoding from ECoG neural signals during preclinical experiments in [START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF]] [Chen et al., 2013[START_REF] Eliseyev | Penalized Multi-Way Partial Least Squares for Smooth Trajectory Decoding from Electrocorticographic (ECoG) Recording[END_REF][START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF][START_REF] Shimoda | Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques[END_REF]. Alternative algorithms based on the PLS such as Multi-way PLS (NPLS), Sobolev NPLS (SNPLS), Polynomial Penalized NPLS (PNPLS) [START_REF] Eliseyev | Penalized Multi-Way Partial Least Squares for Smooth Trajectory Decoding from Electrocorticographic (ECoG) Recording[END_REF], Higher Order Partial Least Squares (HOPLS2) [Zhao et al., 2013a], kernel tensor partial least squares (KTPLS) [Zhao et al., 2013b] and generalized additive model PLS (GAM-PLS) [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF] were designed and tested in offline 3D arm trajectory reconstruction preclinical BCI studies.

While the majority of the BCI algorithms assumed a linear relationship between continuous output variables and neural signal features, this assumption may be limited for accurate decoding. Therefore, several non-linear algorithms were evaluated [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. The application of non-linear models such as Wiener, cascade Wiener [START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF]] [Schaeffer and Aksenova, 2016a] [Suminski et al., 2013], Support Vector Machine Regression (SVR) [K. H. Kim et al., 2006[START_REF] Mehring | Inference of hand movements from local field potentials in monkey motor cortex[END_REF],

Piecewise probabilistic decoding (PPD) [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF], Piecewise Linear Model (PLM) [START_REF] Willett | A Comparison of Intention Estimation Methods for Decoder Calibration in Intracortical Brain-Computer Interfaces[END_REF], neural networks [START_REF] Pandarinath | Inferring single-trial neural population dynamics using sequential auto-encoders[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF][START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF], Bayesian maximum-likelihood estimation [START_REF] Ludwig | Use of a Bayesian maximumlikelihood classifier to generate training data for brain-machine interfaces[END_REF], Gaussian process regression (GPR) [START_REF] Wang | Decoding Finger Flexion from Electrocorticographic Signals Using a Sparse Gaussian Process[END_REF][START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF] were reported for offline ECoG [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF][START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF]] [Schaeffer and Aksenova, 2016a[START_REF] Wang | Decoding Finger Flexion from Electrocorticographic Signals Using a Sparse Gaussian Process[END_REF] and online ECoG [START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF] or MEA neural signal decoding [START_REF] Pandarinath | Inferring single-trial neural population dynamics using sequential auto-encoders[END_REF][START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF]] [Suminski et al., 2013] [Willett et al., 2018]. The Additive Models (AMs), Generalized Linear Models (GLMs) and Generalized Additive Model (GAM) encompass the strategies dedicated to change a linear model into a non-linear one using specific well-known non-linear link functions: (e.g. Gaussian, binomial distribution, etc.) [Yun [START_REF] Gao | A quantitative comparison of linear and non-linear models of motor cortical activity for the encoding and decoding of arm motions[END_REF]. The generalized models were tested in offline preclinical epidural and subdural ECoG studies on monkeys [Eliseyev and Aksenova, 2014[START_REF] Engel | Kernel-Based NPLS for Continuous Trajectory Decoding from ECoG Data for BCI Applications[END_REF][START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF].

Commonly, non-linear models outperformed linear ones. The Non-linear SVM and MLP provided better decoding than linear algorithms in [K. H. Kim et al., 2006] whereas the Gaussian process models outperformed the pace regression algorithm in [START_REF] Wang | Decoding Finger Flexion from Electrocorticographic Signals Using a Sparse Gaussian Process[END_REF]. Additionally, PLS linear algorithms highlighted lower performance than PPD [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF], kernel PLS [START_REF] Engel | Kernel-Based NPLS for Continuous Trajectory Decoding from ECoG Data for BCI Applications[END_REF] and AM, GLM, GAM-PLS [Eliseyev and Aksenova, 2014] algorithms. The multiplicative recurrent neural network (MRNN) outperformed the state of the art closed-loop online continuous decoding algorithms REFIT-Kalman Filter during preclinical experiments [START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF].

The non-linear models may be more qualified to estimate the complex relationship between the neural signals and the decoded kinematic signals [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF] [ [START_REF] Wang | Decoding Finger Flexion from Electrocorticographic Signals Using a Sparse Gaussian Process[END_REF]. However, the non-linear model superiority does not make consensus. The PPD outperformed the PLS but the GAM-PLS highlighted poor decoding performance compared to PLS algorithms in [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF]. Additionally, the Kernel-PLS in [START_REF] Engel | Kernel-Based NPLS for Continuous Trajectory Decoding from ECoG Data for BCI Applications[END_REF] was slightly above PLS accuracy but did not show a significant improvement.

The non-linear decoders have several drawbacks to consider before their integration into real-life BCI applications. Firstly, they are more complex and sensitive to highdimensional space and a small training dataset which may lead to overfitting issues [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2018] [Wang et al., 2010]. Secondly, the non-linear model optimization algorithms are time-consuming and require generally more computational resources than linear models [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2018] [Wang et al., 2010] which is a constrain to their use in BCI applications [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2018] [Wang et al., 2010].

Linear models highlighted up to 10D and 3D control using MEA and ECoG recording systems respectively [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF]. To introduce nonlinear algorithms into online BCI applications, they need to stress far superior decoding performance than simple linear models to counter the significant disadvantages related to their complex non-linear model calibration procedure.

Previously presented regression algorithms are static decoders which assume that continuous output variable 𝐲 𝑡 ∈ ℝ 𝐽 1 at time t, is only related to the instantaneous neural signal activity 𝐱 𝑡 ∈ ℝ 𝐼 1 . However, in the BCI field, this assumption is often violated and dependence of 𝐱 𝑡 with previous input 𝐱 𝑡-1 and output variables 𝐲 𝑡-1 is expected. Dynamic models consider the previously predicted variables to evaluate a more reliable estimation of the desired variable. Like the HMM for the discrete models, dynamic models were applied during online BCI experiments and offline studies.

Kalman filter (KF) is the most widespread dynamic decoder in the BCI field. KF is a linear stochastic state-space model with a recursive Bayesian estimation procedure evaluating an unknown kinematic variable 𝐲 𝑡 as it evolves over time. Given noisy neural signal observations 𝐱 𝑡 and the previously observed cursor variable 𝐲 𝑡-1 [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF][START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF] following the equations:

𝐲 𝑡 = 𝐀𝐲 𝑡-1 + 𝒘 𝑡 , 𝒘 𝑡 ~𝒩(0, 𝐖) 𝐱 𝑡 = 𝐂𝐲 𝑡-1 + 𝒗 𝑡 , 𝒗 𝑡 ~𝒩(0, 𝐕)
where 𝒘 𝑡 and 𝒗 𝑡 are additive Gaussian noise terms with covariance matrices 𝐖 and 𝐕, respectively [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF]. The KF applications were reported in various MEA [START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF][START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF]] [Kim et al., 2011, 2008[START_REF] Perge | Intra-day signal instabilities affect decoding performance in an intracortical neural interface system[END_REF]] [Simeral et al., 2011] and ECoG [START_REF] Kellis | Decoding hand trajectories from micro-electrocorticography in human patients[END_REF] [ [START_REF] Marathe | Decoding continuous limb movements from high-density epidural electrode arrays using custom spatial filters[END_REF] online BCI experiments as well as in MEA [START_REF] Willett | A Comparison of Intention Estimation Methods for Decoder Calibration in Intracortical Brain-Computer Interfaces[END_REF] and ECoG [Eliseyev andAksenova, 2016] [Pistohl et al., 2008] [P. T. Wang et al., 2013] offline trajectory reconstruction studies. The Unscented Kalman Filter (UKF) is a non-linear version of the KF which uses a gaussian variable estimation with a deterministic sampling technique known as the unscented transformation [START_REF] Haykin | The Unscented Kalman filter, in: Kalman Filtering and Neural Networks[END_REF]. The MEA neural signal decoding using UKF was reported in preclinical online [Hotson et al., 2016b[START_REF] Ifft | A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys[END_REF] [Z. Li et al., 2009] and offline [START_REF] Ma | Decoding Lower Limb Muscle Activity and Kinematics from Cortical Neural Spike Trains during Monkey Performing Stand and Squat Movements[END_REF][START_REF] Tseng | Decoding Movements from Cortical Ensemble Activity Using a Long Short-Term Memory Recurrent Network[END_REF]] BCI studies. In [START_REF] Ifft | A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys[END_REF], Ifft used a UKF for the online decoding of bimanual movements from monkeys' neural signals recorded with MEA.

Other alternative dynamic decoding algorithms such as Particle, Point-Process or Laplace-Gaussian Filtering [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF], Long short-term memory (LSTM), Recurrent Neural Networks (RNN) [START_REF] Du | Decoding ECoG Signal with Deep Learning Model Based on LSTM[END_REF][START_REF] Tseng | Decoding Movements from Cortical Ensemble Activity Using a Long Short-Term Memory Recurrent Network[END_REF][START_REF] Xie | Decoding of finger trajectory from ECoG using deep learning[END_REF] were applied to offline BCI experiments.

Dynamic decoders highlighted relevant results to decode continuous variables from brain neural signals in every type of recording and experimental paradigms. Only few static and dynamic algorithm comparative studies were reported. Nevertheless, some conclusions emerged from the state of the art. Over the last few years, Kalman filter families hogged the field of BCI continuous decoding based on MEA recording system [START_REF] Cunningham | A closed-loop human simulator for investigating the role of feedback control in brain-machine interfaces[END_REF][START_REF] Bibliography Fan | Intention estimation in brain-machine interfaces[END_REF][START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF], 2012[START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF]] [Hotson et al., 2016b[START_REF] Ifft | A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys[END_REF][START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF]] [Kim et al., 2011, 2008] [Z. Li et al., 2009] [Ma et al., 2017[START_REF] Orsborn | Closed-Loop Decoder Adaptation on Intermediate Time-Scales Facilitates Rapid BMI Performance Improvements Independent of Decoder Initialization Conditions[END_REF][START_REF] Perge | Intra-day signal instabilities affect decoding performance in an intracortical neural interface system[END_REF][START_REF] Shenoy | Combining Decoder Design and Neural Adaptation in Brain-Machine Interfaces[END_REF]] [Simeral et al., 2011[START_REF] Tseng | Decoding Movements from Cortical Ensemble Activity Using a Long Short-Term Memory Recurrent Network[END_REF][START_REF] Vaskov | Cortical Decoding of Individual Finger Group Motions Using ReFIT Kalman Filter[END_REF]] [Willett et al., 2018]. Kalman filter algorithms highlighted good SUA and MUA spike decoding in offline and online BCI experiments, outperforming other algorithms in preclinical and clinical open-loop and closed-loop experiments [START_REF] Kim | Neural control of computer cursor velocity by decoding motor cortical spiking activity in humans with tetraplegia[END_REF][START_REF] Koyama | Comparison of braincomputer interface decoding algorithms in open-loop and closed-loop control[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. They are suitable for online spike decoding and do not require large computational burden. However, the overwhelming superiority of the Kalman filter family was not extended to other recording systems. The benefits of dynamic decoders compared to static ones are still ambiguous for ECoG decoding [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]. The Kalman filter algorithm was outperformed by static decoders in [START_REF] Eliseyev | Penalized Multi-Way Partial Least Squares for Smooth Trajectory Decoding from Electrocorticographic (ECoG) Recording[END_REF] whereas the Kalman filter provided better decoding performance in [START_REF] Marathe | Decoding continuous limb movements from high-density epidural electrode arrays using custom spatial filters[END_REF]. From epidural ECoG dataset, static based continuous decoders outperformed a Switching Kalman Filter (SKF) and provided similar results than a Wiener filter [Schaeffer and Aksenova, 2016a]. A deeper investigation on the relevance of dynamic decoders still needs to be led for BCI based on neural signal population recordings.

Hybrid decoders

Regression decoder extracts from the neural signals kinematic output variables. However, classic continuous decoder presented limitations for asynchronous BCI experiments which require to switch between control and intended idle phases.

Traditional decoders reported non-zero velocity during intended idle state resulting in inappropriate movements of the prosthesis [START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF][START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF]. In the case of multi-limb effector control, neuro-prostheses may benefit from distinguishing between periods of control with only part of the prostheses available (one limb) to avoid false activations of the other limbs of the effector which may be disturbing and stressful for the patient.

One possible approach is to evaluate a decoder for continuous kinematic variable decoding and another for mental state decoding to extract complementary information and combine their outputs. Hybrid discrete/continuous decoders involve a classifier that inhibits or enhances the continuous decoding output variables depending on the activated mental state [Schaeffer and Aksenova, 2016a].

Various hybrid models were integrated into preclinical experiments, commonly to detect idle versus active state during continuous effector control experiments or to discriminate finger activations during alternative finger movement experiments. Hybrid decoders using LDA classifier to detect idle versus active movement state combined with a Laplace Gaussian Filter [START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] or a Kalman filter [START_REF] Aggarwal | State-based decoding of hand and finger kinematics using neuronal ensemble and LFP activity during dexterous reach-tograsp movements[END_REF] for continuous movement decoding were reported in offline MEA preclinical experiments. A similar decoder using LDA coupled with a kinematic decoder was integrated into online MEA closed-loop preclinical experiments in [START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF]. Offline finger movement decoding from ECoG neural signals was performed based on a SVM classifier and a linear regression decoder [START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF]. During an offline clinical ECoG study, a linear Bayesian classifier was applied to estimate the posterior probabilities of the idle and the movement states before applying a Kalman filter-based trajectory decoder if the movement state was more likely to be activated [P. T. Wang et al., 2013] Among the hybrid models, the Mixture of experts (ME) architecture supposes that (multiple) intended idle and active states are associated to specific movements or actions that can be independently shaped by regression models called "experts". The ME output variables are estimated with the best expert or the combination of multiple experts. The expert selection/combination is performed by a so-called discrete "gating" model [START_REF] Jacobs | Adaptive Mixtures of Local Experts[END_REF].

ME architecture was integrated into online finger decoding preclinical experiments to decode MEA neural signals using neural networks (NN) as gating and experts [START_REF] Aggarwal | Asynchronous Decoding of Dexterous Finger Movements Using M1 Neurons[END_REF]. Offline preclinical 3D reaching trajectory reconstruction from MEA neural signals using normalized Least Mean square regressions experts and MLP gating was reported in [START_REF] Kim | Divideand-conquer approach for brain machine interfaces: nonlinear mixture of competitive linear models[END_REF]. In ECoG offline study, Flamary et al mix linear ridge regression gating and linear regression experts to decode finger movements [START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF]. Additionally, Bundy combined a Logistic Regression gating model with PLS experts for offline 3D trajectory reconstruction from ECoG recording of epileptic patients to take into account the intended idle state into the decoding process [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF]. Furthermore, Choi, in an offline preclinical study, mixes a LDA classifier and PLS experts for unimodal and bimanual 3D hand trajectory reconstructions from the ECoG recordings of a non-human primate [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]. Finally, a ME architecture, built with a mixture of multi-layer perceptrons, was tested in offline noninvasive EEG studies [START_REF] Ebrahimpour | EEG-based motor imagery classification using wavelet coefficients and ensemble classifiers[END_REF][START_REF] Kheradpisheh | Mixture of feature specified experts[END_REF].

Previously presented hybrid decoders employed the state of the art static and dynamic regression algorithms. However, all of them relied on static classifiers as gating models.

The integration of dynamic classifiers into hybrid decoder as dynamic gating models was reported in offline and online clinical ECoG [START_REF] Wang | Prior Knowledge Improves Decoding of Finger Flexion from Electrocorticographic Signals[END_REF] and preclinical MEA [START_REF] Achtman | Free-paced highperformance brain-computer interfaces[END_REF][START_REF] Darmanjian | Bimodal brain-machine interface for motor control of robotic prosthetic[END_REF]] [Kao et al., 2017] based experiments. HMM was combined with a moving average model [START_REF] Darmanjian | Bimodal brain-machine interface for motor control of robotic prosthetic[END_REF]] and a REFIT-Kalman Filter [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF] to discriminate active and idle states. Finally, a ME with dynamic gate estimation was reported by Schaeffer who combined a dynamic HMM classification with PLS experts to decode monkeys 3D arm trajectory [Schaeffer and Aksenova, 2016a] and Hotson who used a HMM with an Unscented Kalman Filter during preclinical experiments [Hotson et al., 2016b].

Hybrid decoder is a simple strategy to create an asynchronous BCI system. Instead of estimating a continuous model which must perform zero-velocity prediction during intended idle state, training a classifier dedicated to idle state detection to bypass the non-zero velocity outputs of the decoder is an easier strategy that highlighted good performance with invasive and non-invasive recording systems during preclinical and clinical online BCI experiments. Moreover, ME structure is a good way to introduce nonlinearity based on linear models. As previously mentioned, non-linear models are commonly related to large computational burden and risk of overfitting. Non-linearity based on multiple linear models may be a solution to reduce the drawbacks of the non-linear models. Nevertheless, only few online experiments with hybrid and/or ME architectures were reported.

Closed-loop decoder adaptation (CLDA) and incremental adaptive decoders

Neural signals are known to be non-stationary signals [START_REF] Shenoy | Towards adaptive classification for BCI[END_REF] with high variability. Brain signal non-stationarity is related to inter (subject to subject) or intra (sessions to session or trials to trials) variability [Clerc et al., 2016b]. Intra-variability clusters short-term variation related to attention, mood and muscle tension and longterm variation caused by patient's learning/adaptation (e.g. new MI strategy, skills, etc.) [START_REF] Mladenović | A generic framework for adaptive EEG-based BCI training and operation[END_REF]. Additionally, neurons firing patterns continuously changed over hours, days and months requiring constant recalibration of the BCI decoders based on MEA recordings [START_REF] Perge | Intra-day signal instabilities affect decoding performance in an intracortical neural interface system[END_REF]] [Simeral et al., 2011].

Several studies reported the poor decoding performance of models calibrated with passive subjects (open-loop procedure) when they were applied in online closed-loop experiments. These variations may be related to the modification of neuronal activity patterns with and without sensory feedback [Clerc et al., 2016b[START_REF] Schlögl | Adaptive Methods in BCI Research -An Introductory Tutorial[END_REF] et al., 2014[START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]. CLDA provides a solution to adapt the model parameters depending on the neural signals related to the patient's feedback creating a closed-loop system where the patient is learning from the model and the model is adapting to the patients.

CLDA models the relationship between the neural features and the subject's motor intention inferred by the training data provided during the model calibration phase.

CLDA can be dissociated into two main strategies, namely the offline and online CLDA.

With the offline CLDA strategy, the model calibration is performed offline after the end of a closed-loop experiment whereas the calibration is achieved during the ongoing experiment with online CLDA procedures.

Generally, an offline or online CLDA procedure follows the same steps. The decoder is firstly initialized with a preliminary open-loop imagery task in which the subject imagines controlling a preprogrammed effector (cursor, robotic arm, etc.) which moves CLDA does not necessarily require online calibration even though online model recalibration procedure benefits from many advantages. Online adaptive decoder shortens and streamlines the CLDA procedure which is a great asset considering that BCI is dedicated to patients who may have trouble keeping engaged and focused during long calibration phases [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]. Conventional offline and online CLDA procedures re-estimate the model parameters by concatenating all the recorded data and re-evaluating the model using a machine learning algorithm on the entire dataset. However, such procedures are not suited for long-term experiments with regular model recalibration and high-dimensional dataflow processing because the model recalibration requires increasing memory consumption and computational power as long as the model is re-estimated and input and output variables are stored.

Incremental adaptive decoders are causal algorithms that update or re-estimate the model parameters in an incremental manner with a continuous or batch learning procedure based on the new incoming data and previously computed models [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Schlögl | Adaptive Methods in BCI Research -An Introductory Tutorial[END_REF]. They generally rely on strategies which do not require saving all the previously recorded samples using basic statistic estimators. This constrain restricts the possible algorithms to more efficient and optimized methods with low computational cost than previously presented offline algorithms. CLDA algorithms are typically divided into two main components, the real-time application of the model to infer a subject's intended movement and the update block to incrementally adapt the model parameters [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF]. The updating rule can rely on batch-based algorithms which update the models during specific timed events occurring every second, every 10 minutes, after each trial, etc. or in a continuous manner after each decoder iteration [START_REF] Shpigelman | Kernel-ARMA for Hand Tracking and Brain-Machine interfacing During 3D Motor Control[END_REF].

This section introduces the various adaptive BCI algorithms and strategies for online and offline CLDA.

CLDA with classifiers

Model adaptation is commonly based on supervised learning, nevertheless, unsupervised strategies to update the classifier parameters are more frequently reported than previously presented study with fixed decoders.

In 2004, an offline open-loop study showed the possible improvements related to online adaptive calibration using means and covariance matrices update [START_REF] Millan | On the need for on-line learning in brain-computer interfaces[END_REF]. In 2006, Shenoy highlighted the benefits of an adaptive classifier to manage the EEG neural signals shifting in offline binary classification study [START_REF] Shenoy | Towards adaptive classification for BCI[END_REF]. The study proposed an offline CLDA procedure with two LDA adaptation strategies: RETRAIN and REBIAS. With the REBIAS strategy, Shenoy underlined the performance improvement of LDA classification just using a bias shifting approach. Since, numerous variations of the adaptive LDA algorithm were reviewed in EEG experiments [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Mladenović | A generic framework for adaptive EEG-based BCI training and operation[END_REF][START_REF] Schlögl | Adaptive Methods in BCI Research -An Introductory Tutorial[END_REF].

Li proposed a non-incremental adaptive decoder named Importance Weighted Linear Discriminant Analysis (IWLDA) with covariate shift adaptation [Y. Li et al., 2009] and tested this algorithm on EEG and ECoG dataset. Another simple approach presented in [START_REF] Mend | Human computer interface with online brute force feature selection[END_REF] was applied during online binary classification experiments.

The BCI system is divided into two parallel processes. The main loop applies the classifier for neural signal decoding. The second loop concatenated the training data (to store the entire dataset) and performed the training procedure on all the cumulated data (the feature selection was performed through cross-validation and LDA re-training). This method had the benefit of being simple and straightforward. Nevertheless, this solution is restricted to small training dataset, simple decoding algorithms and is highly dependent on the computing power of the computer.

SVM algorithm is a popular algorithm for offline neural state classification. Numerous studies extended the SVM classifier to adaptive algorithms [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Mladenović | A generic framework for adaptive EEG-based BCI training and operation[END_REF]. The Adaptive mixture of Relevance Vector Machine (a sparse version of the SVM algorithm) [START_REF] Nguyen | Adaptive multi-degree of freedom Brain Computer Interface using online feedback: Towards novel methods and metrics of mutual adaptation between humans and machines for BCI[END_REF] was tested in an online 4 states classification experiments from online EEG neural signals. Online BCI application with offline CLDA using linear kernel SVM with covariate shift detection was reported to control an exoskeleton hand for rehabilitation application [START_REF] Chowdhury | Online Covariate Shift Detection based Adaptive Brain-Computer Interface to Trigger Hand Exoskeleton Feedback for Neuro-Rehabilitation[END_REF]. Another offline study reported the application of adaptive Kernel Fisher SVM (KF-SVM) [START_REF] Yang | Adaptive KF-SVM Classification for Single Trial EEG in BCI[END_REF].

Other less common classifiers were reported in the BCI field to handle non-stationary neural signals. Among these decoders, an incremental adaptive probabilistic neural network was designed for online binary classification [START_REF] Hazrati | An online EEG-based brain-computer interface for controlling hand grasp using an adaptive probabilistic neural network[END_REF], the Extended Kalman filter (EKF) was tested during offline binary classification studies [Ji Won [START_REF] Yoon | Adaptive Classification by Hybrid EKF with Truncated Filtering: Brain Computer Interfacing[END_REF]] [Yoon et al., 2008] and the semi-supervised adaptive Naïve Bayesian Parzen window (NBPW) classifier was tested offline in an EEG study [START_REF] Ang | Filter Bank Common Spatial Pattern (FBCSP) algorithm using online adaptive and semi-supervised learning[END_REF].

The Online Dictionary Learning using Correlation-Based Least Squares Update (CBLSU) was reported for the offline discrimination of 3 classes based on EEG dataset [START_REF] Sharghian | Online Dictionary Learning for Sparse Representation-Based Classification of Motor Imagery EEG[END_REF].

Finally, offline CLDA integrating the patient feedback was performed with a neural network combining Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM) to decode various hand shapes based on LFP recordings (MEA). The decoder allows a tetraplegic patient to control his forearm thanks to functional electrical stimulation (FES). The network parameters were re-trained during an offline supervised or unsupervised procedure using the past training dataset concatenated with new closed-loop data [START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF].

Incremental CLDA classifiers

The previously presented adaptive decoder generally concatenated the stored training dataset to recalibrate the model with the entire dataset and improve the model performance. However, this strategy might be limited if the model recalibration requires a long training procedure (for example to control complex effector with high DoF) as it requires storing all the data to re-evaluate the model without losing information.

Another solution tested during BCI applications is to incrementally update the model with new data or incrementally update necessary and sufficient variables to re-evaluate the model. The algorithms referred to as an incremental adaptive decoders perform a CLDA procedure with a reduced computational time and computer memory which suit them for online BCI application.

Vidaurre tested offline four supervised and unsupervised incremental adaptive LDA algorithms (adaptive mean, adaptive covariance matrices, etc.) using EEG neural signals from four different dataset [START_REF] Vidaurre | Toward Unsupervised Adaptation of LDA for Brain-Computer Interfaces[END_REF]. Additionally, incremental adaptive Chapter 3 : Transducers in BCI Kalman-LDA was tested in offline EEG studies [START_REF] Hsu | EEG-based motor imagery classification using enhanced active segment selection and adaptive classifier[END_REF]. Incremental adaptive Kalman-LDA was compared to an incremental adaptive version of the QDA algorithm referred to as the ADaptive Information Matrix (ADIM) which performed incremental adaptive non-linear classification [START_REF] Vidaurre | Study of On-Line Adaptive Discriminant Analysis for EEG-Based Brain Computer Interfaces[END_REF][Vidaurre et al., , 2006b]]. Online adaptive binary classification from EEG neural signals was reported using incremental LDA\QR [START_REF] Wen | Online motor imagery BCI based on adaptive and incremental linear discriminant analysis algorithm[END_REF] and incremental Kalman adaptive LDA [START_REF] Vidaurre | Study of On-Line Adaptive Discriminant Analysis for EEG-Based Brain Computer Interfaces[END_REF][START_REF] Woehrle | An Adaptive Spatial Filter for User-Independent Single Trial Detection of Event-Related Potentials[END_REF]. During online EEG based-experiments, an adaptive SVM classifier was applied for BCI speller applications using incremental SVM [START_REF] Ma | Online learning using projections onto shrinkage closed balls for adaptive brain-computer interface[END_REF][START_REF] Vo | Subject-Independent ERP-Based Brain-Computer Interfaces[END_REF] [ [START_REF] Woehrle | An Adaptive Spatial Filter for User-Independent Single Trial Detection of Event-Related Potentials[END_REF] and iterative semi-supervised SVM [START_REF] Long | Semi-supervised joint spatio-temporal feature selection for P300-based BCI speller[END_REF].

Finally, less conventional algorithms achieving incremental adaptive model calibration procedures were reported. An incremental adaptive classifier named adaptive Extreme machine learning was tested in an offline EEG-based study for binary classification [START_REF] Bamdadian | Improving session-to-session transfer performance of motor imagery-based BCI using adaptive extreme learning machine[END_REF]. Incremental adaptive fuzzy models were reported for offline 2class motor imagery task discrimination and offline 4-state classification from EEG neural signal using first-order Takagi-Sugeno fuzzy model [Rong et al., 2018b] and selfregularized supervised Gaussian fuzzy adaptive system adaptive resonance theory [START_REF] Jafarifarmand | Real-time multiclass motor imagery brain-computer interface by modified common spatial patterns and adaptive neuro-fuzzy classifier[END_REF] respectively. Additionally, several incremental adaptive Riemannian classifiers based on minimum distance to the mean (MDM) were designed and tested in offline 3 and 4-states classification studies in [START_REF] Kumar | Towards Adaptive Classification using Riemannian Geometry approaches in Brain-Computer Interfaces[END_REF].

Unsupervised CLDA classifiers

Unsupervised learning is a more common strategy for model re-estimation than for model calibration. With unsupervised adaptation, the labels associated with the recorded neural signals are unknown. Unsupervised model adaptation which only required neural signals features were reported in BCI experiments.

For example, the global mean of neural signal features was adapted to overcome the shift between the distributions estimated during the calibration procedure and the test period in offline EEG studies [START_REF] Vidaurre | Toward Unsupervised Adaptation of LDA for Brain-Computer Interfaces[END_REF] and in online EEG experiments for upperlimb control through functional electrical stimulations [START_REF] Vidaurre | EEG-based BCI for the linear control of an upper-limb neuroprosthesis[END_REF]. A similar strategy with global mean and global covariance matrix adaptations was tested in [START_REF] Vidaurre | Toward Unsupervised Adaptation of LDA for Brain-Computer Interfaces[END_REF].

Another unsupervised learning strategy referred to as a semi-supervised adaptation in [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]] combined both the model initially calibrated with labeled data and the new incoming neural signals with unknown labels to perform the model retraining/adaptation.

One solution is to use the classification outputs of the current decoder and the associated neural signals as new training data for model re-estimation. This easy to implement strategy was reported in [START_REF] Hasan | Unsupervised adaptive GMM for BCI[END_REF] for offline EEG classification based on incremental adaptive unsupervised Gaussian Mixture Models with a sequential expectation-maximization procedure. Similar learning procedures using unsupervised incremental adaptive SVM were reported for offline and online MEG experiments [Spüler et al., 2012b]. Spüler introduced a threshold criterion to only select the samples which were most likely well classified. As a wrong prediction could lead to degrading the classifier performance, only the samples with high probabilities of being correct were stored for incremental learning [Spüler et al., 2012b].

Another proposed solution was to integrate other stereotyped neurophysiological signals to infer the neural signals labels. Some semi-supervised strategies relied on Error Potentials (ERP) detection [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. ERP detection provides a good estimation of the correctly and badly classified unlabeled data. This strategy was tested offline in EEG based classification study using Weighted Majority Voting (WMV) based on adaptive SVM [START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF] and was reported several times for offline and online P300 BCI applications [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF].

CLDA with classifiers: conclusion

The integration of a CLDA procedure in decoders dedicated to BCI application is a poorly explored area where most of the proposed adaptive classifiers were tested during offline and online EEG experiments or in offline studies based on dataset acquired with invasive recording systems. Moreover, the reported closed-loop decoder was not all based on incremental adaptive strategies which might be an obstacle for real-time classier updates due to hardware limitations (increasing memory, computing time and resources required, etc.). Clinical and pre-clinical experiments using invasive recording systems with real-time incremental CLDA remained unexplored in the BCI domain.

Additionally, to our knowledge, dynamic adaptive classification was weakly studied.

Whereas the application of HMM was reported in BCI experiments for reaching and grasping control of a robotic arm [START_REF] Fifer | Simultaneous Neural Control of Simple Reaching and Grasping With the Modular Prosthetic Limb Using Intracranial EEG[END_REF], classification of finger movements [Hotson et al., 2016a], detection of evoked neural signals [START_REF] Lisi | Markov Switching Model for Quick Detection of Event Related Desynchronization in EEG[END_REF][START_REF] Moses | Real-time classification of auditory sentences using evoked cortical activity in humans[END_REF] or prediction of the idle state during trajectory decoding [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF], no study on the online calibration of HMM in the BCI domain was reported. In other research fields, a HMM with low computational cost and potential for adaptive classifier calibration was reported in [START_REF] Chan | Continuous myoelectric control for powered prostheses using hidden Markov models[END_REF] to command prostheses through myoelectric control. Additionally, several theoretical articles proposed strategies to extended online HMM to the adaptive classifier domain based on adaptive expectation-maximization procedure, gradient descent techniques [START_REF] Cappé | Online EM Algorithm for Hidden Markov Models[END_REF][START_REF] Chis | Adapting Hidden Markov Models for Online Learning[END_REF][START_REF] Digalakis | Online adaptation of hidden Markov models using incremental estimation algorithms[END_REF] [ [START_REF] Khreich | A survey of techniques for incremental learning of HMM parameters[END_REF][START_REF] Mongillo | Online Learning with Hidden Markov Models[END_REF][START_REF] Stiller | Online estimation of hidden Markov models[END_REF], etc. Adaptive dynamic classifier exploitation for real-time BCI decoding is an interesting research field to explore.

CLDA for continuous outputs

This section introduced the CLDA procedure and the adaptive algorithms designed to decode continuous variables and control continuous effectors.

The first offline CLDA approach to control a 2D cursor movement was reported by Wolpaw [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF] during online EEG experiments. The control of the cursor was provided to four healthy subjects by a linear equation which combined the weighted amplitude of the µ and β frequency bands (one for the vertical and one for the horizontal displacement). After each trial, the weights were adapted based on the past trials using an offline least-mean squares algorithm [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF].

In other experiments, based on Kalman filter algorithms, two tetraplegic patients controlled a 2D cursor through MEA neural signals decoding [START_REF] Kim | Neural control of computer cursor velocity by decoding motor cortical spiking activity in humans with tetraplegia[END_REF] where the model was updated at the end of each closed-loop block (a group of several trials).

Since, other algorithms with offline CLDA were proposed for communication BCI applications to tetraplegic and ALS patients based on MEA neural signals [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF][START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF]] [Kim et al., 2011]. Jarosiewicz proposed a real-time point and click cursor control for virtual typing applications [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]. They used a Kalman filter to control the cursor movement whereas the LDA classifier detected the click activation. Both decoders were recalibrated between blocks of trials. Participants were able to pause the BCI application by their brain signals. During those breaks, the last N blocks were used to recalibrate the BCI decoder which lasts between 1-3 min before restoring the cursor control to the patient [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF]. Similar clinical experiments with real-time MEA neural signal decoding based on offline CLDA procedure using Kalman filter and univariate Gaussian classifier with dynamic/temporal post-processing were reported in [START_REF] Kim | Point-and-Click Cursor Control With an Intracortical Neural Interface System by Humans With Tetraplegia[END_REF].

Another offline CLDA procedure was tested during preclinical MEA experiments by Gilja in [START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF]. This study proposed a decoder named the recalibrated feedback intention Kalman filter (ReFit-KF) which performed a discrete batch-based calibration algorithm to update the decoder 10-15 min after the initial seeding [START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF][START_REF] Orsborn | Closed-Loop Decoder Adaptation on Intermediate Time-Scales Facilitates Rapid BMI Performance Improvements Independent of Decoder Initialization Conditions[END_REF]. Additionally, Gilja proposed a recursive procedure to enhance the model decoding accuracy which re-calibrated the ReFit-KF based on new neural signals and shifted closed-loop decoder outputs. After the estimation of the initial parameters through an open-loop calibration, the non-optimal model was used for closed-loop cursor control. The neural signal recorded during the closed-loop experiment and an "intended estimate" of the cursor velocity was used for the model recalibration (Figure 3-2). The "intended estimate" of the cursor velocity was generated by rotating the decoded velocity vectors such that they pointed straight towards the targets. This velocity estimation assumed that the monkey was constantly intending to move directly towards the target [START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF]Carmena, 2014]. ReFit-KF performed a batch-based CLDA applying one discrete decoder update 10-15 min after the initial seeding [START_REF] Orsborn | Closed-Loop Decoder Adaptation on Intermediate Time-Scales Facilitates Rapid BMI Performance Improvements Independent of Decoder Initialization Conditions[END_REF]. A single ReFit-KF CLDA update was sufficient to achieve a significant improvement in reach kinematics [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF]. However, the procedure could be repeated several times to improve the decoding performance. This innovation highlighted significant continuous decoding performance improvements. Refit-KF was used in many applications to decode MEA neural signals and highlighted good continuous performance in many online preclinical [START_REF] Bibliography Fan | Intention estimation in brain-machine interfaces[END_REF][START_REF] Vaskov | Cortical Decoding of Individual Finger Group Motions Using ReFIT Kalman Filter[END_REF] and clinical [START_REF] Willett | A Comparison of Intention Estimation Methods for Decoder Calibration in Intracortical Brain-Computer Interfaces[END_REF]

Incremental CLDA continuous decoders

Other decoders inspired by the previously presented "intended estimate" ("cursorgoal") procedure of Gilja were designed. They relied on incremental batch-online CLDA with a medium update frequency ("adaptation on intermediate Time-Scales") [START_REF] Orsborn | Closed-Loop Decoder Adaptation on Intermediate Time-Scales Facilitates Rapid BMI Performance Improvements Independent of Decoder Initialization Conditions[END_REF][START_REF] Orsborn | Exploring time-scales of closed-loop decoder adaptation in brain-machine interfaces[END_REF]. For example, an alternative of the ReFit-KF calibration procedure named "SmoothBatch" CLDA was proposed to incrementally update the Kalman filter decoder during online batch recalibration (1-2 min time scale) using exponentially weighted sliding average and recursive maximum likelihood algorithm [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF] [ [START_REF] Dangi | Continuous Closed-Loop Decoder Adaptation with a Recursive Maximum Likelihood Algorithm Allows for Rapid Performance Acquisition in Brain-Machine Interfaces[END_REF]] [Orsborn et al., 2014, 2012]. The observed neural activity and intended cursor kinematics were collected over one batch period before to compute a weighted average of the current parameters with those estimated from the new batch of neural activity [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF]. SmoothBatch-KF showed good results in online 2D cursor control based on monkeys implanted with MEA recording devices [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF][START_REF] Dangi | Continuous Closed-Loop Decoder Adaptation with a Recursive Maximum Likelihood Algorithm Allows for Rapid Performance Acquisition in Brain-Machine Interfaces[END_REF]] [Orsborn et al., 2014, 2012].

Finally, CLDA can be performed online in shorter time scales going from a recalibration at each iteration (at every new sample) or in the order of few seconds. An incremental adaptive version of the Kalman algorithm was designed such as adaptive Kalman filter (AKF) [START_REF] Dangi | Adaptive Kalman filtering for closed-loop Brain-Machine Interface systems[END_REF], Unscented KF with unsupervised Bayesian regression updater [START_REF] Li | Adaptive Decoding for Brain-Machine Interfaces Through Bayesian Parameter Updates[END_REF] or Gaussian Process regression Discriminative KF (GP-DKF) [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]. AKF performed the model parameter recalibration at each decoder iteration. AKF was only tested on simulated data [START_REF] Dangi | Adaptive Kalman filtering for closed-loop Brain-Machine Interface systems[END_REF] and was compared to SmoothBatch-KF during online preclinical MEA neural signals decoding in [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF].

The incremental adaptive Unscented KF and the GP-DKF algorithms updated their parameters every few seconds and performed, based on MEA neural signals, online 2D pursuit task using monkey [START_REF] Li | Adaptive Decoding for Brain-Machine Interfaces Through Bayesian Parameter Updates[END_REF] and 2D cursor center-out real-time decoding from three tetraplegic patients [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF], respectively. Other dynamic decoders were adapted for online incremental model calibration. Online preclinical 2D random pursuit task experiments were reported in [START_REF] Suminski | Online adaptive decoding of intended movements with a hybrid kinetic and kinematic brain machine interface[END_REF] using an adaptive Wiener filter with incremental gradient descent for MEA neural signal decoding.

Based on MEA recordings of non-human primates, Shanechi designed the Point Process Filter (PPF) which updated its parameters at every spike event and used an optimal feedback-control model to infer the velocity intention during adaptation [START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF][START_REF] Shanechi | Robust Brain-Machine Interface Design Using Optimal Feedback Control Modeling and Adaptive Point Process Filtering[END_REF][START_REF] Shanechi | High-performance brainmachine interface enabled by an adaptive optimal feedback-controlled point process decoder[END_REF]. This decoder outperformed the SmoothBatch-Kalman filter and highlighted robust and high online decoding performance in 2D cursor control preclinical experiments [START_REF] Shanechi | Robust Brain-Machine Interface Design Using Optimal Feedback Control Modeling and Adaptive Point Process Filtering[END_REF][START_REF] Shanechi | High-performance brainmachine interface enabled by an adaptive optimal feedback-controlled point process decoder[END_REF]. Additionally, PPF allowed studying the impact of the decision and feedback rates to control neuroprosthetic devices [START_REF] Shanechi | Rapid control and feedback rates enhance neuroprosthetic control[END_REF].

With the exception of Wolpaw's study [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF], all the presented articles were performed based on MEA recording devices. Other MEA-based CLDA procedures were reviewed in [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF]. Eliseyev presented an incremental adaptive algorithm referred to as the Recursive Exponentially Weighted N-way Partial Least Square (REW-NPLS) [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF] for ECoG neural signal decoding. REW-NPLS is a PLS algorithm extended to input and output tensor variables that perform model recalibration in few seconds (10 seconds). This algorithm highlighted good results in offline 3D arm monkey movement reconstruction study based on ECoG neural signals decoding as well as during offline fingers movement reconstruction study from four healthy subjects using MEG recordings [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. Moreover, REW-NPLS, in the CLINATEC "BCI and Tetraplegia" clinical trial, provided to a tetraplegic patient, based on wireless ECoG recordings, the 3D hand translation control of an exoskeleton during real-time point-to-point pursuit task experiments [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]. et al., 2013]. This difference may be related to the neural signals generated by the sensory feedbacks added during closed-loop experiments or the co-adaptation of the user and the decoder. The CLDA with "human-in-loop" training highlighted better decoding performances than open-loop calibrated models.

CLDA for continuous outputs: Conclusion

However, as depicted, numerous CLDA procedures were reported with diverse "timescale of adaptation" [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF]. The update might be performed at the end of every session [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF][START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF], trials [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF], 15 min [START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF][START_REF] Orsborn | Closed-Loop Decoder Adaptation on Intermediate Time-Scales Facilitates Rapid BMI Performance Improvements Independent of Decoder Initialization Conditions[END_REF], 1 min [START_REF] Dangi | Closed-loop decoder adaptation algorithms for brain-machine interface systems[END_REF], few seconds [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]] [Eliseyev et al., 2017], every sample [START_REF] Dangi | Adaptive Kalman filtering for closed-loop Brain-Machine Interface systems[END_REF], etc. Fast update calibration provides quicker feedback to the patient and decreases the time required to converge to an adequate neural control [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]. A balance should be found between model accuracy, model complexity and decoder update rate.

Most of the reported adaptive regression decoders were based on clinical or preclinical MEA neural signals processing or offline studies using EEG dataset. With the exception of [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF], ECoG-based adaptive decoders were not reported. Further investigation on the use of CLDA using an incremental adaptive decoder should be carried out.

Adaptive feature selection

Commonly, non-stationarity adaptation and neural signals related to patient's feedback are integrated into the BCI system through the BCI decoder adaptation with an adaptive classifier or an adaptive regression model. Another proposed solution was to modify the features extraction step in order to modify the computed feature depending on the brain variability [START_REF] Mladenović | A generic framework for adaptive EEG-based BCI training and operation[END_REF]. As an example of simple feature non-stationarity tracking, Jarosiewicz designed a communication BCI for a tetraplegic patient using MEA neural signals which recomputed the raw average firing rate of the neurons using an offline batch update procedure to maintain an accurate baseline [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF] and avoid bias.

Similarly to the adaptive decoders, adaptive feature selection algorithms generally relied on state of the art feature selection algorithms. Several independent component algorithms were reported in [START_REF] Hsu | Real-Time Adaptive EEG Source Separation Using Online Recursive Independent Component Analysis[END_REF] to manage brain signal non-stationarity or to perform online adaptive denoising/artifact detection. For example, the online recursive independent component analysis (ORICA) [START_REF] Chen | A reliable brain-computer interface based on SSVEP using online recursive independent component analysis[END_REF][START_REF] Ho | A SOC Design of ORICA-based Highly Effective Real-time Multi-channel EEG System[END_REF]] [Hsu et al., 2016, 2015] [Wang et al., 2018] was tested during offline simulations based on artificially noised EEG neural signals. Online calibrated common spatial pattern algorithms were tested offline in EEG motor imagery studies using an incremental or an adaptive common spatial pattern (ICSP and ACSP) [START_REF] Costa | An adaptive CSP filter to investigate user independence in a 3-class MI-BCI paradigm[END_REF][START_REF] Mobaien | ACSP: Adaptive CSP filter for BCI applications[END_REF][START_REF] Song | Improving brain-computer interface classification using adaptive common spatial patterns[END_REF]] [Zhao et al., 2008]. Vidaurre in [Vidaurre et al., 2006b] proposed a fully online adaptive BCI using adaptive autoregressive (AAR) feature selection and an ADIM classifier (the incremental adaptive version of the non-linear classifier QDA) during offline and online EEG neural signals experiments. An extension of the AAR to the multivariate case was tested during an offline analysis of EEG recordings in [START_REF] Hettiarachchi | Multivariate Adaptive Autoregressive Modeling and Kalman Filtering for Motor Imagery BCI[END_REF]. Incremental learning of axDawn spatial filter algorithm was applied online during single-trial detection of ERPs [START_REF] Woehrle | An Adaptive Spatial Filter for User-Independent Single Trial Detection of Event-Related Potentials[END_REF].

Finally, other less conventional feature selection algorithms such as adaptive spatial filter [START_REF] Morales-Flores | Non-supervised technique to adapt spatial filters for ECoG data analysis[END_REF] or brute force algorithm [START_REF] Mend | Human computer interface with online brute force feature selection[END_REF] were tested offline through ECoG and EEG neural signal decoding studies respectively.

Adaptive feature selection is a poorly explored area and was mainly evaluated during offline EEG studies. This strategy highlighted interesting results to handle brain nonstationarity, enhance signal-to-noise ratio and remove artifacts. However, deeper investigations on the impact of feature adaptation on the decoder performance should be carried out. Indeed, with a fixed decoder, modifying the input feature space can deteriorate the decoding accuracy. Vidaurre in [Vidaurre et al., 2006b] adapted both the feature extraction block and the model parameters nevertheless more investigations on this type of "fully" adaptive decoder must be achieved.

Alternative strategies

It must be mentioned that Sussillo proposed an alternative or complementary algorithm to the presented adaptive decoders in order to handle brain non-stationarity across time.

The algorithm, named Multiplicative Recurrent Neural Network (MRNN) [START_REF] Sussillo | Making brain-machine interfaces robust to future neural variability[END_REF] handled brain signal non-stationarity by concatenating multiple models calibrated with diverse "recording conditions". RNN "remembered" the state over time to handle dynamic complex and time-varying relationship between neural signals and movements. The "multiplicative" structure allowed the neural signal features to influence the recurrent weights. MRNN learned a "library" of various neural-kinematic mappings representing the natural dynamic of the RNN and various neural recording conditions [292]. Combining the "library" architecture to handle various neural conditions and the adaptive decoder to manage the natural drifting of the neural signals could be an interesting solution to test for future BCI experiments. This algorithm was evaluated during online closed-loop center-out experiments with MEA neural signal decoding from a monkey and outperformed the state-of-the-art REFIT-KF algorithm.

Post-processing

Post-processing techniques reported in BCI applications relied on prior knowledge about the desired output variables, the restrictions related to the effector or the subject's safety. They were integrated into BCI systems to smooth the output of the decoder or to apply modifications to the decoder output such as output thresholding, enable state transition for a defined time period [START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF], etc. Jarosciewicz applied an adaptive bias correction to the decoded cursor velocity using an exponentially weighted mean of the velocity outputs exceeding a predefined speed threshold during online BCI communication experiments based on the MEA neural signals of tetraplegic patients [START_REF] Jarosiewicz | Virtual typing by people with tetraplegia using a selfcalibrating intracortical brain-computer interface[END_REF]. Similarly, to reduce click errors related to noisy classification from MEA neural signals of a tetraplegic patient, Kim averaged the click state predictions over a period of time (typically 500 ms) [START_REF] Kim | Point-and-Click Cursor Control With an Intracortical Neural Interface System by Humans With Tetraplegia[END_REF]. To enable a monkey to control in real-time a robotic arm from intracortical recordings, Velliste applied to the continuous output variables a 5 to 11-sample temporal filter to smooth the decoded trajectory [START_REF] Velliste | Cortical control of a prosthetic arm for self-feeding[END_REF]. The Graz-BCI system integrated a post-processing step based on a dwell time [START_REF] Pfurtscheller | Self-Paced Operation of an SSVEP-Based Orthosis With and Without an Imagery-Based "Brain Switch:" A Feasibility Study Towards a Hybrid BCI[END_REF][START_REF] Townsend | Continuous EEG classification during motor imagerysimulation of an asynchronous BCI[END_REF] and refractory period [START_REF] Townsend | Continuous EEG classification during motor imagerysimulation of an asynchronous BCI[END_REF] for online synchronous 2D classification [START_REF] Townsend | Continuous EEG classification during motor imagerysimulation of an asynchronous BCI[END_REF] and SSVEP-Based Orthosis control [START_REF] Pfurtscheller | Self-Paced Operation of an SSVEP-Based Orthosis With and Without an Imagery-Based "Brain Switch:" A Feasibility Study Towards a Hybrid BCI[END_REF] experiments from EEG neural signals. The dwell time was a fixed period of time in which the decoder output variables should be above a defined threshold to be considered as a valid event detection [START_REF] Pfurtscheller | Self-Paced Operation of an SSVEP-Based Orthosis With and Without an Imagery-Based "Brain Switch:" A Feasibility Study Towards a Hybrid BCI[END_REF][START_REF] Townsend | Continuous EEG classification during motor imagerysimulation of an asynchronous BCI[END_REF]. The refractory period was the duration after a detected valid event during which new valid events were ignored [START_REF] Townsend | Continuous EEG classification during motor imagerysimulation of an asynchronous BCI[END_REF].

Other possible post-processing strategies based on the detection of neurophysiological patterns related to BCI errors were reported. The detection of decoding errors from the neural signals was used to automatically correct the decoded output variables. This strategy was commonly used for EEG-based BCI applications with Error Related Neuronal Response (ERNR) detection algorithms to correct decoder mistakes. For example, the incorrect decoded letters from an EEG-based BCI P300 speller were automatically deleted if an ERNR was detected in [Spüler et al., 2012a]. An extensive review of the integration of ERNR-based correction for BCI application, particularly focused on EEG neural signals decoding, was proposed in [START_REF] Chavarriaga | Errare machinale est: the use of error-related potentials in brain-machine interfaces[END_REF].

Additionally, few studies highlighted error-related neural signal modulations from invasive recording systems. Shenoy's team proved that error-related neural signals could be extracted from the premotor and primary motor cortices using MEA recordings and could be evaluated to improve the BCI decoding performances [Even- Chen et al., 2018[START_REF] Chen | A reliable brain-computer interface based on SSVEP using online recursive independent component analysis[END_REF]. They stressed relevant results in preclinical online experiments [Even- [START_REF] Chen | A reliable brain-computer interface based on SSVEP using online recursive independent component analysis[END_REF] and clinical offline study [Even- Chen et al., 2018]. ERNR patterns were detected from ECoG motor cortex neural signals in [START_REF] Milekovic | Detection of Error Related Neuronal Responses Recorded by Electrocorticography in Humans during Continuous Movements[END_REF] but were not integrated in ECoG-based BCI systems to improve the decoding performance.

Other post-processing strategies based on moving average, confidence measures of the classification outputs and blocking state transitions approaches were reported in EEG experiments [Bashashati et al., 2007a] [Schaeffer and[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF].

Prediction filtering across time is often applied as a post-processing step to smooth the decoder outputs [START_REF] King | The feasibility of a braincomputer interface functional electrical stimulation system for the restoration of overground walking after paraplegia[END_REF]. The impact of error magnitude, smoothness, prediction delay and velocity on the closed-loop continuous movement decoding accuracy (Figure 3-3) was investigated in [START_REF] Marathe | The impact of command signal power distribution, processing delays, and speed scaling on neurally-controlled devices[END_REF]. The conclusion of the study stressed that smoothing the predictions reduced the prediction errors but added supplementary delays. Additionally, they highlighted that minimizing the prediction delay was highly relevant especially when decoding accuracy was poor [START_REF] Marathe | The impact of command signal power distribution, processing delays, and speed scaling on neurally-controlled devices[END_REF]]. This study is relevant for post-processing filtering tuning, as 100 Chapter 3 : Transducers in BCI higher temporal filtering may improve the decoder prediction but also increase the prediction delay. Generally, during BCI online closed-loop experiments, for the control of real effectors, physical boundaries were added via a post-processing step to restrict the possible position, movements, speed, etc. of the effector and assure the patient's safety. For example, in [START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF], a tetraplegic patient controlled in real-time a DEKA prosthetic arm through the decoding of her MEA neural signals. The workspace of the prosthetic arm was limited with virtual boundaries to avoid any collision with the tabletop, support stand and the patient. This type of post-processing is highly dependent on the controlled effector.

Post-processing is a powerful tool to reduce the weaknesses of the online decoders using post-processing techniques/rules that would be too complex to integrate into the model calibration procedure. Additionally, post-processing allows integrating prior knowledge on the desired task. However, post-processing must be carefully integrated into a closed-loop BCI system. A trade-off between performance improvement related to the post-processing step and delay added to the prediction must be considered to not disrupt patient's feedback and inevitably deteriorate the decoded predictions.

Conclusion

In this chapter, the principal blocks of a BCI transducer were described and the main approaches of the state-of-the-art BCI studies were presented. After a preprocessing step, relevant information is extracted from the brain neural signals to allow a decoder to estimate a specific neural state (classifier) or the patient's intended continuous outputs (continuous decoder) via a neural signal decoder. Exhaustive reviews referring to less common methods used in BCI are available in [Bashashati et al., 2007a[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF].

The most common studies are classification experiments based on EEG recording systems. Continuous decoding is generally performed with invasive recording systems which highlighted higher decoding performance and a higher number of controlled DoF than non-invasive ones. For online application, linear models are favored by their ease of use and calibration compared to more complex models.

Asynchronous BCIs are more realistic and representative of daily life applications than synchronous BCIs. Asynchronous BCIs introduce an "idle" state detection in which no command/action is performed by the effector [START_REF] Han | Brain-Switches for Asynchronous Brain-Computer Interfaces: A Systematic Review[END_REF]. However, asynchronous BCIs are more complex than synchronous ones and present higher false-positive detection rates (detection of an active state instead of idle one) which is highly problematic for many applications and may be disturbing for the users. BCI state of the art with the highest number of controlled DoF are synchronous [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF] et al., 2011]. Only a few EEG studies integrated idle detection in more complex tasks [START_REF] Chae | Toward brain-actuated humanoid robots: Asynchronous direct control using an EEG-Based BCI[END_REF]] [Li et al., 2013[START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF]. Asynchronous decoding, while more prevalent in the EEG studies, were reported in some preclinical ECoG [START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Schaeffer and Aksenova, 2016b], preclinical MEA [START_REF] Achtman | Free-paced highperformance brain-computer interfaces[END_REF][START_REF] Ludwig | Use of a Bayesian maximumlikelihood classifier to generate training data for brain-machine interfaces[END_REF][START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF][START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] and clinical [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF] [P. T. Wang et al., 2013] ECoG studies which combined idle state detection and continuous movement predictions. Nevertheless, with the exception of the MEA preclinical experiments [START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF], all the invasive asynchronous BCI studies were tested offline. Asynchronous BCIs is a major condition for the majority of the BCI daily life applications and should always be integrated for the control of complex effectors.

Hybrid (discrete/continuous) is one of the solutions tested for asynchronous BCI application as well as mixing discrete and continuous outputs such as moving a cursor to a target and activate a "clicking" state. Additionally, hybrid decoders highlighted promising results for multi-limbs control experiments in ECoG online experiments [Hotson et al., 2016a] and offline studies [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]. However, the multi-limbs studies with hybrid decoders were mainly restricted to individual finger movement detection [START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF]] [Hotson et al., 2016a[START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]. Further investigation on the benefits of hybrid (discrete/continuous) decoders in the case of complex multi-limb or bimanual effector control should be carried out.

Dynamic classifiers and regression decoders were only restricted to few studies. With the exception of Kalman filters which are prevalent for continuous decoding, only a few other dynamic continuous decoders were employed in online applications. Time dependency is often integrated through post-processing. Dynamic classifiers were tested using complex algorithms such as LSTM and CRF during offline studies and outperformed HMM classifiers. However, the computational burden of these algorithms is superior to HMM decoders which is problematic for online BCI applications. HMM performance improvements compared to static decoders do not reach a consensus in the BCI field and are highly application dependent [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. Few offline and online preclinical experiments [Kao et al., 2017] [Schaeffer andAksenova, 2016a] stressed the benefits of HMM for idle state detection in asynchronous BCI applications. Further investigation on the advantages of dynamic models for online closed-loop BCI applications must be achieved.

CLDA integrates the neural signal modulations related to the patient's feedback in the model calibration procedure. CLDA highlighted benefits for closed-loop BCI decoding performance as well as robustness to brain the neural signal non-stationarity. CLDA procedure should be achieved to enhance closed-loop BCI applications. However, offline CLDA algorithms require numerous training and calibration sequences to estimate a model which can be time-consuming and increase patient's mental load. Incremental online CLDA algorithms train and apply the model at the same time which presents various advantages for real-time BCI applications. However, to this date, incremental online CLDA algorithms were understudied. On the one hand, EEG-based adaptive decoders were mainly focused on adaptive feature extraction algorithms and adaptive classifiers. On the other hand, MEA-based decoders were more focused on continuous movement decoding and highlighted interesting approaches to re-calibrate the models at various time scales. Nonetheless, incremental adaptive EEG-based-classifiers were mainly tested during offline studies whereas incremental adaptive MEA basedcontinuous decoder performance evaluations were generally limited to online preclinical center-out experiments. Moreover, with the exception of the REW-NPLS algorithm [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF], no adaptive algorithms were reported with ECoG recordings. Deeper investigations on such adaptive incremental decoders and experiments should be carried out.

Chapter 4

Online incremental adaptive multilinear switching model This chapter introduces the new Recursive Exponentially Weighted Markov Switching multi-Linear Model (REW-MSLM) designed for online closed-loop adaptive decoder calibration and asynchronous multi-limb effector control based on ECoG recordings. To control a multi-limb effector and handle stable idle state decoding, this algorithm relies on a Mixture of Experts architecture. This new fully adaptive decoder was derived from two algorithms named recursive exponentially weighted n-way partial least squares (REW-NPLS) [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF] and Markov-Switching linear model (MSLM) [Schaeffer and Aksenova, 2016b]. The first one is an adaptive tensor based decoder for closed-loop adaptive calibration. The second is a Mixture of Experts algorithm which integrates a dynamic classifier to enhance the discrimination between the active states and improve the idle state detection. Before further description of the REW-MSLM, the Mixture of Experts, the MSLM and the REW-NPLS algorithms are described.

Mixture of experts

Mixture of Experts (ME) can be described as the parallel computation of several predictions from different regression models named "experts" that are weighted (enhanced or inhibited) according to the input variables using a classifier referred as "gate" model [START_REF] Jacobs | Adaptive Mixtures of Local Experts[END_REF][START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF]. Gate and experts terminologies have been firstly introduced by Jacobs in [START_REF] Jacobs | Adaptive Mixtures of Local Experts[END_REF].

Let 𝐱 𝑡 ∈ 𝑋, 𝑋 ⊂ ℝ 𝑚 and 𝐲 𝑡 ∈ 𝑌, 𝑌 ⊂ ℝ 𝑛 be the explanatory and the response variables, respectively, indexed by 𝑡 ∈ ℕ, where 𝑚 and 𝑛 are the feature space dimensions. ME assumes [START_REF] Jacobs | Adaptive Mixtures of Local Experts[END_REF][START_REF] Waterhouse | Classification using hierarchical mixtures of experts[END_REF][START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF] that the feature space of independent variables is sub-divided into 𝐾 (possibly overlapped) regions 𝑋 = ⋃ 𝑋 𝑘 𝐾 𝑘=1

, and that 𝑋 is mapped to 𝑌 with the set of 𝐾 ∈ ℕ local linear or nonlinear functions Φ = {𝜑 𝑘 : 𝑋 𝑘 → 𝑌, 𝑘 ∈ [1; 𝐾]} called experts:

𝐲 𝑡 = ∑ 𝛿 𝑘,𝑧 𝑡 𝜑 𝑘 (𝐱 𝑡 ) 𝐾 𝑘=1 + 𝛆 𝑘,𝑡 .
Here, 𝑧 𝑡 represents the selected expert at time 𝑡, 𝛿 𝑘,𝑧 𝑡 is the Kronecker delta (where 𝛿 𝑘,𝑧 𝑡 = 1 if 𝑘 = 𝑧 𝑡 and 𝛿 𝑘,𝑧 𝑡 = 0 otherwise) and 𝜀 𝑘,𝑡 is the observation noise (possibly) related to the k th expert, generally, supposed to be independent and identically distributed (iid). The vector 𝐲 𝑡 is predicted from the input variable 𝐱 𝑡 , using the estimated expert 𝜑 ̂𝑘 fitted at the corresponding region of the neural space: = 1 [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF].

𝐲 ̂𝑡 = ∑

106

Chapter 4 : Online incremental adaptive multilinear switching model ME is suited for naturally sub-divided dataset due to its ability to train each expert independently to a specific pattern [START_REF] Jacobs | Adaptive Mixtures of Local Experts[END_REF][START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF] which fit with multi-limb effectors. It introduces non-linearity to the model by mixing the multiple linear (or not) regression models [START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF]. ME architecture is commonly used across divers fields [Carvalho andTanner, 2003] [Yuksel et al., 2012] with many applications in finance [Carvalho andTanner, 2003] [Hoang and[START_REF] Hoang | A mixture of global and local gated experts for the prediction of high frequency foreign exchange rates[END_REF]] [START_REF] Yu | Prediction of Bank Telemarketing with Co-training of Mixture-of-Experts and MLP[END_REF],

weather study [START_REF] Jeffries | EEG Classification of Forearm Movement Imagery Using a Hierarchical Flow Convolutional Neural Network[END_REF], bioinformatics [START_REF] Lê Cao | Integrative mixture of experts to combine clinical factors and gene markers[END_REF][START_REF] Qi | A mixture of feature experts approach for protein-protein interaction prediction[END_REF], facial recognition [START_REF] Ebrahimpour | Improving mixture of experts for view-independent face recognition using teacher-directed learning[END_REF][START_REF] Gutta | Mixture of experts for classification of gender, ethnic origin, and pose of human faces[END_REF], etc.

Various regressions and classification algorithms were developed to improve ME gate and experts performance [START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF]. ME was built based on Support Vector Machine (SVM) [START_REF] Elgharabawy | Decoding of finger movement using kinematic model classification and regression model switching[END_REF]Wahed, 2016] [Gutta et al., 2000], multilayer perceptrons (MLP) [START_REF] Kim | Divideand-conquer approach for brain machine interfaces: nonlinear mixture of competitive linear models[END_REF], Gaussian mixture model (GMM) [START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF], etc. Finally, ME was modified to have specific features for each expert [START_REF] Kheradpisheh | Mixture of feature specified experts[END_REF] enhancing the feature space sub-division of the original algorithm [START_REF] Jacobs | Adaptive Mixtures of Local Experts[END_REF][START_REF] Waterhouse | Classification using hierarchical mixtures of experts[END_REF][START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF].

In motor BCI research, hybrid decoders, defined in the manuscript as a combination of discrete (e.g. classifiers) and continuous (e.g. regression) decoders, were widely spread in the case of asynchronous experiments particularly to distinguish the idle and control periods [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF][START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] [P. T. Wang et al., 2013]. Hybrid models were reported for MEA-based real-time 3D reaching task preclinical experiments in which the discrete predictions of a Linear discriminant analysis (LDA) classifier were cascaded with a Population Vector Algorithm (PVA) [START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF] or a Gaussian Filter to predict zero velocity movements during idle state periods [START_REF] Suway | Resting state detection for gating movement of a neural prosthesis[END_REF]. Similar offline studies were reported in ECoG-driven clinical and preclinical BCI [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF] [P. T. Wang et al., 2013]. A LDA classifier was mixed with a PLS decoder to decode idle state, alternative or simultaneous bimanual movements from Non-Human Primate (NHP) [START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]. The ECoG neural signals of able-bodied patients were decoded offline using the discrete predictions of a linear Bayesian classifier combined to a Kalman filter in order to inhibit the trajectory predictions during idle state [P. T. Wang et al., 2013]. Among the hybrid decoders, ME architecture was applied for several offline studies. EEG offline studies based on motor imagery control strategy decoded thanks to MLPs for both gating and experts decoders were reported in [START_REF] Ebrahimpour | EEG-based motor imagery classification using wavelet coefficients and ensemble classifiers[END_REF][START_REF] Kheradpisheh | Mixture of feature specified experts[END_REF]. MEA-based 3D reaching tasks was completed on monkeys using a linear filter and MLP [START_REF] Kim | Divideand-conquer approach for brain machine interfaces: nonlinear mixture of competitive linear models[END_REF] While many of motor BCI articles stressed the interest of dynamic modeling to take into account the temporal dependencies in the data to enhance the prediction performances [Saa et al., 2016] [Schaeffer and[START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF] [P. T. Wang et al., 2013] [Williams et al., 2013[START_REF] Wu | Modeling and Decoding Motor Cortical Activity Using a Switching Kalman Filter[END_REF], the majority of the mentioned ME decoders applied in the BCI fields were static. To provide temporal information to the model, static ME decoders can be coupled with pre-processing as smoothed auto-regressive features [START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF] or post-processing moving average [Hotson et al., 2016a].

Additionnaly, Dynamic models can be applied to integrate directly temporal dependencies in ME gating.

Sequential/Dynamic hybrid decoders

Dynamic gating of hybrid models or mixture of experts algorithm is not a broad area of research in the BCI field. Only few examples were applied to BCI preclinical and clinical experiments. Achtman [START_REF] Achtman | Free-paced highperformance brain-computer interfaces[END_REF] designed a free-paced system which discriminated the idle, preparation and action states using the instantaneous state estimates and past classification to estimate the state transition whereas another classifier evaluated the reached target. The results of the study were obtained with offline MEA preclinical experiments analysis but the system was suitable to online application [START_REF] Achtman | Free-paced highperformance brain-computer interfaces[END_REF]. A dynamic hybrid decoder using a pace-regression combined with a switching non-linear dynamic system was reported for offline finger trajectory reconstruction from ECoG signals of five epileptic patients [START_REF] Wang | Prior Knowledge Improves Decoding of Finger Flexion from Electrocorticographic Signals[END_REF].

Additionaly, a first-order Markovian transition probability was coupled to LDA classifier to discriminate active versus idle state during clinical offline finger trajectory reconstruction from ECoG signals [Hotson et al., 2016a].

HMM is a well know dynamic classifier used in many fields. This algorithm had already been integrated into hybrid decoder as dynamic gating algorithm and commonly used for asynchronous BCI system. Hybrid decoder coupling moving average models with HMMs was designed to detect idle versus active states for 3D arm trajectory offline reconstruction experiments from MEA signals of a monkey [START_REF] Darmanjian | Bimodal brain-machine interface for motor control of robotic prosthetic[END_REF]. A ReFIT-KF and a Hidden Markov Model (HMM) were coupled to decode 2D cursor trajectory during asynchronous online preclinical experiments with MEA recording system implanted on three rhesus macaques [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF]. The ReFIT-KF controlled the velocity of the cursor while the HMM indicated whether or not to move the cursor (classification between idle and movement state) [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF]

Markov Switching Linear Models

Among the algorithms based on ME architecture, dynamic gating is an extension which already provided promising results in motor BCI studies. M.C Schaeffer developed the Markov Switching Linear Model (MSLM) [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2016b], a variant of the ME, which uses linear experts and a first-order Hidden Markov Model (HMM) for dynamic gating (Figure 4-1).

Figure 4-1: Markov Switching Linear Model (MSLM) structure [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF].

This method assumes that the expert selection (among the 𝐾 ∈ ℕ expert) is conditioned on unobserved discrete state 𝑧 𝑡 , which depends exclusively on the past state 𝑧 𝑡-1 , 𝑝(𝑧 𝑡 |𝑧 1:𝑡 ) = 𝑝(𝑧 𝑡 |𝑧 𝑡-1 ). In [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF], the estimation of MSLM parameters was considered for fully supervised and unsupervised (according to the states sequence) training strategies. The PhD research was only focused on the fully supervised training.

The Maximum Likelihood (ML) estimate of MSLM parameters (experts and gate) using training data {𝐗, 𝐘, 𝐳} was considered [Schaeffer and Aksenova, 2016b]. Here, 𝐗 ∈ ℝ 𝐿×𝑚 , 𝐘 ∈ ℝ 𝐿×𝑛 , 𝐳 ∈ ℕ 𝐿 are the observation matrices of explanatory, response and state sequence variables respectively, 𝐿 is the number of samples.

The linear experts were independently trained with subsets of 𝐗 and 𝐘 generated according to the state labels: expert k was trained with a cluster of training samples belonging to the state 𝑧 𝑡 = 𝑘. Gate parameters were estimated using 𝐗 and 𝐳 variables. The transition matrix 𝐀 was computed by counting transition between 𝑧 𝑡-1 and 𝑧 𝑡 .

In the BCI experiments, HMM emission probability 𝑝(𝐱 𝑡 |𝑧 𝑡 ) was usually estimated using generative state decoders based on classical distributions, e.g. Gaussian models parameterized by an approximation of mean and covariance based on observation samples {𝑐 𝑘 } 𝑘=1 𝐾 = {𝜇, Σ} [START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF][START_REF] Pfeiffer | Hidden Markov model based continuous decoding of finger movements with prior knowledge incorporation using bi-gram models[END_REF][START_REF] Wissel | Hidden Markov Model and Support Vector Machine based decoding of finger movements using Electrocorticography[END_REF]. Nevertheless, discriminative classifiers can also allow to estimating 𝑝(𝑧 𝑡 |𝐱 𝑡 ) and infer the emission probability 𝑝(𝐱 𝑡 |𝑧 𝑡 ) relying on the Bayes' theorem [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. This alternative highlighted relevant results in many applications (using e.g. support vector machine [START_REF] Antelis | Decoding Upper Limb Movement Attempt From EEG Measurements of the Contralesional Motor Cortex in Chronic Stroke Patients[END_REF] or logistic regression [Schaeffer and Aksenova, 2016b]).

After a comparative study, gate model was evaluated based on a discriminative classifier before to be integrated to HMM state decoder for gating emission probability estimation. Discriminative parameters of the classifier were estimated using PLS-based dimension reduction and Logistic Regression algorithm. Parameters of the experts were identified using Partial least Square (PLS) regressions. Hyperparameter optimization of PLS algorithm was performed using 6-fold cross-validation procedure and Wold's R criterion on the cross-validated PRESS statistic [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF].

MSLM was tested using asynchronous pre-clinical and clinical dataset based on ECoG recordings. MSLM performance was evaluated with 3D arm trajectory reconstruction of monkey arm and finger movements from epileptic patients. The MSLM results were compared to a Switching Kalman Filter and a Markovian post-processed Wiener Filter [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]. MSLM outperformed both algorithms in term of state detection Chapter 4 : Online incremental adaptive multilinear switching model accuracy and highlighted better trajectory reconstruction than the Switching Kalman Filter [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]Aksenova, 2016b].

Even though MSLM comparative study presented relevant results in subdural and epidural ECoG signal decoding in preclinical (3D reaching tasks in non-human primates) and clinical (finger movements decoding) data demonstrating, in particular, strong idle state support, offline trajectory decoding was only reported. Additionally, the experts and gate algorithms as well as the cross-validation optimization method employed to evaluate the models were computationally demanding in term of required computation time and resources. Therefore, MSLM might not be suited for online adaptive close-loop decoding. The classical PLS regression algorithm is an offline procedure based on the iterative projection of input 𝐱 𝑡 ℝ 𝑚 and output 𝐲 𝑡 ℝ 𝑛 variables into latent variables spaces of dimension 𝑓 (𝑓 is referred as the PLS "hyperparameter"). Projectors are estimated by maximizing the covariance between the input and the output latent variables [START_REF] Wold | The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses[END_REF]. Subspace dimension 𝑓 is typically determined through cross-validation.

Recursive Exponentially

Generally, iterative algorithms are not well adapted to large number of training samples requiring heavy computations. To overcome this limitation, Lindgren [START_REF] Lindgren | The kernel algorithm for PLS[END_REF] and Dayal [Dayal andMacGregor, 1997a, 1997b] developed the kernel PLS algorithms. These methods, based on kernel matrices, allowed memory saving and faster computations in the estimation of the model parameters.

Conventional PLS [START_REF] Wold | The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses[END_REF] and kernel PLS [Dayal andMacGregor, 1997a] [Lindgren et al., 1993] are offline algorithms. Recursive PLS (RPLS) algorithms for online modeling of data flow were firstly introduced by Helland [START_REF] Helland | Recursive algorithm for partial least squares regression[END_REF] and Qin [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF][START_REF] Qin | Partial least squares regression for recursive system identification[END_REF]. These algorithms update the loading matrices of the previous model using online incoming data [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF]. The Recursive Exponentially Weighted PLS Recursive Exponentially Weighted Nway Partial Least Squares (REW-NPLS) 111 (REW-PLS), proposed by Dayal [Dayal and MacGregor, 1997b], is based on a more efficient kernel algorithm. Although these algorithms performed adaptive decoding, the hyperparameter 𝑓 still needed to be optimized offline in a preliminary study using crossvalidation algorithm.

A generalization of the previously described conventional PLS algorithms to tensor data, N-way Partial Least Square (NPLS) algorithm, was proposed by Bro [START_REF] Bro | Multi-way analysis in the food industry: models, algorithms, and applications[END_REF][START_REF] Bro | Comparative study of band-power extraction techniques for Motor Imagery classification[END_REF]. A tensor is a generalization of a matrix to higher order dimensions, also known as ways or modes. Vector and matrices are special cases of tensors with one and two modes respectively [Bro, 1998] [Kolda and[START_REF] Kolda | Tensor Decompositions and Applications[END_REF]. Tensor-based algorithms emerged as a promising strategy for brain signal processing. In the BCI field, the method allowed simultaneous treatments of high-dimensional data in the temporal, frequency and spatial domains [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF]] [Eliseyev et al., 2017]. NPLS algorithm projects input and output tensors into low dimensional space of latent variables using low rank tensor decomposition. This offline method improved the model stability and robustness compared to the classic unfold PLS leading to more accurate and interpretable predictions [START_REF] Bro | Multi-way analysis in the food industry: models, algorithms, and applications[END_REF][START_REF] Bro | Comparative study of band-power extraction techniques for Motor Imagery classification[END_REF]. NPLS combines the robustness of PLS regression with the ability to preserve the structure of the data, which is lost in vector-oriented approaches [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. Additionally, the preservation of the data structure without unfolding optimizes the data information to provide a more robust estimate of the loading vector in the case of small training dataset [START_REF] Bro | Comparative study of band-power extraction techniques for Motor Imagery classification[END_REF].

Dimensional reduction strategies are solutions to handle high dimensional feature space. Embedded feature selection techniques were developed for NPLS algorithm. L1penalized NPLS algorithm generates more generalized and interpretable models by creating sparse model with the less relevant and noisy parameters fixed to zero. L1penalized NPLS sparse solution can improve the decoding performance and reduce the computational burden. This algorithm was used during offline preclinical BCI analysis [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Foodeh | Regularized Partial Least Square Regression for Continuous Decoding in Brain-Computer Interfaces[END_REF] and in other domain [START_REF] Hervás | Sparse N-way partial least squares by L1-penalization[END_REF].

Nevertheless, the solution proposed in [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF] was not suited for online realtime model update.

For online modeling of tensor data flow, Recursive N-way PLS (RNPLS), a generalization of RPLS to tensor variables, was proposed [START_REF] Eliseyev | Recursive N-way partial least squares for brain-computer interface[END_REF]. Similar to generic RPLS, RNPLS still required fixing the hyperparameter 𝑓 from offline preliminary study. Based on more computationally efficient kernel REW-PLS, REW-NPLS algorithm was proposed by Eliseyev [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. In addition to recursive online tensor-based linear regression identification, the Recursive-Validation (RV) procedure for online hyperparameter 𝑓 optimization was introduced. RV allowed the REW-NPLS algorithm to be a fully adaptive algorithm entirely tuned in real-time.

REW-NPLS algorithms

REW-NPLS algorithm adaptively update a set of 𝐹 (𝐹 ∈ ℕ * is the fixed upper bound latent space dimension) models {𝐁𝐞𝐭𝐚 𝑢 𝑓 , 𝐛𝐢𝐚𝐬 𝑢 𝑓 } 𝑓=1 𝐹 using the current block tensor of observation {𝐗 𝑢 , 𝐘 𝑢 } and previously computed models weighted with the forgetting factor 𝜇 1 . Here 𝑢 ∈ ℕ is the update iteration number, 𝐁𝐞𝐭𝐚 𝒖 𝑓 ∈ ℝ (𝐼 1 ×…×𝐼 𝑀 )×(𝐽 1 ×…×𝐽 𝑁 ) , 𝐛𝐢𝐚𝐬 𝑢 𝑓 ∈ ℝ 𝐽 1 ×…×𝐽 𝑁 are the current update of models' coefficients, and 𝐗 𝑢 ∈ ℝ ∆𝐿×𝐼 1 ×…×𝐼 𝑀 , 𝐘 𝑢 ∈ ℝ ∆𝐿×𝐽 1 ×…×𝐽 𝑁 are the input and output 𝑀 ∈ ℕ and 𝑁 ∈ ℕ order tensors of observations with ∆𝐿 ∈ ℕ * the number of samples recorded between the two update blocks 𝑢 -1 and 𝑢.

REW-NPLS models are adaptively updated using covariance tensors 𝐗𝐗 𝑢 ∈ ℝ (𝐼 1 ×…×𝐼 𝑀 )×(𝐼 1 ×…×𝐼 𝑀 ) and 𝐗𝐘 𝑢 ∈ ℝ (𝐼 1 ×…×𝐼 𝑀 )×(𝐽 1 ×…×𝐽 𝑁 ) following the equations:

𝐗𝐗 𝑢 = 𝜇 1 𝐗𝐗 𝑢-1 + 𝐗 𝑢 × 1 𝐗 𝑢 , 𝐗𝐘 𝑢 = 𝜇 1 𝐗𝐘 𝑢-1 + 𝐗 𝑢 × 1 𝐘 𝑢 ,
Where "× 𝑘 " is the k-mode tensor product and 𝜇 1 is a forgetting factor, and 𝑢 ∈ ℕ is the update iteration number.

PARAllel FACtor analysis (PARAFAC) tensor decomposition algorithm is used to extract a set of projectors {𝐰 𝑓 1 ∈ ℝ 𝐼 1 , … , 𝐰 𝑓 𝑀 ∈ ℝ 𝐼 𝑀 } 𝑓=1 𝐹 from 𝐗𝐘 𝑢 . The new {𝐁𝐞𝐭𝐚 𝑢 𝑓 } 𝑓=1 𝐹 are estimated based on these projectors and the couple scores/loadings from the models estimated with lower latent space dimensions {1, … , 𝑓 -1} [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. Finally, the information decoded by 𝐁𝐞𝐭𝐚 𝑢 𝑓 are removed from 𝐗𝐘 𝑢 using deflation procedure to evaluate 𝐁𝐞𝐭𝐚 𝑢 𝑓+1 in the next loop . PARAFAC based tensor decomposition is a key element in the evaluation of the REW-NPLS models. This method is described in detail in the next sections. As a brief description, PARAFAC aims to decompose a tensor into a linear combination of vectors (the projectors) outer products. In parallel to parameters update, Recursive-Validation (RV) procedure estimates the optimal dimension of the space of latent variables based on incoming data and previously computed models [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF] Recursive algorithms are a promising tool in motor BCI applications to integrate the subject's feedback directly in the model training phase [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]] [Orsborn et al., 2014] and improve the model robustness to the brain neural signal non-stationarity [Clerc et al., 2016b]. The consequences of such brain variability can lead to suboptimal offline-tuned models and thus lower decoding performances than expected during online decoding experiments compared to adaptive models [START_REF] Jarosiewicz | Advantages of closed-loop calibration in intracortical brain-computer interfaces for people with tetraplegia[END_REF]. On the contrary, closed-loop decoders highlighted relevant improvements in trajectory decoding and robustness over time [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF][START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]] [Orsborn et al., 2014]. The REW-NPLS algorithm was designed for real-time adaptive incremental modelling and computation. The model showed relevant results in offline BCI trajectory decoding using epidural ECoG recordings from NHP and using MEG recordings from able-bodied subjects during finger tapping experiments [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. However, such regression model is not consistent to control complex or multilimb effectors which required strong idle state support. In the case of bimanual, and even more, whole body effector, independent control of limbs with zero-velocity movement decoding of the non-activated limbs is mandatory. These tasks are not specially addressed by single model such as REW-NPLS.

Recursive Exponentially Weighted Markov Switching multi-Linear Model (REW-MSLM) 115

Recursive Exponentially Weighted Markov Switching multi-Linear Model (REW-MSLM)

The Recursive Exponentially Weighted Markov Switching multi-Linear Model (REW-MSLM) is a piecewise linear model based on mixture of experts (ME) architecture. This new fully adaptive decoder was derived from two algorithms. On the one hand, the REW-MSLM has a hybrid Mixture of Expert (ME) structure similar to that of the Markov-Switching linear model (MSLM) [Schaeffer and Aksenova, 2016b], mixing discrete and continuous decoding. On the other hand, the REW-MSLM integrates the recursive exponentially weighted n-way partial least squares (REW-NPLS) [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF] for experts and gating model estimations. REW-NPLS is an online adaptive algorithm that handles tensor high-dimensional data flow and handle closed-loop decoder adaptation.

REW-MSLM description

The recursive exponentially weighted Markov-switching multi-linear model (REW-MSLM) is an online tensor-based fully adaptive mixture of multi-linear expert algorithm.

The REW-MSLM inherits the Markov-switching linear model (MSLM) [Schaeffer and Aksenova, 2016a] mixture of experts (ME) structure, generalizing the MSLM model to the online incremental CLDA algorithm family, tensor-input-tensor-output variables and introduces the recursive model parameter identification procedure inspired by the REW-NPLS method [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF].

In our specific case of application, ME is an approach which mixes or switches independent decoders, called "experts". Experts might be associated with a particular control tasks. For example, an independent expert decoder may be related to the control of specific movements of an effector, such as continuous left hand translation, right hand translation, left wrist or right wrist rotation. Such mixture of expert structure was implemented in the present study. Expert models, determined using the online adaptive REW-NPLS algorithm, are linearly mixed according to the gating model which estimates the probability of an expert to be active. This probability is used to compute gating coefficients to weight all the experts' outputs. REW-MSLM output is the linear combination of all the weighted expert outputs. State equiprobability leads to the activation/mixing of all the experts, e.g. both hands translation and wrists rotation. Whereas if the probability for one limb is 1, the probability of the others is 0, this limb will be the only activated. The gating procedure applied in the article is referred as "soft gating" in contrast to "hard gating" which only selects the limb with the highest gating probability (limits the movement to one limb). In addition, fully adaptive REW-MSLM algorithm inherits the dynamic gating of the generic MSLM using hidden Markov model (HMM) for state sequence estimation in order to improve the decoder classification robustness (Figure 4-4). multilinear switching model architecture. The REW-MSLM includes a mixture of experts model, which can be described as the parallel computation of several predictions from different regression models (experts) that are weighted (enhanced or inhibited) according to the input variables using a classifier (gate). We hypothesize that the input feature space 𝑋 can be divided into several specific local regions 𝑋 𝑘 and that each sub-space can be fitted using local multilinear functions 𝜑 𝑘 associated with an expert. Multilinear functions 𝜑 𝑘 are estimated using 𝑘 independent REW-NPLS models. The selected expert is determined based on the dynamic gating model. The gating model is a hidden Markov model (HMM) which computes the probability 𝛾 𝑘 for each expert to be activated. Commands are decoded by the REW-MSLM and sent to the effector to provide visual feedback to the patient.

The REW-NPLS discriminative decoder is also embedded into the HMM-based gating process to evaluate state probability. REW-NPLS was used because of its online adaptive characteristics and relevance for high dimensional input variable decoding. Moreover, discriminative decoders were selected instead of generative one due to their benefits for high dimensional and complex dependencies of features [Schaeffer, 2017] [Sutton and[START_REF] Sutton | An Introduction to Conditional Random Fields[END_REF].

Basic assumption of ME approach is that each expert decodes his own specific region of feature space. Given 𝐗 𝑡 ∈ 𝑋 ⊂ ℝ 𝐼 1 ×…×𝐼 𝑀 and 𝐘 𝑡 ∈ 𝑌 ⊂ ℝ 𝐽 1 ×…×𝐽 𝑁 the independent and dependent 𝑀 and 𝑁 order tensor variables at time 𝑡 respectively. The feature space of independent variables is supposed to be partitioned into 𝐾 possibly overlapping regions 𝑋 = ⋃ 𝑋 𝑘 𝐾 𝑘=1

. It is assumed that the space of input variables is mapped to the space of output variables using 𝐾 local multilinear functions Φ = {𝜑 𝑘 : 𝑋 𝑘 → 𝑌, 𝑘 = 1,2, … , 𝐾}. Let 𝑧 𝑡 [1; 𝐾] ⊂ ℕ * be a latent state variable defining the selected local multilinear function at time 𝑡 such as 𝐘 𝑡 = 𝜑 𝑧 𝑡 (𝐗 𝑡 ).

Recursive Exponentially Weighted Markov Switching multi-Linear Model (REW-MSLM) 117 Dynamic gating is introduced using a first-order HMM [Schaeffer and Aksenova, 2016a]. Let 𝑧 𝑡 be a latent state variable following the first-order Markovian assumption, which states that the dependence of 𝑧 𝑡 is limited to the past state 𝑧 𝑡-1 . 𝐘 𝑡 is estimated as follows:

𝐘 ̂𝑡 = ∑ 𝛾 𝑘,𝑡 (𝐁𝐞𝐭𝐚 𝑘 𝐗 𝑡 + 𝐛𝐢𝐚𝐬 𝑘 ).

𝐾 𝑘=1

Here, 𝐁𝐞𝐭𝐚 𝑘 ∈ ℝ 𝐼 1 ×…×𝐼 𝑀 ×𝐽 1 ×…×𝐽 𝑁 and 𝐛𝐢𝐚𝐬 𝑘 ∈ ℝ 𝐽 1 ×…×𝐽 𝑁 are the k th expert tensor parameters and its associated bias. is the set of parameters employed to estimate the conditional emission probability of the observed variables 𝑝(𝐗 𝑡 |𝑧 𝑡 ), and 𝝅 ∈ ℝ 𝐾 is the initial state probability vector at 𝑡 = 0. As mentioned, REW-MSLM is using dynamic HMM gating. The equivalent mixture of expert algorithm using static gating (without HMM) is referred as REW-SLM in the manuscript. REW-SLM is used to highlight the performance enhancement related the implementation of HMM (further details on the comparative study between REW-MSLM and REW-SLM are available in the Chapter 9). REW-SLM gating is computed with the REW-NPLS algorithm calibrated on the explanatory variables and the latent states followed by the softmax function 𝑝(𝑧 𝑡 |𝐗 𝑡 ). In the static REW-SLM classifier, the HMM forward algorithm is not applied. In the opposite, the REW-MSLM algorithm, in order to integrate the Markov process for the prediction of the discrete state, applies the forward algorithm.

REW-MSLM online incremental training

Decoder application

The proposed REW-MSLM algorithm is used in closed-loop multi-limb experiments. New input data 𝐗 𝑡 are recorded, and each expert model is applied based on the decoding model {𝐁𝐞𝐭𝐚 𝑘 , 𝐛𝐢𝐚𝐬 𝑘 } 𝑘=1 𝐾 . Finally, the dynamic gating coefficients 𝛾 𝑘,𝑡 are estimated using the latent state variable estimator 𝒛 ̂𝑡 (equation (4.1)) post-processed with a softmax function (equation (4.2)) [START_REF] Yuksel | Twenty Years of Mixture of Experts[END_REF] (equivalent to REW-SLM classifier output) followed with the HMM forward algorithm [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF] (equations (4.3) and (4.4)).

The forward algorithm evaluates 𝛾 𝑘,𝑡 by considering the past and current observations:

𝒛 ̂𝑡 = 𝐁 𝐗 𝑡 + 𝐛, ( 4

Conclusion

In this chapter, a new decoder is introduced for online adaptive decoding of multi-limb effector control from ECoG neuronal activity recording. The REW-MSLM conjugates the benefits of generic REW-NPLS and MSLM algorithms. On the one hand, REW-MSLM uses the MSLM mixture of expert structure to control complex effector such as exoskeleton or it avatar. In the case of multi-limb effector, an expert model can be associated to each limb allowing to control several limbs and complete complex actions with a single REW-MSLM decoder. A dynamic gating is introduced to improve expert classification as well as strong idle state discrimination. On the other hand, the experts and gate models are calibrated with several independent REW-NPLS algorithms to incrementally update the REW-MSLM during online closed-loop experiments.

REW-MSLM online adaptive multi-limb decoder was designed to integrate quickly the patient's feedback neural activity in a "human-in-loop" calibration procedure to create an adaptive closed-loop decoder where the patient and model fits together.

Chapter 5

Online incremental group-wise sparse REW-MSLM BCI experiments highlighted high intra and inter-subjects variability in the BCI decoders. Although BCI model is generally relying on neurological markers generalizable on the majority of subjects [Clerc et al., 2016b], it requires to generate a wide range of neural features to include possible neurophysiological patterns and select among them the most relevant subject-specific features. Thus, high dimensional feature space is commonly used in BCI to decode patient's action from the brain neural signals.

However, the processing of noisy and high dimensional features, such as brain signals, brings several challenges to overcome. They can be grouped as: the model calibration issue, the model generalization and interpretation problems and the hardware related obstacles [START_REF] Bellman | Adaptive Control Processes: A Guided Tour[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF] [Nicolas-Alonso and Gomez-Gil, 2012] [ [START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF]. Firstly, with the exception of algorithms specifically oriented to this problematic, in machine learning, higher dimensional models require more training data set. Nevertheless, real-time BCI experiments are performed during rare and brief sessions due to the reduced ability of disabled patients to remain focused in prolonged calibration sequences [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]. Therefore, generally, calibration sessions are too short for complex high dimensional parameter identification and may lead to the classical "curse of dimensionality" problem related to uninformative or correlated features and small training dataset compared to feature space dimension. Additionally, high dimensional feature spaces and related models with high number of parameters are more complicated to interpret than low dimensional one. Finally, high dimensional feature space computation and high dimensional model evaluation require high computational power and time for neural signal processing, model calibration and application. These hardware considerations are key characteristics in the case of realtime embedded/portable BCI application which have limited computational resources.

In this chapter, two online adaptive group-wise sparse decoders are presented to reduce the feature space dimension employed for BCI decoding and improve the model interpretability. The proposed decoders were designed to be integrated in the REW-MSLM algorithms as sparse gating and/or experts models and create BCI systems with low computational cost, suited for portable applications. Firstly a brief review on the different feature dimension reduction strategies is introduced, then, in order to understand the proposed algorithms, the PARAFAC procedure employed in the REW-NPLS algorithm is detailed. Next, the new Lp-Penalized REW-NPLS algorithm (PREW-NPLS) for online sparse model identification is proposed. Finally, an extension of the PREW-NPLS algorithm, including online automatic regularization parameter tuning named Automatic Lp-Penalized REW-NPLS (APREW-NPLS) is presented.

Context related feature dimension reduction

High dimensional feature space may lead to numerous issues such as high computing power requirements, high computing time, and "curse of dimensionality" problem in Chapter 5 : Online incremental groupwise sparse REW-MSLM the decoder training and application steps [START_REF] Bellman | Adaptive Control Processes: A Guided Tour[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF]. Furthermore, high dimension feature space generally presents useless features with irrelevant and/or redundant information which disrupt the model training and application. The presented high dimensional feature space limitations are recurrent complications in the BCI field, especially in the case of real-time BCI experiments.

To prevent these issues, dimensional reduction algorithms decreasing the feature space dimension were employed to create the BCI model. Reduction of the feature space dimension may improve the decoding performance and drastically reduce the computing time and resources required. In the case of daily life BCI applications with high dimensional data flow processing, computing time and resources management is a crucial aspect to consider [START_REF] Haufe | Dimensionality reduction for the analysis of brain oscillations[END_REF]. Dimensional reduction algorithms are dissociated into projection and feature selection algorithm families. Both dimensional reduction algorithm types were applied to online BCI experiments and offline studies.

Projections algorithms were often used in BCI studies [START_REF] Bousseta | EEG Based Brain Computer Interface for Controlling a Robot Arm Movement Through Thought[END_REF]] [Bundy et al., 2016[START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Eliseyev et al., 2017] [Haufe et al., 2014] [Hsu et al., 2016[START_REF] Jafarifarmand | Real-time multiclass motor imagery brain-computer interface by modified common spatial patterns and adaptive neuro-fuzzy classifier[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF] [S. P. Kim et al., 2006[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Marathe | Decoding continuous limb movements from high-density epidural electrode arrays using custom spatial filters[END_REF][START_REF] Palmer | Independent Component Analysis (ICA) Features for Electro-corticographic (ECoG) Brain-Machine Interfaces (BMIs). 臨床神経生理[END_REF][START_REF] Sannelli | Ensembles of adaptive spatial filters increase BCI performance: an online evaluation[END_REF]] [Schaeffer and Aksenova, 2016b[START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Sreenath | Classification of denoising techniques for EEG signals: A review[END_REF].

They project the feature space into a subspace of lower dimension by linear or non-linear combination of the initial feature space components (more details are available in the Chapter 3). This family clusters the principal and independent component analysis (PCA and ICA), spatio-spectral decomposition (SSD), common spatial pattern (CSP) or partial least squares (PLS) [START_REF] Bousseta | EEG Based Brain Computer Interface for Controlling a Robot Arm Movement Through Thought[END_REF]] [Bundy et al., 2016[START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Eliseyev et al., 2017] [Haufe et al., 2014] [Hsu et al., 2016[START_REF] Jafarifarmand | Real-time multiclass motor imagery brain-computer interface by modified common spatial patterns and adaptive neuro-fuzzy classifier[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF] [S. P. Kim et al., 2006[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Marathe | Decoding continuous limb movements from high-density epidural electrode arrays using custom spatial filters[END_REF][START_REF] Palmer | Independent Component Analysis (ICA) Features for Electro-corticographic (ECoG) Brain-Machine Interfaces (BMIs). 臨床神経生理[END_REF][START_REF] Sannelli | Ensembles of adaptive spatial filters increase BCI performance: an online evaluation[END_REF]] [Schaeffer and Aksenova, 2016b[START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Sreenath | Classification of denoising techniques for EEG signals: A review[END_REF], etc. However, such methods may not improve the computing time as they does not optimize feature extraction step. The irrelevant features are still computed.

Feature selection family regroups filter-based, wrapper-based and embedded techniques [Bolón-Canedo et al., 2013] [Khaire and[START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF]. Filter-based methods rank and select independently the features which cluster the most information without consideration of the trained decoder. This method is effective in computation time and have good generalization capacity. However, these methods tend to select highly correlated (redundant) features.

In the opposite, wrapper-based techniques incorporate supervised learning algorithms to evaluate the possible interactions between the features. Wrapper methods add iteratively new features to the subset of selected features space and evaluate the Context related feature dimension reduction 125 performance of the selected subset combined with the trained decoder [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. These methods are efficient, nevertheless, they are costly in terms of computing time and may lead to overfitting.

Embedded techniques regroups the strategies were the feature selection steps is directly integrated into the decoding algorithm to combined the benefits of both previous methods: keeping the advantages of wrapper while decreasing computational complexity [START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF]. Features selection is performed directly within the model learning process. For example, BCI embedded techniques regroups decision tree, and regularization algorithms. Regularization strategies add penalty on the model parameter optimization to reduce the freedom of the model. Numerous regularization are named depending on the applied penalization norm/pseudo-norm: L0, L1 (Lasso), L2 (Ridge) or elastic net regularization algorithms etc. The L1 regularization adds a penalty term equal to the sum of the absolute value of the coefficients whereas L2 regularization integrates a penalty equal to the sum of the squared value of the coefficients and elastic net regularization is defined as the combination of both L1 and L2 penalization [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Lp regularization algorithms with 0 ≤ 𝑝 ≤ 1 discard irrelevant features promoting sparse solution [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Hastie | Statistical Learning with Sparsity : The Lasso and Generalizations[END_REF]. Sparse solution is efficient to avoid overfitting and may lead to reduction in computing time.

Regularization algorithms were commonly applied in the BCI field for feature selection or to improve neural signal decoding such as L0 [START_REF] Sreeja | Weighted sparse representation for classification of motor imagery EEG signals[END_REF], L1 [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] López-Larraz | Continuous decoding of movement intention of upper limb self-initiated analytic movements from pre-movement EEG correlates[END_REF][START_REF] Lotte | Regularizing Common Spatial Patterns to Improve BCI Designs: Unified Theory and New Algorithms[END_REF] [Y. Zhang et al., 2013], L2 [START_REF] Cincotti | High-resolution EEG techniques for brain-computer interface applications[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF], elastic net [START_REF] Kim | Elastic net ensemble classifier for event-related potential based automatic spelling[END_REF][START_REF] Peterson | A penalized time-frequency band feature selection and classification procedure for improved motor intention decoding in multichannel EEG[END_REF] norm penalization or other regularization strategies such as regularization algorithms using, polynomial regression [START_REF] Eliseyev | Penalized Multi-Way Partial Least Squares for Smooth Trajectory Decoding from Electrocorticographic (ECoG) Recording[END_REF], sparse regularization based on automatic relevance determination (ARD) parameters [START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Toda | Reconstruction of two-dimensional movement trajectories from selected magnetoencephalography cortical currents by combined sparse Bayesian methods[END_REF], Kullback-Leibler regularization in the Riemannian mean [START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF], etc.

Generally, regularization algorithms is performed in single-wise manner, they evaluate independently the contribution of each model parameter before to apply a constrain in order to regulate the amplitude of each parameter weight. Each feature is regularized independently and is not evaluated as belonging to a group of features to be penalized. Therefore, in the case of tensor input features, each tensor component is set to zero independently to each other. Such element-wise component regularization of tensor features may lead to more complicated interpretation of the results and extraction of the relevant features (Figure 5-1A). However, there are many applications with structurally grouped input features where it may be of interest to set simultaneously to zero or nonzero value the features within a pre-determined group [START_REF] Hastie | Statistical Learning with Sparsity : The Lasso and Generalizations[END_REF].

Group-wise regularization performs selection by grouping the relevant features and applying the penalization to the groups of features at once (Figure 5 et al., 2013]. Grouping can cluster features over different modalities such as the electrodes, the frequency bands [START_REF] Van Gerven | Interpreting single trial data using groupwise regularisation[END_REF], etc. Group-wise sparse regularization promotes the model convergence to sparse solution (in a group-wise level), simplifies the model interpretation and is suited to naturally structured features. Moreover, group-wise selection discards group of variables from the signal processing workflow (electrode or frequency) reducing the computational cost and the required computing time for real-time applications. Group-wise penalization was rarely applied to the BCI field [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Motrenko | Multi-way feature selection for ECoG-based Brain-Computer Interface[END_REF][START_REF] Van Gerven | Interpreting single trial data using groupwise regularisation[END_REF]] [Wu et al., 2019]. Regularized PARAFAC and Tucker decomposition are two algorithms designed for group-wise tensor penalization. In these algorithms, tensors are expressed as a linear combination of vectors which are independently regularized. Regularized tensor decomposition leads to a slice-wise tensor penalization creating more easily interpretable solution than element-wise regularization strategy (Figure 5-1). These approaches were used in few offline BCI studies [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Martínez-Montes | PENALIZED PARAFAC ANALYSIS OF SPONTANEOUS EEG RECORDINGS[END_REF] and in other research fields [START_REF] Giordani | Constrained Candecomp/Parafac via the Lasso[END_REF][START_REF] Hervás | Sparse N-way partial least squares by L1-penalization[END_REF][START_REF] Kim | Robust iteratively reweighted Lasso for sparse tensor factorizations[END_REF][START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF]. In BCI studies, most of the presented feature dimensional reduction algorithms were tested during offline studies [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF][START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF][START_REF] Jafarifarmand | Real-time multiclass motor imagery brain-computer interface by modified common spatial patterns and adaptive neuro-fuzzy classifier[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF] [S. P. Kim et al., 2006[START_REF] Kumar | An improved discriminative filter bank selection approach for motor imagery EEG signal classification using mutual information[END_REF][START_REF] López-Larraz | Continuous decoding of movement intention of upper limb self-initiated analytic movements from pre-movement EEG correlates[END_REF][START_REF] Lotte | Regularizing Common Spatial Patterns to Improve BCI Designs: Unified Theory and New Algorithms[END_REF][START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF][START_REF] Motrenko | Multi-way feature selection for ECoG-based Brain-Computer Interface[END_REF][START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF][START_REF] Palmer | Independent Component Analysis (ICA) Features for Electro-corticographic (ECoG) Brain-Machine Interfaces (BMIs). 臨床神経生理[END_REF][START_REF] Peterson | A penalized time-frequency band feature selection and classification procedure for improved motor intention decoding in multichannel EEG[END_REF][START_REF] Robinson | Multi-class EEG classification of voluntary hand movement directions[END_REF]] [Schaeffer and Aksenova, 2016b[START_REF] Schroder | Automated EEG feature selection for brain computer interfaces[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Sreeja | Weighted sparse representation for classification of motor imagery EEG signals[END_REF][START_REF] Toda | Reconstruction of two-dimensional movement trajectories from selected magnetoencephalography cortical currents by combined sparse Bayesian methods[END_REF][START_REF] Van Gerven | Interpreting single trial data using groupwise regularisation[END_REF]] [Wu et al., 2019] [Y. Zhang et al., 2013]. Nevertheless, some of them were applied in online applications. Generally, feature selection was performed in offline preliminary studies before to apply the set of selected feature during online clinical or preclinical BCI experiments [START_REF] Bousseta | EEG Based Brain Computer Interface for Controlling a Robot Arm Movement Through Thought[END_REF][START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Huang | EEG-based online two-dimensional cursor control[END_REF]] [Kim et al., 2018[START_REF] Marathe | Decoding continuous limb movements from high-density epidural electrode arrays using custom spatial filters[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF]] [Spüler et al., 2012b].

Online adaptive dimensional reduction strategies have several advantages for online adaptive BCI. Majority of decoders trained in real-time are sensitive to overfitting due to the lack of training data. Moreover, reduced feature space dimensions may reduce the required computing resources to apply the model in real-time with faster data flow analysis.

The adaptive dimensionality reduction algorithms commonly applied in the BCI [START_REF] Ang | Filter Bank Common Spatial Pattern (FBCSP) algorithm using online adaptive and semi-supervised learning[END_REF][START_REF] Chen | A reliable brain-computer interface based on SSVEP using online recursive independent component analysis[END_REF]] [Hsu et al., 2016[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Mobaien | ACSP: Adaptive CSP filter for BCI applications[END_REF][START_REF] Sannelli | Ensembles of adaptive spatial filters increase BCI performance: an online evaluation[END_REF][START_REF] Song | Improving brain-computer interface classification using adaptive common spatial patterns[END_REF][START_REF] Woehrle | An Adaptive Spatial Filter for User-Independent Single Trial Detection of Event-Related Potentials[END_REF]] [Zhao et al., 2008] were based on projection strategies such as adaptive CSP, PCA, ICA or xDAWN algorithms. However, all of them were only tested during offline studies. Few of the dimensional reduction algorithms [Vidaurre et al., 2006b] were integrated into a BCI software made of an adaptive dimensional reduction procedure and an adaptive classifier/regression decoder.

Few adaptive feature selection algorithms were applied in the motor BCI field during online experiments. Filter methods were tested on BCI simulation using Mutual Information [START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF] or during online BCI experiments based on Fisher score [START_REF] Faller | Autocalibration and Recurrent Adaptation: Towards a Plug and Play Online ERD-BCI[END_REF]. Wrapper strategy was optimized using parallel computation for online BCI classifier [START_REF] Mend | Human computer interface with online brute force feature selection[END_REF] whereas embedded methods using semisupervised feature selection [START_REF] Long | Semi-supervised joint spatio-temporal feature selection for P300-based BCI speller[END_REF] and weighting features algorithm [START_REF] Andreu-Perez | A Self-Adaptive Online Brain-Machine Interface of a Humanoid Robot Through a General Type-2 Fuzzy Inference System[END_REF] were designed and used during online BCI applications. Adaptive Genetic Algorithm was proposed for adaptive channel selection in [START_REF] Moro | Towards adaptive brain-computer interfaces: Improving accuracy of detection of event-related potentials[END_REF]. Nevertheless, all these algorithms were applied to simple online binary classification BCI experiments.

Regularized algorithms trained offline were applied during online BCI experiments in [START_REF] Cincotti | High-resolution EEG techniques for brain-computer interface applications[END_REF]] [Ma et al., 2020[START_REF] Shin | Noise robustness analysis of sparse representation based classification method for non-stationary EEG signal classification[END_REF]. Adaptive regularized algorithms with fixed penalization hyperaparemeter were tested using offline dataset but none of these algorithms have been applied to real-time BCI experiments [START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF][START_REF] Roijendijk | Classifying Regularized Sensor Covariance Matrices: An Alternative to CSP[END_REF][START_REF] Sharghian | Online Dictionary Learning for Sparse Representation-Based Classification of Motor Imagery EEG[END_REF]. Adaptive algorithm with a L1-norm regularization strategy were reported in other domains with an adaptive logistic regression [START_REF] Sheikhattar | Adaptive sparse logistic regression with application to neuronal plasticity analysis[END_REF], a Kernel least squares [START_REF] Yang | Online sequential echo state network with sparse RLS algorithm for time series prediction[END_REF] and a recursive least squares algorithms [B. Chen et al., 2012].

Only few dimensional reduction methods were integrated into adaptive algorithms for online incremental calibration during real-time BCI experiments and were generally restricted to EEG-based experiments [Andreu-Perez et al., 2018[START_REF] Faller | Autocalibration and Recurrent Adaptation: Towards a Plug and Play Online ERD-BCI[END_REF]] [Long et wise sparse REW-MSLM al., 2011[START_REF] Mend | Human computer interface with online brute force feature selection[END_REF]] [Moro et al., 2017]. Computational complexity and difficulty to integrate dimensional reduction methods into real-time algorithms may explain the lack of proposed solutions. Moreover, dimensional reduction methods often rely on hyperparameters which required to be tuned to optimize the decoding performances. This hyperparameter optimization problem may be another explanation of the lack of regularized adaptive decoder in the BCI fields.

In motor BCI field, the L1-Regularized N-way PLS algorithm developed by Eliseyev [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF] and the Regularized PLS proposed by Foodeh [START_REF] Foodeh | Regularized Partial Least Square Regression for Continuous Decoding in Brain-Computer Interfaces[END_REF] outperformed their non-penalized version thanks to the suppression of noisy/nonrelevant electrodes. However, these algorithms were not adapted to online adaptive decoding, required preliminary studies to fixed the hyperparameters and were only tested offline on NHP using ECoG [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Foodeh | Regularized Partial Least Square Regression for Continuous Decoding in Brain-Computer Interfaces[END_REF] and rats using LFP [START_REF] Foodeh | Regularized Partial Least Square Regression for Continuous Decoding in Brain-Computer Interfaces[END_REF].

In the next section, the new Lp-Penalized REW-NPLS (PREW-NPLS) is proposed. PREW-NPLS is a new regularized recursive exponentially weighted N-way PLS designed for online adaptive decoding promoting group-wise (slice-wise) sparsity generalized to L0, L0,5 and L1 norm regularization. PREW-NPLS rely on the REW-NPLS algorithms. The crucial REW-NPLS tensor decomposition procedure inspired from PARAFAC algorithm is modified to estimate a sparse L0, L0,5 or L1 PARAFAC tensor decomposition. PREW-NPLS is an incremental adaptive regression algorithm which incrementally estimates a sparse L0, L0,5 and L1 solution with a fixed penalization hyperparameter. As mentioned, hyperparameter generally requires preliminary offline studies to be optimized which is counterintuitive for algorithm dedicated to incremental online closed-loop calibration. Therefore, a new upgraded version referred as Automatic Lp-Penalized REW-NPLS (APREW-NPLS) is introduced in order to automatically optimize the penalization hyperparameter during the online incremental calibration of the model using a reinforcement learning strategy. The PREW-NPLS and APREW-NPLS algorithms were designed to be integrated in the REW-MSLM algorithms as sparse gating and/or experts models for portable applications. Both algorithms are described in the next sections but firstly require to introduce in details the PARAFAC decomposition procedure.

PARAFAC procedure

REW-NPLS algorithm evaluates a set of projectors from the covariance matrix 𝐗𝐘 𝑢 using a rank one decomposition to evaluate the model parameters. Several tensor decomposition strategies were designed such as the PARAFAC, Tucker and multilinear SVD decomposition. The tensor decomposition employed in REW-NPLS algorithm is based on PARAllel FACtor analysis (PARAFAC) tensor decomposition procedure. It is described in further detail in the next section.

Parallel Factor analysis (PARAFAC) or CANDECOMP/PARAFAC (CP) also known as polyadic decomposition (PD) can be considered as the generalization of principal component analysis (PCA) and singular value decomposition (SVD) to the tensor case [Cichocki et al., 2015] [Kolda and[START_REF] Kolda | Tensor Decompositions and Applications[END_REF]. This method represents a 𝑀-order tensor 𝐕 ∈ ℝ 𝐼 1 ×…×𝐼 𝑀 as the linear combination of vector outer products (rank-one tensors) such as :

𝐕 = ∑ 𝜌 𝑟 𝐰 𝑟 1 ∘ 𝐰 𝑟 2 ∘ … ∘ 𝐰 𝑟 𝑀 𝑅 𝑟=1 + 𝐄, with 𝑟, 𝑚 ∶ ‖𝐰 𝑟 𝑚 ‖ = 1.
Here, 1 ≤ 𝑚 ≤ 𝑀 corresponds to the m th mode/dimension of the tensor variable, "∘" is the (vector) outer product of the decomposition projectors 𝐰 𝑟 𝑚 ∈ ℝ 𝐼 𝑚 , 𝑅 ∈ ℕ is the fixed number of rank-one tensors used to decompose the original tensor variable, 𝜌 𝑟 is the weight associated to each rank-one tensor of the decomposition and 𝐄 ∈ ℝ 𝐼 1 ×…×𝐼 𝑀 is the tensor of residuals [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]]. An example of three-order tensor decomposition based on the linear combination of 𝑅 outer products of three vectors is showed in the Figure 5-2.

PARAFAC decomposition computation

Tensor decomposition is an appealing tool since the last twenty years in various fields (audio, image, video processing, biomedical applications, etc.) due to the rising of high dimensional data [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF]. Nevertheless, no specific algorithm determining the rank of tensor has been defined [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]. Consequently, the number of rank-one tensor decomposition 𝑅 is set to a sub-optimal value [START_REF] Kolda | Tensor Decompositions and Applications[END_REF].

Fixing 𝑅 leads to solve a low-rank approximation problem which is an ill-posed problem [Pereira Da [START_REF] Silva | An iterative deflation algorithm for exact CP tensor decomposition[END_REF]. Numerous algorithms has been designed to locally solve this problem.

Chapter 5 : Online incremental groupwise sparse REW-MSLM Most of the solutions can be grouped into two families: direct methods regrouping Alternating least square (ALS), direct tri-linear decomposition (DTLD) and iterative nonleast square methods such as self-weighted alternating tri-linear decomposition (SWATLD) or alternating slice-wise diagonalization (ASD). Hessian and gradient based methods regroup Newton-based algorithms, damped Gauss-Newton with compression (dGNc), positive matrix factorization for 3-way arrays (PMF3) and high-order singular value decomposition (HOSVD) [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]Bader, 2009] [Tomasi, 2006]. No agreements on the best solution has been found on the literature but ALS seems to generally leads to good quality decomposition even though it is slower than numerous algorithms such as ASD [START_REF] Faber | Recent developments in CANDECOMP/PARAFAC algorithms: a critical review[END_REF][START_REF] Tomasi | Practical and Computational Aspects in Chemometric Data Analysis[END_REF][START_REF] Tomasi | A comparison of algorithms for fitting the PARAFAC model[END_REF].

Alternating least square (ALS) method is the most popular algorithm for PARAFAC decomposition [Faber et al., 2003] [Kolda and[START_REF] Kolda | Tensor Decompositions and Applications[END_REF] due to the ease of implementation. Nevertheless, this algorithm has many drawbacks. ALS method can be long to converge without guarantee of finding a global minimum [Bilian. Chen et al., 2012] [Cichocki et al., 2015[START_REF] Kolda | Tensor Decompositions and Applications[END_REF]] [Silva et al., 2015]and is dependent on the initialization of the decomposition vectors [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]. Several methods have been design to improve ALS performances depending on the decomposition quality, computing resources, computation time [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF][START_REF] Faber | Recent developments in CANDECOMP/PARAFAC algorithms: a critical review[END_REF]] such as Tikhonov regularization, maximum block improvement method [Bilian. Chen et al., 2012], coupled-eigenvalue (CE) post-processing [Pereira Da [START_REF] Silva | An iterative deflation algorithm for exact CP tensor decomposition[END_REF], etc.

The dGNc and PFM3 algorithm showed better results than ALS in [START_REF] Tomasi | Practical and Computational Aspects in Chemometric Data Analysis[END_REF] but were more computationally expensive. CE post-processing improved the decomposition of truncated HOSVD whereas the Sequential rank-one approximation (SeROA), presented in [START_REF] Silva | An iterative deflation algorithm for exact CP tensor decomposition[END_REF], highlighted good results that should be compared to ALS. An interesting solution proposed in [START_REF] Tomasi | Practical and Computational Aspects in Chemometric Data Analysis[END_REF] was to combine the different algorithms in order to exploit the benefits of each one. SWATLD algorithm could be used to initialize the decomposition factors of the rank-one tensor decomposition for PMF3, dGN or ALS algorithms before to apply CE post-processing [Pereira Da [START_REF] Silva | An iterative deflation algorithm for exact CP tensor decomposition[END_REF][START_REF] Tomasi | Practical and Computational Aspects in Chemometric Data Analysis[END_REF]. However, there is no consensus on the advantages of the proposed alternative compared to ALS algorithms [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]Bader, 2009] [Tomasi and[START_REF] Tomasi | A comparison of algorithms for fitting the PARAFAC model[END_REF] Next section is mainly focused on the most widespread ALS algorithm employed in the REW-NPLS for the PARAFAC tensor decomposition.

ALS based PARAFAC decomposition

Alternating least square (ALS) method optimizes one projector (𝐰 𝑟 𝑚 ∈ ℝ 𝐼 𝑚 ) at a time and fixes the others [START_REF] Cichocki | Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis[END_REF][START_REF] Kolda | Tensor Decompositions and Applications[END_REF][START_REF] Pereira Da Silva | Rank-1 Tensor Approximation Methods and Application to Deflation[END_REF]. In the next section, PARAFAC decomposition is considered in the specific case of threeorder tensor decomposition to simplify the notation and to be closer to the BCI application presented in the next chapters. However, all the presented equations are generalizable to N-order tensor decomposition procedure.

Let 𝐕 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 be a third order tensor which undergoes PARAFAC decomposition.

The aim is to find a tensor 𝐕 ̂∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 equal to the linear combination of 𝑅 ∈ ℕ outer product of three normalized projectors

𝐰 𝑟 1 ∈ ℝ 𝐼 1 ,𝐰 𝑟 2 ∈ ℝ 𝐼 2 , 𝐰 𝑟 3 ∈ ℝ 𝐼 3 weighted with the coefficient 𝜌 𝑟 ∈ ℝ: min 𝐗 ̂‖𝐕 -𝐕 ̂‖, 𝐕 ̂= ∑ 𝜌 𝑟 𝐰 𝑟 1 ∘ 𝐰 𝑟 2 ∘ 𝐰 𝑟 3 𝑅 𝑟=1 , ‖𝐰 𝑟 1 ‖ = ‖𝐰 𝑟 2 ‖ = ‖𝐰 𝑟 3 ‖ = 1.
The factor matrices refers to the concatenation of the decomposition factors 𝐖 1 ∈ ℝ 𝐼 1 ×𝑅 ,𝐖 2 ∈ ℝ 𝐼 2 ×𝑅 , 𝐖 3 ∈ ℝ 𝐼 3 ×𝑅 with 𝐖 𝑖 = [𝐰 1 𝑖 𝐰 2 𝑖 … 𝐰 𝑅 𝑖 ] and 𝑖 = 1,2,3. From the factor matrices and the weighting vector 𝛒 ∈ ℝ 𝑅 , PARAFAC decomposition can be expressed with the unfolded tensor shape [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]:

𝐕 ̂(1) = 𝐖 1 𝛒(𝐖 3 ⨀𝐖 2 ) T , 𝐕 ̂(2) = 𝐖 2 𝛒(𝐖 3 ⨀𝐖 1 ) T , 𝐕 ̂(3) = 𝐖 3 𝛒(𝐖 2 ⨀𝐖 1 ) T .
The ALS is an iterative procedure which reduces the optimization problem to smaller sub-problems [START_REF] Tomasi | A comparison of algorithms for fitting the PARAFAC model[END_REF]. Each step of the ALS solves a linear regression problem with one vector feature. At each step ALS fixes two of the three matrices 𝐖 1 , 𝐖 2 and 𝐖 3 and reduce the problem to a linear least-squares optimization. For example, in an ALS algorithm iteration, 𝐖 2 and 𝐖 3 are fixed to solve 𝐖 1 then 𝐖 2 is solved by fixing 𝐖 1 and 𝐖 3 and, finally, the same operation is realized for 𝐖 3 . Firstly, 𝐖 2 and 𝐖 3 are fixed which leads to

min 𝐖 ̂1 ‖𝐕 (1) -𝐖 ̂𝛒 1 (𝐖 3 ⨀𝐖 2 ) T ‖,
where 𝐖 ̂1 ∈ ℝ 𝐼 1 ×𝑅 is the estimated factor matrix following the first decomposition dimension with [START_REF] Kolda | Tensor Decompositions and Applications[END_REF]:

𝐖 ̂𝛒 1 = 𝐖 ̂1𝛒. (5.2.1)
Minimum is achieved for

𝐖 ̂𝛒𝟏 = 𝐕 (1) [(𝐖 3 ⨀𝐖 2 ) T ] † ,
which simplifies due to the Khatri-Rao pseudoinverse properties ((0.1) to

𝐖 ̂𝛒 1 = 𝐕 (1) (𝐖 3 ⨀𝐖 2 )(𝐖 3 T 𝐖 3 * 𝐖 2 T 𝐖 2 ) † .
𝐖 2 and 𝐖 3 are estimated following the same steps by fixing 𝐖 1 = 𝐖 ̂1 using columnwise normalization with (5.2.1) resulting in: wise sparse REW-MSLM

𝐖 ̂1 = 𝐕 (1) (𝐖 3 ⨀𝐖 2 ) (𝐖 3 T 𝐖 3 * 𝐖 2 T 𝐖 2 ) † , 𝐖 ̂2 = 𝐕 (2) (𝐖 3 ⨀𝐖 1 ) (𝐖 3 T 𝐖 3 * 𝐖 1 T 𝐖 1 ) † , 𝐖 ̂3 = 𝐕 (3) (𝐖 2 ⨀𝐖 1 ) (𝐖 2 T 𝐖 2 * 𝐖 1 T 𝐖 1 ) † .
This procedure is repeated until a specified condition is reached (fixed number of iteration, convergence criterion, etc.). The projection matrices can be initialized with random values, values estimated in previous iteration of the ALS algorithm or values determined using another algorithm such as DTLD [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF][START_REF] Faber | Recent developments in CANDECOMP/PARAFAC algorithms: a critical review[END_REF][START_REF] Kolda | Tensor Decompositions and Applications[END_REF][START_REF] Tomasi | A comparison of algorithms for fitting the PARAFAC model[END_REF].

PARAFAC decomposition in the REW-NPLS algorithm.

REW-NPLS algorithm integrated a PARAFAC-based decomposition algorithm to extract the set of projectors of 𝐗𝐘 𝑢 . In the next section, the PARAFAC decomposition problem of the REW-NPLS algorithm is considered in the specific case of three order tensor decomposition 𝐗𝐘 𝑢 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 , ‖𝐗𝐘 𝑢 ‖ = 1 to simplify the notation and to be closer to the BCI application considered in the PhD thesis. Nevertheless, all the presented results can be generalized to the n order tensor decomposition.

At each iteration (𝑓 is current iteration number) of REW-NPLS algorithm, one iteration of PARAFAC algorithm is used (with a rank one approximation, 𝑅 = 1) to decompose tensor 𝐗𝐘 𝑢 and to estimate the projectors 𝐰 𝑓 1 , 𝐰 𝑓 2 , 𝐰 𝑓 3 :

min 𝐗𝐘 ̂𝑢 ‖𝐗𝐘 𝑢 -𝐗𝐘 ̂𝑢‖, 𝐗𝐘 ̂𝑢 = 𝜌 𝑓 𝐰 𝑓 1 ∘ 𝐰 𝑓 2 ∘ 𝐰 𝑓 3 , ‖𝐰 𝑓 1 ‖ = ‖𝐰 𝑓 2 ‖ = ‖𝐰 𝑓 3 ‖ = 1.
Here, ‖•‖, as a reminder, always referred to the L2 norm (Frobenius, Euclidian norm depending on the variable dimensions). Equally:

min 𝐗𝐘 ̂𝑢 ‖𝐗𝐘 𝑢 -𝐗𝐘 ̂𝑢‖ 2 (5.2.2) 𝐗𝐘 ̂𝑢 = 𝜌 𝑓 𝐰 𝑓 1 ∘ 𝐰 𝑓 2 ∘ 𝐰 𝑓 3 , ‖𝐰 𝑓 1 ‖ = ‖𝐰 𝑓 2 ‖ = ‖𝐰 𝑓 3 ‖ = 1.
As only one iteration of REW-NPLS algorithm is considered, iteration index 𝑓 is discarded in the section for the purpose of simplification.All the presented steps are repeated for each latent space dimension 𝑓 with 𝑓 = 1, … , 𝐹.

This problem is no longer an ill-posed problem [START_REF] Silva | An iterative deflation algorithm for exact CP tensor decomposition[END_REF]. ALS algorithm guarantees to converge [START_REF] Wang | On the Global Convergence of the Alternating Least Squares Method for Rank-One Approximation to Generic Tensors[END_REF]. In the REW-NPLS algorithm, PARAFAC decomposition is solved using ALS algorithm [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. It optimizes sequentially

min 𝐰 1 ‖𝐗𝐘 𝑢 (1) -𝐰 1 (𝐰 3 ⊗ 𝐰 2 ) T ‖ 2 ,
(5.2.3)

min 𝐰 2 ‖𝐗𝐘 𝑢 (2) -𝐰 2 (𝐰 3 ⊗ 𝐰 1 ) T ‖ 2 ,
(5.2.4)

min 𝐰 3 ‖𝐗𝐘 𝑢 (3) -𝐰 3 (𝐰 2 ⊗ 𝐰 1 ) T ‖ 2 (5.2.5)
until convergence [START_REF] Uschmajew | A new convergence proof for the higher-order power method and generalizations[END_REF]. In the case of three-order tensor, Least Square (LS) solutions for each step are expressed:

𝐰 𝜌 1 = 𝐗𝐘 𝑢 (1) (𝐰 3 ⨀𝐰 2 )(𝐰 3 T 𝐰 3 * 𝐰 2 T 𝐰 2 ) † .
As 𝐰 𝑖 ∈ ℝ 𝐼 𝑖 , the solution can be simplified using:

(𝐰 𝑓 3 T 𝐰 𝑓 3 * 𝐰 𝑓 2 T 𝐰 𝑓 2 ) = ‖𝐰 ̂𝑓2 ‖ 2 * ‖𝐰 ̂𝑓3 ‖ 2 = ‖𝐰 𝑓 3 ⨂𝐰 𝑓 2 ‖ 2 ∈ ℝ,
and

(𝐰 𝑓 3 ⨀𝐰 𝑓 2 ) = (𝐰 𝑓 3 ⨂𝐰 𝑓 2 ).
To obtain the least square solution:

𝐰 𝜌 1 = 𝐗𝐘 𝑢 (1) (𝐰 3 ⊗ 𝐰 2 ) ‖𝐰 3 ⊗ 𝐰 2 ‖ 2 .
(5.2.6)

Normalization allows the estimation of parameter 𝜌 𝑓 and 𝐰 1 with ‖𝐰 1 ‖ = 1. The same procedure is repeated to evaluate both 𝐰 2 and 𝐰 3 :

𝐰 𝜌 2 = 𝐗𝐘 𝑢 (2) (𝐰 3 ⊗ 𝐰 1 ) ‖𝐰 3 ⊗ 𝐰 1 ‖ 2 ,
(5.2.7)

𝐰 𝜌 3 = 𝐗𝐘 𝑢 (3) (𝐰 2 ⊗ 𝐰 1 ) ‖𝐰 2 ⊗ 𝐰 1 ‖ 2 .
(5.2.8)

Each one is normalized to evaluate 𝜌 and 𝐰 1 , 𝐰 2 , 𝐰 3 with ‖𝐰 1 ‖ = ‖𝐰 2 ‖ = ‖𝐰 3 ‖ = 1. These three solutions are successively computed until a convergence or maximum iteration number criterion is reached (Figure 5-3). wise sparse REW-MSLM Figure 5-3: PARAFAC-inspired tensor decomposition used in the REW-NPLS algorithm using alternative least square (ALS) algorithm. At each iteration (𝑓 is current iteration number), the tensor 𝑿𝒀 is decomposed into three vectors. In the considered case of this PhD manuscript, the decomposition factor 𝒘 1 , 𝒘 2 , 𝒘 3 , are attributed to the time, frequency and spatial domain. Each of the decomposition factor is evaluated alternatively by fixing two of them to estimate the third. This operation is repeated for each factor until a convergence criterion is reached.

Lp-Penalized REW-NPLS (PREW-NPLS)

PREW-NPLS algorithm exploited a penalized version of the PARAFAC algorithm to create group-wise sparse solution. This algorithm is an online adaptive algorithm which introduced Lp penalization with p being the classic norm regularization (L1) or less conventional norm and pseudo-norm penalization type such as L0 and L0.5. This section describes the penalized PARAFAC procedure and its integration into the REW-NPLS algorithm to build the new online adaptive sparse PREW-NPLS algorithm.

Penalized PARAFAC procedure

In the PARAFAC-based algorithm used in REW-NPLS, ALS strategy fixes all projectors except one at each step of the algorithm. Consequently, each step of the ALS solved a linear regression with one vector feature. In this section, L0, L0,5 and L1 regularized linear regression are simplified to be applied in online PARAFAC subroutine of REW-NPLS.

The following equation will be presented in the case of three-order tensor and rank one (𝑅 = 1) PARAFAC decomposition to simplify the notations but can be generalized to Norder tensor.

Given a three order-tensor 𝐕 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 to decompose using regularized PARAFAC with ALS strategy and 𝐰 𝑖 ∈ ℝ * 𝐼 𝑖 with 𝑖 = 1,2,3 the decomposition factors estimated by the PARAFAC. Let us consider the unfolded tensor 𝐕 (𝑖) with 𝐕

(𝑖) = (𝐯 1 1  … 𝐯 1 𝐼 1 ) ∈ 𝑹 𝐼 1 ×𝐼 2 𝐼 3
where 𝐯 𝑖 𝑗 are the rows of matrix 𝐕 (𝑖) . Taking into account that (𝐰 2 ⊗ 𝐰 1 ) T ∈ 𝑹 𝐼 1 𝐼 2 , (𝐰 3 ⊗ 𝐰 1 ) T ∈ 𝑹 𝐼 1 𝐼 3 and (𝐰 3 ⊗ 𝐰 2 ) T ∈ 𝑹 𝐼 2 𝐼 3 are vectors, the optimization tasks (5.2.3)- (5.2.5) are separated into element-wise optimization: ( 5.3.3) Here w 𝑗 𝑖 are the projector elements of vectors 𝐰 1 = (w 1 1 , … , w 𝐼 1 1 ) 𝑇 ∈ ℝ * 𝐼 1 , 𝐰 2 = (w 1 2 , … , w 𝐼 2 2 ) 𝑇 ∈ ℝ * 𝐼 2 , and 𝐰 3 = (w 1 3 , … , w 𝐼 3 3 ) 𝑇 ∈ ℝ * 𝐼 3 estimated by the PARAFAC. The (5.2.6)-( 5.2.8) least squares (LS) solutions may be written as: (5.3.4) (5.3.6) Sparse L p (p = 0, 1 2 , 1) norm/pseudo norm penalization including a protection variable characteristic is proposed to be integrated to the cost function of REW-NPLS procedure to provide a group-wise sparse solutions, namely, solutions sparse by slices. Optimization task ( 5

min w 𝒋 𝟏 ‖𝐯 1 𝑗 -w 𝑗 1 (𝐰 3 ⊗ 𝐰 2 ) T ‖ 2 𝑗 = 1, … , 𝐼 1 , (5.3.1) min w 𝒋 𝟐 ‖𝐯 2 𝑗 -w 𝑗 2 (𝐰 3 ⊗ 𝐰 1 ) T ‖ 2 𝑗 = 1, … , 𝐼 2 , ( 5
(w 𝑗 1 ) 𝐿𝑆 = 𝐯 1 𝑗 (𝐰 3 ⊗ 𝐰 2 ) ‖𝐰 3 ⊗ 𝐰 2 ‖ 2 , 𝑗 = 1, … , 𝐼 1 ,
(w 𝑗 2 ) 𝐿𝑆 = 𝐯 2 𝑗 (𝐰 3 ⊗ 𝐰 1 ) ‖𝐰 3 ⊗ 𝐰 1 ‖ 2 , 𝑗 = 1, … , 𝐼 2 (5.3.5) (w 𝑗 3 ) 𝐿𝑆 = 𝐯 3 𝑗 (𝐰 2 ⊗ 𝐰 1 ) ‖𝐰 2 ⊗ 𝐰 1 ‖ 2 , 𝑗 = 1, … , 𝐼 3 .
P(𝐰 1 , 𝐰 2 , 𝐰 3 ) = 𝜆 1 ‖𝐰 1 ‖ 𝑞,ℒ 1 + 𝜆 2 ‖𝐰 2 ‖ 𝑞,ℒ 2 + 𝜆 3 ‖𝐰 3 ‖ 𝑞,ℒ 3 , ‖𝐰 1 ‖ = ‖𝐰 2 ‖ = ‖𝐰 3 ‖ = 1.
Where ‖𝐰 𝑖 ‖ 𝑝,ℒ 𝑖 for 𝑝 = 0, 1 2 , 1 and 𝑖 = 1, 2, 3 is denoted as :

‖𝐰 𝑖 ‖ 0,ℒ 𝑖 = ∑ (1 -𝛿 0,w 𝑘 𝑖 ) 𝑘∈ℒ 𝑖 , ‖𝐰 𝑖 ‖ 1,ℒ 𝑖 = ∑ |w 𝑘 𝑖 | 𝑘∈ℒ 𝑖 , ‖𝐰 𝑖 ‖1 2 ,ℒ 𝑖 = ∑ √|w 𝑘 𝑖 | 𝑘∈ℒ 𝑖 .
Here, the regularization functions may only regularize a part of the indices (projector elements). The indices of the potentially penalized projector elements are defined by a set ℒ 𝑖 ⊂ {1,2, … , 𝐼 𝑖 } with 𝑖 = 1, 2, 3 while the other elements not included in ℒ 𝑖 are "protected" and cannot be penalized. ℒ 𝑖 may vary depending on the REW-NPLS iteration. 0 < 𝜆 𝑖 ≤ 1 are the penalization hyperparameters. The Kronecker delta 𝛿 0,w 𝑘 𝑖 = 1 if w 𝑘 𝑖 = 0, 𝛿 0,w 𝑘 𝑖 = 0 otherwise.

The same ALS strategy (5.2.3)- (5.2.5) than the procedure used in conventional REW-NPLS is proposed to be applied for the optimization of (5.3.7). ALS fixed all projectors except one at each step of the algorithm, leading to the three successive optimization tasks:

min 𝐰 1 (‖𝐕 (1) -𝐰 1 (𝐰 3 ⊗ 𝐰 2 ) T ‖ 2 + 𝜆 1 ‖𝐰 1 ‖ 𝑞,ℒ 1 ), min 𝐰 2 (‖𝐕 (2) -𝐰 2 (𝐰 3 ⊗ 𝐰 1 ) T ‖ 2 + 𝜆 2 ‖𝐰 2 ‖ 𝑞,ℒ 2 ), min 𝐰 3 (‖𝐕 (3) -𝐰 3 (𝐰 2 ⊗ 𝐰 1 ) T ‖ 2 + 𝜆 3 ‖𝐰 3 ‖ 𝑞,ℒ 3 ).
The solutions of non-regularized problem (5.3.4)- (5.3.6) are used as initial approximation and are referred as the Least Square (LS) solution noted 𝐰 𝐿𝑆 𝑖 .

Previously, similar penalized ALS was considered in [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF]. However the study was limited to L 1 -norm and did not consider additional protection variables ℒ 𝑖 . Moreover, the problem was solved using non-adaptive NPLS regression for offline classification preclinical experiments and highlighted non-viable solution for real-time processing if more than 14 electrodes were considered [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF]. More general case of L p (p = 0,

𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 𝐿 0 = √𝜆 1 ‖𝐰 3 ⊗ 𝐰 2 ‖ .
In the case of 𝑳 𝟎.𝟓 penalization, and considering one of the optimization step, e.g. (5.3.11) of ALS optimization. Based on (5.3.11) and (5.3.14), the function to minimize 𝐹𝑜𝑝𝐿 0.5 takes the form: (5.3.15) or equivalently: .3.16) The solution to this minimization problem is: 𝑥(1 -𝑥) 2 = 𝐶 (5.3.17)

𝐹𝑜𝑝𝐿 0.5 (w 𝑗 1 ) = ‖𝐯 1 𝑗 -w 𝑗 1 (𝐰 3 ⊗ 𝐰 2 ) T ‖ 2 + 𝜆 1 √|w 𝑗 1 |,
𝐹𝑜𝑝𝐿 0.5 (w 𝑗 1 ) = ‖𝐰 3 ⊗ 𝐰 2 ‖ 2 ((w 𝑗 1 ) 𝐿𝑆 -w 𝑗 1 ) 2 + 𝜆 1 √|w 𝑗 1 |. ( 5
𝑤𝑖𝑡ℎ 𝑥 = w 𝑗 1 (w 𝑗 1 ) 𝐿𝑆 𝑎𝑛𝑑 𝐶 = 𝜆 1 2 16‖𝐰 3 ⊗ 𝐰 2 ‖ 4 ((w 𝑗 1 ) 𝐿𝑆 ) 3 .
To summarize, in the case 𝐶 > 4 27

, (w 𝑗 1 ) 𝐿 0.5 = 0 whereas in the case 𝐶 ∈ [0, 4 27 ], by the properties of the cubic polynomial function (Figure 5-4), the biggest root of (5.3.17 Finally, in the case of 𝑳 𝟏 penalization, considering one of the optimization step, e.g. (5.3.11) of ALS optimization, the solution turns out to be an element-wise softthresholding of the least square solution (w 𝑗 1 ) 𝐿𝑆 𝑗 = 1, … , 𝐼 1 leading to [START_REF] Moly | Online adaptive group-wise sparse NPLS for ECoG neural signal decodin[END_REF]:

(w 𝑗 1 ) 𝐿 1 = { 0 , if 𝑗 ∈ ℒ 1 and (w 𝑗 1 ) 𝐿𝑆 ≤ 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑𝐿 1 𝑠𝑖𝑔𝑛 ((w 𝑗 1 ) 𝐿𝑆 ) (|(w 𝑗 1 ) 𝐿𝑆 | -𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑𝐿 1 ) , if 𝑖 ∈ ℒ 1 and (w 𝑗 1 ) 𝐿𝑆 > 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑𝐿 1 (w 𝑗 1 ) 𝐿𝑆 otherwise , 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑𝐿 1 = 𝜆 1 ‖𝐰 3 ⊗ 𝐰 2 ‖ 2 .

Penalized PARAFAC in the PREW-NPLS algorithm

Penalized PARAFAC based tensor decomposition is integrated into REW-NPLS algorithm to extract iteratively the set of penalized projectors

{𝐰 𝑓 1 ∈ ℝ 𝐼 1 , 𝐰 𝑓 2 ∈ ℝ 𝐼 2 , 𝐰 𝑓 3 ∈ ℝ 𝐼 3 } 𝑓=1 𝐹
from 𝐗𝐘 𝑢 for each latent space dimension 𝑓 ⊂ {1,2, … , 𝐹}.

For 𝑓 = 1, all the projector elements can be potentially penalized. Therefore, the protection set is initialized to ℒ 𝑖,1 ⊂ {1,2, … , 𝐼 𝑗 } as each projector element can be penalized. For any 𝑓, after that the PARAFAC convergence criteria are reached, indices with non-zero elements of 𝐰 𝑓 𝑖 (non-penalized projector elements) are removed from ℒ 𝑖,𝑓 resulting in the protection set for the next iteration ℒ 𝑖,𝑓+1 ⊂ ℒ 𝑖,𝑓 . Therefore, the nonpenalized elements at the iteration 𝑓 cannot be penalized anymore for the next iterations 𝑓 + 1, 𝑓 + 2, … , 𝐹.

The protection variable is introduced because REW-NPLS model is estimated via an incremental procedure, the model at iteration 𝑓 + 1 contains information extracted at iteration 𝑓. Therefore, if a decomposition factor has a non-zero value at iteration 𝑓, it must be considered at iteration 𝑓 + 1. A scheme representing the basic steps of the PREW-NPLS main loop for a specific 𝑓 is represented in the case of spatial L1 penalization with a penalization hyperparameter 𝜆 in The previously presented PREW-NPLS based on the regularized PARAFAC procedure allows to perform group-wise parameter penalization for a fixed penalization hyperparameter 𝜆. The selection of this hyperparameter influences greatly the sparsity of the solution and the global performance of the algorithm. The selection of the 𝜆 hyperparameter may be a complex task and is often optimized based on random or grid search using cross-validation strategy. However such strategy cannot be applied for online decoding because they require high computing resources, too long computing time and are not suited to data-flow processing. Therefore, during online experiments, penalization hyperparameter 𝜆 is commonly fixed using prior knowledge or preliminary offline studies whereas the optimal penalization hyperparameter might be different in offline and online closed-loop experiments.

To overpass this drawback, an upgrade of the PREW-NPLS algorithm named Automatic Lp-PREW-NPLS (APREW-NPLS) is proposed. APREW-NPLS is an adaptive penalized REW-NPLS which automatically evaluate the performance of several penalization hyperparameters. Chapter 5 : Online incremental groupwise sparse REW-MSLM

Automatic 𝝀 penalized REW-NPLS (APREW-NPLS)

Although machine learning aims to optimize numerous parameters to automatically find a model that fits a problem, some specific parameters need to be fixed before the learning process begin. These parameters are named "hyperparameters" and can be found in all the machine learning strategies (supervised, reinforcement, unsupervised). Hyperparameters govern numerous aspects of machine learning algorithms [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF] [Jia Wu and Jia Wu, n.d.] [ [START_REF] Kuhn | Applied Predictive Modeling[END_REF]. They can define the model architecture (neural networks, mixture of experts structure), the applied kernel functions (nonlinear SVM), the regularization term (L1 or L2 norm penalization), the learning rates (KNN clustering convergence speed), the number of decomposition factors (PCA, PARAFAC, ICA, etc.), the dimension of the latent variable space (PLS) [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Flamary | Statistical Learning for BCIs[END_REF]] [Hutter et al., 2011a[START_REF] Kuhn | Applied Predictive Modeling[END_REF] [L. Li et al., 2018] etc. Hyperparameters highly influence the generalization of a model. Consequently, it is required to find the optimal hyperparameters. However, hyperparameter dynamic is not well understand and may be difficult to optimize [Jia Wu and Jia Wu, n.d.] [ [START_REF] Keerthi | An Efficient Method for Gradient-Based Adaptation of Hyperparameters in SVM Models[END_REF] [L. Li et al., 2018].

APREW-NPLS is an incremental adaptive penalized algorithm which aims to select the best penalization hyperparameter 𝜆, introduced by the PREW-NPLS algorithm. Before to introduce the APREW-NPLS method to evaluate the algorithm performance with various penalization hyperparameter 𝜆, a brief review on the state of the art of the hyperparameter optimization strategies is proposed.

Hyperparameter optimization

Hyperparameter automatic optimization is an important field of a more general research domain named Automated Machine Learning (AutoML) gathering various techniques and strategies for algorithm selection, model selection, hyperparameter optimization, etc. [START_REF] Feurer | Efficient and Robust Automated Machine Learning[END_REF][START_REF] Hutter | Automatic Machine Learning: Methods, Systems, Challenges[END_REF]. Nevertheless, most of the reported studies were theoretical and needed to be tested in real life applications.

With the exception of some EEG studies oriented on feature selection [START_REF] Andreu-Perez | A Self-Adaptive Online Brain-Machine Interface of a Humanoid Robot Through a General Type-2 Fuzzy Inference System[END_REF][START_REF] Corralejo | Feature selection using a genetic algorithm in a motor imagerybased Brain Computer Interface[END_REF][START_REF] Faller | Autocalibration and Recurrent Adaptation: Towards a Plug and Play Online ERD-BCI[END_REF][START_REF] Flamary | Statistical Learning for BCIs[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF][START_REF] Long | Semi-supervised joint spatio-temporal feature selection for P300-based BCI speller[END_REF][START_REF] Mend | Human computer interface with online brute force feature selection[END_REF][START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF][START_REF] Schroder | Automated EEG feature selection for brain computer interfaces[END_REF], a deep neural network stochastic gradient descent optimization [START_REF] Shojaedini | A New Method for Detecting P300 Signals by Using Deep Learning: Hyperparameter Tuning in High-Dimensional Space by Minimizing Nonconvex Error Function[END_REF] or a dynamic stopping calibration criterion procedure [START_REF] Schreuder | Optimizing eventrelated potential based brain-computer interfaces: a systematic evaluation of dynamic stopping methods[END_REF], AutoML is highly uncommon in the BCI field. The only hyperparameter optimization procedure reported in online adaptive experiments were limited to adaptive feature selection algorithms for P300 or binary classification [START_REF] Andreu-Perez | A Self-Adaptive Online Brain-Machine Interface of a Humanoid Robot Through a General Type-2 Fuzzy Inference System[END_REF][START_REF] Faller | Autocalibration and Recurrent Adaptation: Towards a Plug and Play Online ERD-BCI[END_REF][START_REF] Long | Semi-supervised joint spatio-temporal feature selection for P300-based BCI speller[END_REF][START_REF] Mend | Human computer interface with online brute force feature selection[END_REF][START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF]. To our knowledge, in the BCI field, no algorithm has been designed for real-time automatic selection of an optimal penalization hyperparameter for online adaptive model. AutoML is a poorly explored field in the domain of motor BCI. However, an overview of AutoML state of the art in the BCI field and other domains is detailed in the next section.

Strategies and algorithms focused on hyperparamer optimization can be clustered in different groups. Firstly, the most common reported strategies were grid (exhaustive) and random search in the hyperparameter space [START_REF] Flamary | Statistical Learning for BCIs[END_REF][START_REF] Schreuder | Optimizing eventrelated potential based brain-computer interfaces: a systematic evaluation of dynamic stopping methods[END_REF]. Grid search tests numerous hyperparameter configurations linearly (or not) distributed over the hyperparameter space whereas random search randomly selects hyperparameter configurations. Among these strategies, studies demonstrated the superiority of random search providing better results with smaller computation time [START_REF] Bergstra | Random Search for Hyper-Parameter Optimization[END_REF]Bengio, 2012] [Hutter et al., 2014]. Although these strategies are simple to understand and implement, they are restricted to a low number of hyperparameter optimization due to the exponentially increasing possible settings with growing dimension of the hyperparameter space. These methods do not find optimal solution, are computationally expensive and time consuming. To handle complex algorithm configuration optimization, subtler methods were designed.

Several studies reported optimization strategies based on Model-free methods. These strategies are quite simple and efficient because they do not have to alternate between fitting and testing a model to find a solution. They are based on stochastic search of optimal hyperparameter configuration. For example, classical gradient descent based strategies were used for hyperparameter optimization [START_REF] Bakhteev | Comprehensive analysis of gradient-based hyperparameter optimization algorithms[END_REF][START_REF] Chapelle | Choosing Multiple Parameters for Support Vector Machines[END_REF][START_REF] Keerthi | An Efficient Method for Gradient-Based Adaptation of Hyperparameters in SVM Models[END_REF]. Racing algorithm family regroups numerous algorithms (ROAR, F-race family, etc) based on competitive learning where bad configurations are iteratively removed from the configuration space [Hutter et al., 2011a]. F-Race algorithms family evaluates the performance of numerous hyperparameter configurations and use a nonparametric Friedman test to identify statistically less efficient configurations before to remove them from the space of possible configurations [Hutter et al., 2011a]. PaRaMILS employs local search procedure to find the best configuration before to undergo perturbations and repeat this procedure [START_REF] Hutter | ParamILS: An automatic algorithm configuration framework[END_REF][Hutter et al., , 2011a]].

Another group of Model-free algorithms named population-based methods clusters the genetic algorithms (GA) such as Tabu_GA, or gender GA [START_REF] Guo | The Tabu_Genetic Algorithm: A Novel Method for Hyper-Parameter Optimization of Learning Algorithms[END_REF]] [Hutter et al., 2011a, 2014]. In a first step, these algorithms starts with a finite set of possible hyperparameter configurations and evaluate the performance of each of them. Next, the set of hyperparameter configurations are mixed ("reproduce") to form a new set of configurations. The hyperparameter configurations which presents the best performance are more likely to be used to create the next generation of hyperparameter configurations [START_REF] Whitley | A genetic algorithm tutorial[END_REF]. GAs have been reported in BCI research for EEG optimal feature selection [START_REF] Corralejo | Feature selection using a genetic algorithm in a motor imagerybased Brain Computer Interface[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF][START_REF] Schroder | Automated EEG feature selection for brain computer interfaces[END_REF]. Another wise sparse REW-MSLM reported Model-free algorithm was based on the common multi-armed bandit problem from the reinforcement-learning field. Multi-Armed Bandit (MAB) problem can be defined as the selection of an action among 𝑁. For each action, a reward from a probability distribution is obtained. MAB solution aims to maximize the expected reward over numerous successive actions or a time period [L. Li et al., 2018[START_REF] Slivkins | Introduction to Multi-Armed Bandits[END_REF][START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF]. In the PhD studied case, the selection of an action is associated to the selection of a hyperparameter configuration and the reward is an evaluation of the performance of the selected configuration. Several MAB algorithms were designed to handle the autoML problems for hyperparameter optimization such as HYPERBAND algorithm (based on pure-exploration non-stochastic infinite-armed bandit problem and racing algorithms) [START_REF] Dôres | Bandit-Based Automated Machine Learning[END_REF] [L. Li et al., 2018], algorithms for simultaneous model and hyperparameter selection [START_REF] Efimova | Reinforcement-Based Simultaneous Algorithm and Its Hyperparameters Selection[END_REF], MASSCAH algorithm [START_REF] Shalamov | Reinforcement-based Method for Simultaneous Clustering Algorithm Selection and its Hyperparameters Optimization[END_REF], Extreme-Region Upper Confidence Bound Bandit (ER-UCB bandit) [START_REF] Hu | Cascaded Algorithm-Selection and Hyper-Parameter Optimization with Extreme-Region Upper Confidence Bound Bandit[END_REF], TUPAQ [START_REF] Sparks | Automating model search for large scale machine learning[END_REF], etc. Q-learning algorithm (reinforcement learning method) was also used for dynamic model selection [START_REF] Feng | Reinforcement Learning based Dynamic Model Selection for Short-Term Load Forecasting[END_REF].

Finally, Model-based approach regroups several methods which recently highlighted interesting results. Strategies based on Sequential Model-Based Optimization (SMBO) alternatively fit the models and apply them in order to select the next hyperparameter configuration to evaluate [Hutter et al., 2011a]. Bayesian optimization methods outperformed or at least provided equivalent performance than other SMBO algorithms [Z. Wang et al., 2016]. Bayesian optimization algorithms estimating the probability 𝑝(𝑦|𝜆) with 𝑦 the performance evaluation and 𝜆 a set of hyperparameter configuration were reported in several studies [L. Li et al., 2018] [Z. Wang et al., 2016]. Bayesian optimization methods aim to optimize the hyperparameter selection in an adaptive procedure using exploration-exploitation strategy to fit 𝑝(𝑦|𝜆). State of the art algorithms employed Gaussian distribution to model 𝑝(𝑦|𝜆) using various algorithms such as random forest algorithms in Sequential Model-based Algorithm Configuration (SMAC) [Hutter et al., 2011a[START_REF] Thornton | Auto-WEKA: Combined Selection and Hyperparameter Optimization of Classification Algorithms[END_REF], random matrix to reduce the hyperparameter configuration space dimension in Random Embedding Bayesian Optimization (REMBO) algorithm [Z. Wang et al., 2016], Tree-structure Parzen Estimator (TPE) [START_REF] Thornton | Auto-WEKA: Combined Selection and Hyperparameter Optimization of Classification Algorithms[END_REF], efficient global optimization (EGO) [Hutter et al., 2011a[START_REF] Jones | Efficient Global Optimization of Expensive Black-Box Functions[END_REF]], Entropy Search algorithm for fast Bayesian hyperparameter optimization (Fabolas) [START_REF] Klein | Fast Bayesian hyperparameter optimization on large datasets[END_REF], etc.

Model-based approach can lead to better hyperparameter configuration optimization [START_REF] Klein | Fast Bayesian hyperparameter optimization on large datasets[END_REF]. However, these algorithms are computationally expensive and are more complicated to implement than model-free approach. Additionally, some limitations of Model-based approach compared to model-free approach were reported [Hutter et al., 2011a[START_REF] Klein | Fast Bayesian hyperparameter optimization on large datasets[END_REF] [L. Li et al., 2018].

Model-based and Model-free strategies can be clustered into two different categories.

The algorithms which are optimizing hyperparameter selection in adaptive manner trying to find the optimal configuration from a specified hyperparameter set of possible configurations (GA, SMBO algorithm) and the ones which are adaptive in computation by eliminating the bad configuration without uniformly training all the configuration (HYPERBAND, racing algorithms etc.).

The presented methods performed the hyperparameter optimizations. Nevertheless, the majority of the proposed algorithms were only tested on simulated data [START_REF] Bakhteev | Comprehensive analysis of gradient-based hyperparameter optimization algorithms[END_REF]] [Hu et al., 2019[START_REF] Hutter | ParamILS: An automatic algorithm configuration framework[END_REF], 2011a[START_REF] Klein | Fast Bayesian hyperparameter optimization on large datasets[END_REF] [Z. Wang et al., 2016] or from various online available dataset repository [START_REF] Chapelle | Choosing Multiple Parameters for Support Vector Machines[END_REF][START_REF] Dôres | Bandit-Based Automated Machine Learning[END_REF][START_REF] Efimova | Reinforcement-Based Simultaneous Algorithm and Its Hyperparameters Selection[END_REF][START_REF] Feng | Reinforcement Learning based Dynamic Model Selection for Short-Term Load Forecasting[END_REF][START_REF] Keerthi | An Efficient Method for Gradient-Based Adaptation of Hyperparameters in SVM Models[END_REF] [L. Li et al., 2018[START_REF] Shalamov | Reinforcement-based Method for Simultaneous Clustering Algorithm Selection and its Hyperparameters Optimization[END_REF][START_REF] Sparks | Automating model search for large scale machine learning[END_REF][START_REF] Thornton | Auto-WEKA: Combined Selection and Hyperparameter Optimization of Classification Algorithms[END_REF]. With the exception of the Genetic algorithms and the Particle Swarn Optimization (PSO) for feature selection optimization [START_REF] Atyabi | Mixture of autoregressive modeling orders and its implication on single trial EEG classification[END_REF][START_REF] Corralejo | Feature selection using a genetic algorithm in a motor imagerybased Brain Computer Interface[END_REF]] [Moro et al., 2017] [Park et al., 2018[START_REF] Schroder | Automated EEG feature selection for brain computer interfaces[END_REF], none of these methods were evaluated in the BCI fields.

Additionally numerous strategies are computationally expensive and/or are optimized using cross-validation procedure [START_REF] Chapelle | Choosing Multiple Parameters for Support Vector Machines[END_REF][START_REF] Flamary | Statistical Learning for BCIs[END_REF][START_REF] Hutter | Automatic Machine Learning: Methods, Systems, Challenges[END_REF][START_REF] Keerthi | An Efficient Method for Gradient-Based Adaptation of Hyperparameters in SVM Models[END_REF]]. These methods were not tested in the scope of online closedloop decoders hyperparameter optimization (even though ROAR may be tested). An online closed-loop automatic decoder should train the decoder, optimize the hyperparameters and apply the decoder with the best hyperparameters in the same time.

The previously proposed Penalized-REW-NPLS algorithm brings penalized solution to online adaptive decoders. However, PREW-NPLS algorithm requires to set a fixed regularization hyperparameter 𝜆 which modulates the sparsity of the model. Selecting a good penalization hypermarameter which preserves the neural signal decoding performance while increasing the solution sparsity is a complicated problem. PREW-NPLS algorithm requires prior offline study to select the most efficient 𝜆 before to apply the selected penalization hyperparameter during online adaptive closed-loop experiments.

We present an Automatic 𝜆 Penalized-REW-NPLS algorithm using model-free algorithm configuration methods inspired of multi-arm bandit problem to train and optimize automatically several models with different penalization hyperparameters. Model-free algorithm configuration methods inspired of multi-arm bandit problem was selected because it is a suited to online adaptive algorithm as it requires less computational resources than model-based algorithms.

Automatic 𝝀 Penalized-REW-NPLS algorithm

Multi-arm bandit problem is a well-known optimization problem. Given an unknown environment in which numerous action can be performed. Each action on the environment emits a reward chosen from an unknown probability distribution which is dependent on the selected action. The objective is to perform the best actions to wise sparse REW-MSLM maximize the expected reward over a time period [Slivkins, 2019] [Sutton and[START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF].

Multi-arm bandit is an exploitation versus exploration algorithm that can be integrated into the decoder calibration procedure to optimize the hyperparameter settings. In the case of APREW-NPLS, multi-arm bandit optimization algorithm was integrated into the Recursive Validation procedure of APREW-NPLS to optimize the penalization hyperparamter 𝜆 during the model calibration steps.

APREW-NPLS principles

Given an update iteration number 𝑢 ∈ ℕ, APREW-NPLS considers a set of penalization hyperparameter 𝛌 = {𝜆 1 , 𝜆 , 𝜆 𝑖 ∈ 𝛌. 𝐹 ∈ ℕ is the fixed upper bound latent space dimension. 𝐁𝐞𝐭𝐚 𝑢 𝑓,𝜆 𝑖 ∈ ℝ (𝐼 1 ×…×𝐼 𝑀 )×(𝐽 1 ×…×𝐽 𝑁 ) , 𝐛𝐢𝐚𝐬 𝑢 𝑓,𝜆 𝑖 ∈ ℝ 𝐽 1 ×…×𝐽 𝑁 are the current models' parameters and associated bias with the penalization hyperparameter 𝜆 𝑖 (see section 4.4) (Figure 56). APREW-NPLS algorithm only updates at each iteration 𝑢 the models that maximizes the expected rewards among all the models. The expected reward set is defined as 𝐐 𝑢 = {𝑄 𝑢 (1), 𝑄 𝑢 ( 2), … , 𝑄 𝑢 (𝑖), … , 𝑄 𝑢 (Λ)}, where 𝑄 𝑢 (𝑖) is the i th expected reward associated to the model with the penalization hyperparameter 𝜆 𝑖 , 1 ≤ 𝑖 ≤ Λ, at the update iteration 𝑢. From the point of view of multiarm bandit optimization, the environment is the neural signal input variables. The selection of the models to update is considered as the action and the decoding performance of the updated models on the incoming new batch of data are considered as the reward used to select the models to recalibrate at the next update iteration.

APREW-NPLS model calibration

At each calibration increment 𝑢, a set of Λ 𝑢𝑝𝑑𝑡 ∈ ℕ + penalized models 𝚯 u (𝛌 𝑢𝑝𝑑𝑡 ) with the associated set of penalization hyperparameters 𝛌 𝑢𝑝𝑑𝑡 ⊂ 𝛌 is selected among the set 𝚯 𝑢 (𝛌). Then, the models contained in the set 𝚯 u (𝛌 𝑢𝑝𝑑𝑡 ) are updated. The training procedure of each selected model is similar to the calibration procedure of the PREW-NPLS algorithm and is based on the current tensor block of observation {𝐗 𝑢 , 𝐘 𝑢 } and previously computed models 𝚯 𝑢-1 (𝛌) weighted with the forgetting factor 𝜇 1 . APREW-NPLS models calibration follows the same strategy than PREW-NPLS algorithm with the exception that, at each iteration 𝑢, Λ 𝑢𝑝𝑑𝑡 models with different penalization hyperparameters are updated instead of only one in PREW-NPLS algorithm.The models not belonging to 𝚯 u (𝛌 𝑢𝑝𝑑𝑡 ) are not updated and fixed to the same parameter weights than the models in 𝚯 𝑢-1 (𝛌). The selection of the models to update, contained in the set 𝚯 u (𝛌 𝑢𝑝𝑑𝑡 ), is performed during the Recursive Validation (RV) procedure using the observation tensors 𝐗 𝑢 and 𝐘 𝑢 in parallel to the selection of the optimal latent space [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. From the performance of each model a ranking of the best models is evaluated depending on a criteria (sparsity, decoding performance, etc.). This model ranking is weighted depending on how long model has not been updated. The models with the best weighted ranks are updated during the next APREW-NPLS algorithm iteration.

APREW-NPLS Recursive Validation strategy

At each iteration 𝑢, before the APREW-NPLS model calibration phase, RV procedure evaluates the performance of a set of penalized models referred as 𝚯 u (𝛌 𝑡𝑒𝑠𝑡 ). A set 𝛌 𝑡𝑒𝑠𝑡 ⊂ 𝛌 of penalization hyperparameters groups the Λ 𝑢𝑝𝑑𝑡 models updated at the previous calibration increment 𝚯 𝑢-1 (𝛌) and Λ 𝑡𝑒𝑠𝑡 ∈ ℕ + other models selected randomly or based on the expected rewards 𝐐 𝑢-1 . The performance of the 𝚯 u (𝛌 𝑡𝑒𝑠𝑡 ) models are evaluated for each latent space dimension similarly to REW-NPLS and PREW-NPLS (see section 4.4.2). The prediction of each penalized model for a specific latent space dimension 𝑓 is noted {𝐘 ̂𝑢 𝑓,𝜃 𝑢,𝜆 𝑖 } 𝑓=1,𝜆 𝑖 ∈𝛌 𝑡𝑒𝑠𝑡

𝐹

. The penalized models prediction for each specific latent space dimension 𝑓 and each 𝜆 𝑖 ∈ 𝛌 𝑡𝑒𝑠𝑡 are compared to the block of training dataset of output observations 𝐘 𝑢 to compute an estimated performance index defined as:

𝑒 𝑢 𝑓,𝜃 𝑢,𝜆 𝑖 = 𝜇 2 𝑒 𝑢-1 𝑓,𝜃 𝑢,𝜆 𝑖 + ε (𝐘 𝑢 , 𝐘 ̂𝑢 𝑓,𝜃 𝑢,𝜆 𝑖 ) , 𝜆 𝑖 ∈ 𝛌 𝑡𝑒𝑠𝑡 ,
where 𝜇 2 is the forgetting coefficient (𝜇 2 ∈ [0; 1]) and ε is a function evaluating the accuracy of the prediction to maximize. For the penalized models that do not belong to the set of tested models 𝚯 u (𝛌 𝑡𝑒𝑠𝑡 ) , 𝑒 𝑢 𝑓,𝜃 𝑢,𝜆 𝑖 = 𝑒 𝑢-1 𝑓,𝜃 𝑢,𝜆 𝑖 , 𝜆 𝑖 ∉ 𝛌 𝑡𝑒𝑠𝑡 . For 𝜆 𝑖 ∈ 𝛌, the estimated optimal latent space dimension 𝑓 𝜃 𝑢,𝜆 𝑖 * of each penalized model 𝜃 𝑢,𝜆 𝑖 is defined as 𝑓 * = argmax 𝑓 (𝑒 𝑢 𝑓,𝜃 𝑢,𝜆 𝑖 ) and the associated optimal performance of the models is noted p 𝑢 (𝜆 𝑖 ) = 𝑒 𝑢 𝑓 * ,𝜃 𝑢,𝜆 𝑖 . From the optimal performance of each model, a reward is granted to each model following the reward function [START_REF] Feng | Reinforcement Learning based Dynamic Model Selection for Short-Term Load Forecasting[END_REF]:

ℛ 𝑢 (𝑖) = 𝑟𝑎𝑛𝑘𝑖𝑛𝑔 (argmax 𝑖 (p 𝑢 (𝜆 𝑖 ))) -𝑟𝑎𝑛𝑘𝑖𝑛𝑔(p 𝑢 (𝜆 𝑖 )), 𝜆 𝑖 ∈ 𝛌, i = {1, … , Λ}.
Other reward function can be selected depending on the objective of the hyperparameter optimization. The presented reward function is focused on the decoding performance of each penalized model without considering the degree of sparsity of each solution. In the case of embedded BCI system where the model sparsity is a key characteristic, a sparsity indicator may be added to the reward function to select the models depending on this criterion.

Most of the time, the multi-arm bandit problem is applied to stationary data where reward probability distributions do not change over time. Nevertheless, numerous studies highlighted the non-stationarity property of the neural signals and the intrasubject variability due to inattention, habituation, etc. [Clerc et al., 2016b[START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF]. Recent rewards are more representative of the current environment (probability distribution) than past rewards. Given 𝜇 3 ∈ [0; 1] a constant forgetting factor, the expected reward 𝑄 𝑢 (𝑖) obtained if the penalized model with a regularization hyperparameter 𝜆 𝑖 and i = {1, … , Λ} is updated, is defined as: wise sparse REW-MSLM

𝑄 𝑢 (𝑖) = (1 -𝜇 3 )𝑄 𝑢-1 (𝑖) + 𝛼ℛ 𝑢 (𝑖) = (1 -𝜇 3 ) 𝑢 𝑄 1 (𝑖) + ∑ 𝛼(1 -𝜇 3 ) 𝑢-𝑖 ℛ 𝑢 (𝑖) 𝑢 𝑖=1 .
Multi-arm bandit is an exploitation versus exploration algorithm. The expected reward estimation algorithm 𝑄 𝑢 (𝑖) is purely exploitation. Nevertheless, it is possible that lessexplored actions lead to better results than the current selected best action. This assumption is even more valid in the case of non-stationary data where each actionreward probability distribution changes. Numerous strategies to include exploration properties were designed. The best-known method is the ε-greedy algorithm which forces the selection of an action among the sub-optimal solution according to 𝑄 𝑢 (𝑖) with a probability ε. The ε-greedy algorithm considers the sub-optimal actions as equivalent and selects one of them randomly without difference between nearly greedy actions and unlikely ones [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF]. The exploitation-exploration expected reward Q 𝑢 selects a sub-optimal action considering the current expected reward as well as the number of iterations since the last time this action has been selected. Therefore, in the current PhD research, the APREW-NPLS expect reward is modified using this upperconfidence bound following the equation:

Q 𝑢 (𝑖) = 𝑄 𝑢 (𝑖) + 𝜐 𝑁 𝑖𝑡 -𝑁 𝑢 (𝑖) 𝑁 𝑖𝑡 , with i = {1, … , Λ}.
Here, 𝜐 ∈ ℝ weights the degree of exploration, 𝑁 𝑖𝑡 ∈ ℕ * is the number of update iteration realized and 𝑁 𝑢 (𝑖) ∈ ℕ is the index of the last update where the model 𝑖 has been selected. The selected models that will be updated in the next update 𝚯 u (𝛌 𝑢𝑝𝑑𝑡 ) are the models which maximize Q 𝑢 . The exploitation-exploration expected reward Q 𝑢 is used to select the models to update at each iteration but numerous other strategies could be implemented to choose the models to update depending on various criteria.

Conclusion

In this chapter adaptive incremental penalized decoders were proposed to estimate a group-wise sparse solution for continuous ECoG decoding namely Lp-Penalized REW-NPLS (PREW-NPLS) and its extension Automatic Lp-Penalized REW-NPLS (APREW-NPLS). PREW-NPLS proposed three different types of norm penalization to estimate a group-wise sparse model during online closed-loop experiments. The created sparse model is more interpretable which is an important feature in the BCI field. Moreover, sparse solution is suited for the integration of models into a system with lower computing power and resources. Sparse models may reduce the calibration period required.

PREW-NPLS requires to fix the type of penalization norm and penalization hyperparameter before the start of the experiment. These mandatory fixed settings are problematic in the case of BCI applications with unknown penalization hyperpamameter and high inter and intra patient variability. Based on autoML strategies, APREW-NPLS algorithm was designed to address the issues raised by PREW-NPLS. Automatic selection of the penalization hyperparameter is performed using a reinforcement learning strategy inspired by the Multi-arm bandit problem. Moreover, APREW-NPLS calibration procedure trains several models in parallel allowing to save time and adapt the optimal penalized model through time/experiments depending on the brain signals changes during online closed-loop experiments.

PREW-NPLS and APREW-NPLS can be integrated in the REW-MSLM algorithms to evaluate the experts and gate models. Integrating such algorithms in the multi-limb REW-MSLM decoder may be interesting to enhance the specificity of each expert model. Indeed, to control a complex multi-limb effector, REW-MSLM decoder uses one REW-NPLS expert per limb to decode the intended movements from the neural signals.

Although these models converge independently, they are all based on the same neural signal feature input variables. APREW-NPLS expert creates more specific model which only takes into accounts the neural features which provide relevant decoding information for the concerned limb.

Chapter 6

Hierarchical HMMinspired gating model for lowlatency state switching

Hierarchical approach in BCI, and motor BCI 155

The Hidden Markov Model was integrated into the REW-MSLM to sustain strong idle state detection and to ensure the patient's asynchronous control of a multi-limb effector. However, with increasing number of states/classes and, potentially, class imbalance, classifier may struggle to detect all the states with high accuracy. Few studies subjected the benefits of hierarchical static classifiers compared to conventional single-layer decoders [START_REF] Abascal | A Hierarchical BCI System Able to Discriminate between Non Intentional Control State and Four Intentional Control Activities[END_REF][START_REF] Dong | Classification of multi-class motor imagery with a novel hierarchical SVM algorithm for brain-computer interfaces[END_REF][START_REF] Salazar-Ramirez | A hierarchical architecture for recognising intentionality in mental tasks on a brain-computer interface[END_REF]. While Hierarchical decoders were poorly tested in the BCI field, they presented some advantages in the case of complex multi-class problem. Hierarchical classifiers highlighted benefits in the case of naturally structured state sequences such as idle versus active state classification before to discriminate the neural signals between multiple active states [START_REF] Abascal | A Hierarchical BCI System Able to Discriminate between Non Intentional Control State and Four Intentional Control Activities[END_REF]] [Bashashati et al., 2007b[START_REF] Dong | Classification of multi-class motor imagery with a novel hierarchical SVM algorithm for brain-computer interfaces[END_REF][START_REF] Gundelakh | Mobile robot control based on noninvasive brain-computer interface using hierarchical classifier of imagined motor commands[END_REF][START_REF] Gupta | A hierarchical meta-model for multi-class mental task based brain-computer interfaces[END_REF]] [Hotson et al., 2016a[START_REF] Jeffries | EEG Classification of Forearm Movement Imagery Using a Hierarchical Flow Convolutional Neural Network[END_REF][START_REF] Kee | A Hierarchical Classification Strategy for Robust Detection of Passive/Active Mental State Using User-Voluntary Pitch Imagery Task[END_REF][START_REF] Murguialday | Brain-Computer Interface for a Prosthetic Hand Using Local Machine Control and Haptic Feedback[END_REF][START_REF] Omedes | Hierarchical decoding of grasping commands from EEG[END_REF]] [Onaran et al., 2011a[START_REF] Salazar-Ramirez | A hierarchical architecture for recognising intentionality in mental tasks on a brain-computer interface[END_REF]. Moreover, as mentioned in previous chapter, dynamic decoding classifiers highlighted interesting properties for EEG, ECoG and MEA neural signals decoding in several online [START_REF] Darmanjian | Bimodal brain-machine interface for motor control of robotic prosthetic[END_REF]] [Hotson et al., 2016a] [Kao et al., 2017[START_REF] Lisi | Markov Switching Model for Quick Detection of Event Related Desynchronization in EEG[END_REF]] [Millan et al., 2004[START_REF] Moses | Real-time classification of auditory sentences using evoked cortical activity in humans[END_REF][START_REF] Obermaier | Hidden Markov models for online classification of single trial EEG data[END_REF][START_REF] Schwemmer | Meeting brain-computer interface user performance expectations using a deep neural network decoding framework[END_REF] and offline [START_REF] Antelis | Decoding Upper Limb Movement Attempt From EEG Measurements of the Contralesional Motor Cortex in Chronic Stroke Patients[END_REF]] [Bashashati et al., 2017] [Bashashati and Ward, 2017] [J. Cano-Izquierdo et al., 2012[START_REF] Delgado Saa | Hidden conditional random fields for classification of imaginary motor tasks from EEG data[END_REF][START_REF] Dobiáš | Movement EEG classification using parallel Hidden Markov Models[END_REF][START_REF] Du | Decoding ECoG Signal with Deep Learning Model Based on LSTM[END_REF]] [Hasan and Gan, 2011a, 2011b[START_REF] Haselsteiner | Using time-dependent neural networks for EEG classification[END_REF]] [Onaran et al., 2011a[START_REF] Pfeiffer | Hidden Markov model based continuous decoding of finger movements with prior knowledge incorporation using bi-gram models[END_REF][START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF], 2012] [Wang et al., 2011[START_REF] Williams | Markov Model-Based Method to Analyse Time-Varying Networks in EEG Task-Related Data[END_REF][START_REF] Wissel | Hidden Markov Model and Support Vector Machine based decoding of finger movements using Electrocorticography[END_REF][START_REF] Xie | Decoding of finger trajectory from ECoG using deep learning[END_REF] BCI experiments.

In this chapter, a new gating model, named H2M2, inspired by Hierarchical Hidden Markov Model (HHMM) is introduced in order to improve the state transition responsiveness of the classifier during complex tasks and to design a lower-latency classifier than HMM classifier. This decoder is inspired by the natural structure of the movement. The first section reminds the state of the art of dynamic hybrid models and hierarchical classifiers in the BCI research field, whereas the rest of the chapter focuses on the HHMM definition and the description of the new H2M2 gating algorithm.

Hierarchical approach in BCI, and motor BCI

The hierarchical decoders were poorly studied in the BCI field. The few reported hierarchical classifiers were mainly focused on idle state detection for asynchronous BCIs. Indeed, hierarchical BCI decoders were typically organized in a two layer structure which firstly isolated the idle state from the active states and then applied another classifier to select one of the available active state [START_REF] Abascal | A Hierarchical BCI System Able to Discriminate between Non Intentional Control State and Four Intentional Control Activities[END_REF]] [Bashashati et al., 2007b] [Hotson et al., 2016a[START_REF] Jeffries | EEG Classification of Forearm Movement Imagery Using a Hierarchical Flow Convolutional Neural Network[END_REF][START_REF] Kee | A Hierarchical Classification Strategy for Robust Detection of Passive/Active Mental State Using User-Voluntary Pitch Imagery Task[END_REF][START_REF] Murguialday | Brain-Computer Interface for a Prosthetic Hand Using Local Machine Control and Haptic Feedback[END_REF][START_REF] Omedes | Hierarchical decoding of grasping commands from EEG[END_REF][START_REF] Salazar-Ramirez | A hierarchical architecture for recognising intentionality in mental tasks on a brain-computer interface[END_REF].

In particular, a hierarchical linear classifier designed to control the closure of a robotic hand based on EEG mu-band power modulation was reported [START_REF] Murguialday | Brain-Computer Interface for a Prosthetic Hand Using Local Machine Control and Haptic Feedback[END_REF].

In [START_REF] Murguialday | Brain-Computer Interface for a Prosthetic Hand Using Local Machine Control and Haptic Feedback[END_REF], a first classifier discriminated the active and idle states whereas a second one selected between three states (release, maintain, or crush an object in the robotic hand). Similarly, hierarchical architecture was considered for a 5-class problem based on offline EEG neural signal analysis to cluster firstly the idle and action states using unsupervised K-mean algorithm and supervised SVM to distinguish left hand, right hand, tongue or foot imaginary movements [START_REF] Salazar-Ramirez | A hierarchical architecture for recognising intentionality in mental tasks on a brain-computer interface[END_REF].

Other less conventional hierarchical decoder architecture were reported in BCI applications in order to combine binary classifier predictions for multi-states classification problems [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. In [START_REF] Dong | Classification of multi-class motor imagery with a novel hierarchical SVM algorithm for brain-computer interfaces[END_REF], hierarchical SVM algorithm was designed for offline 4-state classification from EEG neural signals. The first layer of the classifier was composed of four "one versus all" SVM algorithms. If no dominant state was apparent, a second layer composed of six "one versus one" SVMs was activated for thinner classification. In [START_REF] Gundelakh | Mobile robot control based on noninvasive brain-computer interface using hierarchical classifier of imagined motor commands[END_REF], online 4-class decoding was reported using EEG recording system. A first layer based on two ANN and two SVM classifiers was applied before to compute a second-classification layer composed of ANN which made the final classification. Finally, 3-class, 4-class, and 5-class offline mental task classifications from EEG signals were tested using Optimal Decision Tree based Support Vector Machine (ODT-SVM) classifier [START_REF] Gupta | A hierarchical meta-model for multi-class mental task based brain-computer interfaces[END_REF]. Based on ECoG dataset, offline classification of finger movements was performed using 10 pair-wise SVM decoders in parallel with hierarchic classification rule [Onaran et al., 2011a].

Dynamic Hierarchical decoders

Dynamic hierarchical decoder family is underrepresented in the BCI field. Only few studies reported the combination of both hierarchical structure and dynamic decoding.

Hotsons' study discriminated flexion of individual finger to control in real-time a hand prosthetic effector from ECoG neural signals of an epileptic patient [Hotson et al., 2016a].

The hierarchical classification was performed with two LDA decoders which classified idle versus movement states and individual finger state detection respectively. The binary movement versus idle LDA classifier was given a first-order Markov chain. Additionally, dynamic hierarchical classifier such as HHMM [START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF][START_REF] Sugiura | A Discriminative Model Corresponding to Hierarchical HMMs[END_REF][START_REF] Suk | Two-Layer Hidden Markov Models for Multi-class Motor Imagery Classification[END_REF] and hierarchical CRF [START_REF] Sugiura | A Discriminative Model Corresponding to Hierarchical HMMs[END_REF] were explored during offline EEG-based motor imagery experiments.

Dynamic hierarchical classifier are more widespread in other fields with more evident data structure such as medical applications with electrocardiography [START_REF] Hu | A Real-Time Cardiac Arrhythmia Classification System with Wearable Sensor Networks[END_REF][START_REF] Liang | A Novel Approach to ECG Classification Based upon Two-Layered HMMs in Body Sensor Networks[END_REF] or muscles recordings classification [START_REF] Malešević | Decoding of individual finger movements from surface EMG signals using vector autoregressive hierarchical hidden Markov models (VARHHMM)[END_REF], motion/gesture/activity recognition [START_REF] Aarno | Layered HMM for motion intention recognition[END_REF][START_REF] Asghari | Online human activity recognition employing hierarchical hidden Markov models[END_REF]] [He et al., 2012[START_REF] Kabir | Two-Layer Hidden Markov Model for Human Activity Recognition in Home Environments[END_REF][START_REF] Kulić | Incremental learning of human behaviors using hierarchical hidden Markov models[END_REF][START_REF] Lee | Layered hidden Markov models to recognize activity with built-in sensors on Android smartphone[END_REF][START_REF] Nguyen | Learning and detecting activities from movement trajectories using the hierarchical hidden Markov model[END_REF][START_REF] Solaimanpour | A layered HMM for predicting motion of a leader in multi-robot settings[END_REF][START_REF] Wei | Layered hidden Markov models for real-time daily activity monitoring using body sensor networks[END_REF][START_REF] Zhu | Online hand gesture recognition using neural network based segmentation[END_REF], handwriting recognition [START_REF] Lee | A Hierarchical HMM Network-Based Approach for On-Line Recognition of Multi-Lingual Cursive Handwritings[END_REF], etc. Some applications rely on the modelling of structured data with different stochastic levels, temporal length scales, complexity and behaviors.

Taking the example of text recognition, punctuation marks, frequent combination of letters, frequent combination of words and endings of phrases may have different time scales and be represented by different temporal models [START_REF] Fine | The Hierarchical Hidden Markov Model: Analysis and Applications[END_REF].

Taking the example of a disabled patient performing a reach-and-grasp task composed of reaching and grasping movements. To complete the task, the following sequential actions from the same body side should be realized: reach the object, open the hand, turn the wrist to match the hand aperture with the object shape and close the hand. During this sequence, the arm, wrist and grasp states of the same body side are more likely to be activated than the arm, wrist and grasp states of the other arm. Such movement behavior may be represented using a classifier with a hierarchical architecture. Such decoder may reduce the number of misclassifications and improve the decoder responsiveness.

Motor cortex activity on the contraparietal side of the intended movement is higher than ipsilateral side which can be considered as a pseudo neurological architecture to exploit. To our knowledge, there is no study which reported an online dynamic adaptive incremental hierarchical decoder.

Based on these considerations, a HHMM-inspired gating classifier referred as H2M2 was designed. H2M2 classifies the left, right body side intended movements and idle states in a first layer before to cluster finer movements in deeper layers. This gating model was created to improve the classification and speed up the decoding transition.

REW-MSLM with HHMM-inspired gating algorithm 6.3.1. Hierarchical Hidden Markov Model (HHMM)

Dynamic decoders (as HMM) describe the extrinsic dynamics of data allowing to model transitions between classes. Hierarchical dynamic decoders represent at the same time the intrinsic structure of each class and their extrinsic dynamic. Hierarchical hidden markov models (HHMM) generalized HMM to a structured multi-level stochastic process [START_REF] Fine | The Hierarchical Hidden Markov Model: Analysis and Applications[END_REF]. Each hidden state is considered to be a self-contained sequential probabilistic model which might generate a sequence of sub-states activation. In other words, each state might activate a sub-HMM with sub-states that might generate another sub-HMM etc. Only specific states or sub-states named production states emit output observations whereas hidden intermediary states (named internal states) are not visible [START_REF] Fine | The Hierarchical Hidden Markov Model: Analysis and Applications[END_REF]. Each sub-HMM has a final state whose activation results in a return to the parent state which activated the sub-model. Transition between states of the same sub-model is referred as horizontal transition whereas diving into lower sub-state or turning back to a parent state is called vertical transition.

Let 𝐷 ∈ ℕ * be the number of hierarchical layers of HHMM, layers are indexed by 𝑑 ∈ {1, … , 𝐷}, 𝐻 𝑑 ∈ ℕ * be the numbers of sub-HMMs of the hierarchical level 𝑑, sub-HMMs of the hierarchical level 𝑑 are indexed by ℎ ∈ {1, … , 𝐻 𝑑 }: HMM ℎ,𝑑 , 𝐾 ℎ,𝑑 ∈ ℕ * be the number of state in each HMM ℎ,𝑑 , 𝑑 ∈ {1, … , 𝐷}, ℎ ∈ {1, … , 𝐻 𝑑 }. States of the HHMM at a hierarchical level 𝑑 in a sub-model ℎ with a sub-index 𝑘 ∈ {1, … , 𝐾 ℎ,𝑑 } are noted as 𝑠 𝑘,ℎ,𝑑 . The set of HMMM states 𝑆 = 𝑆 𝑖𝑛𝑡 ∪ 𝑆 𝑝𝑟𝑜𝑑 is the union of the set of production state 𝑆 𝑝𝑟𝑜𝑑 , and the set of internal states 𝑆 𝑖𝑛𝑡 .

An example of HHMM is shown in Figure 6-1 with 3 layers, 𝐷 = 3, single sub-HMM at the two first layers 𝐻 1 = 𝐻 2 = 1, and two sub-HMMs at the third layer, 𝐻 3 = 2. In this example, the set of production states is 𝑆 𝑝𝑟𝑜𝑑 = {𝑠 1,1,1 , 𝑠 4,1,2 , 𝑠 3,1,3 , 𝑠 1,2,3 } whereas others states are grouped in the set of internal states 𝑆 𝑖𝑛𝑡 .

The next section introduced the variant of Hierarchical HMM structure, referred as H2M2, proposed for hierarchical gating in REW-MSLM algorithm.

General H2M2 parameters description

Similarly to conventional HHMM, each state of H2M2 is considered to be a selfcontained sequential probabilistic model which might generate a sequence of sub-state activations. Each state might activate a sub-HMM with sub-states that might generate another sub-HMM etc. Output observations are only emitted by production states and internal states do not emit visible observation.

Contrary to conventional HHMM, in H2M2, each sub-HMM is independent and change with the brain neural signals. Additionally, every state from the last layer is considered as a production state. Therefore, at each time step, vertical and horizontal transition probabilities inside each sub-HMM are evaluated.

At each time step, one of the production states is activated and emits the output observations. Similarly to conventional single level HMM, active production state is defined by latent variable 𝑧 𝑡 ∈ 𝑆 𝑝𝑟𝑜𝑑 .

For each sub-HMM, denoted as HMM ℎ,𝑑 , 𝑑 is hierarchical level, ℎ is the index of sub-HMM at a given hierarchical level, a state transition probability matrix is noted as 𝐀 ℎ,𝑑 = (𝑎 𝑖𝑗 ℎ,𝑑 ) ∈ ℝ 𝐾 ℎ,𝑑 ×𝐾 ℎ,𝑑 . Here 𝑎 𝑖𝑗 ℎ,𝑑 = 𝑝(𝑠 𝑖,ℎ,𝑑 |𝑠 𝑗,ℎ,𝑑 ) is the probability of making a horizontal transition from the 𝑖 𝑡ℎ state to the 𝑗 𝑡ℎ [START_REF] Fine | The Hierarchical Hidden Markov Model: Analysis and Applications[END_REF]]. Equivalently to HMM, the initial state probability of each state is defined as 𝜋 𝑘,ℎ,𝑑 . Finally, for the production states, {𝑐 𝑠 } 𝑠 ∈ 𝑆 𝑝𝑟𝑜𝑑 is the set of parameters employed to estimate the conditional emission probability of the observed variables 𝑝(𝐗 𝑡 |𝑧 𝑡 ).

At time 𝑡, the active production state is defined by the variable 𝑧 𝑡 ∈ 𝑆 𝑝𝑟𝑜𝑑 . Other states (intern) may be active at precedent layers. A function Ψ: 𝑆 → 𝑆 designed to determine the active inner states in the upper layers leading to the active production state is defined as follows. For every active state 𝑠 𝑘,ℎ,𝑑 at the layers 𝑑 > 1, Ψ(𝑠 𝑘,ℎ,𝑑 ) is the active state at the precedent layer 𝑑 -1 considered in the HMM ℎ,𝑑-1 leading to 𝑠 𝑘,ℎ,𝑑 . For the first layer states: Ψ(𝑠 𝑘,ℎ,1 ) = 𝑠 𝑘,ℎ,1 if 𝑑 = 1. For active production state 𝑧 𝑡 ∈ 𝑆 𝑝𝑟𝑜𝑑 at time 𝑡, the states Ψ(𝑧 𝑡 ) , Ψ 2 (𝑧 𝑡 ) = Ψ(Ψ(𝑧 𝑡 )), Ψ 3 (𝑧 𝑡 ), etc. form a whole set of active sates at time 𝑡:

𝒵 𝑡 = ⋃ {Ψ 𝑗 (𝑧 𝑡 )} 𝐷 𝑗=0 = {𝑧 𝑡 , Ψ(𝑧 𝑡 ), Ψ 2 (𝑧 𝑡 ), … , Ψ 𝐷 (𝑧 𝑡 )}. Here Ψ 0 (𝑧 𝑡 ) = 𝑧 𝑡 .
The set 𝒵 𝑡 describes the path of all the activated intern states leading to the production states 𝑧 𝑡 ∈ 𝑆 𝑝𝑟𝑜𝑑 at time 𝑡. For example, for a production state 𝑧 𝑡 at the third layer, 𝑑 = 3, 𝒵 𝑡 = {𝑧 𝑡 , Ψ(𝑧 𝑡 ), Ψ 2 (𝑧 𝑡 )}, 𝑐𝑎𝑟𝑑(𝒵 𝑡 ) = 3. For a production state 𝑧 𝑡 at the first layer, 𝑑 = 1, 𝒵 𝑡 = {𝑧 𝑡 }, 𝑐𝑎𝑟𝑑(𝒵 𝑡 ) = 1. The function 𝑐𝑎𝑟𝑑() is the function estimating the cardinality of the set 𝒵. The cardinality is defined as the "number of elements" for a finite set.

For H2M2 the activation probability of the production state 𝑧 𝑡 = 𝑠 is defined as :

H2M2 online incremental training

Similarly than the HMM gating parameters training in the section 4.5.2, at each update 𝑢, H2M2 gating parameter estimation is updated based on the update block dataset {𝐗 𝑢 , 𝐳 𝑢 }

where 𝐗 𝑢 ∈ ℝ Δ𝐿×𝐼 1 ×…×𝐼 𝑀 , 𝐳 𝑢 = (𝑧 𝑡 1 , … , 𝑧 𝑡 1 +Δ𝐿 ) 𝑇 ⊂ ℕ * Δ𝐿 and Δ𝐿 the update block size.

H2M2 transition matrix 𝐀 1,1 , 𝐀 1,2 and 𝐀 2,2 are approximated by counting the successive transition of states in 𝐳 𝑢 and the transition matrix estimated during the previous updates weighted with the forgetting factor 𝜇 𝑔 , 0 ≤ 𝜇 𝑔 ≤ 1.

For the matrix 𝐀 1,1 which contains internal (non-production) states 𝑆 𝑖𝑛𝑡 = {𝑠 2,1,1 , 𝑠 3,1,1 }, it is considered that the internal states 𝑠 2,1,1 or 𝑠 3,1,1 are activated and considered for the gating parameter estimation if 𝑠 2,1,1 ∈ 𝒵 𝑡 or 𝑠 3,1,1 ∈ 𝒵 𝑡 respectively.

H2M2 conditional emission probability 𝑝(𝐗 𝑡 |𝑧 𝑡 ) is inferred through the combination of 𝑝(Ψ 𝑗 (𝑧 𝑡 )|𝐗 𝑡 ) and their class prior 𝑝 (Ψ 𝑗 (𝑧 𝑡 )) using Bayes' theorem [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF], with 𝑗 = 0 … 𝐷. Three REW-NPLS discriminative decoders are embedded into the H2M2based gating process to evaluate each sub-model probability . Each sub-model is trained independently on the observation tensor of input variables 𝐗 𝑢 and the latent state dummy variable matrix 𝐙 𝑢 ∈ {0,1} 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )×Δ𝐿 where the column-wise (single) nonzero element depicts the activated state for each sample.

H2M2 conditional emission probability is inferred similarly than HMM in the section 4.5.2. However, instead of calibrating one set of 𝐹 multilinear models θ 𝑔 𝑓 = {𝐁 𝑢 𝑓 , 𝐛 𝑢 𝑓 } 𝑓=1 𝐹 to evaluate the conditional emission probabilities 𝑝(𝐗 𝑡 |𝑧 𝑡 ), a model is evaluated for each sub-HMM: HMM ℎ,𝑑 . In the presented case, a group of three multilinear models are calibrated θ 𝑔 = {θ 𝑔 𝑓,1,1 , θ 𝑔 𝑓,1,2 , θ 𝑔 𝑓,2,2 } (Figure 6-3). The REW-NPLS discriminative algorithm computes the three set of 𝐹 multilinear models θ 𝑔 𝑓,ℎ,𝑑 = {𝐁 𝑢 𝑓,ℎ,𝑑 , 𝐛 𝑢 𝑓,ℎ,𝑑 } 𝑓=1 𝐹 , where 𝐁 𝑢 𝑓,ℎ,𝑑 and 𝐛 𝑢 𝑓,ℎ,𝑑 are the tensor of parameters and related bias of the sub-model with index ℎ in the 𝑑 layer.

The Recursive-Validation procedure selects the best models based on the estimated optimal gating hyperparameter (the latent variable space dimension 𝑓) for each submodel 𝑓 𝑔 ℎ,𝑑 * ≤ 𝐹 and defines the optimal gating sub-models as θ 𝑔 ℎ,𝑑 = {𝐁 ℎ,𝑑 , 𝐛 ℎ,𝑑 } = {𝐁 𝑢 𝑓 𝑔 ℎ,𝑑 * ,ℎ,𝑑 , 𝐛 𝑢 𝑓 𝑔 ℎ,𝑑 * ,ℎ,𝑑 } for dynamic gating weight 𝛾 𝑠,𝑡 estimation.

Figure 6-3: Illustration of the H2M2 sub-models. Each sub HMM: 𝐻𝑀𝑀 ℎ,𝑑 is considered as independent model to evaluate the conditional emission probability for each state of the submodels. The 𝐻𝑀𝑀 ℎ,𝑑 conditional emission probability are estimated based on REW-NPLS algorithms. The optimal latent space dimensions are found for each sub-HMM independently using the Recursive Validation procedure of REW-NPLS algorithm.

Online H2M2 application

The variable 𝛾 𝑠,𝑡 ∈ ℝ determines how likely the production state 𝑠 ∈ 𝑆 𝑝𝑟𝑜𝑑 is generated based on the current observation variable 𝐗 𝑡 . Let define 𝒔 ̂𝑘,ℎ,𝑑 the prediction of the REW-NPLS discriminative sub-models θ 𝑔 ℎ,𝑑 and 𝑝(Ψ 𝑑 (𝑧 𝑡 )|𝐗 𝑡 ) the activation probability of each sub-state in the considered sub-HMM: HMM ℎ,𝑑 . 𝑝(Ψ 𝑑 (𝑧 𝑡 )|𝐗 𝑡 ) is evaluated independently for the three sub-HMM HMM ℎ,𝑑 before to compute 𝑝(Ψ 𝑑 (𝑧 𝑡 )|𝐗 1:𝑡 ) using H2M2 forward algorithm:

𝑗 = 0, … , 𝐷, 𝒔 ̂ℎ,𝑑 = 𝐁 ℎ,𝑑 𝐗 𝑡 + 𝐛 ℎ,𝑑 , 𝑝(Ψ 𝑗 (𝑧 𝑡 ) = 𝑠 𝑘,ℎ,𝑑-𝑗 |𝐗 𝑡 ) = exp (𝑠̂𝑘 ,ℎ,𝑑-𝑗 ) ∑ exp (𝑠̂𝑖 ,ℎ,𝑑-𝑗 ) 𝐾 ℎ 𝑖=1
, 𝑝(Ψ 𝑗 (𝑧 𝑡 ) = 𝑠 𝑘,ℎ,𝑑-𝑗 , 𝐗 1:𝑡 ) = 𝑝(𝐗 𝑡 |Ψ 𝑗 (𝑧 𝑡 ) = 𝑠 𝑘,ℎ,𝑑-𝑗 ) ∑ 𝑎 𝑘,𝑖 ℎ,𝑑 𝑝(Ψ 𝑗 (𝑧 𝑡-1 )|𝐗 1:𝑡-1 )

𝐾 ℎ 𝑖=1 , Conclusion 163
𝑝(Ψ 𝑗 (𝑧 𝑡 ) = 𝑠 𝑘,ℎ,𝑑-𝑗 |𝐗 1:𝑡 ) = 𝑝(Ψ 𝑗 (𝑧 𝑡 ) = 𝑠 𝑘,ℎ,𝑑-𝑗 , 𝐗 1:𝑡 ) ∑ 𝑝(Ψ 𝑗 (𝑧 𝑡 ) = 𝑠 𝑖,ℎ,𝑑-𝑗 , 𝐗 1:𝑡 ) 𝐾 ℎ 𝑖=1
.

Finally, the production state probability 𝛾 𝑠,𝑡 = 𝑝(𝑧 𝑡 = 𝑠 𝑘,ℎ,𝑑 |𝐗 1:𝑡 ) with 𝑠 𝑘,ℎ,𝑑 ∈ 𝑆 𝑝𝑟𝑜𝑑 is evaluated by mixing the sub-state probabilities of the internal states 𝑆 𝑖𝑛𝑡 = {𝑠 2,1,1 , 𝑠 3,1,1 } with the related production states probabilities estimated in the sub-models. Following the equation ( 6.3.1), 𝛾 𝑠,𝑡 can be expressed for each production state as:

𝛾 𝑠 1,1,1 ,𝑡 = 𝑝(Ψ 0 (𝑧 𝑡 ) = 𝑠 1,1,1 |𝐗 1:𝑡 ), 𝛾 𝑠 1,1,2 ,𝑡 = 𝑝(𝑧 𝑡 = 𝑠 1,1,2 |𝐗 1:𝑡 ) = 𝑝(Ψ 0 (𝑧 𝑡 ) = 𝑠 1,1,2 |𝐗 1:𝑡 )𝑝(Ψ 1 (𝑧 𝑡 ) = 𝑠 2,1,1 |𝐗 1:𝑡 ), 𝛾 𝑠 2,1,2 ,𝑡 = 𝑝(𝑧 𝑡 = 𝑠 2,1,2 |𝐗 1:𝑡 ) = 𝑝(Ψ 0 (𝑧 𝑡 ) = 𝑠 2,1,2 |𝐗 1:𝑡 )𝑝(Ψ 1 (𝑧 𝑡 ) == 𝑠 2,1,1 |𝐗 1:𝑡 ), 𝛾 𝑠 1,2,2 ,𝑡 = 𝑝(𝑠𝑧 𝑡 = 1,2,2 |𝐗 1:𝑡 ) = 𝑝(Ψ 0 (𝑧 𝑡 ) = 𝑠 1,2,2 |𝐗 1:𝑡 )𝑝(Ψ 1 (𝑧 𝑡 ) = 𝑠 3,1,1 |𝐗 1:𝑡 ), 𝛾 𝑠 2,2,2 ,𝑡 = 𝑝(𝑧 𝑡 = 𝑠 2,2,2 |𝐗 1:𝑡 ) = 𝑝(Ψ 0 (𝑧 𝑡 ) = 𝑠 2,2,2 |𝐗 1:𝑡 )𝑝(Ψ 1 (𝑧 𝑡 ) = 𝑠 3,1,1 |𝐗 1:𝑡 ).

H2M2 gating integration in REW-MSLM

The dynamic gating introduced with REW-MSLM is replaced by a dynamic gating evaluated with the H2M2 algorithm. Therefore, 𝐘 𝑡 is estimated as follows:

𝐘 ̂𝑡 = ∑ 𝛾 𝑠,𝑡 (𝐁𝐞𝐭𝐚 𝑠 𝐗 𝑡 + 𝐛𝐢𝐚𝐬 𝑠 ).

𝑠∈𝑆 𝑝𝑟𝑜𝑑

Here, 𝐁𝐞𝐭𝐚 𝑠 and 𝐛𝐢𝐚𝐬 𝑠 are the expert tensor parameters related to the production state 𝑠 ∈ 𝑆 𝑝𝑟𝑜𝑑 and its associated bias. 𝛾 𝑠,𝑡 = 𝑝(𝑧 𝑡 = 𝑠|𝐗 1:𝑡 ) is the dynamic gating weight coefficient at time 𝑡 of the expert assigned to the production state 𝑠 ∈ 𝑆 𝑝𝑟𝑜𝑑 . REW-MSLM models are entirely defined through the experts' parameters θ 𝑒 = {𝐁𝐞𝐭𝐚 𝑠 , 𝐛𝐢𝐚𝐬 𝑠 } 𝑠∈𝑆 𝑝𝑟𝑜𝑑 and the H2M2 models' parameters θ 𝑔 .

As the application and the incremental training of the expert models are strictly identical than the training procedure described in section 4.5, experts evaluation is not detailed in the following section which are more focus on the evaluation of the HHMM parameters.

Conclusion

In this chapter, a new gating decoder has been introduced to improve the decoding performance of the REW-MSLM gate in the case of asynchronous complex state experiments using the natural prior knowledge related to movement discrete state sequences. The new gating model inspired by the Hierarchic Hidden Markov Model (HHMM) relies on the natural movement structure to improve the state classification and recognition. Instead of modelling the state probability with one model, state probability estimation is divided into sub-models to represent more accurately the different stochastic levels and time scales of the brain neural signals. This new gating model is based on the combination of adaptive incremental linear models. H2M2 was designed for real-time calibration and application during online closed-loop experiments.

The performance of the algorithms were evaluated online with multiple closed-loop experiments performed during the CLINATEC clinical trial or using pseudo-online simulations using the dataset recorded during previous closed-loop experiments. All the experiments were designed to stress the asynchronous multi-limb decoding performance and stability of the designed algorithms compared to other decoders based on chronic ECoG neural signals. This chapter firstly presents the BCI platform set up used during the experiments. Then it introduces the data analyzed in this study.

Recording set up

WIMAGINE is an active implantable medical device able to record epidural ECoG on 64 electrodes. The digitized epidural ECoG data were radiotransmitted to a custom designed base station connected to a computer [START_REF] Mestais | WIMAGINE: Wireless 64-Channel ECoG Recording Implant for Long Term Clinical Applications[END_REF]. During the experimental sessions, 32 electrodes for each implant were selected in a checkerboard pattern because of limited data rates, caused by restricted radio link [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]] (Figure 7-1). Epidural ECoG signals were recorded at a sampling rate of fs= 586 Hz.

Figure 7-1: WIMAGINE is an active implantable epidural ECoG recording system composed of 64 electrodes. Two WIMAGINE were implanted above specific region of the patient's motor cortex. Due to limited data rate, half of the electrodes for each implant were selected in a checkerboard pattern to cover a large brain area. The selected electrodes for the neural signal decoding experiments are colored in green whereas not recorded ones are shown in purple. The same electrodes were selected for all the experiments presented in this manuscript.

Effector control

During laboratory experiments, the patient was strapped into the enhancing mobility (EMY) exoskeleton. A computer station receiving ECoG radio-emitted signals was embedded in the back of the exoskeleton. The neural signals samples were decoded and translated into incremental end-point-control commands through the decoding software. Finally, the control commands were converted into joints movement by the exoskeleton control system activating the limbs and producing the appropriate movements. The virtual avatar was a virtual replica of the exoskeleton and was used for home-based training. For both laboratory and home-based experimental sessions, the patient was sitting down in the exoskeleton or his wheelchair respectively. During the laboratory experiments, the LED panel was placed in front of the patient to provide him the task instructions. At home, a television broadcasted the instructions and the virtual avatar movements. The virtual environment and avatar were presented with a first person view. The task success feedback was provided to the patient. During laboratory experiments, the lightened LEDs showing the target to reach were manually switched off by the experimenter when the task was completed. For wrist rotation tasks, clockwise or counter clockwise successive flashing LEDs informed the patient with the task to achieve. Flashing LEDs were manually turned off when the task was completed. During the experiments with the virtual environment, the target to reach during an arm translation task automatically turned from red or blue colour (left or right arm) to green colour when the task was completed. During a wrist rotation task continuous feedback was provided to the patient through a gauge which turned green for successful tasks. The patient was allowed to move and talk freely during the training and test experiments in order to create models robust artefacts related to muscular activities.

Experiments design

Numerous experiments with various tasks were carried out during the "BCI and Tetraplegia" clinical trial. This section is only focused on the experiments performed within the framework of the PhD research.

The experiments studied in the manuscript were performed between March 5 th , 2018 and January 17 th , 2020. Each experiment/session was composed of successive tasks decided by an experimenter. All the experiments related to the PhD thesis were asynchronous alternative multi-limb/bimanual experiments and relied on the same experimental structure. Each task corresponded to one of the available state between the idle state (IS) and the active states (AS). During IS, no target was presented to the patient. The patient had to remain in a non-active state until a new task started. AS tasks regroup all the intended movement tasks that should be performed by the user. Depending on the experiments, diverse AS tasks with various complexity (controlled dimension) were proposed and executed. The AS tasks controlled in the experiments presented in this manuscript were the translation of the left (ASLH) and right (ASRH) hand in the 3D space and the 1D angular rotation of the left (ASLW) and right (ASRW) wrist (Figure 7-2). Each task was made of several successive trials where the patient attempted to reach a target location set sequentially with the left or right hand or to rotate the left or right wrist until a given angle. During a session, the hand position was not reset by the system between state, task and trials. For a given AS, the starting position of the hand for a trial was the position of the hand at the end of the previous trial of the same AS. An example of a session with three tasks IS, ASLH and ASRH and 4D continuous decoding (2D for each AS) is shown in Figure 7-3. In the Figure 7-3, the AS tasks are represented in 2D space during an alternative 2D left and right hand translation experiment for ease of understanding but the same experimental paradigm can be generalized to 3D space environment and tasks requiring 3D movements. Seven trials from two non-consecutive ASLH are represented with the position of the cursor at the beginning of the second ASLH task equal to the position of the cursor at the end of the first ASLH task. . Each active task is composed of several trials in which the 2D cursor must reach the proposed targets. The index of the AS tasks is noted with a superscript index, the first ASLH task is noted ASLH 1 whereas the second is referred as ASLH 2 . The cursor position is not reset between tasks, during task and during idle state. The last cursor position is not reset and correspond to the position of the cursor before the patient changes its mental state and the model decodes the state transition.

The session example depicted in Figure 7-3 is referred as an asynchronous alternative bimanual point-to-point pursuit experiments. A point-to-point pursuit task is more complicated than the commonly reported center-out experiment which reset the cursor to an initial stating position at the end of each trial. Pursuit task is characterised by a better exploration of the control space with multiple possible starting points.

All the closed-loop experiments presented in the manuscript were based on this experimental paradigm (in the 3D space). During a session, the patient aimed to reach the proposed targets or rotate the wrist to specific angle by controlling an exoskeleton or a virtual avatar. 22 targets were 3D symmetrically distributed in two cubes in front of the patient with 11 targets per hand for the exoskeleton reaching tasks and were virtually reproduced for the avatar training and testing sessions.

Dataset specification

All the experiments were based on multiple alternative pursuit tasks sessions. However, several specific experiments were designed to highlight the benefits of the proposed algorithms. This section introduced all the experiments used for offline and online performance evaluation. During these experiments, an additional specification to the experimental paradigm was added. Before any transition from an AS task to another AS task (not between trials from the same AS task), an IS task was always imposed. This paradigm forced the patient to control with high accuracy the idle state and enhanced the asynchronous characteristic of proposed BCI experiments. All the achieved BCI experiments were in closed-loop sessions. Therefore for each dataset, the online decoder used to decode the patient's neural signals during the online closed-loop experiments is specified.

Online closed-loop 6D experiments using REW-NPLS decoder

Alternative bimanual 3D left and right hand translation pursuit tasks using the virtual avatar was achieved during the intermediate stage of the clinical trial. During these experiments, the patient controlled in real-time 6 dimensions (6D) clustered in 𝑧 = 3 states: idle (IS), left (ASLH) and right (ASRH) hand control states using the REW-NPLS incremental adaptive decoder. All the experiments were closed-loop sessions recorded between March and June 2018. Three different training/testing paradigms were appraised during these online closed-loop experiments.

First, the series of sessions titled A (𝑛 = 5) was carried out to evaluate the performance of the algorithm with a small training dataset. The sessions from the series A were selfcontained experiments. The decoding models were independently created (initialized to zero), trained and tested during the same experiment. The sessions of the series B (𝑛 = 4) were used to evaluate the importance of cross-session training. The models were initialized to zero in the first session. Then, the models created during the previous sessions were used to initialize the model parameters of the next experiment. Finally, the experiments of the C series (𝑛 = 5) were performed to evaluate the robustness of a model calibrated using cross-session training. The experiments from C series were carried out from 9 days to 28 days after the last experiments of the series B and thus, after the last model calibration. The model estimated during the last experiment of the B series was used as neural signal decoder in the C series. A representation of the three training/testing paradigms is shown in Figure 7-4. The registered dataset of the three series A, B, C were used in order to perform further algorithms performance comparison in pseudo-online studies. Particularly, these series were used to compare the decoding performance of the REW-NPLS algorithm and REW-MSLM.

Online closed-loop 8D experiments using REW-MSLM decoder

The patient performed real-time asynchronous closed-loop 8D experiments using the REW-MSLM incremental adaptive closed-loop decoder. The session clustered 3D alternative two-handed reaching tasks (ASLH and ASRH), 1D wrist rotation movements for each hand (ASLW and ASRW) and idle state (IS) for a total of 𝑧 = 5 states and 8 continuous dimensions [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]. The number of experts was fixed to N=2 with one expert associated to left body-side limb decoding whereas the other estimated the right body-side limb model. The hand and wrist continuous movements from the same body side were decoded in the same expert.

This 8D experiment paradigm was achieved using both the virtual avatar and the exoskeleton effector for the sessions at home and inside the laboratory, respectively. 8D experiments with virtual avatar or exoskeleton control were performed independently.

The models created with one effector were not used to control the other one. The dataset obtained from the virtual avatar control are referred as series D of experiments whereas dataset with exoskeleton effector control is named series E of experiments.

During a session, the patient aimed to reach the proposed targets or rotate the wrist to specific angle. 22 targets were 3D symmetrically distributed in two cubes in front of the patient (11 targets per hand) for the exoskeleton reaching tasks and were virtually reproduced for the avatar training sessions (Figure 7-5). Sessions last in average 29 ± 8 min and 20 ± 6 min using the virtual avatar or the exoskeleton effector, respectively.

For the exoskeleton, a REW-MSLM decoder was recursively trained in real-time during 6 closed-loop experiments distributed over 2 months and was not reupdated since. For the virtual avatar control, 6 closed-loop experiments were achieved in late September for incremental real-time REW-MSLM adaptation. The total calibration time of the models for virtual avatar was 3 hours and 37 minutes with a total of 189, 194, 181 and 218 trials for the left and right hand translation and left and right wrist rotation tasks, respectively. 3 hours and 33 minutes calibration time was performed to train the model dedicated to the exoskeleton control with a total of 180, 184, 188 and 226 trials for the left and right hand translation and left and right hand rotation control.

The performance of the models were evaluated during 37 avatar experiments distributed over 5 to 203 days after the last model update session and 468 to 666 days after the recording system implantation. For the exoskeleton control sessions, 15 test experiments distributed over 0 to 167 days after the last model update session and 531 to 698 days after implantation were performed. Five exoskeleton experiments conducted between the 62 nd and 63 rd days were excluded due to patient health issues unrelated to the study. Timeline representing the model calibration and test during the virtual avatar and exoskeleton based experiments across time is shown in Figure 7-5. The real-time closed-loop experiments with the avatar and the exoskeleton were achieved to evaluate the performance of the REW-MSLM algorithm with an online CLDA protocol during real-time experiments. The dataset D using the virtual avatar effector is composed of 43 experiments including the calibration and test sessions. Pseudo-online studies using the same procedure than during the online closed-loop 8D virtual avatar control experiments (Pre-processing, buffer size, batch training, number of training experiments, etc.) were carried out to evaluate the performance of the PREW-NPLS and APREW-NPLS algorithms.

Online closed-loop full-state transition 4D experiments using REW-MSLM decoder

The real-time closed-loop experiments described in 7.4.1 and 7.4.2 were alternative pursuit tasks from AS states with mandatory IS transition between each AS states. Nevertheless, for daily life application, the subject can sequentially switch from one active state to another without idle state transition multiplying the possible state transitions.

New online adaptive closed-loop experiments were recorded using the virtual avatar effector between late October 2019 and mid-January 2020. The patient controlled 1D continuous movements of the left and right hands translation (vertical displacement) and 1D left and right wrists rotation tasks. This paradigm lead to a 4D continuous and 𝑧 = 5 discrete states control problem (IS, ASLH, ASRH, ASLW, ASRW). The model was

To evaluate the performance of the proposed online closed-loop adaptive REW-MSLM decoder, it is mandatory to perform online closed-loop BCI experiments. Therefore, REW-MSLM was integrated into the software chain inside the Adaptive Brain Signal Decoder (ABSD) software for real-time decoding and online incremental CLDA. The BCI platform is presented in Figure 8-1. The ABSD software processes the ECoG neural signals in order to create a command to an effector (virtual avatar, exoskeleton, etc.). ABSD performs the classic signal processing steps (pre-processing, feature extraction, decoding, post-processing). The main loop achieved ECoG neural signal decoding at a 10 Hz frequency rate. Additionally, in parallel to the decoding main loop, ABSD was designed to run a second loop which executes the incremental batch update of the decoder at an update frequency rate fixed between 0.066Hz and 0.1Hz (every 10 to 15 seconds).

The following section introduced the signal processing steps performed in the main loop of the ABSD software before to describe the integration of REW-MSLM decoder and the interaction between the application and update loops to carry out incremental online CLDA. In the PhD research, ABSD and REW-MSLM algorithm were integrated into a DELL 7810 computer with Windows 10 operating system, Intel Xeon E5-2637 V3, 3,5GHz (2 processors) and 64 Gb of RAM.

Pre-processing

The pre-processing steps are directly integrated inside the implantable wireless recording system WIMAGINE. WIMAGINE is an active implantable medical device able to record epidural ECoG on 64 electrodes.

The ECoG signals were low and high pass filtered in a bandwidth from 0.5Hz to 300Hz, amplified and cleaned thanks to filters embedded into the implant hardware [START_REF] Mestais | WIMAGINE: Wireless 64-Channel ECoG Recording Implant for Long Term Clinical Applications[END_REF]. The digitized epidural ECoG data were radiotransmitted to a custom designed base station connected to a computer [START_REF] Mestais | WIMAGINE: Wireless 64-Channel ECoG Recording Implant for Long Term Clinical Applications[END_REF]. epidural ECoG neural signals from 32 electrodes were recorded at a sampling rate of fs= 586 Hz. Data were sent to the decoding software in batch of 100 ms. Finally, before the feature extraction step, aberrant values were removed from the dataflow at time 𝑡 and replaced by the recorded value at the precedent time step 𝑡 -1. Aberrant values clustered non-numeric values or artefacts.

If the neural signal variation between the time step 𝑡 -1 and 𝑡 were above a fixed threshold, the neural signals at time 𝑡 were labeled as artefacts and replaced by the neural signal at time 𝑡 -1. BCI Adaptive platform

Feature extraction

During the experimental sessions, at each time step 𝑡, the epidural ECoG neural signal epochs for all the electrodes, 𝐗 𝑡 ℝ 586x64 , were generated using a ∆𝑡 = 1 s window with a 100 ms sliding step [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. ECoG epochs were mapped to the temporal frequency space using complex continuous wavelet transform (CCWT) (Morlet) with a The REW-MSLM gate and expert models were estimated using a supervised learning strategy. Supervised learning CLDA required the neural signals and associated labels for the estimation of both the gate and expert models. Therefore, continuous and discrete output features were estimated.

The movement output features dedicated to asynchronous multi-limb control at the time 𝑡 were characterized by the optimal continuous movement 𝐲 𝑡 and the discrete state label Similarly to the calibration hypothesis introduced by Gilja [START_REF] Gilja | A high-performance neural prosthesis enabled by control algorithm design[END_REF], the assumption that the intended movement of the patient always followed the most efficient trajectory towards the target was made. Thus, 𝐲 𝑡 𝐿𝑡𝑟 and 𝐲 𝑡 𝑅𝑡𝑟 were defined as the 8-2) and the optimal kinematic features 𝒚 𝑡 . The optimal kinematic features 𝒚 𝑡 were defined as the 3D Cartesian vector between the current position and the target position for the 3D hand translation and as 1D angular vector between the current angle and the target angle for 1D wrist rotation. The discrete state labels was noted 𝑧 𝑡 . The prediction from the current model 𝒚 ̂𝑡, the optimal prediction 𝒚 𝑡 according to the current position and the associated state 𝒛 𝑡 were recorded as movement features. 𝐗 𝑡 , 𝒚 𝑡 and 𝒛 𝑡 were stored in a buffer until the next update (every 15s) to update the REW-MSLM decoder.

𝑧 𝑡 [1; 𝐾] ⊂ ℕ * ,

Post-processing

Speed limit post-processing was applied to 𝐲 𝑡 𝑎𝑠𝑠𝑖𝑠𝑡 . If the Euclidian norm of 𝐲 𝑡 𝑎𝑠𝑠𝑖𝑠𝑡 was above a defined speed limit threshold, the command sent to the effector was limited to:

𝐲 𝑡 𝑒𝑓𝑓 = 𝑚𝑖𝑛(𝑆𝑝𝑒𝑒𝑑𝐿𝑖𝑚𝑖𝑡, ‖𝐲 𝑡 𝑎𝑠 ‖ ) 𝐲 𝑡 𝑎𝑠 ‖𝐲 𝑡 𝑒𝑓𝑓 ‖ .
𝑆𝑝𝑒𝑒𝑑𝐿𝑖𝑚𝑖𝑡 is an experiment fixed threshold and ‖. ‖ is the Euclidian norm.

Finally, the decoded incremental endpoint-control commands 𝐲 𝑡 𝑒𝑓𝑓 were converted into joints movement by exoskeleton control system activating the limbs and producing the appropriate movements through the activation of the motors of the exoskeleton or the displacement of the virtual avatar.

8.4. CLDA procedure integration in the BCI adaptive platform 8.4.1. Features labelling for CLDA.

In order to perform online decoding with online CLDA, the main application loop for the online decoding and the adaptation loop for the update of the REW-MSLM submodels were split and implemented in two independent processes while communicating through shared memory.

The REW-MSLM decoder is structured with a discrete gating model and several continuous expert models. All the models are independently and incrementally updated in real-time on different batch of data.

During the online closed-loop experiments, in order to incrementally update the REW-MSLM decoder, the input and output features were stacked in buffers before to be sent to the ABSD second loop for gate and expert models update. To perform the incremental batch learning of the gate and expert models, the data were stored in buffers. In the considered application case, the buffer was defined as a memory storage used to temporarily store data while it is being moved from the application loop to the adaptation loop. The process is represented in Figure 8-4.

In The size of the gate buffer was fixed to cluster 150 samples before to launch the model calibration procedure.

The filling of the gate buffer operated as an updating threshold to start the calibration procedure. If the k th expert buffer stacked more than 150 samples when the gate buffer was full, the corresponding expert model was updated in the same time than the gate model (Figure 8-4B). The gate model buffer acted as a computer clock cycle, the buffer was full every 150 samples and was updated with an expert if the expert buffer collected a sufficient amount of data. The full buffers (the buffer gate and optionally the dataset recorded for an expert if enough data were collected) were sent to the ABSD calibration loop in order to achieve the gate model update and optionally one of the expert model update. After the update of the models on the new batch of data, the calibration loop transferred the updated models to the ABSD main loop in order to apply the updated gate and optionally expert models to the incoming neural signals.

REW-MSLM initialization.

REW-MSLM has a specific architecture with independent experts which estimate continuous outputs before to be enhanced or inhibited by the gating classifier predictions. As mentioned in previous chapters, each REW-MSLM (gate and experts) can be initialized from scratch or initialized from a previously trained model. However, a REW-MSLM dedicated to a specific experimental paradigm can also be initialized from several REW-MSLMs previously trained on other different experimental paradigms. For ease of understanding, an example is shown in Figure 8-5.

Let's consider a first REW-MSLM model trained on an asynchronous multi-limb experimental paradigm referred as 𝑃𝑟𝑑𝑔 1 composed of three tasks (IS, ASLH, ASLW) with one expert associated to each task and a second REW-MSLM calibrated with another experimental paradigm, titled 𝑃𝑟𝑑𝑔 2 , including four tasks (IS, ASRH, ASLW, ASRW) with one expert trained for each task.

In order to create a REW-MSLM for a new paradigm 𝑃𝑟𝑑𝑔 3 clustering previously trained tasks (IS, ASLH, ASRH, ASLW, ASRW), the expert sub-models of the REW-MSLM dedicated to 𝑃𝑟𝑑𝑔 3 can be initialized using the expert models from the 𝑃𝑟𝑑𝑔 1 and 𝑃𝑟𝑑𝑔 2 .

In the example presented in Figure 8 In order to compare the decoder performance, an offline pseudo-online comparative study was undertaken before to integrate REW-MSLM into the BCI platform. Pseudoonline experiments are offline simulations conducted using the same parameters as those used for the online experiments. Pre-processing, buffer size, batch-wise training and application of the model are performed following the same procedure as that used for online real-time experiments to reproduce the online experiment conditions. Pseudoonline comparison is not fully generalizable for the online case. Nevertheless, it allows characterising the studied algorithms before an integration into the clinical BCI decoding platform. The REW-MSLM is a hybrid decoder which mixes discrete and continuous decoding. Therefore, evaluations of the performance related to the classifier (the gate) and the regression algorithms (the experts) must be carried out.

The first section of this chapter introduces the performance evaluation procedure followed to evaluate the REW-MSLM performance during offline studies and online closed-loop experiments. Then, the performance evaluation procedures of the new decoders PREW-NPLS, APREW-NPLS and the new classifier H2M2 are described.

REW-MSLM performance evaluation

REW-MSLM offline comparative study and online evaluation

Before to be evaluated during real-time closed loop experiments, REW-MSLM performance evaluation was achieved during offline studies. The datasets used for the pseudo-online comparison were recorded during online closed-loop experiments using REW-NPLS algorithms previously performed in the clinical trial (dataset A, B and C). REW-MSLM and the decoders used for performance comparison were recomputed in pseudo-online manner. Three key features of the REW-MSLM were evaluated.

Firstly, REW-MSLM integrates a gate which discriminates several neural states to weight the experts' output. We first highlighted the benefit of integrating discrete multi-state classifier for an asynchronous multi-limb control paradigm to switch between active states (AS) and handle robust idle state (IS) support. REW-MSLM performance comparison was focused on the REW-NPLS algorithm because this algorithm is a state-of-the-art online adaptive tensor-input tensor-output algorithm which have been previously employed during online and offline clinical and preclinical ECoG-based BCI experiments [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]] [Eliseyev et al., 2017]. Furthermore, as described in the previous chapters, REW-MSLM was designed based on REW-NPLS decoder. Therefore, it is relevant to evaluate the performance differences between both algorithms. To clarify the importance of the REW-MSLM ME model structure, which combines a classifier (state classification) with a continuous decoder, and the importance of dynamic vs. static gating, the REW-MSLM was compared to the state-of-the-art adaptive algorithms with three database using simulated pseudo-online experiments. The series A evaluated the performance of the algorithms with short training dataset. Then, the series B was carried out to stress the cross-session training. Finally, the C series provided information on the models robustness across time.

Although offline pseudo-online studies give an initial overview of the potential REW-MSLM decoding performance and benefits, they are not generalizable. No definitive conclusion can be extracted from these studies. Online experiment is the only solution to appraise the model robustness and to analyse the neural signal patterns modulation of the patient. Therefore, online closed-loop experiments integrating REW-MSLM as neural signal decoder were achieved. Here, the weighting coefficient 𝛽 was set to one, the true positives 𝑡𝑝 𝑘 were considered for samples labelled as belonging to state 𝑘, and the true negatives 𝑡𝑛 𝑘 included those from all the other states (one versus all analysis). 𝐾 was set to 𝐾 = 3 for the pseudo-online comparative study, and 𝐾 = 5 for the 8D online experiments.

Accuracy is a commonly reported indicator in the BCI for binary and multi-state classification [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF]] [Hotson et al., 2016a] [Nguyen et al., 2019] [Schaeffer and Aksenova, 2016b] [Vidaurre et al., 2006b] and is useful for performance comparison due to its ease of computation and interpretation. Nevertheless, as accuracy presents weaknesses in analysis the case of highly unbalanced class, F1-score was also computed to evaluate classification performance.

The previously described state decoding indicators are sample-based performance estimators. They do not reflect the dynamic behaviour of the misclassified samples. Therefore, supplementary indicators were introduced to quantify the performance of the multi-state classification (Figure 9-3). First, the latency between the instruction and the estimated state transition was computed to evaluate the combined response time of the patient and the model. The estimated state transition was considered valid only when the decoded state was stable for 1s (10 samples). The transition had to be achieved in the 5s following the instruction state transition for it to not be counted as an incorrectly labelled state. Samples belonging to the transition/latency period were not considered in the other discrete performance indicators. Finally, the block of errors defined as consecutive misclassified samples were counted (Figure 9-3B) to evaluate the block error rate determined as the number of error blocks divided by the length of the experiments. The block error rate was then converted in averaged error blocks per minute. The averaged duration of the block of errors (Figure 9-3C) was also evaluated. In this PhD study, it was considered that several consecutive misclassification of the same class was potentially less disturbing than badly labelled samples switching at a high frequency which may lead to jerky effector command. Significance of the differences between the three decoders were computed for datasets A and C. Significance analysis on the B series was excluded because of the low number of sessions (n=4). The Mann-Whitney U test with Bonferroni corrections (𝛼 𝑚𝑢𝑙𝑡𝑖-𝑐𝑙𝑎𝑠𝑠 = 0.0167) was computed in the multi-class comparisons. Otherwise, 𝛼 = 0.05.

Continuous performance indicators

As mentioned above, the predicted trajectories performed during the online closed-loop experiments are related to the decoding model currently used during the experiments and the patient's feedback. Therefore, trajectory decoding performance indicators cannot be used to evaluate the performance of different algorithms in pseudo-online experiments. A sample-based indicator is introduced to compare the continuous predictions of several algorithms (Figure 9-4A). The dot product indicator 𝐷𝑜𝑡𝑃, known in other field as the cosine similarity, is based on the comparison between the predicted directions 𝐲 ̂𝑡 and the optimal prediction defined as the 3D Cartesian vector between the current position and the target 𝐲 𝑡 for 3D translation tasks using the scalar product. After normalization:

𝐷𝑜𝑡𝑃 = 1 𝑇 ∑ 𝐲 𝑡 • 𝐲 ̂𝑡 ‖𝐲 𝑡 ‖‖𝐲 ̂𝑡‖ 𝑇 𝑡=1 ,
where " • "defined the dot product, 𝐷𝑜𝑡𝑃[-1,1], 𝑇 is the number of samples recorded for a specific limb (right or left hand). The average dot product over time provided an indicator of the algorithm global static prediction. To our knowledge, this indicator was only referenced in three articles in which EEG neural signals were analysed [START_REF] Olcay | Evaluation of synchronization measures for capturing the lagged synchronization between EEG channels: A cognitive task recognition approach[END_REF][START_REF] Rashid | An EEG Experimental Study Evaluating the Performance of Texas Instruments ADS[END_REF]] [Xu et al., 2019]. This indicator was often used in the information retrieval, text mining and data mining fields [START_REF] Rani | Perspectives of the performance metrics in lexicon and hybrid based approaches: a review[END_REF][START_REF] Schenker | Comparison of Distance Measures for Graph-Based Clustering of Documents[END_REF][START_REF] Umakanth | Classification and ranking of trending topics in twitter using tweets text[END_REF]. target is considered as reached if the cursor is inside a sphere with the target coordinates as center and 2.5cm as radius. The R-ratio is the ratio between the distance travelled by the cursor during the task and the minimal distance travelled to reach the target.

Online performance indicators

Discrete performance indicators

For the online discrete performance, the same static indicators than the one computed for the pseudo-online studies were computed. The accuracy (𝑎𝑐𝑐) and F-score (𝑓𝑠𝑐) indicators are defined by the equations (9.1 and (9.2 respectively.

Continuous performance indicators

For the evaluation of the online closed-loop experiment performance, the success rate (SR) [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF] set as the percentage of targets hit, and the R-ratio [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF], defined as the ratio between the distance travelled by the effector to reach a target and the distance from the initial position of the effector to the target location were computed (Figure 9-4B). R-ratio [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF] was also named distance ratio in [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] and was equivalent to the inverse of the individual path efficiency defined in [START_REF] Collinger | High-performance neuroprosthetic control by an individual with tetraplegia[END_REF][START_REF] Wodlinger | Tendimensional anthropomorphic arm control in a human brain-machine interface: difficulties, solutions, and limitations[END_REF] for each task. The SR and R-ratio performance indicators were defined in the same way for the evaluation of the wrist rotation performance. The target was considered to have been hit when the 1D angular vector between the wrist position and the target was null.

Stability indicators

Additionally, the evolution of the performance indicators across experiments (across time) was evaluated. The linear fit with a 95% confidence interval was estimated for each indicator testing the zero slope hypothesis and evaluating the performance stability of the REW-MSLM decoder across time.

Chance level comparative study

An additional study was performed to evaluate the discrete and continuous performance indicators chance level during online closed-loop 8D experiments. Discrete states are not uniformly distributed, with a higher prior probability for idle and hand translation tasks than wrist rotations tasks. During exoskeleton-based experiments, idle, left and right hand, left and right wrist states represented 26%, 36%, 27%, 6%, 5% of the discrete state distribution, respectively. For the SR and R-ratio, 𝑛 = 100 random hit simulations were repeated. Simulation of random movement reaching tasks were performed with the same target locations as those used during the exoskeleton-based experiments. A 3D randomly moving cursor must reach a randomly selected target within a fixed duration (defined as 99% of the cumulative distribution of the experimental time used during the exoskeleton-based experiments). At each time step, the cursor moved in a random direction with a speed fixed to the maximal speed of the exoskeleton. The target was considered to have been reached if the distance between the cursor and the target was less than 5 cm. These random sessions highlighted an averaged SR of 7.1 ± 5.5% (R-ratio: 24 ± 14) for the left hand translation, 9.5 ± 6.6% (R-ratio: 33 ± 19) for the right hand translation, 40 ± 7.1% (R-ratio: 15 ± 4.6) for the left hand rotation and 33 ± 4.9% (R-ratio: 12 ± 2.7) for the right hand rotation tasks.

Decoding model influence analysis

In order to evaluate the parameter estimated by the REW-MSLM algorithm which had the most influence on the neural signal decoding, an analysis of the experts and gating models was carried out.

The gating emission probability model is defined by the couple {𝐁, 𝐛} where 𝐁 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 ×𝐾 and 𝐛 ∈ ℝ 𝐾 are the tensor of the model parameters and its related bias, 𝐼 𝑖 ∈ ℕ * with 𝑖 ∈ {1,2,3} are the tensor dimensions and 𝐾 ∈ ℕ * is the number of possible states to predict. In the specific case of the experiments proposed in the PhD manuscript, 𝐼 1 = 10, 𝐼 2 = 15 and 𝐼 3 = 64 are the temporal, frequency and spatial neural feature dimensions and, generally, 𝐾 = 5 (IS, ASLH, ASLW, ASRH, ASRW). The influence of the gating model parameter weights for the activation of each state on the temporal, frequency or spatial modality was estimated following the equation:

𝑃𝑟𝑜𝑗(𝐼 1 , 𝑖, 𝑘) = ∑ ∑ 𝑎𝑏𝑠(𝑏 𝑖𝑗𝑙𝑘 ) 𝐼 3 𝑙 𝐼 2 𝑗 ∑ ∑ ∑ 𝑎𝑏𝑠(𝑏 𝑖𝑗𝑙𝑘 ) 𝐼 3 𝑙 𝐼 2 𝑗 𝐼 1 𝑖 .
Here, 𝑃𝑟𝑜𝑗(𝐼 1 , 𝑖, 𝑘) is considered as the influence of the i th (𝑖 ∈ {1, … , 𝐼 1 }) parameter weight on the first dimension (the size of the dimension is 𝐼 1 ) for the 𝑘 𝑡ℎ expert.

Similarly, the influence of the expert model parameters on the continuous predictions of the REW-MSLM was estimated for each expert on each modality. The proposed decoding model influence analysis allowed to provide an interpretation of the REW-MLSM parameter weights. However, this analysis had some limitations. 𝑃𝑟𝑜𝑗(𝐼 1 , 𝑖, 𝑘) is computed based on summation of the absolute value of the REW-MLSM parameter weights which does not allow to conclude on the positive or negative influence of each parameter weight on the predictions.

Neural signal modulation analysis

Different mental tasks may lead to different neural activation. The averaged timefrequency responses during the attempted discrete state activation was evaluated for each electrode using the online 8D closed-loop experiments performed with the exoskeleton. The neural activity modulation related to left and right hand translation state activations and left and right wrist rotation state activations (ASLH, ASLW, ASRH, ASRW) were compared to the averaged time-frequency response during idle state (IS) to highlight the neural signal time-frequency modulations after a state instruction and during the task activation.

The neural signals modulation was evaluated on the series of experiments E composed of 20 experiments where 8D exoskeleton control was achieved (training and testing analysis sessions were considered). These sessions were online closed-loop experiments performed with the REW-MSLM decoder.

In all the experiments, the neural signals related to an active states (ASLH,ASLW,ASRH,ASRW) were extracted in an 8 sec window around the AS activation corresponding to the decoder correct classification of the active state (Figure 9-5A). Additionally, the latency (delay) between the task instruction and the activation of an AS was computed.

Neural signals related to idle state (IS) period were also extracted. All the neural signals labeled as IS were stacked with the exception of the 2 seconds before or after any state transition with an AS (AS towards IS and IS towards AS).

For the neural signals windows related to IS, the 2 sec before or after any activation/transition from or to an AS were not considered (Figure 9-5B).

To obtain more relevant and cleaner neural signal modulations, the extracted AS and IS windows containing false activations (𝑓𝑝 and 𝑓𝑛 classification) of the non-selected state as well as non-stable activation of the intended state were excluded. Additionally, extracted AS windows which presented latency state activation considered as outliers (latency above 10 seconds) were removed.

Both AS and IS remaining neural signal windows were mapped to the time-frequency space based on the complex continuous wavelet transform (Morlet) similarly as for feature extraction step of the ABSD BCI system previously described (details in the Chapter 8). The absolute value of the IS and AS windows was computed before to be log-transformed (Figure 9-5).

All the remaining IS neural time-frequency windows were concatenated and averaged to obtain an IS baseline. Then, the IS baseline was subtracted to every remaining AS timefrequency window. Finally, the AS vs IS time-frequency windows were averaged. This procedure was achieved independently for the 64 electrodes (Figure 9-5).

PREW-NPLS and APREW-NPLS performance evaluation 9.2.1. Algorithm comparison

The PREW-NPLS and APREW-NPLS were designed to integrate the REW-MSLM algorithm as sparse expert or gate sub-models. PREW-NPLS and APREW-NPLS algorithms performance were evaluated during pseudo-online experiments based on the 8D dataset D. Both algorithms are a penalized version of the REW-NPLS algorithms which may be integrated into the REW-MSLM as regularized experts. Therefore the performance of these algorithms were compared to their non-penalized version (REW-NPLS) trained with the same expert-subset calibration strategy than REW-MSLM. PREW-NPLS and APREW-NPLS algorithms were particularly evaluated on the left and right 3D hand translation tasks of the D series of experiments. Similarly as during the online closed-loop experiments of the D series, the PREW-NPLS and APREW-NPLS models were calibrated on the first 6 sessions and were tested on the remaining 37 experiments.

Lp-PREW-NPLS was presented in the previous section for 𝑝 = 0, 1 2 , 1. The three type of penalization were tested during the pseudo-online studies. PREW-NPLS penalized models required to fixed a supplementary hyperparameter named the penalization hyperparameter λ. To evaluate the influence of λ on the model performance and on the model sparsity, 31 models were evaluated with the penalization hyperparameter λ going from 0 to 0.6 with a 0.02 steps. In the case of the L0-PREW-NPLS studies, preliminary analysis results highlighted that the studied λ range was not relevant. Therefore, additional models with the penalization hyperparameter λ going from 0 to 0.06 with a 0.002 steps were estimated (Figure 9-6A).

APREW-NPLS similarly than PREW-NPLS can be evaluated with three different types of penalization. However, the preliminary pseudo online study was only focused on the L1 norm penalization. The study was limited to L1 norm penalization due to the limitations of the L0 and L0.5 norm/pseudo norm penalization which were highlighted in the results chapter. APREW-NPLS selected the most relevant penalization coefficients among a list of possible λ value. All the lambda of the set 𝛌 were tested and evaluated during the same pseudo-online experiments contrarily to PREW-NPLS which required one pseudo-online experiment per penalization coefficient. Six models with λ going from 0.1 to 0.6 with a 0.1 steps were estimated and compared during the pseudo-online experiments (Figure 9-6B). H2M2 gating performance evaluation 201

Performance indicators

Similarly to the pseudo-online evaluation of the REW-MSLM performance, the scalar product (cosine similarity) 𝐷𝑜𝑡𝑃(𝑡) was computed. The median, 95% confidence interval of the median, 25th and 75th percentiles of the 𝐷𝑜𝑡𝑃(𝑡) were estimated for each model.

The PREW-NPLS and APREW-NPLS algorithms converged into sparse solutions by fixing non-relevant (non-informative / noisy) electrodes to exactly 0. Direct decoding performance was therefore not the only relevant indicator. A sparse decoder with the same performance than a "classic" decoder might lead to faster model application and better generalization of the decoded neural signals. Here, 𝛿 is the Kronecker symbol.

For the PREW-NPLS algorithm, significance of the differences between the cosine similarity of REW-NPLS and PREW-NPLS algorithm was computed for the left and right hand translation studies and for each penalization hyperparameter 𝜆. The statistical analysis was performed with the non-parametric paired Wilcoxon signed rank test with (𝛼 𝑚𝑢𝑙𝑡𝑖-𝑐𝑙𝑎𝑠𝑠 = 0.00161) and without (𝛼 = 0.05) the Bonferroni correction.

H2M2 gating performance evaluation

The H2M2 algorithms was designed to integrate the REW-MSLM algorithm as gating model. The performance improvement related to H2M2 gating model was evaluated using pseudo-online simulations.

Algorithm comparison

H2M2 is a generalization of the HMM to a hierarchical structure and sub-HMM models to decode the discrete latent variables. To stress the benefits of H2M2, the classifier was compared to two HMMs with specific prior paradigm.

202

Chapter 9 : Performance evaluation and analysis

Firstly, H2M2 was compared to a HMM with a limited number of available transition.

The architecture was similar to the HMM computed during the online closed-loop 8D experiments with left and right hand translation and wrist rotation states (series D and E of experiments). During the 8D experiments, the consecutive pursuit tasks were always separated by short or long period of idle state. Therefore, the created HMM had limited number of transition (Figure 9-7A). This HMM architecture, referred as HMMlimited, had a limited number of transition which induced delay to the transition between two AS. However, HMMlimited architecture reduced the number of false activation. Additionally, H2M2 was compared to HMMfull with all the transition available (Figure 9-7B).

The H2M2 required to fix a prior architecture of the state. Knowing the number and the possible transition between the state is useful to reduce the calibration procedure. Figure 9-7C shows the H2M2 architecture tested during the pseudo-online study and compared to HMMlimited, and HMMfull. During the pseudo-online simulations, the H2M2 was structured into two layers 𝐷 = 2. The first layer clustered three states defined as idle state (IS), the left body side movement state (ASL) and the right body side movement state (ASR). IS was a production state (emit output observations) whereas both AS states were defined as autonomous sub-HMM. ASL and ASR were both composed of two states defined as the left and right hand translation and wrist rotations states respectively (ASLH, ASRH, ASLW, ASRW). Therefore the set of production state was 𝑆 𝑝𝑟𝑜𝑑 = {IS, AS LH , AS RH , AS LW , AS RW } and the set of internal state was 𝑆 𝑖𝑛𝑡 = {AS L , AS R }.

The pseudo-online analysis was performed on the F series of experiments where the REW-MSLM with HMMfull gating model was calibrated and applied during online closed-loop 4D continuous and 5 discrete state experiments with the virtual avatar effector. All the presented HMMs (H2M2, HMMlimited, HMMfull) were calibrated and tested offline using pseudo-online simulations. During the pseudo-online simulations, each HMM (H2M2, HMMlimited, and HMMfull) was integrated as REW-MSLM gating model. Each HMM calibration was performed on the same sessions as during the online experiments.

Performance indicators

Similarly, as for the discrete REW-MSLM gating performance evaluation, static (samplebased) and dynamic performance indicators were evaluated. Latency, frequency rate and length or error blocks were computed as well as numerous static multi-class indicators. The samples labeled as belonging to transition/latency period were not considered for the performance indicator evaluation (with the exception of the latency score). For further investigation on the state classification, other static performance indicators in addition to the Accuracy ((9.1), F-score ((9.2), Precision and Recall ((9.3), were evaluated. Let's defined the indicators of the production state 𝑠 ∈ 𝑆 𝑝𝑟𝑜𝑑 with 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 ) the number of production states, computed using the number of correctly classified samples from one state (true positives, 𝑡𝑝 𝑠 ), the incorrectly labelled samples in one state (false negatives, 𝑓𝑛 𝑠 ), the correctly classified samples not belonging to the state (true negatives, analysis .

All the described indicators were used to evaluate the pseudo-online performance of the three REW-MSLM gating models. Multi-class classification performance description is a highly complex task. Each indicator provides different information of the classification performance. For example, Kappa score measures the agreement between the accuracy and the chance level whereas Matthews correlation coefficient is a discretized version of the classic Pearson correlation which computes the balance ratios of the 𝑡𝑝, 𝑡𝑓, 𝑓𝑝 and 𝑓𝑛 in one formula. Bookmaker mixes the probability to correctly classified the selected state (recall score) and the one related to the other classes (specificity score) etc. A comparative study on each indicator was proposed in [START_REF] Martel | Evaluation criteria for closed-loop adaptive dynamic discrete-continuous brain-computer interfaces: clinical study case with tetraplegic patient[END_REF].

Bookmaker, Gmean, Kappa and Matthews Correlation Coefficient were reported to be good performance indicators especially for imbalanced dataset. However, Accuracy and F-score were comparison indicators commonly reported in BCI studies. In order to Conclusion 205 compare the classification performance of the algorithms with other reported articles, accuracy and F-scores indicators were still evaluated.

Conclusion

In this chapter, it was proposed to compare the new mixture of expert decoder REW-MSLM with other online incremental state of the art decoder. Before to integrate the REW-MSLM into the online adaptive BCI platform of the clinical trial, pseudo-online study was carried out based on previously recorded online closed-loop BCI experiments. However, although pseudo-online studies were mandatory, they were not fully representative of the possible benefits of the new designed adaptive incremental decoders during online closed-loop experiments as the patient's feedback and adaptation were not considered in such offline simulations. Therefore, in a second step, online closed-loop experiments integrating REW-MSLM were performed for 8D exoskeleton or virtual avatar control.

In order to improve the continuous and discrete performance of the REW-MSLM gating and expert models, new continuous and discrete decoders were proposed. The recorded 8D experiments using REW-MSLM to control the virtual avatar were exploited for pseudo-online simulations of sparse PREW-NPLS and APREW-NPLS algorithms.

Additionally, new online experiments were performed with REW-MSLM in order to compare the new dynamic classifier to the current gating model of the REW-MSLM.

Various performance indicators were selected to evaluate the effectiveness of the algorithms. Static and dynamic indicators were estimated for discrete performance evaluation whereas an indicator based on the dot product was computed in the pseudoonline study. During the online experiments, more straightforward indicators were presented such as the number of hit targets. PREW-NPLS and APREW-NPLS required to evaluate the "classic" decoding performance as well as the sparsity of the estimated model. Finally, in order to fully describe H2M2 classification differences with other dynamic classifiers, numerous discrete indicators were estimated.

Chapitre 10

Results

REW-MSLM

To validate the REW-MSLM algorithm benefits before an integration into the clinical trial BCI platform (ABSD), it was compared to the state-of-the-art adaptive algorithms by testing their discrete and continuous decoding performance. To this aim, series of closed loop 6D alternative two-handed reaching experiments where offline analysed in pseudo-online manner. We next present online closed-loop 8D alternative two-handed reaching and rotating clinical trial performances using the exoskeleton and virtual avatar effectors over 6 months.

Offline comparative study

The REW-MSLM mixes discrete and continuous decoding. The discrete multi-state decoding performance for an asynchronous control paradigm evaluating the accuracy of switching between all active states (AS) and, especially, the robustness of idle state (IS) support was firstly evaluated. We demonstrated that HMM dynamic gating enhanced classification performance compared to classic sample-wise gating. Supplementary latency which may be induced by HMM classifier is evaluated. To be made, REW-MSLM was firstly compared to the continuous REW-NPLS model [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF] thresholded in post-processing (referred to as REW-NPLST) to stress the interest of the ME structure which dedicates a specific model to discrete state decoding. Next, the REW-MSLM was compared to its own variant without HMM (called REW-SLM) to determine the benefits of dynamic HMM gating. The performances of the three algorithms were evaluated using three different experimental paradigms. The session of experiments A was designed to test the models in all-in-one experiments with small training dataset whereas the sessions of experiments B and C were achieved to evaluate the performance of the model with cross-session training and the stability of the neural signal decoding without recalibration.

REW-MSLM discrete pseudo-online performance.

The REW-MSLM demonstrated strong discriminative abilities (Figure 10-1A) between all states (𝑎𝑐𝑐 = 93 ± 1.8%, 𝑓𝑠𝑐 = 86 ± 3%), between IS and AS (𝑎𝑐𝑐 = 91 ± 3%,𝑓𝑠𝑐 = 84 ± 5%) and between ASLH and ASRH (𝑎𝑐𝑐 = 99 ± 0.8%, 𝑓𝑠𝑐 = 99 ± 0.8%) regardless of the experimental paradigm. The same performance indicators lead to 𝑎𝑐𝑐 = 87 ± 2%, 𝑓𝑠𝑐 = 76 ± 3% between all states, 𝑎𝑐𝑐 = 86 ± 2%, 𝑓𝑠𝑐 = 75 ± 3% between IS and AS and 𝑎𝑐𝑐 = 93 ± 0.3%, 𝑓𝑠𝑐 = 93 ± 0.2% between ASLH and ASRH for REW-SLM algorithm whereas REW-NPLS performs 𝑎𝑐𝑐 = 62 ± 2%, 𝑓𝑠𝑐 = 36 ± 5% between all states, 𝑎𝑐𝑐 = 70 ± 7%,𝑓𝑠𝑐 = 49 ± 0.6% between IS and AS and 𝑎𝑐𝑐 = 59 ± 8%, 𝑓𝑠𝑐 = 57 ± 9% between ASLH and ASRH (Figure 10-1A).

The REW-MSLM strongly discriminated each state with a particularly robust distinction between the left and right hand states. Significant improvements compared to REW-NPLST and REW-SLM were found in the majority of the performance indicators (Figure 10 -1A). No significant differences between the performance in the experimental sessions B and C were found (𝑝 > 0.1), indicating the model stability in session C, even though the model was not recalibrated in these experiments.

The latency of the switching state averaged over the three experimental paradigms (A, B and C) was higher for the REW-MSLM than for the REW-SLM: 𝑙𝑎𝑡 = 2.05 ± 0.059 s versus 𝑙𝑎𝑡 = 1.46 ± 0.31 (Figure 10-1B). Similarly, the block error duration increased with the REW-MSLM decoders. The HMM state decoder error lasted 𝐸𝑟𝑟𝐵 𝑡𝑖𝑚𝑒 = 4.31 ± 0.88 s, whereas the discrete static decoder error duration of the REW-SLM was 𝐸𝑟𝑟𝐵 𝑡𝑖𝑚𝑒 = 0.49 ± 0.024 s. However, the error block frequency decreased considerably with the REW-MSLM decoders: the error block frequency for the REW-SLM was high (𝐸𝑟𝑟𝐵 𝑟𝑎𝑡𝑒 = 20.7 ± 1.95 error blocks per minute), whereas the REW-MSLM error block frequency was reduced to 𝐸𝑟𝑟𝐵 𝑟𝑎𝑡𝑒 = 1.6 ± 0.26 blocks per minute (Figure 10-1B). Chapter 10 : Results

Next, the continuous decoding of the REW-MSLM experts are compared to the REW-NPLS model trained on the entire dataset to evaluate expert-specific subset training strategy. The decoding performances was compared to REW-NPLS algorithm because this algorithm is a state-of-the-art online adaptive tensor-input tensor-output algorithm which has been previously employed for closed-loop ECoG-based BCI [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF]] [Eliseyev et al., 2017]. Although pseudo-online experiments allow appraising the proposed decoder performance and stability, these results cannot be generalized to online closed-loop experiments due to the lack of patient's feedback. The purpose of CLDA procedure is to integrate the patient's feedback and related neural signals into the model calibration to perform both patient and model learning/adaption simultaneously. This behaviour cannot to be computed and evaluated in pseudo-online studies. Therefore, the performances of the REW-MSLM algorithm for 8D realtime closed-loop experiments using a virtual avatar or exoskeleton effectors over several months were appraised.

Online closed-loop 8D results

The patient achieved online asynchronous closed-loop 8D control of the effectors performing 3D alternative two-handed reaching tasks and 1D wrist rotation movements [START_REF] Benabid | An exoskeleton controlled by an epidural wireless brain-machine interface in a tetraplegic patient: a proof-of-concept demonstration[END_REF] using the virtual avatar or the exoskeleton for the sessions at home and inside the laboratory, respectively. Sessions were composed of successive tasks selected by the experimenter (idle state IS or the left ASLH, right hand ASRH, right wrist ASLW, and left wrist ASLW activation states). Each task was made of several successive trials where the patient attempted to reach target locations set sequentially with the left or right hand or to rotate the left or right wrist until a given angle. The following section presents the results obtained firstly during the 37 home sessions controlling the virtual avatar and then the 15 exoskeleton experiments. The experiments were performed over 5 to 203 days and 0 to 167 after the last model recalibration for the virtual avatar and exoskeleton experiments respectively. The experiments were carried out 468 to 666 days and 531 to 698 days after patient's surgery for the virtual avatar and exoskeleton experiments respectively

Online closed-loop virtual avatar experiments

Classification decoding performance between the five states (idle, left and right hands translation and left and right wrists rotation) was demonstrated with the REW-MSLM algorithm across all the experiments (Figure 10-3A) with an average (averaged across states and experiments) F-score of 𝑓𝑠𝑐 = 76 ± 9% and accuracy of 𝑎𝑐𝑐 = 93 ± 3% (Figure 10-4A).

The hit performance demonstrated a right hand translation SR of 53 ± 15% (R-ratio:5.4 ± 3.5) and a left hand translation SR of 55 ± 18% (R-ratio: 5.2 ± 3.1), whereas the average wrist rotation SR was 95 ± 8.2% (R-ratio: 3.6 ± 3.3) across all the experiments. Considering the prior probability of the idle, hand translations and wrist rotations states as the chance level to activate each state. The discrete state chance level of IS, ASLH, ASRH, ASLW, ASRW were estimated at 26%, 36%, 27%, 6%, 5% respectively. The classification performances of REW-MSLM were estimated higher than the chance level for every state and every experiment across the 203 days. Similarly, the continuous decoding performance were above the averaged SR of 7.1 ± 5.5% (R-ratio: 24 ± 14) for the left hand translation, 9.5 ± 6.6% (R-ratio: 33 ± 19) for the right hand translation, 40 ± 7.1% (R-ratio: 15 ± 4.6) for the left hand rotation and 33 ± 4.9% (Rratio: 12 ± 2.7) for the right hand rotation tasks estimated during the chance level study.

An analysis on the model decoding performance stability across time and experiments without model recalibration was carried out. The zero slope hypothesis was not rejected for 16 of the 18 indicators. It was rejected for the left wrist rotation R-ratio, which increased by 0.014 daily, and the right hand translation SR, which reduced daily by 0.07%. The zero slope analysis of the performance indicators are summarized in Table 2. These results highlighted the stability of the REW-MSLM over 6 months using a virtual avatar effector during 8D experiments, even though the model did not undergo a long training/calibration period and no model recalibration was carried out.

Online closed-loop exoskeleton experiments

The Discrete decoding performances of 8D experiments yielded relevant and stable discrete decoding performance results across the 167 days (Figure 10-3B). The REW-MSLM gating yielded an average F-score of 75 ± 12% and accuracy of 92 ± 4% with high distinctiveness between the classification of the left and right sides of the body (less than 1% misclassified samples) and strong idle state decoding with an average of 85% accurately classified idle state samples.

Left hand translation demonstrated an average SR of 69 ± 13% with an R-ratio of 6.7 ± 5.4.

Right hand translation showed similar SR but higher standard deviation than left hand translation, with an average SR of 65 ± 29% and an R-ratio of 13 ± 4.5 (Figure 10-4B). The decoding for both wrist rotation tasks showed an average right and left wrist rotation task completion rate of 93 ± 12% with a low R-ratio (2.9 ± 2.4).

It is worth to note, that for the period 0 to 37 days after the last decoder calibration session, the online sessions using the exoskeleton yielded a decoding accuracy of 94% averaged across the five classes. Additionally, on the same period, 8D control with an average SR (for both hands) of 83% and 97% with an average R-ratio of 6.4 ± 2.3 and 3.3 ± 1.7 for the 3D hand translation and 1D wrist rotation was reported. This 0 to 37 days period corresponds or overpasses the time interval reported generally in ECoG-based BCI studies. Commonly, ECoG based clinical trials last from several days to 1 or 2 weeks (less than 28 days) of research with an implantation from 3 to 35 days [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Leuthardt | A brain-computer interface using electrocorticographic signals in humans[END_REF][START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF][START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF], 2007[START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] [W. Wang et al., 2013[START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF].

All the 18 performance indicators had values higher than those obtained by our chance level studies for all the experiments. The discrete state chance level of IS, ASLH, ASRH, ASLW, ASRW were estimated at 26%, 36%, 27%, 6%, 5% whereas the continuous decoding performance of the chance level study were evaluated at an averaged SR of 7.1 ± 5.5% (R-ratio: 24 ± 14) for the left hand translation, 9.5 ± 6.6% (R-ratio: 33 ± 19) for the right hand translation, 40 ± 7.1% (R-ratio: 15 ± 4.6) for the left hand rotation and 33 ± 4.9% (R-ratio: 12 ± 2.7) for the right hand rotation tasks.

Similarly to the 8D experiments with the virtual avatar, the decoding stability was evaluated with linear regression fitting analysis (Table 3). The zero slope hypothesis was not rejected for 12 of the 18 indicators. The right side of the body seemed to have a slow performance decrease across experiments, gathering 5 of the 6 diminishing indicators. The linear fits demonstrated significant reduction in the right limb performance for the discrete (-0.25% F-score and -0.04% accuracy per day) right wrist rotation indicators and for the right hand translation F-score (-0.17%), SR (-0.42%) and R-ratio (+0.24). However, the outliers of the right hand translation R-ratio indicator of the day 167 might bias the analysis. Significant decreases were found in the left hand SR (-0.18% per day). The left hand SR seemed to decay in the first experiments before stabilizing.

REW-MSLM parameters visualization

The gating model of REW-MSLM used for exoskeleton control is represented on the spatial, frequency and temporal modality in Figure 10-6.

On the spatial modality, the contralateral electrodes presented a higher influence on the left and right hand (translation and rotation) state estimation. Spatial modality presented higher parameter weights on the contralateral electrode array for the left and right hand (translation and rotation) state estimation. Additionally, translation and rotation from the same hand seemed to activate nearby but distinct electrodes. The parameters weights of the frequency band between 20-30Hz (β-band) and 80Hz-120Hz (γ-band) showed higher influence for the left and right hand translation state discrimination. The same frequency bands were relevant for rotation and idle state classification, nevertheless, lower frequency bands (<20 Hz) significantly contributed to the decoding, especially for idle state decoding. Finally, parameter weights in the temporal modalities between -0.1s and -0.8s before the event were dominant. Temporal parameter weights were similar for all states.

Expert models used for exoskeleton control are represented on the spatial, frequency and temporal modality in Figure 10-7. Spatial modality presented heavy parameter weights on the contralateral electrode array for left hand (translation and rotation) continuous model. All the left hand 3D translation parameters presented similar model with dominant frequency band between 80Hz-120Hz (γ-band). Left wrist rotation parameters showed dominant frequency band between 20-30Hz (β-band). Right hand continuous parameters were more complex to analyse. However, β and γ frequency bands were dominant in each model. Finally, parameter weights in the temporal modalities increased with temporal parameters closer to the movements. ASRW presented a higher modulation on the upper electrodes whereas for ASRH the main activated electrodes were located in the lower electrodes of the implants.

Discussion

The REW-MSLM decoder was proposed to address the poorly explored field of asynchronous multi-limb effector control. ME architecture was employed to handle numerous dimensions and to decode the robust idle state. To allow cross-session training of the decoder with multiple recording conditions during closed loop BCI effectors control experiments directly, an adaptive/incremental learning algorithm was designed. Dynamic expert gating using a HMM was added to ME decoder to ensure the robustness of idle state support.

To clarify the importance of the ME model structure, which combines discrete decoding (state classification) and continuous trajectory decoding, and the importance of dynamic vs. static gating, REW-MSLM was compared to the the state-of-the-art adaptive algorithms with 3 databases using simulated pseudo-online experiments.

For discrete decoding, the REW-MSLM outperformed alternative approaches in discrete classification regardless of the dataset and the paradigm (all states decoding, IS against AS, and ASLH-ASRH switching) with an averaged F-score improvement across all paradigms of 39±4% and 8.3±2% compared to REW-NPLS and REW-SLM respectively. These results sustain the benefits to train a specific model dedicated to state classification and the improvements related to dynamic classification. The switching state latency study related to the state transition delay between the instruction and the discrete decoding response demonstrated an average increase in duration by 0.45 s, 0.87 s and 0.38 s (over 3 datasets) between the discrete decoder with and without dynamic HMM processing. However, the REW-MSLM results show a drastic 92% decrease in the block error rate between the discrete decoder with and without dynamic HMM processing, overcoming the high frequency misclassified sample issue of static classifier. For physical effectors, such as an exoskeleton, which are in direct contact with the patient and has a latency of mechanical activation/deactivation of up to a few seconds, it is mandatory that false activation blocks remain exceptionally rare events.

For continuous pseudo-online experiments, REW-MSLM experts highlighted slight improvement or similar performance compared to REW-NPLS whereas the training datasets were different. REW-MSLM allows experts training using independent data sets. This may be highly profitable for progressive BCI decoder training increasing tasks complexity. In addition, considering a specific task (e.g. left hand translation), the developed REW-MSLM and the state of the art REW-NPLS performed similar left hand movement decoding. However, numerous non-desired movements (non-zero velocity predictions) was decoded for the other limb (e.g. right hand translation) with the REW-NPLS algorithm. In contrast, REW-MSLM performed similar trajectory decoding performance for the required limb without unintentional movements from the other limbs thanks to accurate state classification provided by the gate. Unintended movements of the not-intentionally controlled limb impede the control of complex effectors such as exoskeleton and all the more in the case of asynchronous control with idle state decoding. The suppression of the unintended movements leads to better visual feedbacks and concentration of the patient which may induce better model calibration.

Finally, the REW-MSLM was integrated into the homemade BCI adaptive brain signal decoder (ABSD) software platform and was used in the "BCI and tetraplegia" clinical trial. This algorithm provided to a tetraplegic patient the control of a virtual avatar and an exoskeleton in real-time with alternating rotation and translation movements of both hands on his own intention, which corresponds to 8D continuous control and supporting 5 discrete states and preserve good decoding performance for 6 months.

Figure 10-6 illustrates the gating model weights in the frequency, temporal and spatial modalities. In the frequency modality the model coefficients are consistent with the previous studies which highlighted the significance of β and high γ-band to decode movements from direct neural signals [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]. As expected, spatial weights were higher in the contralateral electrodes of the realized movement for both left and right hand translation and rotation which is corroborated by previous studies [START_REF] Fukuma | Closed-Loop Control of a Neuroprosthetic Hand by Magnetoencephalographic Signals[END_REF][START_REF] Jerbi | Inferring hand movement kinematics from MEG, EEG and intracranial EEG: From brain-machine interfaces to motor rehabilitation. IRBM[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF].

Finally, the neural signal modulations computed between the idle state and the active states represented in the Figure 10 The 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 of the L0, L0.5 and L1 models had similar patterns. Three distinct phases can be extracted. If the penalization coefficient 𝜆 was small, the sparsity index 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 was close to 0. This behavior is shown in the Figure 10-11 with 𝜆 < 0.01, 0.1 and 0.1 for the L0, L0.5 and L1 penalization norm respectively. In the opposite, for a high penalization coefficient 𝜆, the models seemed to reach a maximum of sparsity for 𝜆 above 0.05, 0.3 and 0.3 for the L0, L0.5 and L1 penalization left hand translation models and 𝜆 > 0.06, 0.36 and 0.36 for the L0, L0.5 and L1 penalization right hand translation models respectively. This models from this phase are referred as "converged" models in the next sections. Between these two phases, the 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 value increased with the penalization hyperparameter 𝜆 whatever the tested penalization norm and the controlled hand.

Higher was the penalization hyperparameter 𝜆, sparser was the estimated models whereas high latent space dimension led to reduce the model sparsity. As an example, in the "stationary phase" with high penalization coefficient 𝜆, 50% of the electrodes were set to zero with a dimension of the latent space fixed to 𝑓 = 34, 40 and 53 whereas only 25% of the electrodes were removed from the models for 𝑓 = 60, 72 and 85 for the L0, L0.5 and L1 penalization right hand translation models respectively.

The models decoding performance and sparsity were highly dependent on 𝑓 and 𝜆 hyperparameters. In the following sections the latent space dimension 𝑓 was set to the optimal value estimated during the pseudo-online model calibration using the conventional Recursive Validation procedure designed in the REW-NPLS algorithm [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. As mentioned, for high penalization hyperparameter 𝜆, the sparsity was stable and the calibration led to equivalent models. Therefore, all the models penalized with a coefficient 𝜆 going from 0 to 0.6 (0.02 steps) are not represented in the next studies for high 𝜆 values.

PREW-NPLS decoding performance

The dot product performance and the sparsity index of the L0, L0.5 and L1 models for the left and the right hand movement tasks are presented depending on the penalization coefficient 𝜆 in the Figure 10-12 and Figure 10-13 respectively. The results are presented using the median, the 25 th (Q 1 ) and the 75 th (Q 3 ) percentiles using the notation: median (Q 1 -Q 3 ). L0 REW-NPLS algorithms (Figure 10-12A) showed relevant performance for different penalization coefficient 𝜆 value. Obviously the 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 indicator increased with higher penalization hyperparameter value. However, the dot product (cosine similarity) highlighted better performance than REW-NPLS algorithm with various 𝜆 value. For 𝜆 = 0.01, the 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 0% but the dot product (cosine similarity) was evaluated at 0.252 (0.165 -0.296). For 𝜆 = 0.026, the cosine similarity was estimated at 0. L0.5 REW-NPLS algorithm (Figure 10-12B), similarly to L0 REW-NPLS algorithms presented equal decoding performance than REW-NPLS algorithm with small decoding performance improvements for some models. For 𝜆 = 0.22 with 18 electrodes parameter weights set to zero (𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 28.13%) the PREW-NPLS model highlighted higher cosine similarity 𝐿 0.5 0.21 0.84 0.39 0.99 0.21 𝟎. 𝟎𝟐𝟓 0.07 0.06 0.47 0.96 𝟎. 𝟎𝟎𝟎𝟑 0.59 0.83 0.18 𝟎. 𝟎𝟎𝟕 𝟎. 𝟎𝟎𝟎𝟓 𝟎. 𝟎𝟎𝟎𝟓 𝟎. 𝟎𝟎𝟎𝟓 𝟎. 𝟎𝟎𝟎𝟓 𝟎. 𝟎𝟎𝟎𝟓 𝜆 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 00.18 0.02 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.038 0.040 𝐿 0 0.22 𝟏𝟎 -𝟒 𝟎. 𝟎𝟏𝟐 0.45 𝟎. 𝟎𝟎𝟎𝟔 0.34 𝟎. 𝟎𝟏 0.11 𝟎. 𝟎𝟎𝟗 0.40 𝟏𝟎 -𝟓 0.10 𝟎. 𝟎𝟒 𝟎. 𝟎𝟎𝟏𝟕 0.13 0.47 𝟎. 𝟎𝟏𝟐 𝟎. 𝟎𝟒𝟒 𝟎. 𝟎𝟏𝟑 𝟎. 𝟎𝟎𝟖 𝜆 0.042 0.044 0.46 0.48 0.50 𝐿 0 𝟎. 𝟎𝟎𝟖 𝟎. 𝟎𝟎𝟖 0.67 0.67 0.67 𝐿 0.5 0.26 0.08 0.07 𝟎. 𝟎𝟏𝟐 𝟎. 𝟎𝟎𝟎𝟏𝟖 0.16 𝟎. 𝟎𝟏𝟑 𝟏𝟎 -𝟒 𝟎. 𝟎𝟎𝟎𝟏 𝟏𝟎 -𝟓 𝟎. 𝟎𝟎𝟕 0.72 0.27 0.15 0.36 0.17 0.455 0.17 0.17 0.17 𝜆 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 00.18 0.02 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.038 0.040 𝐿 0 0.88 𝟎. 𝟎𝟏𝟓 𝟎. 𝟎𝟎𝟏 𝟏𝟎 -𝟔 𝟏𝟎 -𝟔 𝟎. 𝟎𝟎𝟗 0.13 𝟎. 𝟎𝟎𝟔 𝟏𝟎 -𝟔 𝟏𝟎 -𝟓 𝟎. 𝟎𝟎𝟎𝟏 𝟏𝟎 -𝟓 0.77 𝟎. 𝟎𝟓 𝟎. 𝟎𝟑 0.41 0.41 0.12 0.

The

Lp REW-NPLS parameters visualization

The REW-NPLS and the Lp REW-NPLS model parameter weights are illustrated on the temporal, frequency and spatial domain in the Figure 10-14 for the left hand translation models and in the Figure 10-15 for the right hand translation models. For easier comparison and selection the presented models are the ones with "converged" penalization hypeparameter 𝜆 > 0.06, 0.4 and 0.4 for L0, L0.5 and L1 REW-NPLS algorithms respectively whereas the latent space dimension 𝑓 was fixed using the Recursive-Validation (RV) of PREW-NPLS algorithm similar to the Recursive-Validation procedure of the REW-NPLS algorithm. Lp REW-NPLS model parameter weights of the left hand translation models (Figure 10-14) in the frequency domain were higher in the low frequency bands than REW-NPLS model. In the opposite, the models estimated on the offline right hand translation study showed dominant parameter weights in the high frequency bands. The temporal parameter weights were similar for all the algorithms. As previously mentioned, the sparsity of the solution was highly dependent on the latent space dimension hyperparameter 𝑓. The L0.5 and L1 REW-NPLS algorithms, with small latent space dimensions (𝑓 = 26 and 𝑓 = 32), led to sparser solution than the L0 REW-NPLS model with a latent space dimension 𝑓 = 52. However, similarities between the dominant electrodes of the Lp REW-NPLS solutions and between the Lp REW-NPLS and REW-NPLS are visible.

For easier visualization, the spatial parameter weights are presented for the left (Figure 10-16) and right (Figure 10-17) hand translation models on a map with the electrode locations relative to the sensory (SS) and motor (MS) sulci. As most of the parameter weights were fixed to zero value in Lp REW-NPLS algorithms, the amplitude of the parameters are much more important than the REW-NPLS model weights which are more balanced between all the electrodes. For both, left and right hand models, Lp, L0.5 and L1 REW-NPLS solutions used electrodes located in similar regions.

For the left hand models (Figure 10-16), the contralateral electrodes (right implant) highlighted strong influence on the movement decoding. In particular, the region located around the electrodes R21, R23, R24, R25, R36 and R47 above the lower part of the MS and the upper part of the SS show large parameter weights for the L0, L0.5 and L1 REW-NPLS algorithms. Additionally, the electrodes positioned on the upper left side of the right MS stressed major influenced in the left arm decoding.

Similarly, for the right hand models (Figure 10-17), the prominence of the contralateral electrodes (left implant) was visible for the REW-NPLS and all the penalized version of REW-NPLS algorithms. Important spatial parameter weights were stressed for the electrodes located in the upper part of the MS named electrodes L05, L03 and L14 as well as the electrodes positioned in the lower part of the MS referred as electrodes L41, L43 and L50. Similarly to left hand models, the important parameter weights were noticeable on the ipsilateral implant.

Discussion

The study was based on pseudo-online decoding of the left or right hand translation movements recorded during the online closed-loop experiments. The dataset D is composed of 43 experiments. The tested models were calibrated during the offline study using the first 6 experiments (recorded in late September 2018) and was tested based on the experiments recorded between early October 2018 and mid-March 2019.

To be as close as possible to the online experiment settings, the penalized models were calibrated on the same experiments which were used during the online-closed loop experiments. The number of training session was small (14%) and focused at the beginning of the series of experiments (no re-calibration period). This may explain the high inter-session variability of the decoding performance for both REW-NPLS and PREW-NPLS algorithms.

However, the L0, L0.5 and L1-PREW-NPLS algorithms highlighted equivalent or better decoding performance than REW-NPLS decoder using sparse solutions with up to 41 and 48 of the electrode parameter weights set to zero value for the left and right hand translation L1 norm penalized models. Decoding performance improvements were more evident for the 3D right hand translation models than the left hand translation model. Sparse solutions allow removing the majority of the electrodes which may reduce the required computational burden for the model recalibration.

Additionally, L0, L0.5 and L1 REW-NPLS algorithms converged to similar solutions with comparable decoding performance. However, the L0.5 REW-NPLS algorithm is looking for a solution of a cubic equation which requires higher computational load to be solved than the calibration procedure with the L0 and L1 norm penalization. Indeed, the equation solving required for the L0.5 pseudo-norm penalization is applied at every loop of the PARAFAC algorithm which is repeated several times for every latent space dimension (𝑓) of the PREW-NPLS model. The repeated equation solving procedure of the L0.5 pseudo-norm penalization is more complicated than the thresholding procedure applied with the L0 and L1 norm penalization. Consequently, L0.5 REW-NPLS algorithm may not be adapted to online CLDA. L0 and L1 REW-NPLS algorithms highlighted decoding performance and computational requirements more adapted for an integration into REW-MSLM experts or gating algorithms. The study was focused on the L1 norm penalization but can be adapted to the L0 and L0.5 norm penalization type.

The Figure 10-18 shows the sparsity of all the models estimated by the APREW-NPLS algorithm during the pseudo-online study depending on the set of available penalization hyperparameter 𝜆 and the latent space dimension hyperparameter 𝑓 without considering the estimated ranking of the models. Similarly to the PREW-NPLS algorithms, the sparsity is small for 𝜆 = 0.1 and is increasing with the penalization factor.

For 𝜆 > 0.3 the models did not converge to similar solutions as the PREW-NPLS algorithms because all the models were not trained with the same number of update increments.

For the left hand decoders with 𝜆 > 0.3, the models converged to a solution with 50% of the electrode parameter weights set to zero with a dimension of the latent space close to 𝑓 = 50 whereas 25% of the electrodes were removed for 𝑓 ≈ 83. The right hand translation models show higher sparsity variability than left hand models for 𝜆 > 0.3.

Example of APREW-NPLS calibration procedure across experiments

The APREW-NPLS algorithm performed the calibration of six penalized models in the same session. To select which models should be re-estimated at each update increment, the Recursive-Validation procedure estimated the rank of each model.

AREW-NPLS parameters visualization

The REW-NPLS and the AREW-NPLS model parameter weights are illustrated on the temporal, frequency and spatial domains in the Figure 10-21 for the left hand translation models and in the Figure 10-22 for the right hand translation models.

Similarly than the Lp REW-NPLS model parameter weights, for easier visualization and comparison, the spatial parameter weights of the APREW-NPLS models are displayed for the left (Figure 10-23) and right (Figure 10-24) hand translation models on a map with the electrode locations relative to the sensory (SS) and motor (MS) sulci.

The model parameter weights estimated with APREW-NPLS algorithms (Figure 10-21 and Figure 10-22) were similar to the parameter weights observed in the PREW-NPLS studies (Figure 10-14 and Figure 10-15). Low frequency band parameters were prevalent for the left hand models whereas both low and high frequency bands are visible for the right hand models.

For the left hand translation spatial parameter weights (Figure 10-23), the electrodes R21, R23, R24, R25, R36 and R47 above the lower part of the MS and the upper part of the SS strongly impact the neural signals decoding similarly to the PREW-NPLS models analyzed previously (Figure 10 -16). Similarities between PREW-NPLS (Figure 10-17) and APREW-NPLS (Figure 10-24) models for the right hand translation models were also discernable for the L05, L03, L14, L41, L43 and L50 electrodes parameter weights.

Discussion

Numerous algorithms applied in the BCI field require to tune a hyperparameter. Generally, the hyperparameter is optimized using a validation dataset, a crossvalidation procedure, a preliminary study etc. However, these strategies are time consuming, and are commonly performed offline. In the case of online closed-loop experiments, the optimal hyperparameter may be different that the one found during offline studies and may vary across time. In order to go beyond these limitations, a decoder which automatically determines the best penalization hyperparameter was proposed and tested in a preliminary pseudo-online study.

The designed APREW-NPLS trained incrementally multiple models with different penalization hyperparameters all at once. In this pseudo-online study, the number of possible penalization hyperparameter was set to 6 but a larger set of hyperparameter can be selected. The models converged to similar parameter weights and performance than REW-NPLS and PREW-NPLS. Even though some models highlighted higher median cosine similarity than the REW-NPLS algorithm.

The APREW-NPLS algorithm selected sparse models as the optimal solutions for both left and right hand translation studies. Similar decoding performance than the REW-NPLS decoder was highlighted by the APREW-NPLS models with up to 47 and 46 electrodes parameter weights set to zero for the left and right hand translation models ranked 1 by the APREW-NPLS models.

The estimated ranking estimated during the calibration procedure highlighted similarities with the ranking of the decoding performance on the test dataset for the left hand translation study. However, the ranking similarities in the right hand translation study were less evident. Across all the analyzed experiments, left hand translation decoding always highlighted better and more stable decoding performance than the decoding of the right hand translation movements. It is likely that the recorded data for the calibration of the left hand translation models are of better quality than the recorded data for the right hand translation models. Longer calibration periods for the estimation of the right hand translation models may lead to better performance.

As mentioned, APREW-NPLS models converged to similar models than PREW-NPLS but trained multiple models in one calibration procedure. APREW-NPLS decoder highlighted slightly smaller average median cosine similarity on the estimated models than PREW-NPLS models with a 6.3% and 2.6% cosine similarity reduction for the left and right hand translation decoding respectively. However, the highest decoding performance obtained with APREW-NPLS models on the test dataset were similar to the H2M2 gating algorithm offline comparative study 257 best decoding performance of the PREW-NPLS models with 3.6% and 2.15% decrease for the left and right hand translation decoding performance.

The presented pseudo-online APREW-NPLS results were limited to a preliminary study. APREW-NPLS algorithm requires deeper investigation. In particular, the PREW-NPLS studies highlighted that models with a L1 norm penalization factor 𝜆 > 0.4 lead to "converged" models with equal parameter weights. Therefore, the APREW-NPLS models with 𝜆 = 0.4, 0.5 and 0.6 should lead to the same models. APREW-NPLS calibration with lower penalization hyperparameter (between 0.1 and 0.4) may lead to better cosine similarity and discrimination between the best and worse models.

Additionally, the model rank estimation procedure used in the APREW-NPLS algorithm to select the next models to update was only based on the model decoding performance.

In the case of portable application, a balance between sparsity and model accuracy may be a more suited solution. Moreover, the model ranks tagged in the Figure 10-19 were only estimated on the last expected reward increment. A better ranking algorithm taking into account other model characteristics may lead to a better estimation of the model to select.

Finally, the APREW-NPLS decoder was tested for continuous 3D left and right hand translation decoding. Nevertheless, as the REW-NPLS algorithm, the APREW-NPLS algorithm can be integrated to estimate the gating model used for discrete classification. Therefore, future research will also test the decoding performance and sparsity of the APREW-NPLS algorithm for discrete decoding.

H2M2 gating algorithm offline comparative study

In order to improve the dynamic gating algorithm used in the REW-MSLM experiments, the H2M2 classifier was designed. The H2M2 dynamic decoder was compared to the HMMlimited classifier originally integrated in the REW-MSLM algorithm and the HMMfull algorithm. The three dynamic classifiers were tested during pseudo-online study with the series F of experiments. In this dataset, the patient controlled the virtual avatar in 4D continuous and 𝑧 = 5 discrete states (IS, ASLH, ASRH, ASLW, ASRW). In contrast to the series of experiments D and E, all the state transitions were achieved during the experiments.

H2M2 classification performance comparison

The discrete decoding performance of the HMMlimited, HMMfull and H2M2 algorithm are presented in the For the static indicators (Figure 10-25), it should be mentioned that the classical F-score and accuracy indicators highlighted high scores for all the models with a Fscore of 91.6% ± 1.7, 92.1% ± 1.8 and 92.4% ± 1.9 and a accuracy of 88.7% ± 3.2, 89.3% ± 3.2

H2M2 parameter weights visualization

H2M2 splits the classification problem into several sub-problems estimated by submodels. This section presents the H2M2 parameter weights projected on the frequency, temporal and spatial modalities (Figure 10-27) compared to the parameter weights of the HMMlimited or HMMfull (Figure 10-28).

As previously mentioned, a HMM model is defined by an initial state probability 𝝅, an emission model {𝐁, 𝐛} and a transition matrix 𝐀. HMMlimited and HMMfull models have the same initial state probability 𝝅 and emission model {𝐁, 𝐛}. However, they are defined with different transition matrices 𝐀 𝑙𝑖𝑚𝑖𝑡𝑒𝑑 and 𝐀 𝑓𝑢𝑙𝑙 . Therefore, the parameter weights represented in the Figure 10-28 are the parameter weights of the HMMlimited and the HMMfull algorithms: 𝐁 = 𝐁 𝑙𝑖𝑚𝑖𝑡𝑒𝑑 = 𝐁 𝑓𝑢𝑙𝑙 .

The projection of the H2M2 in the first line of Figure 10-27 showed the model estimated for IS state and the two internal states: the left body side movement state and the right body side movement state noted ASL and ASR respectively. Frequency bands around 10Hz and more particularly around 20Hz exhibited great influence on the predictions of the two internal states ASL and ASR. IS state was not as clear as active states but showed important influence of the 10Hz frequency band. Similarly to the HMM models, H2M2 exhibited large value for the parameter weights related to high frequency bands above 130Hz. The parameter weights projected on the temporal modalities highlighted the typical curved shape that was found with the online closed-loop 8D REW-MSLM gating model (Figure 10-6).

The projection of the H2M2 in the second and third line of Figure 10-27 show the models estimated to discriminate the hand and wrist states from the same body side. The projection of the frequency modality exhibited large parameter weights below 20Hz, between 50Hz and 90Hz and above 130Hz. The H2M2 frequency parameter weights were similar to the HMM parameters.

For ease of visualization and interpretation, the H2M2 and HMM parameter weights projected on the spatial modalities are presented in the Figure 10-29A and Figure 10-29B respectively with the location of the electrodes compared to MS and SS. For H2M2, whereas the parameter weights discriminating the ASL and ASR seems more diffused on all the electrodes with a small superiority of contralateral electrodes, the models classifying the binary problem hand versus wrist are more focused on specific areas. For the left hand versus wrist model, the higher parameter weights were located on the electrodes R14, R16, R25 and R32 around the MS. For the right hand versus wrist model, the electrodes L03, L05, L12, L14 and L21 showed stronger parameter weights.

The H2M2 and HMM parameter weights were likely to be similar especially the model related to right hand versus wrist model of H2M2 compared to the ASRH and ASRW models estimated by HMM. For both models, majority of the most impactful electrodes are located above the MS and the upper part of the SS.

Discussion

The aim of the H2M2 was to exploit the hierarchical structure of the proposed control tasks to improve the gating classifier. The pseudo-online study using the 4D continuous and 𝑧 = 5 discrete states experiments highlighted better static and dynamic performance indicator using the H2M2 decoder.

Highest static decoder improvements were stressed on the Recall-based indicators (HF Difference, Bookmaker and Gmean indicators). From these results, it can be concluded that H2M2 algorithm enhanced the true positive (𝑡 𝑝 )-false negative (𝑓 𝑛 ) ratio. In the same time, the precision and specificity indicators remained constant highlighting stable true positive (𝑡 𝑝 )-false positive (𝑓 𝑝 ) and true negative (𝑡 𝑛 )-false positive (𝑓 𝑝 ) ratio. H2M2 presented lower transition latency between states and shorter error blocks but more frequent error blocks.

The H2M2 decoding performance were evaluated in a preliminary pseudo-online study with small training and testing dataset. New online closed-loop experiments integrating directly H2M2 algorithm as gating classifier of the REW-MSLM mixture of expert algorithm must be performed to confirm the results stressed during this pseudo-online study.

The incremental adaptive H2M2 classifier may be a relevant solution to improve the responsiveness of the REW-MSLM. Nevertheless, it is important to notice that in our application case, the H2M2 trained three classifiers instead of one which increase the computational loadings. H2M2 algorithm may be more relevant in the case of more complicated hierarchical state classification task with higher number of mental state to discriminate. A trade-off between the responsiveness, the block error rate and the computational loading must be considered for the integration of H2M2 in the REW-MSLM depending on the BCI application and the control task to realize.

Conclusion

The Recursive Exponentially Weighted Markov Switching multi-Linear Model was designed to control multi-limb effectors during online closed-loop experiments using an incremental CLDA procedure. The decoding performance of REW-MSLM were firstly evaluated during a pseudo-online study where the gating benefits and the interest of cross-session training were stressed. Finally, the REW-MSLM was integrated in the online clinical BCI and tested during online closed-loop 8D experiments with 3D left and right hand translation and 1D left and right wrist rotation tasks. The REW-MSLM decoder highlighted good performance and stability across time with a good decoding performance without any model recalibration during 6 months.

The pseudo-online studies of the incremental adaptive PREW-NPLS and APREW-NPLS algorithms highlighted the benefits of group-wised penalized solutions promoting the sparsity in the case of small calibration dataset. PREW-NPLS decoders highlighted at least similar decoding performance than REW-NPLS algorithm with decreasing dimension of the features in the spatial modality. APREW-NPLS was designed to automatically set the penalization hyperparameter during the online closed-loop experiments. APREW-NPLS showed promising results but require deeper test, training and investigations.

H2M2 was designed as new adaptive gating model for the REW-MSLM algorithm. In a pseudo-online study, the H2M2 algorithm presented a better responsiveness and shorter error blocks than HMM models at the cost of higher error block frequency. A trade-off between false positive detection and decoding latency must be considered depending on the BCI application.

PREW-NPLS, APREW-NPLS and H2M2 are supplementary algorithms which were designed to be easily integrated into the REW-MSLM algorithm in order to adapt the mixture of expert decoder depending on the BCI application. All the algorithms were evaluated during pseudo-online experiments. While the offline studies were mandatories in order to evaluate the performance of the algorithms, the results cannot be fully generalized to the online closed-loop experiments due to the lack of patient's feedback. Therefore new online closed-loop experiments integrating the new decoders into the REW-MSLM algorithm must be carried out.

Chapter 11

Discussion Limitations Perspectives

Discussion

The clinical trial of Clinatec provided the proof of concept that a tetraplegic patient can control a four-limb neuroprosthetic exoskeleton through ECoG neural signal decoding. The current Ph.D. research focused on several technical challenges namely, online decoding, direct neural decoding control strategy from neural population recordings, asynchronous multi-limb decoding, complex pursuit task completion and closed-loop decoder adaptation.

In order to control an effector, the BCI decoder must be sufficiently optimized and efficient to decode neural signals in real-time. However, most of the algorithms tested in offline studies do not take into consideration the computational requirements of the proposed solutions and generally required high computational loading to estimate a decoding model and to apply it.

In order to make more transparent control of an effector performing complex tasks, the direct neural decoding control strategy seemed more adapted than the motor imagery (MI) control strategy. Somatotopic remapping control strategy may lead to easier neural signal decoding, nevertheless, EEG studies reported that 10 to 30% of the users were unable to control MI-BCIs [START_REF] Jeunet | Chapter 1 -Advances in user-training for mental-imagery-based BCI control: Psychological and cognitive factors and their neural correlates[END_REF]. The control of complex effectors required numerous different MI strategies which are by definition in limited number. Moreover, MI may create a high mental load to the user. Besides, direct neural decoding is mandatory for other applications such as rehabilitation applications.

Additionally, in order to translate the BCIs from the laboratory to real-life applications, common challenges to overcome were reported, namely, the high-dimensional control of effectors, experiments closer to real life behavior, and the ability of the asynchronous BCI system to act as a stand-alone device. Daily life actions often require multi-limb and/or more complex actions than the one tested during clinical trial experiments. Generally, BCI performance are evaluated through center-out tasks [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Hochberg | Neuronal ensemble control of prosthetic devices by a human with tetraplegia[END_REF][START_REF] Mcfarland | ELECTROENCEPHALOGRAPHIC (EEG) CONTROL OF THREE-DIMENSIONAL MOVEMENT[END_REF] [W. Wang et al., 2013[START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF]. However, these tasks are simple and do not look like the complex everyday life actions that should be performed by the patient. Pursuit tasks were used for decoding performance evaluation as they seem more similar to the daily-life patient-environment interactions than center-out tasks.

Patients must be able to control freely an effector to perform several tasks from the same decoder and have strong idle state control.

Finally, the decoding performance of the patient should be optimized and not degrade across time. This challenge is particularly difficult knowing the brain signal variability across time. Moreover, during online closed-loop experiments, neural signals are highly dependent on the sensory feedback provided to the patient through the control of the effector. Therefore, the patient should be integrated into the model calibration procedure ("human-in-loop") in order to improve the decoding performance. A strategy which already highlighted good results for stable and accurate neural signal decoding is the adaptation of the decoder during closed-loop experiments. Closed-loop decoder adaptation (CLDA) leads to different model parameter convergence, better performance compared to decoders trained offline during open-loop experiments [START_REF] Lebedev | Brain-Machine Interfaces: From Basic Science to Neuroprostheses and Neurorehabilitation[END_REF], 2006[START_REF] Murphy | Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients[END_REF]] [Orsborn et al., 2014], easier/faster training procedure [START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF] and allows the model adaptation to the variations of the neural signals across time.

Based on the first successful long-term (more than 36 months) chronic exploitation of bilateral epidural ECoG recordings in a tetraplegic individual, new BCI decoders were designed. The Recursive Exponentially Weighted Markov Switching Model (REW-MSLM), the Penalized Recursive Exponentially Weighted n-way Partial Least Square (PREW-NPLS), the Automatic 𝜆 PREW-NPLS (APREW-NPLS) and the Hierarchical structured Hidden Markov Model (H2M2) are incremental adaptive decoders created in order to control in real-time an asynchronous multi-limb effector.

The REW-MSLM is a piecewise linear model based on a mixture of experts (ME) architecture composed of several continuous "expert" models decoding continuous movements from the neural signals and a dynamic "gating" model activating or inhibiting the expert continuous outputs. The REW-MSLM algorithm is an adaptive mixture of experts algorithm where every expert and gate models are independently and incrementally updated to perform an online closed-loop decoder adaptation and allow the decoder and the patient to learn from each other (human-in-loop calibration).

The REW-MLSM has a specific architecture, each expert calibration procedure is independent. This highly flexible REW-MSLM structure allowed to perform the expert parameter weight estimation with different algorithms for each expert in order to fit the best model to each task. Therefore, new algorithms were designed in order to be integrated into the REW-MSLM algorithm and provide new properties to the gate and expert models of REW-MSLM.

PREW-NPLS and APREW-NPLS are two adaptive group-wise sparse decoders designed to reduce the feature space dimension, improve the model interpretability, create low computational cost models suited for portable applications and be integrated in the REW-MSLM algorithms as sparse gating and/or expert models. The Lp-PREW-NPLS is a new regularized recursive exponentially weighted N-way PLS designed for online adaptive decoding promoting group-wise (slice-wise) sparsity generalized to L0, L0,5 and L1 norm/pseudo-norm regularization. Lp-PREW-NPLS group-wise sparse regularization was proposed to prevent overfitting, to improve the decoding performance and to simplify the model interpretation compared to the REW-NPLS algorithms. The Automatic PREW-NPLS (APREW-NPLS) was designed in order to estimate the penalization hyperparameter during the incremental online calibration. APREW-NPLS overcame the drawbacks of PREW-NPLS which required to determine the best penalization hyperparameter during offline preliminary studies before its use into online closed-loop experiments.

Finally, H2M2 dynamic classifier was designed to create a gating model with a high decoding responsiveness and low latency state transitions.

Offline evaluation of the developed algorithms

Before to integrate the new decoders into the ABSD software chain of the clinical trial to perform online adaptive multi-limb experiments, the developed algorithms were evaluated in several pseudo-online comparative studies.

REW-MSLM offline evaluation

In the REW-MSLM offline pseudo-online study, the interests of adding a dynamic gating model in order to inhibit or enhance the output variables of the continuous decoders were demonstrated. Continuous REW-NPLS decoder was not able to perform non-zero velocity predictions as numerous regression algorithms. The integration of a gating model allowed avoiding the non-intended movements from the non-controlled limb which should significantly reduce the stress and mental task complexity of the subject. Moreover, the significant result variations between static and dynamic gating in term of pure decoding performance (accuracy and F-score) and in term of misclassified sample distribution (the frequency rate of the error blocks) attested the needs to integrate a dynamic classifier into the decoder in order to perform asynchronous multi-limb experiments.

In the REW-MSLM offline study, the results induced the benefits of cross-session training in order to obtain a better decoder more robust to the variation of the neural signals and the experimental condition variability. Indeed, the results from dataset C highlighted stable performance whereas the model was trained based on cross-session calibration procedure from dataset B recorded 9 to 28 days before.

Previous ECoG-based direct neural decoding state-of-the-art studies highlighted a correlation between the neural signals and the cursor velocity of 0.48 ± 0.09 [START_REF] Schalk | Decoding two-dimensional movement trajectories using electrocorticographic signals in humans[END_REF], and 0.41 ± 0.14 [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF]. Nevertheless, these results are not fully comparable to the present study. Firstly, the reported experiments were based on subdural ECoG recordings [Bundy et al., 2016[START_REF] Schalk | Decoding two-dimensional movement trajectories using electrocorticographic signals in humans[END_REF] while the presented Ph.D. study was based on epidural ECoG. Epidural ECoG reduces invasiveness and the potential impact of surgical site infection. However, a significant decrease in the decoding performance was highlighted in experiments with non-human primates (NHP) [START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF]Erfanian, 2018] [Schaeffer andAksenova, 2016a] using epidural ECoG recordings compared to subdural ECoG. Finally, state-of-the-art experiments were based on able-bodied epileptic patients performing 2D/3D center-out experiments using actual arm movements [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF] or 2D circular movements based on actual joystick control [START_REF] Schalk | Decoding two-dimensional movement trajectories using electrocorticographic signals in humans[END_REF]. The recordings were processed offline. In the present study, a tetraplegic patient performed online control of a real effector to achieve alternative multiple point-to-point pursuit tasks. Pursuit tasks explore the entire 3D space and are harder than generic center-out task.

The benefits of the REW-MSLM induced during the pseudo-online studies were confirmed during the online-adaptive closed-loop experiments using the online incremental adaptive REW-MSLM decoder.

The PREW-NPLS offline pseudo-online study exhibited the potential benefits of penalized models to decode the neural signals using sparse solutions.

APREW-NPLS offline evaluation

To evaluate the Lp-PREW-NPLS model performance compared to REW-NPLS algorithm, a grid search was performed to create lots of penalized models using three penalization norms/pseudo-norm (L0, L0.5 and L1) and increasing penalization hyperparameter 𝜆. This Lp-PREW-NPLS offline grid analysis results in a total of 558 pseudo-online calibration procedures to achieve for each 3D hand translation study (left and right). This study was highly time consuming and required high computational resources. Moreover, there is no evidence demonstrating that the optimal hyperparameter extracted from the offline pseudo-online study of PREW-NPLS does not change over time and is not influenced by the patient's feedback during the online closed-loop experiments.

To go beyond the presented limitations, the Automatic PREW-NPLS (APREW-NPLS) algorithm was designed. The APREW-NPLS algorithm tests online a set of different PREW-NPLS model configurations (with different hyperparameters) and incrementally updates the models that are most likely to be the best decoding models. With this competition procedure, models with poor decoding performance are not often updated whereas decoders with good performance are put forward. To evaluate the behavior and decoding performance of the APREW-NPLS, a preliminary study was carried out. APREW-NPLS was tested offline for the neural signal decoding of the 3D left and right hand translation task with a set of six possible hyperparameter values. All the six models were trained in the same time on the six training dataset recorded from the online closed-loop calibration procedure achieved for the REW-MSLM estimation. APREW-NPLS models showed similar decoding performance than the original REW-NPLS algorithm with a sparsity index up to 78.13% and 71.88% for the left and right hand models respectively. Additionally, the APREW-NPLS models converged to sparse solutions with parameter weights similar to the one estimated in the offline PREW-NPLS study. The optimization of the model hyperparameters during the online experiments is a powerful tool which can be applied in various BCI applications with numerous different hyperparameters.

The presented APREW-NPLS performance evaluation relies on a preliminary study which requires deeper investigation on, the set of tested penalization hyperparameter, the model ranking algorithm, the calibration procedure, etc. Nevertheless, it is likely that a larger number of hyperparameters to tune may require larger training dataset. A trade-off between the number of penalization hyperparameters and the length of the training dataset must be reached.

H2M2 offline evaluation

Finally, in order to integrate a low latency state transition gating model in REW-MSLM, a new dynamic decoder, named H2M2, inspired by HHMM, was proposed. This algorithm breaks down the classification problem into sub-problems with one classifier dedicated to each sub-problem. A preliminary pseudo-online study based on online closed-loop 4D continuous movements and 5 discrete state virtual avatar experiments were carried out. H2M2 highlighted better responsiveness with a lower latency between the task instruction and the decoding than the HMM gating model. However, the block error rate increased. The H2M2 gating is a model to consider in the case of complex classification problems where the system responsiveness is a high priority characteristic. The benefits of H2M2 algorithm integrated in REW-MSLM as a gating model is highly dependent on the classification problem requirements. A trade-off between the responsiveness, the block error rate and the computational loading must be considered.

Real-time closed-loop BCI experiments

The REW-MSLM decoder highlighted promising results in multiple offline pseudo-online studies. Therefore, the REW-MSLM decoder was integrated into the online neural signal decoding platform ABSD to control complex virtual or real effectors. Using the REW-MSLM algorithm, a tetraplegic patient performed the online highdimensional control of an exoskeleton and a virtual avatar. The patient achieved 8D continuous control including alternative 3D hand reaching tasks and 1D wrist rotation for each hand with 5 discrete states: idle state, left and right hands translation and left and right wrists rotation. The discrete and continuous decoding performance highlighted stable results over 6 months of clinical experiments after the last model recalibration for both effectors.

During the online closed-loop 8D experiments, the REW-MSLM 8D models were trained for each effector based on cross-session calibration procedure during 6 experiments for approximately 3.5 h (with an average of 195 training trials per task). These models training periods and the global number of trials seem moderate considering the number of dimensions to control and the performance obtained compared to those in similar studies [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] [W. Wang et al., 2013]. More training data may lead to a more generalized model and thus, better results.

The online alternative multi-limb 8D pursuit tasks proposed to the patient to control a complex effector are more complicated tasks than the usual state-of-the-art. Generally, 3D arm control is evaluated based on classical center-out experiments [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] [W. Wang et al., 2013[START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF]. Center-out tasks request to go from the center of a workspace to one of the targets localized at equal distance from the initial hand location. During center-out tasks, after each trial (succeeded or failed), the hand position is reset to the initial position after few seconds of rest. In the alternative multi-limb point-to-point pursuit task experiments proposed in the CLINATEC clinical trial, the patient controlled the effector all along the session and no takeover was provided by the system to reset the hand position or to propose a rest period to the patient. Consequently, the control task is more complex because the initial position of the hand in a trial changes constantly, the entire 3D control space is explored and decoding mistake/drifting of the hand from one trial is affecting the following ones.

In the presented online 8D (5 discrete states) asynchronous alternative bimanual experiments, the control of both virtual avatar and exoskeleton was maintained far above the chance level, without recalibration, over 167 days and 203 days for the exoskeleton and virtual avatar effectors, respectively. The 8D experiments were carried out between 468 and 698 days after the implantation of the WIMAGINE electrodes. These results highlighted the stability of both the REW-MSLM decoder and the neural activity recording method with the two WIMAGINE epidural ECoG recording implants [START_REF] Mestais | WIMAGINE: Wireless 64-Channel ECoG Recording Implant for Long Term Clinical Applications[END_REF]. Additionally, these results demonstrated that CLINATEC epidural ECoG-driven BCI outperformed the state-of-the-art ECoG-based BCIs, gets closer to MEAs in terms of decoding performance and outpaced both the state-of-the-art ECoG and MEAs-based BCIs in terms of decoder stability.

Generally ECoG-driven BCI studies were mainly performed on temporary ECoG grid implantation from 3 to 35 days [START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Leuthardt | A brain-computer interface using electrocorticographic signals in humans[END_REF][START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF][START_REF] Schalk | Two-dimensional movement control using electrocorticographic signals in humans[END_REF], 2007[START_REF] Schalk | Brain-Computer Interfaces Using Electrocorticographic Signals[END_REF][START_REF] Volkova | Decoding Movement From Electrocorticographic Activity: A Review[END_REF] [W. Wang et al., 2013[START_REF] Yanagisawa | Electrocorticographic control of a prosthetic arm in paralyzed patients[END_REF]. The online closed-loop success rate (SR) for both effectors from 0 to 37 days after the last model calibration highlighted an averaged hit score of 83% and 97% for the 3D hand translation and 1D wrist rotation tasks (averaged on both hands). The 3D hand translation results are similar to the best patient's 3D decoding performance of the current ECoG-driven BCI state of the art [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF]. However, the decoding performance of the referred ECoG-driven BCIs state of the art were evaluated during online 3D center-out experiments which are easier to complete than the alternative point-to-point pursuit tasks used to evaluate the REW-MSLM decoding performance. Additionally, after the completion of the calibration experiments, the REW-MSLM models used to control the virtual avatar and the exoskeleton were fixed for 167 days and 203 days without any model recalibration. In the opposite, in the mentioned state-of-the-art BCI experiments [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF], the model was often recalibrated between test experiments.

The online closed-loop results presented a high stability level and was far above the realized chance level study across all experiments for both effectors. For the exoskeleton experiments, the left hand translation SR seemed to decay between the 37 th and the 104 th day and stabilize until the end, whereas the right hand translation SR showed higher variability in the performance (between 17% and 100%). For discrete decoding, it was noticeable that switching from the left arm control to the right arm control (and vice versa) represented less than 1% of the errors. Most of the decoding misclassifications were related to two issues. First, the majority of the mistakes were related to false positive idle state activation. Secondly, the decoders struggled to differentiate rotation and translation from the same body side. This difficulty may be related to the similarity of both tasks and consequently lead to brain neural signal pattern activations within a close proximity.

The results seem to demonstrate higher average performance using the exoskeleton than using a virtual avatar effector. Nevertheless, it is worth mentioning that the calibration and test procedures of the models dedicated to the virtual avatar and the exoskeleton control were not performed in the same manner. The 6 calibration experiments of the virtual avatar model were performed during the same week whereas test experiments of the virtual avatar model were carried out weekly at a high frequency. Conversely, the calibration experiments of the model dedicated to the exoskeleton control were distributed in 2 months and the test experiments were less common than virtual avatar test experiments. The lower frequency of exoskeleton experiments compared to virtual avatar experiments may explain the higher variability in the performance using the exoskeleton compared to the virtual avatar. It is likely that a higher frequency of experiments is beneficial for patient's training and control. Finally, all the pseudo offline and online closed-loop BCI experiments previously presented confirmed in a long term study that direct neural decoding is not limited to individual neuron action potential driven (MEA-based) BCIs and can be achieved from population neuron recordings, particularly from epidural ECoG neural signals. This result challenges the empirical evidence that population neuron recordings are limited to the control of fewer dimensions partly due to lower spatio-temporal resolution and the restricted number of possible somatotopic remapping combination [START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF].

Limitations

While the presented Ph.D. results relied on the proof of concept that a tetraplegic patient can control a complex multi-limb effector using a direct neural decoding strategy, several limitations of the Ph.D. research should be mentioned.

Firstly, all the experiments were carried out on a single patient. The results obtained with one patient must be generalized to other disabled subjects.

The current manuscript reports long-term stable performance of high dimensional control of multi-limb exoskeleton and its avatar. While the experiments demonstrated encouraging results, only alternative bimanual control was performed due to the experimental paradigm.

While the current study reported an experimental paradigm with a better exploration of the 3D control space and less restrictive experimental conditions than the traditional center-out tasks, experiments closer to domestic, urban, and professional environments should be designed to move further the technology from clinical trials toward to daily life applications.

Although the Ph.D. research was focused on online closed-loop brain signals decoding, PREW-NPLS, APREW-NPLS and H2M2 algorithms were only tested offline using pseudo-online procedure. Pseudo-online procedures are a good way to test algorithms dedicated to incremental closed loop decoder adaptation, nevertheless, the results obtained are not fully generalizable and the algorithms must be tested during online experiments to bring definitive conclusions of the decoding performance. Indeed, algorithms designed for "human-in-loop" integration in the calibration process will always be difficult to analyze offline.

Finally, it is worth mentioning that the majority of the Ph.D. manuscript was focused on one specific algorithm family. Indeed, The REW-MSLM gating and expert models were estimated with the REW-NPLS algorithm. Other online adaptive decoders reported in BCI studies may be evaluated to be integrated into the REW-MSLM sub-models such as adaptive SVM or LDA for the gate and online adaptive Kalman filter for the experts.

Limitations 279

Moreover, the pseudo-online comparative studies performed to appraise the decoding performance improvements of the new decoders were limited to performance comparison with decoders from the NPLS algorithm family. In order to stress the benefits of the PREW-NPLS, APREW-NPLS and H2M2 algorithms, a more representative comparative study with other adaptive real-time decoders should be considered.

During the 8D online closed-loop experiments, the REW-MSLM calibration duration was empirically determined. The calibration phase ended when the experimenter decided that the decoding performance was visually sufficiently high. Therefore, the calibration time was fixed based on subjective criteria and was likely not optimal. The performance differences between 8D virtual avatar and exoskeleton control might be related to the variation in the model calibration procedure. It is likely that the model's calibration during the online experiments and during the pseudo-online experiments were undertrained.

While long-term stable performance of high dimensional (8D) control of multi-limb exoskeleton and its avatar were obtained with REW-MSLM decoder, the 3D left and right hand translation tasks highlighted slow and curved reaching trajectories. These results matched the pseudo-online studies where the cosine similarities of both hand translation tasks were low.

Considering the pseudo-online PREW-NPLS study, the best penalization hyperparameter and electrodes to penalized are probably different during offline and online calibration. Therefore, it is difficult to conclude on the best penalization hyperparameter to select. Moreover, the study was only focused on the spatial feature regularization whereas time and frequency features could be also evaluated.

The APREW-NPLS evaluation was limited to a preliminary study with a reduced number of penalization hyperparameters. Moreover, a better selection of the tested penalization hyperparameters could have been done. Indeed, the L1-PREW-NPLS study highlighted that the penalized models with 𝜆 = 0.4, 0.5 and 0.6 should lead to identical models. Therefore, the ranking estimated between the models in APREW-NPLS might be biased.

Similarly to APREW-NPLS, H2M2 was limited to a pseudo-online preliminary study.The study was carried out with small training and testing dataset limiting the possible interpretation of the results. Moreover, the state distribution in the training and testing dataset was very different which might affect the decoding performance.

Perspectives

From the hereby-reported studies, several investigations should be carried out to address the previously presented limits and respond to the questions raised in the study.

While the presented results were focused on only one patient, the CLINATEC clinical trial "BCI and Tetraplegia" is still ongoing. The "BCI and Tetraplegia" protocol planned the inclusion in the clinical trial of a total of five patients. A new subject was included in the clinical trial protocol in late 2019 and was implanted the November 19 th 2019. Since, the new patient started training and his control performance will be evaluated in future studies.

The 8D multi-limb experiments performed with the REW-MSLM algorithm highlighted encouraging results but were limited to alternative bimanual control as previously mentioned. However, simultaneous bimanual control is theoretically possible thanks to REW-MSLM soft gating strategy: the gating is not a selection of one limb among the others but the mixing of all of them depending on the limb activation probability computed by the HMM gating. REW-MSLM is not limiting to perform simultaneous bimanual effector control. This bimanual experimental paradigm is the nearest perspective of the study. In a more general guideline perspective, the future experimental paradigms should attempt to get closer to domestic, urban, and professional applications with new experiments including bimanual control, grasping control etc.

In the near future, PREW-NPLS, APREW-NPLS and H2M2 algorithms should be tested during online closed-loop experiments to evaluate their performance with an incremental CLDA procedure.

Offline comparative study between PLS algorithm and other state-of-the-art decoders were reported based on able-bodied subjects' and NHPs' ECoG neural signals [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]. For the continuous decoding, PLS was compared to Principal Component Regressions (PCR) with different settings (a hyperparameter defined the percentage of input variable variance explained 20%, 40%, 60% and 80%) and penalization regression algorithms (LASSO) [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]. The comparative study highlighted that PLS and LASSO algorithms outperformed PCR and that LASSO training was computationally more expensive than the PLS algorithm [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]. Therefore, in the reported study [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF], PLS was preferred for continuous output variable decoding. For the discrete decoder comparative study, LDA, QDA, Logistic Regression (LR), linear and non-linear SVM coupled with PCA or PLS dimension reduction algorithms were tested. PLS-based decoders outperformed the decoders using PCA dimensional reduction algorithms. In the preclinical dataset, PLS coupled with LR outperformed the other decoders. In the clinical dataset based on able-bodied subjects' ECoG neural signals, PLS-LR decoder was outpaced by the LASSO-LR algorithm but required less computational resources. Although the comparative study [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF] provides an initial overview on the expected decoding performance of the PLS algorithm family, it cannot be fully transposed to the Ph.D. application case. Therefore, new comparative studies between state-of-the-art decoders should be performed [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF].

Considering the current REW-MSLM algorithm integrated in the online ECoG-drive BCI of the clinical trial, several opportunities for improvements should be investigated. As mentioned, the previously trained models were fixed without determining an optimal training time and were probably undertrained. The model should be trained for a longer time in the future to accumulate more information and evaluate the impact of a larger dataset on online closed-loop performance. Additionally, studies evaluating the impact of experiment frequency on the decoding performance, the model stability and the patients' adaptations should be performed for a better understanding of the model calibration procedure. Model interpretation and convergence will be further investigated too.

Additionally, each expert is trained with independent dataset which allow removing, changing or adding new experts to the REW-MSLM structure. This model structure enable adding the experts from multiple models calibrated with different experiments in order to simplify the model initialization and the initial control provided to the patient. Gathering the experts from different experiments and models may shorten the model calibration procedure, as it only requires updating the gating model to enable the switch to the newly added expert. Additionally, gate or expert models calibrated to perform neural signal decoding for the same task may be merged to create a new more general model with higher stability. The benefits that the REW-MSLM flexible structure could bring to the calibration procedure using strategies such as grouping models from different tasks or merging sub-models calibrated on the same task should be evaluated.

To improve the decoding performance with faster and less curved reaching trajectories, the integration of other algorithms in the REW-MSLM may be investigated. Various post-processing strategies could be integrated in ABSD to achieved smoother and more straightforward trajectory decoding. Additionally, variations of the current REW-NPLS algorithm integrating penalization or non-linear kernels should be tested in order to improve the decoding performance.

Several articles highlighted in preclinical experiments the benefits of using several linear models calibrated on different movement phases to improve the reaching decoding performance [START_REF] Kang | Decoding of finger, hand and arm kinematics using switching linear dynamical systems with pre-motor cortical ensembles[END_REF]] [Kim et al., 2003] [Yu et al., 2007]. As an example of future REW-MSLM improvements, an alternative Mixture of Expert architecture with states associated to movement phases could be explored [START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF].

PREW-NPLS pseudo online study was limited to spatial feature penalization. In order to evaluate the potential decoding performance improvements of the decoder with a norm penalization on other modalities, PREW-NPLS algorithm with group-wised Chapter 11 : Conclusion, Limitations, Perspectives regularizations applied to the frequency and temporal modalities should be studied.

Additionally, the created models should be interpreted in order to extract relevant information on the prevalent frequency bands for neural signal decoding.

New settings as well as the ranking and exploitation-exploration algorithms must be explored to enhance the APREW-NPLS performance. The APREW-NPLS algorithm methodology can be modified depending on the BCI system requirements. The presented preliminary study focused on the decoding performance to evaluate the best model of the set of hyperparameters, nevertheless, other indicators such as the sparsity index can be integrated as criteria in the model ranking algorithm. Moreover, the exploitation-exploration algorithm selecting the models to calibrate during the next update increment can be tuned or totally changed to obtain more or less greedy algorithms. Another possibility is to remove from the possible set of tested hyperparameters the worse configurations to evaluate new settings. Such procedure had already been reported with racing/competitive algorithms (ROAR, F-race family, PaRaMILS, etc.) in other fields than the BCI research area [Hutter et al., 2011b[START_REF] Hutter | ParamILS: An automatic algorithm configuration framework[END_REF]. With the discarding procedure, the set of tested hyperparameters is not limited to a finite number of configurations and can explore continuously different hyperparameter settings. Indeed, the algorithm can be extended to an infinite number of penalization hyperparameter with the only restriction than λ ∈ [0,1]. A set of finite penalization hyperparameter could be selected at the model initialization. Using the presented performance/ranking procedure, a new penalization hyperparameter set can be computed from the best penalization hyperparameters through optimization strategy (e.g. gradient descent). The best penalized models could be preserved whereas penalized models with low performance could be discarded from the explored set of possible penalization hyperparameters. Additionally, models with L1, L0.5 and L0 norm penalization can be calibrated and compared in the same APREW-NPLS algorithm as the penalization type is not constrained during the selection. Finally, at the end of the calibration phase, the APREW-NPLS algorithm disposed of several models with different performance and rank. Instead of using only the best model to decode the neural signals, a prediction of the intended movements can be estimated from all the penalized models. The output variables of each model could be weighted depending on their rank in order to create a prediction based on soft voting ensemble.

Finally, a deeper examination of the benefits of H2M2 classifier will be conducted. Larger training and testing dataset will be acquired in order to perform more reliable performance comparison in offline pseudo-online studies before integrating the H2M2 classifier in REW-MSLM as the gating model. Additionally, H2M2 should be evaluated on more complex tasks with numerous possible state transitions and on applications where the classifier responsiveness has priority compared to the error block frequency rate.

In three years of experiments with chronic ECoG neural signals, CLINATEC built a unique database of BCI experiments. During these experiments, numerous decoding models were calibrated. All the previously trained REW-MSLM models should be analyzed in order to extract information of the general convergence of the models which could be used for prior knowledge initialization of the models.

While the somatotopic remapping (MI) approach may be sufficient for BCI functional compensation, direct neural decoder is mandatory for BCIs dedicated to the rehabilitation of individuals suffering from severe motor disabilities. Direct neural control based on semi-invasive recording systems (epidural ECoG recordings) may open new perspectives for medical BCI applications.

Finally, the REW-MSLM decoder translates the motor cortex activity into commands based on ECoG neural signals recorded with the WIMAGINE implants. The application of the decoder could be considered for other applications possibly requiring the recording of the neural signals from other brain regions. The decoder could also be applied in other medical fields such as, for instance, seizure detection for epileptic patients.

Innovative decoding algorithms for Chronic ECoG-based Brain Computer Interface (BCI) for motor disabled subjects in laboratory and at home

Brain-computer interfaces (BCIs) are systems that allow the control of external devices from the brain neural signal recordings without neuromuscular activation. Among the various applications, functional compensation and rehabilitation of individuals suffering from severe motor disabilities (motor BCIs) has always been a focus for BCI research. Relying on the "BCI and Tetraplegia" clinical trial of CEA/LETI/CLINATEC, the present doctoral thesis aim to address the challenges related to motor BCIs control of multi-limb effectors namely asynchronous multi-limb BCI and online closed-loop decoder adaptation. The algorithm Recursive exponentially weighted Markov switching multi-linear model (REW-MSLM) integrating an online incremental closedloop decoder adaptation procedure was designed to handle high dimensional multi-limb effector control. REW-MSLM is based on a Mixture of Experts (ME) architecture composed of several continuous "expert" models decoding continuous movements from the neural signals and a dynamic "gating" model activating or inhibiting the expert continuous outputs. The continuous expert models were evaluated using the Recursive Exponentially Weighted N-way Partial Least Squares (REW-NPLS) algorithm whereas the gating model is a Hidden Markov Model (HMM). REW-MSLM allows a tetraplegic patient, who underwent bilateral epidural electrocorticographic arrays (ECoG) implantation of chronic wireless implants (WIMAGINE), to perform the 8D control of a whole body exoskeleton over 6 months without model recalibrations. During this period, the patient was able to perform alternative 3D left and right hand translations and 1D left and right wrist rotations. For the experiments carried out from 0 to 37 days after the last model calibration experiments, the decoding performance highlighted a hit score of 71 ± 12% and 99 ± 2% for the 3D hand translation and 1D wrist rotation tasks whereas the dynamic classifier showed a five-state classification F-score performance of 77 ± 14% . For the experiments performed from 0 to 167 days, the decoding performance highlighted hit scores of 67 ± 21% and 93 ± 12% for the hand translation and wrist rotation tasks whereas the dynamic classifier demonstrated a fivestate classification F-score performance of 75 ± 12% . Additionally, other algorithms were proposed and tested offline in order to be potentially integrated into the REW-MSLM, namely the (Automatic) Penalized REW-NPLS algorithms (PREW-NPLS and APREW-NPLS) and a version of Hierarchical Hidden Markov Model (H2M2). PREW-NPLS is an incremental adaptive regularized NPLS algorithm promoting group-wise sparse solutions using Lp (p=0,0.5,1) norm/pseudo-norm penalization. APREW-NPLS is an automatic version of PREW-NPLS. It compares, based on reinforcement learning strategies, several penalized models with different penalization hyperparameters during the closed-loop experiments in order to optimize the best penalization hyperparameter in real-time. Both PREW-NPLS and APREW-NPLS algorithms were designed to reduce the feature space dimension and to improve the decoding performance. H2H2 is a dynamic classifier with a hierarchical architecture designed to promote decoding responsiveness and low latency state transitions. The proposed algorithms allowed performing asynchronous online direct neural decoding from epidural ECoG population recording system.

Such results may open new application perspectives.

Une interface cerveau-machine basée sur des algorithmes de décodage innovants pour le contrôle d'effecteurs complexes en vue d'un usage au quotidien par des patients en situation de handicap moteur Lp-PREW-NPLS est un algorithme incrémental adaptatif permettant une pénalisation par groupes du modèle de décodage suivant la norme/pseudo norme Lp=0, 0.5 ou 1. Pour une utilisation en temps réel, l'algorithme PREW-NPLS nécessite de déterminer avec une étude offline préliminaire l'hyperparamètre de pénalisation optimal. APREW-NPLS permet de comparer en temps réel plusieurs modèles avec des hyperparamètres de pénalisation différents afin de l'optimiser en temps réel durant l'expérience. Les algorithmes PREW-NPLS et APREW-NPLS ont été créés pour réduire la dimension de l'espace des caractéristiques et améliorer les performances de décodage. H2M2, quant à lui est un classifieur dynamique similaire aux modèles de type HMM mais avec une structure hiérarchique. La structure hiérarchique est répartie en couches avec les états des couches inférieures dépendants des états des couches supérieures. L'algorithme H2M2 a été conçu dans le but d'améliorer la réactivité du modèle de classification de REW-MSLM (gating). Les résultats de décodage neural direct des signaux épiduraux ECoG obtenus nous poussent à diversifier l'utilisation de ces algorithmes à d'autres domaines.

Mots-clés :

Figure 1

 1 Figure 1-1: Definiton of a tensor. A) Example of third order tensor: 𝑿 ∈ ℝ 𝐼 1 ×𝐼 2 ×𝐼 3 . B) Rank one 3mode tensor example.

  Chapitre 1 INTRODUCTION.................................................................................................... 1.1. Basic Neuroscience for BCI ........................................................................................ 1.2. BCI components .......................................................................................................... 1.2.1. Acquisition systems .................................................................................................. 1.2.2. Neural signal processing ....................................................................................... 1.2.3. Effectors .................................................................................................................... 1.2.4. Feedback ................................................................................................................... 1.3. BCI control strategies: somatotopic remapping versus direct neural decoding 19 1.3.1. Exogenous BCIs....................................................................................................... 1.3.2. Endogenous BCIs .................................................................................................... 1.4. Motor BCI requirements for daily life applications ............................................. 1.4.1. Acquisition system requirements ......................................................................... 1.4.2. control system requirements ................................................................................. 1.4.3. Decoder requirements ............................................................................................ 1.5. BCI project at CLINATEC........................................................................................ 1.5.1. Concept and context ............................................................................................... 1.5.2. Principles / Approaches ......................................................................................... 1.5.3. Project specific requirements ................................................................................ 1.6. PhD goals ................................................................................................................... 1.6.1. Real-time application ............................................................................................. 1.6.2. Asynchronous multi-limbs decoder ..................................................................... 1.6.3. Online incremental adaptive decoder .................................................................. 1.7. Manuscript organization ......................................................................................... Chapter 2 Scientific context: BCI and Tetraplegia clinical trial .......................................... 2.1. Inclusion criteria of the clinical trial ....................................................................... 2.2. Participants of the clinical trial ............................................................................... loop full-state transition 4D experiments using REW-MSLM decoder ................................................................................................................ Chapter 8 Algorithms integration into BCI Adaptive platform ....................................... 8.1. Pre-processing ......................................................................................................... 8.2. Feature extraction ................................................................................................... 8.3. Post-processing........................................................................................................ 8.4. CLDA procedure integration in the BCI adaptive platform ............................. 8.

Figure 1 - 1 :Figure 1 - 2 :

 1112 Figure 1-1: Definiton of a tensor. ........................................................................................ XXXVIII Figure 1-2:Example of third-order tensor unfolding ..............................................................XXXIX Figure 1-1 : Evolution of the BCI research field across years and domain of application. ............

Figure 1 - 2 :

 12 Figure 1-2: Illustration of the common neuron structure .............................................................

Figure 1 - 3 :

 13 Figure 1-3: Brain anatomy and functional topographic organization of the motor cortex ...........

Figure 1 - 4 :

 14 Figure 1-4: Common Brain Computer Interface (BCI) architecture ...............................................

Figure 1 - 5 :

 15 Figure 1-5: Invasiveness and resolution of BCI recording systems ................................................

Figure 1 - 6 :

 16 Figure 1-6: Summary of the advantages and disadvantages of the BCI recording systems ........

Figure 1 - 7 :

 17 Figure 1-7: BCI signal transducer sub-steps .................................................................................

Figure 1 - 8 :

 18 Figure 1-8: Example of motor imagery strategy carried out during clinical BCI experiments ....

Figure 1 - 9 :

 19 Figure 1-9: Motor imagery strategy achieved during online BCI clinical experiments using ECoG recording system .........................................................................................................................

Figure 1 - 10 :

 110 Figure 1-10: Direct neural decoding principles ............................................................................

Figure 1 - 11 :

 111 Figure 1-11: average spectrogram of center out task based on LFP, ECoG, EEG and MEG recordings during direct neural decoding experiment ................................................................

Figure 1 -

 1 Figure 1-12: Synchonous and Asynchronous BCI system principle .............................................

Figure 1 -

 1 Figure 1-13: Clinatec BCI platform with the four main pillars of the project : imagine, monitor, decode and control [Eliseyev et al., 2014]...................................................................................

Figure 1 -

 1 Figure 1-14: Main requirements of the Clinatec BCI project "BCI and Tetraplegia". ..................

Figure 2 -

 2 Figure 2-1: CLINATEC "BCI and Tetralegia" specific BCI platform ................................................

Figure 2 - 2 :

 22 Figure 2-2: Typical sheet for American Spinal Injury Association Impairment (ASIA) score evaluation and typical muscular group functions with the corresponding vertebrae ................

Figure 2 - 3 :

 23 Figure 2-3: Offline study achieved to localize the sensory motor cortex of the patient before the surgery ...................................................................................................................................

Figure 2 - 4 :

 24 Figure 2-4: details of the ASIC CINESIC32 integrated into the wireless implant WIMAGINE ......

Figure 2 - 5 :

 25 Figure 2-5: Global recording chain composed of the helmet worn by the patient .....................

Figure 2 - 6 :

 26 Figure 2-6: Clinatec BCI platform for real-time BCI experiments ................................................

Figure 2 - 7 :

 27 Figure 2-7: Real and virtual effectors available in the BCI and tetraplegia project .....................

Figure 2 -Figure 2 - 9 :

 229 Figure 2-8: Evolution of the experiment paradigm and controlled DoF of the second patient across time ...................................................................................................................................

Figure 2 - 10 :

 210 Figure 2-10: Experimental paradigm for online BCI control of the exoskeleton in a sitting position ........................................................................................................................................

Figure 3 - 1 :

 31 Figure 3-1: Feature selection family schematics .........................................................................

Figure 3 - 2 :

 32 Figure 3-2: The different closed loop decoder adaptation (CLDA) procedure reported in the BCI field ..............................................................................................................................................

Figure 3 - 3 :

 33 Figure 3-3: Impact of the error magnitude, error smoothness, prediction velocity and prediction delay on the closed-loop continuous decoding performance .................................

Figure 4 -

 4 Figure 4-1: Markov Switching Linear Model (MSLM) structure ................................................

Figure 4 - 2 :

 42 Figure 4-2: REW-NPLS main loop principles ..............................................................................

Figure 4 - 3 :

 43 Figure 4-3: Recursive Validation procedure is performed to evaluate to optimal latent space dimension ..................................................................................................................................

Figure 4 - 4 :

 44 Figure 4-4: Recursive exponentially weighted Markov-switching linear model (REW-MSLM) architecture ...............................................................................................................................

Figure 5 - 1 :

 51 Figure 5-1: Difference between element-wise and group-wise regularization .........................

Figure 5 - 2

 52 Figure 5-2 Example of a PARAFAC decomposition of a 3-order tensor .....................................

Figure 5 - 3 :

 53 Figure 5-3: PARAFAC-inspired tensor decomposition used in the REW-NPLS algorithm using alternative least square (ALS) algorithm ...................................................................................

Figure 5 - 4 :

 54 Figure 5-4 : Cubic polynomial function related to the evaluation of 𝑤𝑗1𝐿0.5 ..........................

Figure 5 - 5 :

 55 Figure 5-5 : Penalized REW-NPLS (PREW-NPLS) algorithm ........................................................

Figure 5 - 6 :

 56 Figure 5-6: Automatic 𝜆 Penalized REW-NPLS (APREW-NPLS) algorithm .................................

Figure 5 - 7 :

 57 Figure 5-7: Recursive-Validation procedure based on multi-arm bandit problem for penalized model selection optimization ....................................................................................................

Figure 6 - 1 :

 61 Figure 6-1: Hierarchic hidden markov model structure ............................................................

Figure 6 - 2 :

 62 Figure 6-2: Structure of the proposed H2M2 algorithm ............................................................

Figure 6 - 3 :

 63 Figure 6-3: Illustration of the H2M2 sub-models ......................................................................

Figure 7 -

 7 Figure 7-1: WIMAGINE is an active implantable epidural ECoG recording system composed of 64 electrodes .............................................................................................................................

Figure 7 - 2 :

 72 Figure 7-2: Available degree of freedom controlled by the patient in the experiments ..........

Figure 7 - 3 :

 73 Figure 7-3: Examples of 4D alternative multi-limb pursuit tasks...............................................

Figure 7 - 4 :

 74 Figure 7-4: Representation of the three paradigms designed for the pseudo-online REW-MSLM decoder evaluation ....................................................................................................................

Figure 7 - 5 :

 75 Figure 7-5: Timeline of the calibration and test sessions ..........................................................

Figure 8 - 2 :

 82 Figure 8-2: Feature extraction procedure .................................................................................

Figure 8 - 3 :

 83 Figure 8-3: Feature extraction for supervised CLDA procedure ................................................

Figure 8 - 4 :

 84 Figure 8-4: Data labelling for CLDA ............................................................................................

Figure 8 - 5 :

 85 Figure 8-5: REW-MSLM architecture exploitation for model initialization ...............................

Figure 9 - 1 :

 91 Figure 9-1: Algorithms compared to evaluate the impact of dynamic gating algorithms for online closed-loop multi-limb experiments...............................................................................

Figure 9 - 2 :

 92 Figure 9-2 : Algorithms compared to evaluate the impact of expert-specific subset training strategy for online closed-loop multi-limb experiments ...........................................................

Figure 9 - 3

 93 Figure 9-3 Three dynamic performance indicators were evaluated .........................................

Figure 9 -

 9 Figure 9-4 : A: online experiment performance indicators .......................................................

Figure 9 - 5 :

 95 Figure 9-5: Averaged time-frequency responses during intended state activation ..................

Figure 9 - 6 :

 96 Figure 9-6: Pseudo-online PREW-NPLS and APREW-NPLS performance studies ......................

Figure 9 - 7 :

 97 Figure 9-7: Pseudo-online evaluated HMMs with different architectures ................................

Figure 10 - 1 :

 101 Figure 10-1 : State decoding results obtained during pseudo online experiments ..................

Figure 10 - 2 :

 102 Figure 10-2: Continuous decoding performance for each hand for datasets A, B and C ..........

Figure 10 - 3 :

 103 Figure 10-3: Discrete decoding performance during the online experiments for the virtual avatar or the exoskeleton effectors ..........................................................................................

Figure 10 - 4 :

 104 Figure 10-4: Online experiment performance across several months for the virtual avatar and exoskeleton effectors, correspondently ....................................................................................

Figure 10 - 5 :

 105 Figure 10-5: Example of session realized 106 days after the last model calibration using exoskeleton effector ..................................................................................................................

Figure 10 - 6 :

 106 Figure 10-6: Example of a gating model ....................................................................................

Figure 10 - 7 :

 107 Figure 10-7: Expert model estimated during online experiments with CLDA ...........................

Figure 10 - 8 :

 108 Figure 10-8: Modulation of the neural signal activity between idle state and Left or right hand translation active states ............................................................................................................

Figure 10 - 9 :

 109 Figure 10-9: Modulation of the neural signal activity between idle state and Left and right wrist rotation active states .................................................................................................................

Figure 10 - 10 :

 1010 Figure 10-10 : Modulation of the neural signal activity between idle state and active states focused on specific electrodes. .................................................................................................

Figure 10 - 11 :

 1011 Figure 10-11 : Sparsity evolution depending on the hyperparameters 𝜆 and 𝑓 .......................

Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 10 - 16 :Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 10 -Figure 11 - 1 :

 101010101016101010101010101010111 Figure 10-12 : The model performance indicators of the Lp REW-NPLS algorithm for left hand movement decoding ..................................................................................................................

Figure 1 - 2 :

 12 Figure 1-2: Illustration of the common neuron structure ["Neuron," 2020]. The neuron is a nerve cell which transmits electrical information through the brain. Previous neuron send chemical components released in the synapse. The receptors of dendritic branches of the postsynaptic cell received the chemical component which create a change of potential in the postsynaptic cell membrane.Depending on the chemical components transmitted as well as the postsynaptic cell receptor type, the probability of generating an action potential in the axon hillock is increased or decreased[START_REF] Purves | Neuroscience, 3rd ed, Neuroscience[END_REF]. If an action potential is generated, it is sent through the axon to the synaptic terminals to transmit the information to the next neurons.

Figure 1 - 3 :

 13 Figure 1-3: Brain anatomy and functional topographic organization of the motor cortex [Purveset al., 2004].

  [START_REF] Georgopoulos | Coding of movements in the motor cortex[END_REF]. Movements generate a modulation of the brain neural activity which can be recorded through various recording systems. The cortical neurons (pyramidal neurons) of the brain are aligned and interconnected into 6 layers which are firing action potentials. Their dendritic trees and axons are in parallel to each other and perpendicular to the cortical surface [Lopes da[START_REF] Lopes Da Silva | EEG and MEG: Relevance to Neuroscience[END_REF]. During a movement, due to the neurons alignments and pyramidal synchronous action potentials patterns, the signals recorded by the population recording system which are equal to the summation of the cortical neurons signals lead to neural signal modulation in specific frequencies. In general, the recorded brain signals are divided into bandwidths following the notation [Lopes da Silva, 2013] [Schaeffer, 2017]: • Delta (δ) for [0.5 Hz -4 Hz] • Theta (θ) for [4 Hz -7 Hz] • Alpha (α) for [8 Hz -13 Hz] • Mu (µ) for [8 Hz -13 Hz] signals frequency in the central and the parietal areas • Beta (β) for [14Hz -30 to 35 Hz]

Figure 1 - 4 :

 14 Figure 1-4: Common Brain Computer Interface (BCI) architecture. Brain neuronal activity is recorded using diverse acquisition systems and is treated to translate it to orders executed by the effector. A feedback (visual, auditory, proprioceptive, etc.) is provided to patient through effectors response.

Chapitre 1 :

 1 Introductionart experiments with at most 40 days between the implantation and explantation of the ECoG grid.

Figure 1 - 6 :

 16 Figure 1-6: Summary of the advantages and disadvantages of the BCI recording systems. EEG, ECoG and MEA are the commonly used systems to record neural signals for clinical and preclinical BCI and motor BCI experiments. The figure is extracted from[START_REF] Stieglitz | Brain-computer interfaces: an overview of the hardware to record neural signals from the cortex[END_REF]

Figure 1 - 7 :

 17 Figure 1-7: BCI signal transducer sub-steps. The transducer block transforms the neural signal recorded from the patient to coherent command for the effect. Transducer block can be split into pre-processing, neural feature extraction, decoding and post-processing steps.

Figure 1 - 8 :

 18 Figure 1-8: Example of motor imagery strategy carried out during clinical BCI experiments. A) mental task strategy with imagined or realized movements performed to control a 2D cursor in [Schalk et al., 2008]. B) Brain motor cortex activation related to tongue or hand real or imagined movement.

Figure 1 - 9 :

 19 Figure 1-9: Motor imagery strategy achieved during online BCI clinical experiments using ECoG recording system. Motor imagery strategy used in[START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF] to performed 3D center out task control with 8 targets. M1 and M2 were used to control the displacement of the target in the X-Y axis whereas M3 was associated to Z-axis (depth axis). "+" represents attempted movements whereas ø" shows relaxation.

Figure 1 - 10 :

 110 Figure 1-10: Direct neural decoding principles. The firing rate of premotor and motor cortex neurons is correlated to specific movement direction. Each neuron is tuned depending on a preferred movement direction. The figure is extracted from Neuroscience 3rd edition [Purves et al., 2004].

Figure 1 - 11 :

 111 Figure 1-11: average spectrogram of center out task based on LFP, ECoG, EEG and MEG recordings during direct neural decoding experiment. The figure is extracted from [Waldert et al., 2009].

  -based offline individual finger movement reconstruction studies.

Figure 1 -

 1 Figure 1-12: Synchonous and Asynchronous BCI system principle.

  to integrate the neural signals related to patient's feedback into the model calibration are based on closed-loop decoder adaptation (CLDA) procedure. CLDA updates the model parameters using closed-loop experiment dataset. Depending on the decoder and the training strategy, CLDA can be carried out in an offline or online manner at different time scales with a model adaptation occurring every sample, second, minute, trial, session, day, etc. The classic strategy to integrate the neural signals related to the patient's feedback into the decoder calibration is divided into three steps. Firstly, the model parameters are estimated based on open-loop experiment dataset collected with the patient. Then, after the model calibration based on the open-loop data, the model is used during closed-loop experiments. Finally, the model is recalibrated based on the new closed-loop experiments[START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF]. The last step can be repeated during several closedloop experiments to improve the decoding performance of the model. This procedure is long and may lead to suboptimal calibration whereas calibration during ongoing use (in closed-loop) optimizes the quality of control during extended use of the patient.

Figure 1 -1. 5 . 3 .

 153 Figure1-13: Clinatec BCI platform with the four main pillars of the project : imagine, monitor, decode and control[Eliseyev et al., 2014].

Figure 1 -

 1 Figure 1-14: Main requirements of the CLINATEC BCI project "BCI and Tetraplegia".

  following doctoral work was completed within the framework of CLINATEC clinical trial "BCI and tetraplegia" and the motor Brain-Computer-Interface (BCI) project. The presented results were obtained based on the online experiments recorded with one of the patients of the clinical trial from mid-2017 to mid-2020. This doctoral work is mainly focused on the development of new innovative decoders which suited the requirements and objectives of the "BCI and Tetraplegia" clinical trial. The proposed decoders were designed in order to create an online adaptive asynchronous algorithm for high multi-limb effectors control and meet the requirements of the BCI clinical trials stressed in the Figure 1-14.

  . The ongoing clinical trial was cataloged the 11/09/2015 in the publically accessible register named ClinicalTrials.gov, under the identifier: NCT02550522 ["ClinicalTrial NCT02550522"] ["ICTRP clinical trial NCT02550522"]. The clinical trial was approved by the French authorities: "Agence Nationale de Sécurité du Médicament et des produits de santé (ANSM)" with the registration Number: 2015-A00650-49 and an ethical committee (Comité de Protection des Personnes -CPP) with the Registration number: 15-CHUG-19. The informed consent for the clinical trial was obtained from the patient as well as the consent to publish the information/image(s) in online open access publications.

Figure 2 - 2 :

 22 Figure 2-2: Typical sheet for American Spinal Injury Association Impairment (ASIA) score evaluation and typical muscular group functions with the corresponding vertebrae. (A) ASIA determines the functional impairment resulting from a spinal cord injury through a myotomalbased motor examination, dermatomal based sensory examination [Roberts et al., 2017]. Motor examination grades five specific muscle groups in the upper body side and five specific muscle groups in the lower body side using a score going from 0 to 5 [Roberts et al., 2017]. (B) Lesion in a vertebrae lead to muscular deficiency. The figure displays for each vertebrae the muscular groups related to.

Figure 2 - 3 :

 23 Figure 2-3: Offline study achieved to localize the sensory motor cortex of the patient before the surgery. A). Localization of the optimal electrode position before surgery through MEG and fMRI motor Imagery study. B). Bilateral implantation using Image Guided Functional NeuroSurgery of two WIMAGINE implants. C) Localization of the electrodes array after surgery compared to the sensory sulcus (SS) and the motor sulcus (MS) represented in yellow and red curves respectively.

Figure 2 - 4 :

 24 Figure 2-4: details of the ASIC CINESIC32 integrated into the wireless implant WIMAGINE[Robinet et al., 2011]. Signal to noise ratio in improved through the steps of High-pass filtering, amplification, low-pass filtering, etc.

Figure 2 - 5 :

 25 Figure 2-5: Global recording chain composed of the helmet worn by the patient. The helmet integrate a high frequency (HF) antenna for transcutaneous remote power supply and wireless data transfer with the WIMAGINE recording implants. The recorded neural signals are transmitted to a control base station which generates the HF field and sends the recorded neural signals to a computer to start the signal processing steps.

Figure 2 - 6 :

 26 Figure 2-6: Clinatec BCI platform for real-time BCI experiments. Global Software chain (shaded in yellow) allowing to tranform ECoG neural signals into commands for the effectors.

  Home and laboratory experiments bring different feedback to the patient which may lead to different model parameters convergence. Therefore, the models calibrated during experiments inside the laboratory were only used during laboratory experiments. Similarly models created during home-based experiments were not used for laboratory experiments. The incremental training strategy as well as the number of continuous DoF controlled by the second patient across the two first years of experiments is presented in Figure 2-8.

Figure 2 - 8 :

 28 Figure 2-8: Evolution of the experiment paradigm and controlled DoF of the second patient across time. The figure is extracted from [Benabid et al., 2019].

  Fixed lightened LED corresponded to a hand reaching task (with left or right hand translation task depending on the side of the lightened led on the panel. Rotating flashing LEDs corresponded to a wrist rotation task (left or right wrist rotation depending on the side of the lightened LEDs on the panel). The direction of rotation to achieve was the same that the order of the flashing LEDs (clockwise or counterclockwise). Virtual avatar and exoskeleton left hand translation and right wrist rotation tasks are represented in the Figure 2-9.

Figure 2 - 9 :

 29  Example of a trial in a continuous left hand movement task and angular wrist movement task using exoskeleton or virtual effector. During a hand movement task with exoskeleton effector, the target to reach in the trial is showed to the patient using a lighted LED. Virtual avatar effector target of the left hand movement are represented with a red cube whereas it is represented with a blue sphere during right hand movement trial. Wrist rotation task is

Figure 2 - 10 :

 210 Figure 2-10: Experimental paradigm for online BCI control of the exoskeleton in a sitting position. A) Representation of the exoskeleton position in the case of 3D continuous left or right hand control tasks. The panel was placed in front of the exoskeleton while the LEDs were located at the end of the blue and yellow tubes displayed on the figure. B) Representation of the theoretical physical limits of the exoskeleton and the saturation limits imposed during the left and right hand translation tasks in order to avoid any uncomfortable arm position for the patient.

3. 4

 4 .2.1. CLDA with classifiers ............................................................................................. 90 Incremental CLDA classifiers .............................................................................................. Unsupervised CLDA classifiers .......................................................................................... CLDA with classifiers: conclusion ...................................................................................... 3.4.2.2. CLDA for continuous outputs ............................................................................... 93 Incremental CLDA continuous decoders .......................................................................... CLDA for continuous outputs: Conclusion....................................................................... 3.4.2.3. Adaptive feature selection ..................................................................................... 97 3.4.2.4. Alternative strategies .............................................................................................. 98 3.5. Post-processing ................................................................................................................. 98 3.6. Conclusion ...................................................................................................................... 100 Pre-processing 61

  Another solution named feature selection estimates the informativeness of the features to select only a subset of the most relevant characteristics and discard the others. Feature selection family regroups filter-based, wrapper-based and embedded techniques (Figure3-1)[START_REF] Bolón-Canedo | A review of feature selection methods on synthetic data[END_REF][START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Rouhi | Feature Selection in High-Dimensional Data[END_REF]. They were applied in combination or instead of projection algorithms.

Figure 3 -

 3 Figure 3-1: Feature selection family schematics. Feature selection family can be divided into filter (left), wrapper (middle) and embedded (right) methods. All of the strategies presents benefits and disadvantages depending on the application. Schematic extracted from [Bolón-Canedo et al., 2013].

  automatically. From this non-perfect open-loop model calibration, closed-loop experiments can be performed and the model can be re-evaluated using the recorded closed-loop data during online or offline (model fixed during the online application) CLDA procedure. Additional recalibration of the model can be performed several times to optimize the closed-loop decoder using an online or offline CLDA procedure (Figure3-2)[START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF] Carmena, 2014].

Figure 3 - 2 :

 32  The different closed loop decoder adaptation (CLDA) procedure reported in the BCI field. Representation of typical BCI with CLDA calibration workflow from classic decoder using open-loop motor imagery model initialization to the entire system calibrated through closed-loop experiments. Hexagonal and rounded squares represents the offline and online steps respectively. No active involvement of the subjects during the offline steps. Black arrows represent the intervention of an engineer in the opposite to white arrows which represent selfmanaged step. Red, yellow and green color represent the initiation, the calibration and the use of the BCI system. The schematic is inspired and extracted from the study[START_REF] Brandman | Rapid calibration of an intracortical brain-computer interface for people with tetraplegia[END_REF].

Figure 3 - 3 :

 33 Figure 3-3: Impact of the error magnitude, error smoothness, prediction velocity and prediction delay on the closed-loop continuous decoding performance. Figure extracted from the closedloop BCI study[START_REF] Marathe | The impact of command signal power distribution, processing delays, and speed scaling on neurally-controlled devices[END_REF]

  Due to the robustness in the computation of high dimensional data, algorithms of Partial Least Squares (PLS) family were frequently used in continuous and discrete BCI decoding. Numerous publications which reported offline ECoG-based hand trajectory decoding[START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF]] [Chen et al., 2013[START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Eliseyev and Aksenova, 2014] [Schaeffer and Aksenova, 2016b], and EEG-based classification or cursor decoding[START_REF] Maleki | Fast and accurate classifier-based brain-computer interface system using single channel EEG data[END_REF][START_REF] Trejo | Brain-computer interfaces for 1-D and 2-D cursor control: designs using volitional control of the EEG spectrum or steady-state visual evoked potentials[END_REF] sustained the interest of such algorithms.

Figure 4 - 2 :

 42 Figure 4-2: REW-NPLS main loop principles.

Figure 4 - 4 :

 44 Figure 4-4: Recursive exponentially weighted Markov-switching linear model (REW-MSLM)architecture. The REW-MSLM includes a mixture of experts model, which can be described as the parallel computation of several predictions from different regression models (experts) that are weighted (enhanced or inhibited) according to the input variables using a classifier (gate). We hypothesize that the input feature space 𝑋 can be divided into several specific local regions 𝑋 𝑘 and that each sub-space can be fitted using local multilinear functions 𝜑 𝑘 associated with an expert. Multilinear functions 𝜑 𝑘 are estimated using 𝑘 independent REW-NPLS models. The selected expert is determined based on the dynamic gating model. The gating model is a hidden Markov model (HMM) which computes the probability 𝛾 𝑘 for each expert to be activated. Commands are decoded by the REW-MSLM and sent to the effector to provide visual feedback to the patient.

  𝛾 𝑘,𝑡 = 𝑝(𝑧 𝑡 = 𝑘|𝐗 1:𝑡 ) is the dynamic gating weight coefficient associated with the k th expert at time 𝑡. REW-MSLM sub-models are entirely defined through the experts' parameters θ 𝑒 = {θ 𝑒 𝑘 } 𝑘=1 𝐾 = {𝐁𝐞𝐭𝐚 𝑘 , 𝐛𝐢𝐚𝐬 𝑘 } 𝑘=1 𝐾 and HMM parameters θ 𝑔 = {𝐀, {𝑐 𝑘 } 𝑘=1 𝐾 , 𝝅}, where 𝐀 is the transition matrix, 𝐀 = (𝑎 𝑖𝑗 ) ∈ ℝ 𝐾×𝐾 , 𝑎 𝑖𝑗 = 𝑝(𝑧 𝑡 = 𝑗|𝑧 𝑡-1 = 𝑖), {𝑐 𝑘 } 𝑘=1 𝐾

  𝑡 = 𝑘, 𝐗 1:𝑡 ) = 𝑝(𝐗 𝑡 |𝑧 𝑡 = 𝑘) ∑ 𝑎 𝑘𝑗 𝛾 𝑘,

Chapter 5 :

 5 -1B)[Eliseyev et al., Online incremental groupwise sparse REW-MSLM 2012][START_REF] Giordani | Constrained Candecomp/Parafac via the Lasso[END_REF][START_REF] Hastie | Statistical Learning with Sparsity : The Lasso and Generalizations[END_REF][START_REF] Martínez-Montes | PENALIZED PARAFAC ANALYSIS OF SPONTANEOUS EEG RECORDINGS[END_REF] [Y. Zhang

Figure 5 -

 5 Figure 5-1: Difference between element-wise and group-wise regularization. Example of third order tensor sparse evaluation using element-wise regularization (A) and group-wise regularization (B).

Figure 5 - 2

 52 Figure 5-2 Example of a PARAFAC decomposition of a 3-order tensor. The tensor is decomposed in linear combination of 𝑅 vectors outer products.

,

  and ℬ is the solution of the cubic polynomial function (Figure5-4):

Figure 5 - 4 :

 54 Figure 5-4 : Cubic polynomial function related to the evaluation of (𝑤 𝑗 1 ) 𝐿 0.5.

Figure 5 -

 5 5A whereas one loop of wise sparse REW-MSLM the penalized PARAFAC estimated with the ALS algorithm integrated in the PREW-NPLS algorithm is shown in Figure 5-5B. With the exception of the penalized PARAFAC decomposition, PREW-NPLS model calibration is similar to the REW-NPLS algorithm (presented in the Chapter 4). At each iteration 𝑢, a set of 𝐹 models are evaluated with a penalization hyperparameter 𝜆 and is noted 𝜃 𝑢,𝜆 = {𝐁𝐞𝐭𝐚 𝑢 𝑓,𝜆 , 𝐛𝐢𝐚𝐬 𝑢 𝑓,𝜆 } 𝑓=1 𝐹 .

Figure 5 - 5 :

 55 Figure 5-5 : Penalized REW-NPLS (PREW-NPLS) algorithm. (A) PREW-NPLS algorithm main steps with penalized PARAFAC decomposition leading to slice-wise sparse model. (B) Example of the L1-PARAFAC decomposition performed in the case of L1-PREW-NPLS penalization on the space (electrodes) domain with the hyperparameter 𝜆. ALS algorithm is used for decomposition factor estimation.

Figure 5 - 6 :

 56 Figure 5-6: Automatic 𝜆 Penalized REW-NPLS (APREW-NPLS) algorithm. APREW-NPLS algorithm main steps with penalized PARAFAC decomposition leading to slice-wise sparse model with incremental optimization of the best penalization hyperparameter.

Chapter 5 :

 5 Online incremental groupwise sparse REW-MSLM dimension 𝑓 * previously explained. The main RV steps selecting the set of models to update at the next calibration increment are represented in Figure 5-7.

Figure 5 - 7 :

 57 Figure 5-7: Recursive-Validation procedure based on multi-arm bandit problem for penalized model selection optimization. A bank of model performance depending of the penalization hyperparameter is incrementally updated using the Recursive-Validation procedure include in the REW-NPLS algorithm[START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF]. From the performance of each model a ranking of the best models is evaluated depending on a criteria (sparsity, decoding performance, etc.). This model ranking is weighted depending on how long model has not been updated. The models with the best weighted ranks are updated during the next APREW-NPLS algorithm iteration.

Figure 6 - 1 :

 61 Figure 6-1: Hierarchic hidden markov model structure. Each state from a layer 𝑑 is a selfcontained sequential probabilistic model. The activation of an internal state leads to a vertical transition toward a lower layer. In a layer, horizontal transition between the state from a sub-HMM are performed similarly as state with HMM. If a production state is activated, as observation is generated and an upward vertical transition is performed to the related internal state. Sub-HMMs are represented with the grey ovoid shapes. Internal states are presented with grey circles whereas production states are in blue. Horizontal and downward vertical transitions are colored in red whereas upward vertical transition are in blue.

Figure 7 - 2 :

 72 Figure 7-2: Available degree of freedom controlled by the patient in the experiments. The possible movements are translation of the left and right hand in the 3D space and the 1D angular rotation of the left and right wrist. An additional idle state is always available in the experiments. All the experiments performed and analyzed in the manuscript are asynchronous alternative multilimb/bimanual experiments relying on the combination of several of the presented available movements.

Figure 7 - 3 :

 73  Examples of 4D alternative multi-limb pursuit tasks. One session is composed of successive tasks. In this 4D session example, the experiments is composed of three different tasks: 2D left and right hands translation task (referred as ASLH and ASRH) and idle task (IS). Each active task is composed of several trials in which the 2D cursor must reach the proposed targets. The index of the AS tasks is noted with a superscript index, the first ASLH task is noted ASLH 1 whereas the second is referred as ASLH2 . The cursor position is not reset between tasks, during task and during idle state. The last cursor position is not reset and correspond to the position of the cursor before the patient changes its mental state and the model decodes the state transition.

Figure 7 - 4 :

 74 Figure 7-4: Representation of the three paradigms designed for the pseudo-online REW-MSLM decoder evaluation. REW-MSLM decoder was evaluated on three different experiment paradigms: model calibrated from scratch at the beginning of each session with small training dataset (paradigm A), model adaptation with multiple recalibration sessions (paradigm B), and fixed model without adaptation using model created with paradigm B (paradigm C). The three paradigms provided indication on the online closed loop behaviour of the model at the beginning (A), during (B) and after (C) model calibration period. The dataset related to the paradigms A, B and C are composed of 5, 4 and 5 experiments respectively.

Figure 7 - 5 :

 75 Figure 7-5: Timeline of the calibration and test sessions. Chronology of the calibration and tests sessions for the 8D online closed-loop experiments based on the virtual avatar or the exoskeleton effector. The models created for the control of the virtual avatar and the exoskeleton were calibrated independently during six real-time closed-loop experiments (in a raw for the virtual avatar control and distributed in two months for the exoskeleton) colored in blue. Test sessions are stressed through the green colored boxes whereas sessions not considered in the evaluation of the decoder performance are shaded in orange and surrounded with red color. The number inside the boxes represents the number the experiments performed weekly.

Figure 8

 8 Figure 8-1: ABSD real-time adaptive BCI platform of the clinical research protocol "BCI and tetraplegia" of CLINATEC®. Two epidural ECoG recording WIMAGINE implants with a 64electrode array [Mestais et al., 2015] are used to record brain signals. Each array provides wirelessly radiotransmitted electrical brain activity to an external processing unit. Implants were placed into the skull in contact with the dura mater above the motor cortex by a craniotomy. ECoG recordings are sent to the BCI decoders that translates neural signals into order (at 10Hz frequency) to control various effectors. Virtual avatar effector is used for patient's training at home whereas exoskeleton effector is used for training in CLINATEC ® Both effectors provides visual feedback to the patient that allows him to adapt and respond in closed-loop to model predictions. Meanwhile, the model is updated (at a 0.07-1Hz frequency) based on supervised learning using ECoG data, movement instructions and movement labels. The ABSD BCI platform allows to generate "humain-in-loop" models where the neural signals related to the patient's feedback are directly integrated into the model calibration procedure.

  10 to 150 Hz (10 Hz step) for all the electrodes. CCWT is a feature extraction strategy that was widely used in the field of BCIs. Its efficiency has previously been demonstrated[START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkey[END_REF][START_REF] Choi | Improved prediction of bimanual movements by a two-staged (effector-then-trajectory) decoder with epidural ECoG in nonhuman primates[END_REF]] [Eliseyev et al., 2017] [Schaeffer and Aksenova, 2016a] [Shimoda et al., 2012]. The absolute value of CCWT was decimated along the temporal modality (by averaging the samples) to obtain a 10-point description of 1s time epoch for each frequency band and for each channel, resulting in the temporalfrequency-spatial neural feature tensor 𝐗 𝑡 ℝ 10x15x64 . All the feature extraction steps are represented in the Figure 8-2.

Figure 8 - 2 :

 82 Figure 8-2: Feature extraction procedure. ECoG neural signals were recorded at a 586 Hz sampling rate. For each electrode, a 100ms batch was extracted and concatenated to previous signals to form a 1 second ECoG epoch. From the raw signals, ECoG epoch spatial-temporalfrequency characteristics of the signals were extracted through continuous complex Morlet wavelet transform between 10 Hz and 150 Hz with a 10Hz step. Absolute values and decimation of the wavelet coefficient were computed to extract the final tensor of observation variables.

3D

  Cartesian vector between the current hand position at the time moment 𝑡 and the target position. 𝑦 𝑡 𝐿𝑟 ℝ and 𝑦 𝑡 𝑅𝑟 ℝ were left and right wrist rotation components of 𝐲 𝑡 , defined as a 1D angle between the current angle position and the target angle position (Figure 8-3). The discrete state 𝑧 𝑡 labels were determined by the task instruction with 𝐾 = 3 in the 6D control experiments (IS, ASLH and ASRH) and 𝐾 = 5 (IS, ASLH, ASRH, ASLW and ASRW,) in the 8D experiments. Output movement features were recorded during experiments at 10 Hz and were used during the model calibration phase.

Figure 8 - 3 :

 83 Figure 8-3: Feature extraction for supervised CLDA procedure. Neural and movement features recorded during the closed-loop experiments were used for the adaptive supervised training procedure based on the temporal-frequency-spatial neural feature tensor 𝑿 𝑡 (computed through the steps detailed in Figure8-2) and the optimal kinematic features 𝒚 𝑡 . The optimal kinematic features 𝒚 𝑡 were defined as the 3D Cartesian vector between the current position and the target position for the 3D hand translation and as 1D angular vector between the current angle and the target angle for 1D wrist rotation. The discrete state labels was noted 𝑧 𝑡 . The prediction from the current model 𝒚 ̂𝑡, the optimal prediction 𝒚 𝑡 according to the current position and the associated state 𝒛 𝑡 were recorded as movement features. 𝐗 𝑡 , 𝒚 𝑡 and 𝒛 𝑡 were stored in a buffer until the next update (every 15s) to update the REW-MSLM decoder.

Figure 8 - 4 :

 84 Figure 8-4: Data labelling for CLDA. (A) The data labeling procedure to stack the data for each expert and prepare the gate and expert models update. The buffer of data dedicated to the gate model stacks all the neural signals 𝑿 𝑡 and the associated discrete label 𝑧 𝑡 in the application loop before sending the batch of data to the update loop. The buffer of the expert 𝑘 only stacks the neural signals and the optimal kinematic features (𝒚 𝑡 𝐿𝑡𝑟 or 𝒚 𝑡 𝑅𝑡𝑟 or 𝑦 𝑡 𝐿𝑟 or 𝑦 𝑡 𝑅𝑟 etc.) related to one specific discrete state 𝑧 𝑡 = 𝑘. (B)The update threshold defined the amount of data stored in the buffer of the gate model before starting the update procedure. If an expert buffer stacks more data than the gate buffer when the gate buffer is full, the gate and the expert buffers are sent to the calibration loop to update the gate and expert models.

Figure 9 - 1 :

 91 Figure 9-1: Algorithms compared to evaluate the impact of dynamic gating algorithms for online closed-loop multi-limb experiments. (A) REW-NPLST is the model evaluated to estimate the benefits of adding a specific model dedicated to state detection. REW-NPLS model is evaluated and continuous model outputs are thresholded to define the active state. (B) REW-SLM is a mixture of expert algorithm similar to REW-MSLM with the exception that the gating model is static. REW-SLM and REW-MSLM comparison is achieved to highlight the benefits of the HMM dynamic gating. (C) Schematic of the REW-MSLM. Red brackets show the specific characteristic in which the analysis was focused on (here, the gating models).

Figure 9 - 2 :

 92 Figure 9-2 : Algorithms compared to evaluate the impact of expert-specific subset training strategy for online closed-loop multi-limb experiments. The REW-MSLM (A) experts' parameter are evaluated on specific subset of data (one for each limb to control) whereas the REW-NPLS continuous model parameters (B) are estimated on the entire dataset. Red brackets show the specific characteristic the analysis is focused on. In this case, the gating model.

Figure 9 - 3

 93 Figure 9-3 Three dynamic performance indicators were evaluated.A) The latency was defined as the delay between the task instruction provided by the experimenter and the correct state activation. This latency encompassed the reaction time of the patient, the system latency as well as the decoding latency. B) Block of errors were defined as the consecutive misclassified samples. From the number of block of error per experiments, a number of block of error per minute was evaluated. C) Block error duration computed the duration T of the block of errors.

Figure 9 -

 9 Figure 9-4 : A: online experiment performance indicators. A) Definiton of the scalar product indicator used to quantify continuous decoding performances in the pseudo-online studies. B) Atarget is considered as reached if the cursor is inside a sphere with the target coordinates as center and 2.5cm as radius. The R-ratio is the ratio between the distance travelled by the cursor during the task and the minimal distance travelled to reach the target.

Figure 9 - 5 :

 95 Figure 9-5: Averaged time-frequency responses during intended state activation. The averaged time-frequency responses were evaluated for each electrode. A) The active states (AS) were extracted in an 8 sec window centered on the AS activation. Additionally, the latency between the task instruction and the AS activation was computed. B) The idle state (IS) periods were extracted with the exception of the 2 seconds before or after any state transition with an AS.

Figure 9 - 6 : 2 , 1 .

 9621 Figure 9-6: Pseudo-online PREW-NPLS and APREW-NPLS performance studies. (A) PREW-NPLS is evaluated for the three penalization type 𝑝 = 0, 1 2 , 1. PREW-NPLS model need to fix the penalization hyperparameter λ. To evaluate the performance the three type of penalization depending on the penalization hyperparameter, 31 λ values (from 0 to 0.6 with a 0.02 steps) were calibrated and tested for each penalization type. (B) APREW-NPLS model was only calibrated ones with the penalization type 𝑝 = 1. As the model optimize the penalization hyperparameter, the calibration was only carried out once but for a smaller subset of possible λ going from 0.1 to 0.6 with a 0.1 steps.

H2M2 gating performance evaluation 203 Figure 9 - 7 :

 20397 Figure 9-7: Pseudo-online evaluated HMMs with different architectures. Three HMMs were integrated into REW-MSLM as gating model and tested to evaluate the performance improvements related to each HMM specificity. (A) The HMM gating of REW-MSLM calibrated during the 8D online closed-loop experiments of the clinical trial (series D andE) was restricted in the achievable state transition. The experimental paradigm of series D and E imposed to return to IS between different AS tasks (not between trials from the same AS task). Therefore, the trained transition were limited to transitions from IS to one of the AS (and vice versa). This HMM is titled HMMlimited in this study. (B) Conversely to the HMMlimited, a HMM was calibrated with all the transition available (including transition from one AS to another). This HMM is referred as HMMfull. (C) The proposed H2M2 gating model is based on a hierarchical architecture. The first layer is composed of three states defined as the IS, the left body side movement state (ASL) and the right body side movement state (ASR) with IS being a production state whereas ASL and ASR states being internal states. ASL is composed of two production states which are the left hand translation and wrist rotations states respectively (ASLH, ASLW). Similarly, ASR lead to two production states which are the right hand translation and wrist rotations states respectively (ASRH, ASRW).

Figure 10 - 1 :

 101 Figure 10-1 : State decoding results obtained during pseudo online experiments. A) Average accuracy and F-score over datasets A, B and C for 3 different analyses: all states (idle state IS, left hand translation active state ASLH and right hand translation active state ASRH) considered independently, IS versus AS (both hand translation states merged) performance and ASLH versus ASRH. B) Time dynamic performance indicators: Latency duration is evaluated as the time required to reach the desired state. Block of error durations shows the average time that last an error block of consecutive misclassified samples. The block error rate represents the occurrence of blocks of wrong detections per minute. Standard deviation is represented for each algorithm and each dataset using a vertical bar. Significance of the differences between the three decoders were computed for datasets A and C (B is excluded because of the sample size) using the Mann-Whitney U test with Bonferroni correction (𝛼 𝑚𝑢𝑙𝑡𝑖-𝑐𝑙𝑎𝑠𝑠 = 0.0167) in the multi-class comparisons. Otherwise, α=0.05. Significant values are indicated by an asterisk.

  To evaluate the expert-specific subset training strategy piece-wise linear continuous REW-MSLM predictions were compared to those of the REW-NPLS decoder trained on the entire data set. Continuous performance median, 25 th and 75 th percentiles are represented in the Figure10-2 for the three experimental paradigms. The results show similar performance in all the paradigms for both hands reaching task in average with (𝐷𝑜𝑡𝑃 𝐿𝐻 = 0.095 ± 0.05,𝐷𝑜𝑡𝑃 𝑅𝐻 = -0.03 ± 0.16) compared to the REW-NPLS model (𝐷𝑜𝑡𝑃 𝐿𝐻 = -0.03 ± 0.14 and 𝐷𝑜𝑡𝑃 𝑅𝐻 = -0.04 ± 0.1) for paradigm A.REW-MSLM left hand translation decoding of experimental sessions B and C (Figure10-2B and C) demonstrated similar average decoding performance (B: 𝐷𝑜𝑡𝑃 𝐿𝐻 = 0.21 ± 0.06 and C: 𝐷𝑜𝑡𝑃 𝐿𝐻 = 0.23 ± 0.13) compared to the REW-NPLS model (B: 𝐷𝑜𝑡𝑃 𝐿𝐻 = 0.18 ± 0.05 and C: 𝐷𝑜𝑡𝑃 𝐿𝐻 = 0.18 ± 0.11). Similar results were observed for the right hand translation decoding with a REW-MSLM average performance of B: 𝐷𝑜𝑡𝑃 𝑅𝐻 = 0.15 ± 0.07 and C: 𝐷𝑜𝑡𝑃 𝑅𝐻 = 0.2 ± 0.03 compared to the REW-NPLS model performance of B: 𝐷𝑜𝑡𝑃 𝑅𝐻 = 0.14 ± 0.09 and C: 𝐷𝑜𝑡𝑃 𝑅𝐻 = 0.19 ± 0.03. No significant difference was evaluated between the REW-NPLS and REW-MSLM performance (Figure10-2). However, significant enhancement of the results were computed between the decoding performance of the left hand translation models of the dataset A and dataset B (𝑝 = 0.0159) and the decoding performance of the right hand translation models of the dataset A and dataset C (𝑝 = 0.0079). The small number of experiments does not allow to confirm other significant changes. The performance improvements between dataset A and datasets B and C highlighted the benefits of cross-session training for increasing both the training data length and robustness to signal variability. No performance differences were visible between datasets B and C, stressing the model robustness.

Figure 10 - 2 :

 102 Figure 10-2: Continuous decoding performance for each hand for datasets A, B and C. Statistics of the scalar product between the predicted hand directions and the optimal prediction (defined as the targetcursor oriented distance) averaged over time and the experiments were evaluated for each dataset. The performance indicators are shown in blue for the state-of-the-art REW-NPLS model and in yellow for the new REW-MSLM. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The whiskers extend to the extreme data.Significance of the differences between the decoders and the dataset were computed using the Mann-Whitney U test (𝛼 = 0.05).

Figure 10 - 3 :

 103 Figure 10-3: Discrete decoding performance during the online experiments for the virtual avatar or the exoskeleton effectors. Average confusion matrices for online test sessions using the virtual avatar (A) or exoskeleton (B) effectors including idle state IS or the left ASLH, right hand ASRH, right wrist ASLW, and left wrist ASLW control states. The diagonals display the correct classification rate for each class.

Figure 10 - 4 :

 104  Online experiment performance across several months for the virtual avatar and exoskeleton effectors, correspondently. Online 8D experiment performance (for each state: idle, left and right hand translation and rotation) using the virtual avatar effector across 203 days after last model calibration (A) or using exoskeleton effector across 167 days after last model calibration (B). F-score and accuracy discrete performance indicators were evaluated for each state. Continuous performances were computed using the success rate (SR) (percentage of targets hit) and the R-ratio (ratio between the distance travelled by the effector to reach a target and the distance from the initial position of the effector to target location). Standard deviation is shown for each algorithm and each dataset using a vertical bar. If the chance levels are not outside the scale of the figure, they are represented for each state or task.

Figure 10 -

 10 5A and Figure 10-5B for the left and right hand translation, respectively. The entire session of the 106 th days is represented in Figure 10-5C. This session is composed of successive tasks with two right hand translation tasks and three idle, left hand translation, left and right hand rotation tasks. Each task was composed of several trials. Trajectories represented in the Figure 10-5A and Figure 10-5B are trials form the first left hand and second right hand translation tasks.

Figure 10 - 5 :

 105 Figure 10-5: Example of session realized 106 days after the last model calibration using exoskeleton effector. A) left hand trajectory across time and trials. These trajectories are extracted from the first left hand task of the session. B) right hand trajectory across time and trials. These trajectories are extracted from the second right hand translation task of the session. C) Movement on X, Y, Z and θ (angle for wrist rotation) across the sessions performed 106 days after the last model calibration. Shaded area color correspond to the task that patient must perform. Colored Lines represent left and right hand coordinates for X, Y and Z-axis and left and right wrist angle for θ axis. Thick line width underline the state controlled by the patient.

Figure 10 - 6 :

 106 Figure 10-6: Example of a gating model. Gating parameter weights (discrete decoding) of the REW-MSLM created using an exoskeleton effector according to the (A) spatial, frequency (B) or temporal (C) modalities for each state: rest state (IS), left hand 3D translation and rotation states (ASLH and ASLW ) or right hand 3D translation and rotation states (ASRH and ASRW). The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively.

Figure 10 - 7 :

 107 Figure 10-7: Expert model estimated during online experiments with CLDA. Experts parameter weights (continuous decoding) of the REW-MSLM decoder used during online clinical 8D alternative bimanual experiments achieved with the exoskeleton according to the (A) spatial, frequency (B) or temporal (C) modalities for the left or right arm continuous movements (hand translation and wrist rotation models). The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively.

Figure 10 - 8 :

 108 Figure 10-8: Modulation of the neural signal activity between idle state and Left or right hand translation active states . (A) Neural signal modulation between left hand translation state and idle state for the electrodes located on the left and right implants selected with a checkerboard pattern. The location of the selected electrodes are represented in the brain schematic at the top. (B) The neural signal modulation between right hand translation state and idle state for the electrodes located on the left and right implants The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively. Dotted lines represent the median latency for each active task and the centre of studied window respectively. The dotted squares highlight the electrodes enlarged on the Figure 10-10.

Figure 10 - 9 :

 109 Figure 10-9: Modulation of the neural signal activity between idle state and Left and right wrist rotation active states . (A) Neural signal modulation between left wrist rotation state and idle state for the electrodes located on the left and right implants selected with a checkerboard pattern. The location of the selected electrodes are represented in the brain schematic at the top. (B) The neural signal modulation between right wrist rotation state and idle state for the electrodes located on the left and right implants The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively. Dotted lines represent the median latency for each active task and the centre of studied window respectively. The dotted squares highlight the electrodes enlarged on the Figure 10-10.

Figure 10 - 10 :

 1010 Figure 10-10 : Modulation of the neural signal activity between idle state and active states focused on specific electrodes. (A) Modulation of the neural signal activity between idle state and left hand translation state (ASLH) focused on the electrodes of the left and right implants L52 and R52. (B) Modulation of the neural signal activity between idle state and right hand translation state (ASRH) focused on the electrodes of the left and right implants L52 and R52. (C) Modulation of the neural signal activity between idle state and left wrist rotation state (ASLW) focused on the electrodes of the left and right implants L54 and R54. (C) Modulation of the neural signal activity between idle state and right wrist rotation state (ASRW) focused on the electrodes of the left and right implants L54 and R54.

  -8, Figure10-9 and Figure10-10 were consistent with knowledge of the state of the art. REW-NPLS (PREW-NPLS) is a penalized version of the REW-NPLS algorithm which estimates group-wised sparse solution. Groups were formed following the feature modalities: grouped by electrode and or frequencies and or time. PREW-NPLS was designed in order to be employed as expert or gate model of REW-MSLM. This section highlighted the performance of Lp-PREW-NPLS algorithms during a pseudo-online study testing the online adaptive decoder with p being the classic L1 norm regularization or less conventional L0 and L0.5 norm penalization. The PREW-NPLS models were tested in a pseudo-online study using the 3D left and right hand translations data of the 8D online closed-loop experiments (sessions from dataset D).The PREW-NPLS algorithm has two hyperparameters, the dimension of the latent space 𝑓 ⊂ {1,2, … , 𝐹} and the penalization parameter 𝜆 ∈ [0; 1]. Even though the latent space dimension hyperparameter is evaluated online during the model calibration using the recursive validation procedure, it is relevant to evaluate the evolution of the models sparsity depending on the two hyperparameter values. The sparsity evolutions of the L0 REW-NPLS, L0.5 REW-NPLS and L1 REW-NPLS models for the left and right hand translation tasks are shown in the Figure10-11A, B and C respectively.

Figure 10 - 11 :

 1011 Figure 10-11 : Sparsity evolution depending on the hyperparameters 𝜆 and 𝑓. Sparsity of the solution is estimated for the L0 REW-NPLS (A), L0.5 REW-NPLS (B) and L1 REW-NPLS (C) algorithms depending on the 𝜆 and 𝑓 where 𝜆 is the penalization coefficient of the model and 𝑓 is the latent space dimension. The sparsity evolution is presented for the decoders of the left or right hand translation continuous movements.

  state of the art REW-NPLS (𝝀 = 𝟎) algorithm performance in the left hand decoding study, presented in the first position of each sub-figure (Figure 10-12A, B and C), highlighted a median = 0.223, a Q 1 = 0.158 and a Q 3 = 0.266 which is noted 0.223 (0.158 -0.266).

Figure 10 -

 10 Figure 10-12 : The model performance indicators of the Lp REW-NPLS algorithm for left hand movement decoding. The cosine similarity and the model sparsity were computed for the L0 REW-NPLS (A), L0.5 REW-NPLS (B) and L1 REW-NPLS (C) algorithms. The cosine similarly performance on each session was summarized using a box plot representation where the red line is the median the blue lines indicate the 25 th and 75 th percentiles (𝑄 1 and 𝑄 3 ). Additionally, the black boundaries show the upper and lower extreme cosine similarity obtained for the experiments. The performance of the REW-NPLS algorithms is presented in the first box plot of each Lp REW-NPLS algorithm sub-plot. The median, 𝑄 1 and 𝑄 3 of the REW-NPLS models are extended using horizontal dotted lines for easier performance comparison. Additionally, the sparsity of each solution is depicted.

Figure 10 -

 10 Figure 10-13: The model performance indicators of the Lp REW-NPLS algorithm (p=0,0.5,1) for right hand movement decoding. The cosine similarity and the model sparsity is computed for the L0 REW-NPLS (A), L0.5 REW-NPLS (B) and L1 REW-NPLS (C) algorithms. The cosine similarly performance on each session is summarized using a box plot representation where the red line is the median the blue lines indicate the 25 th and 75 th percentiles (𝑄 1 and 𝑄 3 ). Additionally, the black boundaries show the upper and lower extreme cosine similarity obtained for the experiments. The performance of the REW-NPLS algorithms is presented in the first box plot of each Lp REW-NPLS algorithm sub-plot. The median, 𝑄 1 and 𝑄 3 of the REW-NPLS models are extended using horizontal dotted lines for easier performance comparison. Additionally, the sparsity of each solution is depicted.

Figure 10 -

 10 Figure 10-15: Parameter weights of the Lp REW-NPLS and REW-NPLS models estimated offline in the right arm decoding study. Model parameter weights of the tested algorithms for 3D right hand translation movements decoding from the D dataset according to the spatial, frequency or temporal modalities. The parameter weights related to the 𝑦 4 , 𝑦 5 and 𝑦 6 axis are represented using blue, orange and yellow lines respectively.

Figure 10 - 16 :

 1016 Figure 10-16 : 3D left hand decoding parameter weights of the three PREW-NPLS models projected on the spatial modality depending on the electrode location on the implant. The optimal latent space dimension 𝑓 estimated using the Recursive-Validation procedure. The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively.

Figure 10 -

 10 Figure 10-17 : 3D right hand decoding parameter weights of the three PREW-NPLS models projected on the spatial modality depending on the electrode location on the implant. The optimal latent space dimension 𝑓 estimated using the Recursive-Validation procedure is used. The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively.

  Automatic L1-Penalized REW-NPLS is a penalized version of the REW-NPLS algorithm which estimates several PREW-NPLS models during the calibration phase, ranks the model depending on the estimated performance and automatically selects the best model with the adequate penalization hyperparameter. Similarly to the PREW-NPLS algorithm, the APREW-NPLS algorithm was tested in a pseudo-online study using the 3D left and right hand translation data of the 8D online closed-loop experiments (sessions from dataset D). The APREW-NPLS calibration procedure was performed on the same experiments than the REW-MSLM during the online closed-loop experiments.

Figure 10 -

 10 Figure 10-18 : Sparsity evolution of the APREW-NPLS models depending on the selected hyperparameters . Sparsity of the solution estimated using the L1 APREW-NPLS algorithm depending on the two hyperparameters 𝜆 and 𝑓 where 𝜆 is the penalization coefficient of the model and 𝑓 is the latent space dimension. The sparsity evolution is presented for the estimated decoders of the left or right hand continuous movements.

Figure 10 -

 10 Figure 10-19: APREW-NPLS model performance. The model performance indicators of the AREW-NPLS algorithm for left hand translation movement decoding (A) and right hand translation movement decoding (B). The cosine similarity and the model sparsity were computed. The cosine similarly performance on the 37 test sessions were summarized using a box plot representation where the red line is the median the blue lines indicate the 25 th and 75 th percentiles (𝑄 1 and 𝑄 3 ). Additionally, the black boundaries show the upper and lower extreme cosine similarity obtained for the experiments. The performance of the REW-NPLS algorithms is presented in the first box plot for left and right hand decoding. The median, 𝑄 1 and 𝑄 3 of the REW-NPLS models are extended with horizontal dotted lines for easier performance comparison. Additionally, the sparsity of each solution is depicted. The penalized models were ranked during the APREW-NPLS model calibration procedure from the worst to the best model. The estimated rank at the end of the incremental model training is displayed for each model.

  The rank was used to evaluate the expected reward of each model. An example of the APREW-NPLS calibration procedure for the left and right hand translation models are shown in the Figure 10-20. More precisely, the dot product and expected reward evolution across update iteration during the pseudo-online incremental model calibration procedure of the left and right hand translation studies are displayed in the Figure 10-20A and Figure 10-20B respectively.The rank of the models displayed on the Figure10-19 was extracted from the expected reward of each model at the last update iteration exhibited in the Figure10-20. The dot product increased with the update iteration whereas the expected reward stabilized for all the models.

Figure 10 -

 10 Figure 10-20: APREW-NPLS model calibration. The dot product and expected reward evolution across update iteration during the pseudo-online incremental model calibration procedure of the left (A) and right (B) hand translation studies. The two indicators were computed using the Recursive Validation algorithm. The dot product indicator as well as the expected reward are displayed for the set of possible penalization hyperparameters. The vertical dotted lines stressed the transition between two pseudo-online calibration sessions.

Figure 10 -

 10 Figure 10-21: Parameter weights of the REW-NPLS and APREW-NPLS models estimated offline in the left hand translation decoding study. Model parameter weights of the tested algorithms for 3D left hand translation decoding from the D dataset according to the spatial, frequency or temporal modalities. The parameter weights related to the 𝑦 1 , 𝑦 2 and 𝑦 3 axis are represented using blue, orange and yellow lines respectively.

Figure 10 -

 10 Figure 10-23 : 3D left hand translation decoding parameter weights of the APREW-NPLS models projected on the spatial modality depending on the electrode location on the implant. All the estimated models with different penalization hyperparameter are represented with the optimal latent space dimension estimated during the Recursive-Validation procedure. The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively.

Figure 10 -

 10 Figure 10-24: 3D right hand translation decoding parameter weights of the APREW-NPLS models projected on the spatial modality represented depending on the electrode location on the implant. All the estimated models with different penalization hyperparameter are represented with the optimal latent space dimension estimated during the Recursive-Validation procedure. The

Figure 10 -

 10 25 and Figure 10-26. The static indicators are shown in the Figure 10-25 whereas dynamic indicators are displayed in the Figure 10-26.

Figure 10 -

 10 Figure 10-26: Estimated dynamic indicators for the three decoder in the pseudo online series F of experiments. A) Latency histograms of the HMMlimited, HMMfull and H2M2 algorithms. B) Estimated latency distribution of the three algorithms: HMMlimited, HMMfull and H2M2. C) The median, the 25 th (𝑄 1 ) and the 75 th (𝑄 3 ) percentiles of the latency, error length and error frequency indicators. The black vertical bars represent the maximum value not considered as outliers.A sample is considered as outlier if the value of the indicator is more than 1.5 times the interquartile range from the 25 th or the 75 th percentile. All the indicators were estimated using all the recorded data (including outliers). Significance of the differences between the three decoders are computed for the three decoders using the Student t-test with a Bonferroni correction (𝛼 𝑚𝑢𝑙𝑡𝑖-𝑐𝑙𝑎𝑠𝑠 = 0.0167). Significant values are indicated by an asterisk.

Figure 10 -Figure 10 -

 1010 Figure 10-27: Parameter weights of the H2M2 gating model estimated offline with the series F of experiments. Model parameter weights of the H2M2 algorithms for the decoding of five states according to the spatial, frequency or temporal modalities. The first line represents the sub-model parameter weights discriminating the Idle state (IS), the left body side movement state referred as (ASL) and the right body side movement state named (ASR). The second line highlights the parameters weights of the sub-model which classifies the states between left hand movements (ASLH) and left wrist rotation state (ASLW). The last sub-model is acting similarly than the second one with the right hand movements (ASRH) and right wrist rotation state (ASRW).

Figure 10 -

 10 Figure 10-29 : Parameter weights of the H2M2 (A) and HMMlimited orHMMfull (B) estimated models following the spatial modality. The sensory sulcus (SS) and motor sulcus (MS) are represented in the spatial domain in yellow and red curves respectively.

  During the clinical trial, different versions of the REW-MSLM algorithm were integrated on the online closed-loop BCI platform ABSD. The evolution of the REW-MSLM was linked to the patient's skills improvements and difficulties of the proposed experiments. Before the first REW-MLSLM version, the REW-NPLS algorithm was used for online closed-loop BCI experiments. During this period, experiments from switching control (one discrete state controlled) to alternative 2D two-hand translation control (4D continuous dimensions and 3 discrete states) were attempted. During these experiments, a strong idle state and independent control of limbs were not achieved. The first integration of REW-MSLM was performed in mid-February 2018 using a dynamic gating model and one expert. This first model achieved good state discrimination and was tested during 6D alternative left and right hand translation (3 discrete states) experiments. The upgraded version of REW-MSLM with the dynamic gating and two experts dedicated to left and right body side decoding was integrated 3 months later. This REW-MSLM version was used to perform the online closed-loop 8D experiments with 5 discrete states presented in this manuscript. Finally, a generalized version of the REW-MSLM using a dynamic gating and one expert for each task (left hand translation, left wrist rotation, right hand translation, etc.) was integrated into the clinical trial BCI platform. Experiments clustering 8D continuous decoding and 6 to 8 discrete state classification tasks were trained. The upgrade milestone of the REW-MSLM algorithm integrated in the BCI platform as well as the evolution of the proposed experimental paradigm are shown in Figure11-1.

Figure 11 -

 11 Figure 11-1: Evolution of the DoF controlled by the patient across the Clinical trial timeline during the online closed-loop experiments. During the clinical trial, more and more complicated tasks and experiments were proposed to the patient depending on the experimental results and the ease of the patient to control the effectors. The red lines represented the modification of the decoder integrated into the online adaptive ABSD platform for online closed-loop experiments. Each experiment paradigm is presented with a colored area showing the period in which the experiments were achieved. The discrete dimension correspond to the number of states 𝐾 which are discriminated by the REW-MSLM gating model whereas the continuous dimension indicator show the sum of independent continuous dimensions controlled during the alternative tasks. For example, an asynchronous experiment characterized by alternative 3D left hand translation, 3D right hand translation, 1D left wrist rotation, 1D right wrist rotations tasks is displayed on the figure in the line corresponding to 𝐾 = 5 states (the four active states and the idle state) and 3 + 3 + 1 + 1 = 8 continuous ouputs

Chapter 11 :

 11 Conclusion, Limitations, Perspectives

  𝐘 𝑥 𝐼 1 2 𝐘 ⋯ 𝑥 𝐼 1 𝐼 2 𝐘 ] = [𝐱 1 ⨂𝐲 1 𝐱 1 ⨂𝐲 2 𝐱 1 ⨂𝐲 3 … 𝐱 𝐼 2 ⨂𝐲 𝐽 2 -1 𝐱 𝐼 2 ⨂𝐲 𝐽 2 ].

	XL	Mathematical operators and notations
		𝑥 11 𝐘 𝑥 12 𝐘 ⋯ 𝑥 1𝐼 2 𝐘
	𝐂 = 𝐗⨂𝐘 =	𝑥 21 𝐘 𝑥 22 𝐘 ⋯ 𝑥 2𝐼 2 𝐘 ⋮ ⋮ ⋱ ⋮
		[ 𝑥 𝐼 1 1

2 𝐽 2 )

is defined by

  𝑦 𝐼 1 1 𝑥 𝐼 1 2 𝑦 𝐼 1 2 ⋯ 𝑥 𝐼 1 𝐼 2 𝑦 𝐼 1 𝐼 2

	𝐂 = 𝐗 * 𝐘 = [	𝑥 11 𝑦 11 𝑥 12 𝑦 12 ⋯ 𝑥 1𝐼 2 𝑦 1𝐼 2 𝑥 21 𝑦 21 𝑥 22 𝑦 22 ⋯ 𝑥 2𝐼 2 𝑦 2𝐼 2 ⋮ ⋮ ⋱ ⋮
		𝑥 𝐼 1 1

  4.1. Features labelling for CLDA. ......................................................................... 8.4.2. REW-MSLM initialization..............................................................................

Chapter 9 Performance evaluation and analysis ................................................................ 9.1. REW-MSLM performance evaluation .................................................................. 9.1.1. REW-MSLM offline comparative study and online evaluation ...............

9.1.2. Performance indicators .................................................................................. 9.1.3. Chance level comparative study ................................................................... 9.1.4. Decoding model influence analysis .............................................................. 9.1.5. Neural signal modulation analysis ............................................................... 9.2. PREW-NPLS and APREW-NPLS performance evaluation .............................. 9.2.1. Algorithm comparison ................................................................................... 9.2.2. Performance indicators .................................................................................. 9.3. H2M2 gating performance evaluation ................................................................. 9.3.1. Algorithm comparison ................................................................................... 9.3.2. Performance indicators .................................................................................. 9.4. Conclusion ............................................................................................................... Chapitre 10 Results ................................................................................................................. 10.1. REW-MSLM ............................................................................................................. 10.1.1. Offline comparative study ................................................................................. 10.1.2. Online closed-loop 8D results ........................................................................... 10.1.3. Discussion ............................................................................................................ 10.2. Lp-Penalized REW-NPLS offline comparative study ........................................ 10.2.1. Sparsity study ...................................................................................................... 10.2.2. PREW-NPLS decoding performance ............................................................... 10.2.3. Lp REW-NPLS parameters visualization .........................................................

Table of contents

 of XLV 10.2.4. Discussion ............................................................................................................ 10.3. Automatic PREW-MSLM offline comparative study ........................................ 10.3.1. Sparsity study ...................................................................................................... 10.3.2. APREW-NPLS decoding performance ............................................................ 10.3.3. Example of APREW-NPLS calibration procedure across experiments....... 10.3.4. AREW-NPLS parameters visualization ........................................................... 10.3.5. Discussion ............................................................................................................ 10.4. H2M2 gating algorithm offline comparative study ........................................... 10.4.1. H2M2 classification performance comparison ............................................... 10.4.2. H2M2 parameter weights visualization ..........................................................

10.4.3. Discussion ............................................................................................................ 10.5. Conclusion ............................................................................................................... Chapter 11 Discussion Limitations Perspectives................................................................ 11.1. Discussion ................................................................................................................ 11.1.1. Offline evaluation of the developed algorithms......................................... 11.1.2. Real-time closed-loop BCI experiments ....................................................... 11.2. Limitations ............................................................................................................... 11.3. Perspectives ............................................................................................................. Bibliography ............................................................................................................................

XLVI

Table of contents

Table of contents XLVII

Table of figures

 of

  Description of the series F of experiments during the calibration and the test phases ...................................................................................................................................................Figure 8-1: ABSD real-time adaptive BCI platform of the clinical research protocol "BCI and tetraplegia" of CLINATEC® .........................................................................................................

	Table of figures	XLIX
	Figure 7-6:

Table 1 :

 1 Several control paradigms and associated controlled dimension used for patient training during the clinical trial. ................................................................................................... 55

Table 2 :

 2 Linear approximation of the performance indicators across time estimated for the 8D virtual avatar experiments. ....................................................................................................... 215

Table 3 :

 3 Linear approximation of the performance indicators across time estimated for the 8D exoskeleton experiments. ......................................................................................................... 218

Table 4 :

 4 Significance of the differences between the REW-NPLS decoder and the L0, L0.5 or L1 PREW-NPLS algorithm in the pseudo-online left hand translation decoding study .................. 238

Table 5 :

 5 Significance of the differences between the REW-NPLS decoder and the L0, L0.5 or L1 PREW-NPLS algorithm in the pseudo-online right hand translation decoding study ............... 239 Basic Neuroscience for BCI ............................................................................................... 1.2. BCI components .................................................................................................................. 1.2.1. Acquisition systems ......................................................................................................... 1.2.2. Neural signal processing ............................................................................................... 1.2.3. Effectors ........................................................................................................................... 1.2.4. Feedback .......................................................................................................................... 1.3. BCI control strategies: somatotopic remapping versus direct neural decoding ...... 1.3.1. Exogenous BCIs ..............................................................................................................

	Chapitre 1
	INTRODUCTION

1.3.2. Endogenous BCIs ............................................................................................................ 1.4. Motor BCI requirements for daily life applications ..................................................... 1.4.1. Acquisition system requirements ................................................................................. 1.4.2. control system requirements ......................................................................................... 1.4.2.1. Degree of freedom and accuracy ........................................................................... 1.4.2.2. Multi-limb control ................................................................................................... 1.4.2.3. Asynchronous BCI with idle state support .......................................................... 1.4.3. Decoder requirements .................................................................................................... 1.4.3.1. Real-time neural signal decoding .......................................................................... 1.4.3.2. Robust and stable decoding over time ................................................................. 1.4.3.3. Online closed-loop adaptive model calibration .................................................. 1.5. BCI project at CLINATEC ............................................................................................... 1.5.1. Concept and context ....................................................................................................... 1.5.2. Principles / Approaches ................................................................................................. 1.5.3. Project specific requirements ........................................................................................ 1.6. PhD goals ........................................................................................................................... 1.6.1. Real-time application ..................................................................................................... 1.6.2. Asynchronous multi-limbs decoder ............................................................................ 1.6.3. Online incremental adaptive decoder ......................................................................... 1.7. Manuscript organization .................................................................................................

  dedicated to BCI speller applications.

	On the one hand, exogenous BCI decoders are easy to train since the evoked potentials
	such as steady state VEP (SSVEP) or P300 VEP are naturally encoded with specific neural
	patterns and do not require extensive recording systems (one EEG channel can be
	sufficient) or patient's training. Moreover, exogenous BCIs can have fast information
	transfer rate. Nevertheless, they require a constant focus of the patient which can induce
	high mental load and tiredness [Nicolas-Alonso and Gomez-Gil, 2012]. On the other hand,
	endogenous BCIs are independent of any stimulus, can control more complex effectors
	with high DoF and are more adapted to continuous effector control and patient with
	affected sensory organs [Nicolas-Alonso and Gomez-Gil, 2012].
	For these reasons, the BCI experiments proposed in this Ph.D. manuscript were
	restricted to endogenous strategy. A special focus on this strategy is carried out in the
	next section. Nevertheless, exogenous BCI is an active field of research and most of the
	decoding algorithms applied in exogenous BCI experiments (generally based on EEG
	neural signal recording system) can be translated to endogenous applications.

  . Neural signals modulation recorded from population recording system are characterized by an increase of slow cortical signals (LFP: ≤13 Hz, ECoG: ≤2 Hz, EEG/MEG≤7 Hz) during movement, amplitude reduction in the LFP: 16-42 Hz, ECoG: 6-30 Hz, EEG/MEG: 10-30 Hz frequency bands and an amplitude rise in the high frequency bands (LFP: 63-200 Hz, ECoG: 34-128 Hz, EEG/MEG: 62-87 Hz). During offline MEG and EEG neural signal analysis of nine healthy subjects, Walder et al [

Table 1 :

 1 Several control paradigms and associated controlled dimension used for patient's training during the clinical trial.

	Paradigm	Type of Control Controlled dimension
	Cartesian left hand translation control	Continuous	1D or 2D or 3D
	Cartesian Right hand translation	Continuous	1D or 2D or 3D
	control
	Angular left wrist rotation control	Continuous	1D
	Angular right wrist rotation control	Continuous	1D
	Left grasping control	Discrete	1D
	Right grasping control	Discrete	1D
	Walking control	Discrete	1D
	Idle/rest control	Discrete	1D

  Contents 3.1. Pre-processing .................................................................................................................. 61 3.2. Neural feature extraction ................................................................................................ 62 3.2.1. Feature generation ...................................................................................................... 62 3.2.1.1. MEA single/multi neuron recording feature extraction .................................... 62 3.2.1.2. Population recording: temporal, frequency and spatial feature extraction .... 63 3.2.2. Feature dimension reduction .................................................................................... 69 3.3. Effector control features .................................................................................................. 74 3.4. Decoder/Model identification ........................................................................................ 75

3.4.1. Offline decoder calibration ........................................................................................ 76 3.4.1.1. Discrete output variables decoding: Classifiers .................................................. 76 3.4.1.2. Continuous output variable decoding ................................................................. 82 3.4.1.3. Hybrid decoders ...................................................................................................... 86 3.4.2. Closed-loop decoder adaptation (CLDA) and incremental adaptive decoders . 88

3.2.1.2. Population recording: temporal, frequency and spatial feature extraction Time features

  are the concatenation of sequential instantaneous neural signal samples to represent the neural signal amplitude variation across time. Time features are highly relevant for time-locked event such as ERPs and were commonly used in P300-based

BCI [Cozza et al., 2020[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Vo | Subject-Independent ERP-Based Brain-Computer Interfaces[END_REF]

. Additionally, EEG neural signals decoding were reported based on Slow Cortical Potentials (SCPs) for virtual cursors control, communication-BCI and rehabilitation-BCI

[START_REF] Hou | Slow cortical potential signal classification using concave-convex feature[END_REF][START_REF] Lazarou | EEG-Based Brain-Computer Interfaces for Communication and Rehabilitation of People with Motor Impairment: A Novel Approach of the 21st Century[END_REF][START_REF] Úbeda | Estimation of Neuromuscular Primitives from EEG Slow Cortical Potentials in Incomplete Spinal Cord Injury Individuals for a New Class of Brain-Machine Interfaces[END_REF][START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF]

etc. SCPs are EEG neural signals below 1-2 Hz which can be selfregulated by patients after training [Nicolas-Alonso and Gomez-Gil, 2012]. Finally, sensorimotor rhythm decoding was commonly reported in BCI experiments. In particular, local motor potentials (LMP) and high gamma neural signals modulation were reported to encode velocity information

[START_REF] Gunduz | Differential roles of high gamma and local motor potentials for movement preparation and execution[END_REF][START_REF] Hammer | The role of ECoG magnitude and phase in decoding position, velocity, and acceleration during continuous motor behavior[END_REF][START_REF] Pistohl | Prediction of arm movement trajectories from ECoG-recordings in humans[END_REF][START_REF] Waldert | A review on directional information in neural signals for brain-machine interfaces[END_REF]

. Therefore, they were often used during offline and online BCI experiments such as ECoG based neural signals decoding of arms, grasping, fingers continuous movement preparation or execution and states classification

  [START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Pistohl | Prediction of arm movement trajectories from ECoG-recordings in humans[END_REF][START_REF] Saa | Asynchronous decoding of finger movements from ECoG signals using long-range dependencies conditional random fields[END_REF][START_REF] Saa | Discriminative Methods for Classification of Asynchronous Imaginary Motor Tasks From EEG Data[END_REF] and EEG studies[START_REF] Andreu-Perez | A Self-Adaptive Online Brain-Machine Interface of a Humanoid Robot Through a General Type-2 Fuzzy Inference System[END_REF][START_REF] Brunner | Online control of a brain-computer interface using phase synchronization[END_REF]] [Ma et al., 2020].

AutoRegressive (AR) method is a parametric model commonly used to extract frequency components from neural signals. AR models the signals as the random output signal of a linear time invariant filter

[START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF]

. Since different tasks produce different brain activity the AR estimated filter coefficients between two tasks are different and can be used as features for BCI decoding or transform into an estimate of the signal power spectrum. Whereas AR had a superior resolution for small timewindows than STFT, it highlighted issues for non-stationary signals

[START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF]

. AR was used in ECoG

[START_REF] Bundy | Decoding three-dimensional reaching movements using electrocorticographic signals in humans[END_REF][START_REF] Degenhart | Remapping cortical modulation for electrocorticographic brain-computer interfaces: a somatotopy-based approach in individuals with upper-limb paralysis[END_REF][START_REF] Farrokhi | A piecewise probabilistic regression model to decode hand movement trajectories from epidural and subdural ECoG signals[END_REF][START_REF] Gunduz | Differential roles of high gamma and local motor potentials for movement preparation and execution[END_REF]

  . Spatial filters can be independent of the recorded signals such as bipolar, Laplacian and surface Laplacian filters. A surface Laplacian filter estimates the radial current of the scalp[START_REF] Andreu-Perez | A Self-Adaptive Online Brain-Machine Interface of a Humanoid Robot Through a General Type-2 Fuzzy Inference System[END_REF] using the recorded neural signals in order to enhance local sources contributions and reduce the contribution from distant sources[START_REF] Mcfarland | The Advantages of the Surface Laplacian in Brain-Computer Interface Research[END_REF]. frequency bands were reported in[START_REF] Ang | Filter Bank Common Spatial Pattern (FBCSP) algorithm using online adaptive and semi-supervised learning[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Chowdhury | Online Covariate Shift Detection based Adaptive Brain-Computer Interface to Trigger Hand Exoskeleton Feedback for Neuro-Rehabilitation[END_REF][START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF][START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF][START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF][START_REF] Tan | Dimensionality reduction in evolutionary algorithms-based feature selection for motor imagery brain-computer interface[END_REF]. Another supervised algorithm named xDAWN filter was designed for ERP classification during EEG experiments[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Rivet | xDAWN Algorithm to Enhance Evoked Potentials: Application to Brain-Computer Interface[END_REF]. Unsupervised spatial filters were designed for feature extractions and broadly used in BCI applications. The classic unsupervised filters are the principal component analysis (PCA) and the independent component analysis (ICA). PCA computes a linear combination of the neural signal features to create a new orthogonal basis maximizing the variance of the projected features. PCA algorithm was applied to MEA[START_REF] Kao | A High-Performance Neural Prosthesis Incorporating Discrete State Selection With Hidden Markov Models[END_REF][START_REF] Velliste | Motor Cortical Correlates of Arm Resting in the Context of a Reaching Task and Implications for Prosthetic Control[END_REF], 2008[START_REF] Wu | Direct feature extraction from multi-electrode recordings for spike sorting[END_REF], ECoG[START_REF] Flint | Continuous decoding of human grasp kinematics using epidural and subdural signals[END_REF]] [Miller et al., 2009[START_REF] Wang | Human Motor Cortical Activity Recorded with Micro-ECoG Electrodes During Individual Finger Movements[END_REF] and EEG[START_REF] Cozza | Dimension Reduction Techniques in a Brain-Computer Interface Application[END_REF] neural signals. ICA method creates a linear combination of the input features in order to build a statistically independent basis. The estimation of the ICA filter weights commonly relies on the minimization of the mutual information or the maximization of the non-Gaussianity of the neural signals[Clerc et al.

	, 2019] [Nicolas-
	Alonso et al., 2015] [Oliver et al., 2013] [Peterson et al., 2019] [Qibin Zhao et al., 2008] [Roijendijk
	et al., 2016] [Rong et al., 2018b] [Sannelli et al., 2016] [Scherer et al., 2015] [Seifzadeh et al.,
	2017] [Sharghian et al., 2019] [Shenoy et al., 2006] [Shin et al., 2015] [Song and Yoon, 2015] [Tan
	et al., 2020] [Vidaurre et al., 2011] [Zhao et al., 2008] and ECoG [Jiang et al., 2017] [Kapeller et
	al., 2015] [Y. Li et al., 2009] [Marathe and Taylor, 2013] [Morales-Flores et al., 2014]

Laplacian filters were applied in numerous non-invasive BCI experiments using EEG

[START_REF] Huang | Adaptive Lasso for sparse high-dimensional regression models[END_REF][START_REF] Kheradpisheh | Mixture of feature specified experts[END_REF][START_REF] López-Larraz | Continuous decoding of movement intention of upper limb self-initiated analytic movements from pre-movement EEG correlates[END_REF][START_REF] Mobaien | ACSP: Adaptive CSP filter for BCI applications[END_REF]

[START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive braincomputer interface in humans[END_REF]

and MEG

[Spüler et al., 2012b]

recording systems. Data-driven spatial filters were also applied during BCI experiments. Depending on the type of spatial filter, the filter weights were estimated using a supervised or unsupervised learning strategy.

The common spatial pattern (CSP) is a supervised algorithm created to optimize a spatial filter which discriminates two classes. The original multichannel neural signals filtered at the frequency of interest are projected into a subspace which maximizes the variance of the filtered neural signals for one class and minimizes it for the other class

[Lotte and Congedo, 2016] [Nicolas-Alonso and[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF]

. The Projection (un-mixing) matrix is estimated by solving the simultaneous diagonalization of the covariance matrices of the two classes through Generalized Eigen Value Decomposition. The eigenvectors with the highest and lowest eigenvalues are the optimal projectors

[START_REF] Lotte | EEG Feature Extraction[END_REF]

[Nicolas-Alonso and Gomez-Gil, 2012]. Numerous variations of CSP were applied to motor BCI systems for binary and multi-state classification during offline and online EEG

[START_REF] Ang | Filter Bank Common Spatial Pattern (FBCSP) algorithm using online adaptive and semi-supervised learning[END_REF][START_REF] Bamdadian | Improving session-to-session transfer performance of motor imagery-based BCI using adaptive extreme learning machine[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Chowdhury | Online Covariate Shift Detection based Adaptive Brain-Computer Interface to Trigger Hand Exoskeleton Feedback for Neuro-Rehabilitation[END_REF]] [Dähne et al., 2014[START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF][START_REF] Kheradpisheh | Mixture of feature specified experts[END_REF][START_REF] Li | A self-adaptive frequency selection common spatial pattern and least squares twin support vector machine for motor imagery electroencephalography recognition[END_REF]

[Y.

Li et al., 2009[START_REF] Lotte | Regularizing Common Spatial Patterns to Improve BCI Designs: Unified Theory and New Algorithms[END_REF]] [Mobaien and Boostani, 2016] [Nguyen et al.

experiments. Among the CSP algorithm possible extensions, sparse CSP versions were designed to avoid overfitting troubles

[START_REF] Lotte | Regularizing Common Spatial Patterns to Improve BCI Designs: Unified Theory and New Algorithms[END_REF][START_REF] Peterson | A penalized time-frequency band feature selection and classification procedure for improved motor intention decoding in multichannel EEG[END_REF][START_REF] Roijendijk | Classifying Regularized Sensor Covariance Matrices: An Alternative to CSP[END_REF]

. Filter-bank CSP (FBCSP) algorithms which apply independent CSP to several , 2016a]

[Naik, 2011] [Nordhausen and[START_REF] Nordhausen | Independent component analysis: A statistical perspective[END_REF]

. ICA hypothesizes that the recorded neural signals are the linear combination of a finite number of independent sources. ICA aims to reconstruct the signals generated by the sources. As PCA, ICA was applied in numerous EEG

[START_REF] Chen | A reliable brain-computer interface based on SSVEP using online recursive independent component analysis[END_REF]] [Moro et al., 2017[START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF]

and ECoG

[START_REF] Palmer | Independent Component Analysis (ICA) Features for Electro-corticographic (ECoG) Brain-Machine Interfaces (BMIs). 臨床神経生理[END_REF] Hirata, 2018] [Rembado et al., 2016

] BCI experiments.

  [START_REF] Iversen | MEG/EEG Data Analysis Using EEGLAB[END_REF] [[START_REF] Kamousi | Classification of motor imagery tasks for brain-computer interface applications by means of two equivalent dipoles analysis[END_REF][START_REF] Kim | Classification of Movement Intention Using Independent Components of Premovement EEG[END_REF]] [Moro et al., 2017[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Ruan | Automatic electroencephalogram channel selection of independent component analysis based motor imagery brain-computer interface[END_REF][START_REF] Senhadji | Sur l'exploitation des approches d'analyse en composantes indépendantes dans les interfaces cerveau machine[END_REF][START_REF] Serby | An improved P300-based brain-computer interface[END_REF]] [Wang et al., 2012[START_REF] Wu | To Explore the Potentials of Independent Component Analysis in Brain-Computer Interface of Motor Imagery[END_REF]. The application of ICA algorithm based on invasive recordings was poorly reported in the BCI field, probably because of the spatial resolution improvements related to this type of recording system.

  The common spatial pattern (CSP) was commonly used in the BCI field for feature extraction and dimension reduction. As mentioned earlier, the eigenvectors with the highest and lowest eigenvalues are the optimal projectors[START_REF] Lotte | EEG Feature Extraction[END_REF] [Nicolas-Alonso and Gomez-Gil, 2012] and the variance of the neural signals is maximized for one class while minimized for the other class. Commonly only few eigenvectors maximizing the variance of each class are selected to create a new subspace with high discriminability between the two classes. Spatial filtering and dimension reduction

procedure are always performed together with CSP algorithm. Numerous variation of CSP were designed such as filter bank CSP

[START_REF] Ang | Filter Bank Common Spatial Pattern (FBCSP) algorithm using online adaptive and semi-supervised learning[END_REF][START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF]

[

[START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF][START_REF] Nicolas-Alonso | Adaptive semisupervised classification to reduce intersession non-stationarity in multiclass motor imagery-based brain-computer interfaces[END_REF][START_REF] Oliver | Online feature selection for Brain Computer Interfaces[END_REF][START_REF] Tan | Dimensionality reduction in evolutionary algorithms-based feature selection for motor imagery brain-computer interface[END_REF]

, common spatial pattern patches (CSPP)

[START_REF] Sannelli | Ensembles of adaptive spatial filters increase BCI performance: an online evaluation[END_REF]

or multiclass versions of the CSP

[START_REF] Jafarifarmand | Real-time multiclass motor imagery brain-computer interface by modified common spatial patterns and adaptive neuro-fuzzy classifier[END_REF]

[

[START_REF] Khan | Multiclass EEG motor-imagery classification with subband common spatial patterns[END_REF][START_REF] Kheradpisheh | Mixture of feature specified experts[END_REF]

, penalized time-frequency band CSP (PTFBCSP)

[START_REF] Peterson | A penalized time-frequency band feature selection and classification procedure for improved motor intention decoding in multichannel EEG[END_REF]

, regularized CSP

[Lotte and Guan, 2011] [Roijendijk et al., 2016]

. Few experiments based on ECoG

[START_REF] Jiang | Characterization and Decoding the Spatial Patterns of Hand Extension/Flexion using High-Density ECoG[END_REF][START_REF] Kapeller | Online control of a humanoid robot through hand movement imagination using CSP and ECoG based features[END_REF][START_REF] Xie | Decoding of finger trajectory from ECoG using deep learning[END_REF]

neural signals were reported but commonly, CSP is applied to EEG decoding problems.

  These methods select the features depending on the performance of a decoder with specific feature subsets[START_REF] Bolón-Canedo | A review of feature selection methods on synthetic data[END_REF][START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Rouhi | Feature Selection in High-Dimensional Data[END_REF]. Wrapper methods add or remove iteratively new features to the subset of selected feature space and evaluate the performance variation of the new subset combined with the trained decoder[START_REF] Bolón-Canedo | A review of feature selection methods on synthetic data[END_REF][START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Rouhi | Feature Selection in High-Dimensional Data[END_REF]. and are decoder-dependent. Particle swarm optimization (PSO)[START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF][START_REF] Kumar | An improved discriminative filter bank selection approach for motor imagery EEG signal classification using mutual information[END_REF][START_REF] Remeseiro | A review of feature selection methods in medical applications[END_REF], sequential floating forward selection (SFFS)

Depending on the performance improvement or decay the added features are kept or removed. These methods are more efficient than Filter-based strategies. Nevertheless, they are costly in terms of computing time (high computational complexity), are sensitive to overfitting

  -1)[START_REF] Bolón-Canedo | A review of feature selection methods on synthetic data[END_REF] [[START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF][START_REF] Rouhi | Feature Selection in High-Dimensional Data[END_REF]. Embedded feature selection algorithms combine the benefits of both previously presented methods: keeping the advantages of wrapper strategies while decreasing the computational complexity[START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF]. In the case of an embedded feature selection algorithm, the features selection procedure is an inseparable part of the model learning/training process. Therefore, the feature selection depends on the selected decoder.

BCI Embedded techniques group the decision tree, regularization algorithms, etc. They were commonly applied during BCI experiments

[START_REF] Cincotti | High-resolution EEG techniques for brain-computer interface applications[END_REF][START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF][START_REF] Eliseyev | Penalized Multi-Way Partial Least Squares for Smooth Trajectory Decoding from Electrocorticographic (ECoG) Recording[END_REF][START_REF] Flamary | Decoding Finger Movements from ECoG Signals Using Switching Linear Models[END_REF][START_REF] Foodeh | Regularized Partial Least Square Regression for Continuous Decoding in Brain-Computer Interfaces[END_REF]] [Kim et al., 2018[START_REF] López-Larraz | Continuous decoding of movement intention of upper limb self-initiated analytic movements from pre-movement EEG correlates[END_REF][START_REF] Lotte | Regularizing Common Spatial Patterns to Improve BCI Designs: Unified Theory and New Algorithms[END_REF][START_REF] Mishra | A Novel Classification for EEG Based Four Class Motor Imagery Using Kullback-Leibler Regularized Riemannian Manifold[END_REF][START_REF] Nagel | Asynchronous non-invasive high-speed BCI speller with robust non-control state detection[END_REF][START_REF] Nakanishi | Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex[END_REF][START_REF] Peterson | A penalized time-frequency band feature selection and classification procedure for improved motor intention decoding in multichannel EEG[END_REF][START_REF] Seifzadeh | Fast and Efficient Four-class Motor Imagery Electroencephalography Signal Analysis Using Common Spatial Pattern-Ridge Regression Algorithm for the Purpose of Brain-Computer Interface[END_REF][START_REF] Sheikhattar | Adaptive sparse logistic regression with application to neuronal plasticity analysis[END_REF][START_REF] Sreeja | Weighted sparse representation for classification of motor imagery EEG signals[END_REF][START_REF] Wen | Review of Sparse Representation-Based Classification Methods on EEG Signal Processing for Epilepsy Detection, Brain-Computer Interface and Cognitive Impairment[END_REF]

[Y.

Zhang et al., 2013]

and in other fields

[START_REF] Hervás | Sparse N-way partial least squares by L1-penalization[END_REF][START_REF] Kalivas | Overview of two-norm (L2) and one-norm (L1) Tikhonov regularization variants for full wavelength or sparse spectral multivariate calibration models or maintenance[END_REF][START_REF] Khaire | Stability of feature selection algorithm: A review[END_REF][START_REF] Muñoz-Romero | Sparse and kernel OPLS feature extraction based on eigenvalue problem solving[END_REF]

.

  and healthy subjects[START_REF] Cantillo-Negrete | Motor imagery-based brain-computer interface coupled to a robotic hand orthosis aimed for neurorehabilitation of stroke patients[END_REF] controlled the opening and closure of a hand orthosis during online experiments based on EEG recordings. Moreover, the lower limb exoskeleton control based on EEG recording and a LDA decoder was reported by a healthy subject during online BCI experiments[START_REF] López-Larraz | Control of an Ambulatory Exoskeleton with a Brain-Machine Interface for Spinal Cord Injury Gait Rehabilitation[END_REF] and by eight paraplegic patients during a longterm neurorehabilitation study

  Artificial Neural Networks (ANN) classifier family regroups all the ANN possible architecture applied to BCI[START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. Multilayer perceptron (MLP) is probably the most popular ANN for BCI classification. ANN are non-linear classifiers which apply successive weighted linear combination and non-linear functions to the input neural signal. ANN presents high flexibility to solve non-linear neural signal problems. ANN was generally applied to offline BCI analysis and algorithms comparison experiments in

fNIRS, EEG and ECoG experiments [AL-Quraishi et al., 2018[START_REF] Faradji | Plausibility assessment of a 2-state self-paced mental task-based BCI using the no-control performance analysis[END_REF][START_REF] Javed | Recognition of finger movements using EEG signals for control of upper limb prosthesis using logistic regression[END_REF][START_REF] Jubien | Decoding of finger activation from ECoG data: a comparative study[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]] [Naseer et al., 2016b[START_REF] Nicolas-Alonso | Brain computer interfaces, a review[END_REF][START_REF] Sakhavi | Learning Temporal Information for Brain-Computer Interface Using Convolutional Neural Networks[END_REF][START_REF] Schaeffer | Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review[END_REF]

[Yang

Chapter 3 :

 3 Transducers in BCI prosthetic arm from EEG neural signals. Numerous models were designed to integrate directly the temporal dependencies into parameters estimation.Hidden Markov Model (HMM) is the most common dynamic decoders in the BCI field. A HMM is a stochastic generative model that deals with observable and latent variables. It assumes that the observable variables at time 𝑡 (e.g. neural signals) are generated by hidden latent variables which follow a N-Markov chain model. At time 𝑡, the latent variable 𝑧 𝑡 is dependent on the previous latent variables 𝑧 𝑡-1 , 𝑧 𝑡-2 … 𝑧 𝑡-𝑁 . Generally, in the BCI field, latent variables are modeled with a N=1 Markov chain and are associated with a specific neural state (e.g. left arm activation, right wrist activation etc). HMM evaluates the dynamic transition of the states.

  [J.-M. Cano-Izquierdo et al., 2012[START_REF] Chiappa | Analysis and Classification of EEG Signals using Probabilistic Models for Brain Computer Interfaces[END_REF][START_REF] Delgado Saa | Hidden conditional random fields for classification of imaginary motor tasks from EEG data[END_REF]] [Hasan and Gan, 2011b[START_REF] Saa | A latent discriminative model-based approach for classification of imaginary motor tasks from EEG data[END_REF][START_REF] Sakhavi | Learning Temporal Information for Brain-Computer Interface Using Convolutional Neural Networks[END_REF][START_REF] Xu | Kernel based hidden Markov model with applications to eeg signal classification[END_REF] [Yang

  . To this date, online BCI experiments which reported the highest number of controlled DoF integrated indirect OLE algorithm for SUA/MUA neural signal decoding. Wodlinger and Collinger highlighted respectively 10 DoF and 7 DoF real-time control of a robotic arm by a tetraplegic patient using MEA recordings. A BCI system using implanted FES, MEA neural signals, a OLE algorithm and a mobile arm support was designed to provide to a tetraplegic patient partially restored reaching and grasping movements during real-time experiments[START_REF] Ajiboye | Restoration of reaching and grasping movements through brain-controlled muscle stimulation in a person with tetraplegia: a proof-of-concept demonstration[END_REF]. Additionally, during closed-loop experiments, two tetraplegic patients implanted with MEA controlled the 3D Cartesian and joint velocity of a virtual arm effector[START_REF] Young | Closed-loop cortical control of virtual reach and posture using Cartesian and joint velocity commands[END_REF]. Finally, Based on indirect OLE algorithm

  experiments based on continuous center-out-back or finger flexion tasks. Kalman filter and Support vector Regression were outperformed by a non-linear model named Kernel-AutoRegressive Moving Average (Kernel-ARMA) in [Shpigelman et al., 2009] during open-loop hand tracking study. Moreover, Kernel-ARMA was applied in an online preclinical 3D cursor experiment using MEA neural signals. Kernel-ARMA performed an online re-estimation of the model based on a limited number of training examples: the oldest training examples are removed from the training dataset.

  CLDA procedure with adaptive decoders offers numerous advantages compared to open-loop calibration procedure and fixed (non-adaptive) decoders to decode brain neural signals during real-time BCI applications. Firstly, adaptive decoders are more robust to brain neural signal non-stationarity. In the case of neural signal patterns changes across time, fixed model performance may degrade whereas adaptive decoder can modify the model parameters to fit the brain neural signals variations and remains effective[START_REF] Li | Adaptive Decoding for Brain-Machine Interfaces Through Bayesian Parameter Updates[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces: a 10 year update[END_REF]. Secondly, numerous articles highlighted that neural signal patterns during open-loop (without feedback) and closed-loop (with feedback) BCI sessions were different and lead to different model parameters[Jarosiewicz

  𝛾 𝑘,𝑡 (𝐱 𝑡 ) 𝜑 ̂𝑘(𝐱 𝑡 )

	𝐾
	.
	𝑘=1
	Generally, 𝐲 ̂𝑡 = 𝐸(𝐲 𝒕 |𝐱 𝒕 ) is the Bayes estimate of the response variables [Bishop, 2006]

[START_REF] Waterhouse | Classification using hierarchical mixtures of experts[END_REF]

, 𝜑 ̂𝑘(𝐱 𝑡 ) = 𝐸(𝐲 𝑡 |𝐱 𝑡 , 𝑧 𝑡 = 𝑘) is the estimate issued by expert 𝑘, and 𝛾 𝑘,𝑡 (𝐱 𝑡 ) = 𝑝(𝑧 𝑡 = 𝑘|𝐱 𝑡 ) is the gating probability distribution of the 𝑘 𝑡ℎ expert at time 𝑡, with 𝛾 𝑘,𝑡 [0; 1] and ∑ 𝛾 𝑘,𝑡 𝐾 𝑘=1

  The neural features 𝐱 𝑡 ∈ 𝑋 are only dependent on the current state 𝑝(𝐱 𝑡 |𝑧 1:𝑡 ) = 𝑝(𝐱 𝑡 |𝑧 𝑡 )[START_REF] Schaeffer | ECoG signal processing for Brain Computer Interface with multiple degrees of freedom for clinical application[END_REF] Aksenova, 2016b].𝐲 𝑡 ∈ 𝑌 is estimated using gating coefficients 𝛾 𝑘,𝑡 (𝐱 𝑡 ) = 𝑝(𝑧 𝑡 = 𝑘|𝐱 1:𝑡 ) and 𝑘 experts: 𝐲 ̂𝑡 = ∑ 𝛾 𝑘,𝑡 (𝐱 𝑡 ) (𝐁𝐞𝐭𝐚 𝑘 . 𝐱 𝑡 + 𝐛𝐢𝐚𝐬 𝑘 )

	Thus, HMM state model is governed by the parameter bunch θ 𝑔 = {𝐀, {𝑐 𝑖 } 𝑘=1 𝐾 , 𝝅}, where 𝐀 is the transition matrix, 𝐀 = (𝑎 𝑖𝑗 ) ∈ ℝ 𝐾×𝐾 , 𝑎 𝑖𝑗 = 𝑝(𝑧 𝑡 = 𝑗|𝑧 𝑡-1 = 𝑖); {𝑐 𝑘 } 𝑘=1 𝐾 is the set of parameters characterizing the conditional distributions of the observed variables 109 𝐾 𝑘=1 , where 𝐁𝐞𝐭𝐚 𝑘 ∈ ℝ 𝑚×𝑛 is the matrix of parameters of the 𝑘 𝑡ℎ linear expert and 𝐛𝐢𝐚𝐬 𝑘 ∈ ℝ 𝑛 its associated bias. Thus, MSLM model is entirely defined by the gating θ 𝑔 and the set 𝑝(𝐱 Markov Switching Linear Models of experts θ 𝑒 = {θ 𝑒 𝑘 } 𝑘=1 𝐾 = {𝐁𝐞𝐭𝐚 𝑘 , 𝒃𝒊𝒂𝒔 𝑘 } 𝑘=1 𝐾 parameters: Θ = {θ 𝑔 , θ 𝑒 }.

𝑡 |𝑧 𝑡 ), and 𝝅 ∈ ℝ 𝐾 is the initial state probability vector at 𝑡 = 0

[START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]

[START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]

.

  The proposed REW-MSLM algorithm recursively estimates 𝚯 = {θ 𝑔 , θ 𝑒 } with a supervised training procedure. At each update 𝑢, the corresponding batch of training dataset {𝐗 𝑢 , 𝐘 𝑢 , 𝐳 𝑢 } is given with 𝐗 𝑢 ∈ ℝ Δ𝐿×𝐼 1 ×…×𝐼 𝑀 , 𝐘 𝑢 ∈ ℝ Δ𝐿×𝐽 1 ×…×𝐽 𝑁 , 𝐳 𝑢 = (𝑧 𝑡 1 , … , 𝑧 𝑡 1 +Δ𝐿 ) Online incremental adaptive multilinear switching model counting the successive transition of states in 𝐳 𝑢 and the transition matrix estimated during the previous updates weighted with the forgetting factor 𝜇 𝑔 , 0 ≤ 𝜇 𝑔 ≤ 1. HMM conditional emission probabilities 𝑝(𝐗 𝑡 |𝑧 𝑡 ) is inferred through the combination of 𝑝(𝑧 𝑡 |𝐗 𝑡 ) and class prior 𝑝(𝑧 𝑡 ) using Bayes' theorem [Bishop, 2006]. The REW-NPLS discriminative decoder is embedded into the HMM-based gating process to evaluate 𝑝(𝑧 𝑡 |𝐗 1:𝑡 ). It is trained on the observation tensor of input variables 𝐗 𝑢 and the latent state dummy variable matrix 𝐙 𝑢 ∈ {0,1} 𝐾×Δ𝐿 where the column-wise (single) nonzero element depicts the activated state for each sample.

	The REW-NPLS discriminative algorithm computes a set of 𝐹 multilinear models
	{𝐁 𝑢 𝑓 , 𝐛 𝑢 𝑓 } 𝑓=1 𝐹	, where 𝐁 𝑢 𝑓 ∈ ℝ 𝐼 1 ×…×𝐼 𝑀 ×𝐾 and 𝐛 𝑢 𝑓 ∈ ℝ 𝐾 are the gating tensor of the parameters
	and its related bias. The Recursive-Validation procedure selects the best model based on
	the estimated gating optimal hyperparameter 𝑓 𝑔 * ≤ 𝐹 and defines the optimal gating
	model as {𝐁, 𝐛} = {𝐁 𝑢 𝑓 𝑔 *	, 𝐛 𝑢 𝑓 𝑔 *
	NPLSe algorithm estimates a set of 𝐹 models for each expert {𝐁𝐞𝐭𝐚 𝑢,𝑘 𝑓 , 𝐛𝐢𝐚𝐬 𝑢,𝑘 𝑓 } 𝑘,𝑓=1 𝐾,𝐹	. 𝐹 ∈
	ℕ * is the fixed highest latent space dimension. The optimal hyperparameter of the k th
	expert 𝑓 𝑘 * ≤ 𝐹 is selected following the Recursive-Validation procedure [Eliseyev et al.,
	2017]. Recursive-Validation exploits the newly available dataset {𝐗 𝑢 𝑘 , 𝐘 𝑢 𝑘 } as testing data
	for the currently available models to evaluate the best hyperperparameters before to use
	them as training dataset for the models updating procedure. The best models are chosen
	independently for each expert and are used for real-time decoding of the neural
	signals: {𝐁𝐞𝐭𝐚 𝑘 , 𝐛𝐢𝐚𝐬 𝑘 } 𝑘=1 𝐾	= {𝐁𝐞𝐭𝐚 𝑢,𝑘 𝑓 𝑘 *	, 𝐛𝐢𝐚𝐬 𝑢,𝑘 𝑓 𝑘 *	𝐾 } 𝑘=1	.

𝑇 ⊂ ℕ * Δ𝐿 and Δ𝐿 the update block size.

The 𝐾 local multilinear functions 𝜑 𝑘 mapping the input variable space to the response variable space are estimated using expert specific samples. The k th expert model parameter update is performed on the training dataset {𝐗 𝑢 𝑘 , 𝐘 𝑢 𝑘 }. 𝐗 𝑢 𝑘 and 𝐘 𝑢 𝑘 are subtensors of 𝐗 𝑢 and 𝐘 𝑢 formed by samples labelled as belonging to state 𝑘. The k th expert model parameters are updated using REW-NPLS algorithm REW-NPLSe = REW-NPLS(𝐗 𝑢 𝑘 , 𝐘 𝑢 𝑘 ) with the forgetting factor 𝜇 𝑘 , 0 ≤ 𝜇 𝑘 ≤ 1. For online optimization latent variable space dimension (hyperparameter 𝑓), the REW-Similarly, at each update 𝑢, the HMM gating parameters estimation is updated based on the update block dataset {𝐗 𝑢 , 𝐳 𝑢 }. The HMM transition matrix 𝐀 is approximated by Chapter 4 : } for dynamic gating weight 𝛾 𝑘,𝑡 estimation. The output variable 𝒛 ̂𝑡 ∈ ℝ 𝐾 determines how likely each hidden state is generated based on 𝐗 𝑡 . From the REW-NPLS discriminative decoder prediction 𝒛 ̂𝑡, 𝑝(𝑧 𝑡 |𝐗 𝑡 ) is evaluated with the softmax function before to compute 𝛾 𝑘,𝑡 = 𝑝(𝑧 𝑡 = 𝑘|𝐗 1:𝑡 ) using HMM forward algorithm.

  .2.2) is replaced by the optimization of the cost function penalized

	136		Chapter 5 : Online incremental group-
				wise sparse REW-MSLM
			min‖𝐕 -𝐕 ̂‖2 + P(𝐰 1 , 𝐰 2 , 𝐰 3 ),	(5.3.7)
	with L p (p = 0,	1 2	, 1) norm/pseudo norms:

  2 , … , 𝜆 𝑖 , … , 𝜆 Λ }, Λ ∈ ℕ + , related to a set of penalized models 𝚯 𝑢 (𝛌) = {𝜃 𝑢,𝜆 𝑖 } 𝜆 𝑖 ∈𝛌 with 𝜃 𝑢,𝜆 𝑖 = {𝐁𝐞𝐭𝐚 𝑢 𝑓,𝜆 𝑖 , 𝐛𝐢𝐚𝐬 𝑢 𝑓,𝜆 𝑖 }

𝑓=1 𝐹

  where 𝐾 is the number of state. Considering the experiments analysed in the PhD research, the optimal continuous movement is defined as 𝐲 𝑡 = ((𝐲 𝑡 𝐿𝑡𝑟 ) 𝑻 , (𝐲 𝑡 𝑅𝑡𝑟 ) 𝑻 ) 𝑻 with 𝐲 𝑡 ℝ 6 for alternative 3D left and BCI Adaptive platform right hand translation tasks (experiment series A, B and C) or 𝐲 𝑡 = ((𝐲 𝑡 𝐿𝑡𝑟 ) 𝑻 , 𝑦 𝑡 𝐿𝑟 , (𝐲 𝑡 𝑅𝑡𝑟 ) 𝑻 , 𝑦 𝑡 𝑅𝑟 ) 𝑻 with 𝐲 𝑡 ℝ 8 if 1D left and right wrist rotations were added (experiment series D and E). Here 𝐲 𝑡 𝐿𝑡𝑟 ℝ 3 and 𝐲 𝑡 𝑅𝑡𝑟 ℝ 3 are the 3D left and right hand translation components of 𝐲 𝑡 whereas 𝑦 𝑡 𝐿𝑟 ℝ and 𝑦 𝑡 𝑅𝑟 ℝ are left and right wrist rotation components of 𝐲 𝑡 .

  order to update the gate model, all the neural signals samples 𝐗 𝑡 as well as all the discrete state samples 𝑧 𝑡 since the last update 𝑢 𝑡-1 were saved in a gate buffer to create the HMM gating parameters update block dataset {𝐗 𝑢 , 𝐳 𝑢 }. For the 𝑘 𝑡ℎ expert model update, only the neural signals and movements features related to the expert 𝑘 were stored in the k th expert buffer in order to create the dataset {𝐗 𝑢 𝑘 , 𝐘 𝑢 𝑘 }. 𝐗 𝑢 𝑘 and 𝐘 𝑢 𝑘 are subtensors of 𝐗 𝑢 and 𝐘 𝑢 formed by samples labelled as belonging to state 𝑘 (Figure 8-4A). BCI Adaptive platform

  -5, expert model parameters associated to the IS and ASLH tasks of 𝑃𝑟𝑑𝑔 3 are initialized with the experts of REW-MSLM from 𝑃𝑟𝑑𝑔 1 whereas other experts are initialized from 𝑃𝑟𝑑𝑔 2 model. It should be noted, that both 𝑃𝑟𝑑𝑔 1 and 𝑃𝑟𝑑𝑔 2 trained an expert model for ASLW task. The ASLW expert model parameters from 𝑃𝑟𝑑𝑔 1 or 𝑃𝑟𝑑𝑔 2 can be used (the selection of the model can be carried out depending on various criterion such performance, stability, etc.). Additionally, new task (and expert model) never trained on any experimental paradigm can be added to the REW-MSLM architecture. This new expert is trained from scratch. Concerning the gate model, the gate model from 𝑃𝑟𝑑𝑔 1 and 𝑃𝑟𝑑𝑔 2 can be exploited to initialize the gate model of the REW-MSLM 𝑃𝑟𝑑𝑔 3 . However, the gate model will require a calibration period to learn the transition from the already trained states to the state associated to the new expert.

9.1.2. Performance indicators 9.1.2.1. Offline performance indicators Discrete performance indicators

  Discrete performance were evaluated based on accuracy (𝑎𝑐𝑐) and F-score (𝑓𝑠𝑐) indicators. These indicators were computed using the confusion matrix, which summarizes the number of correctly classified samples from one state (true positives, 𝑡𝑝), incorrectly labelled samples in one state (false negatives, 𝑓𝑛), correctly classified samples not belonging to the state (true negatives, 𝑡𝑛) and incorrectly labelled samples not belonging to the state (false positives, 𝑓𝑝):

	𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 =	1 𝐾	∑ 𝐾 𝑘=1	𝑡𝑝 𝑘 +𝑡𝑛 𝑘 𝑡𝑝 𝑘 +𝑡𝑛 𝑘 +𝑓𝑝 𝑘 +𝑓𝑛 𝑘	,	(9. 1)
	𝐹𝑠𝑐𝑜𝑟𝑒 =	1 𝐾	. ∑ 𝐾 𝑘=1	(𝛽 2 +1) 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑘 𝑅𝑒𝑐𝑎𝑙𝑙 𝑘 𝛽 2 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑘 +𝑅𝑒𝑐𝑎𝑙𝑙 𝑘	,	(9. 2)
	𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑘 =	𝑡𝑝 𝑘 𝑡𝑝 𝑘 +𝑓𝑝 𝑘	, 𝑅𝑒𝑐𝑎𝑙𝑙 𝑘 =	𝑡𝑝 𝑘 𝑡𝑝 𝑘 +𝑓𝑛 𝑘	.	(9. 3)

  ×…×𝐼 𝑀 )×(𝐽 1 ×…×𝐽 𝑁 ) estimated using the PREW-NPLS or APREW-NPLS algorithms with the group-wise penalization restricted to the 𝑚 𝑡ℎ dimension of size 𝐼 𝑚 . This model was The model sparsity indicator was defined by the number of element w 𝑗,𝑓 𝑚 of 𝐰 𝑓 𝑚 ∈ ℝ 𝐼 𝑚 fixed to zero. The 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 of the model 𝜃 𝑢,𝜆 𝑖 following the 𝑚 𝑡ℎ dimension of size

	Considering a penalized model 𝜃 𝑢,𝜆 𝑖 = {𝐁𝐞𝐭𝐚 𝑢 𝑓,𝜆 𝑖 , 𝐛𝐢𝐚𝐬 𝑢 𝑓,𝜆 𝑖 } 𝑓=1 𝐹	with 𝐁𝐞𝐭𝐚 𝑢 𝑓,𝜆 𝑖 ∈
	ℝ (𝐼 1 computed from the set of penalized projectors {𝐰 𝑓 1 ∈ ℝ 𝐼 1 , … , 𝐰 𝑓 𝑀 ∈ ℝ 𝐼 𝑀 } 𝑓=1 𝐹	evaluated
	with the penalized PARAFAC decomposition.
	𝐼 𝑚 is defined as:
	𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥(𝜃 𝑢,𝜆 𝑖 , 𝑚) =	∑ 𝛿 w 𝑗,𝑓 𝑚 ,0 𝐼 𝑚 𝑗=1 𝐼 𝑚	.

  𝑡𝑛 𝑠 ) and the incorrectly labelled samples not belonging to the state (false positives, 𝑓𝑝 𝑠 ). 𝑡𝑛 𝑠 -𝑓𝑝 𝑠 × 𝑓𝑛 𝑠 √(𝑡𝑝 𝑠 + 𝑓𝑝 𝑠 )(𝑡𝑝 𝑠 + 𝑓𝑛 𝑠 )(𝑡𝑛 𝑠 + 𝑓𝑝 𝑠 )(𝑡𝑛 𝑠 + 𝑓𝑛 𝑠 )

	Then,
		𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 =	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	∑ 𝑠∈𝑆 𝑝𝑟𝑜𝑑	𝑡𝑛 𝑠 𝑡𝑛 𝑠 + 𝑓𝑝 𝑠	,
	𝐵𝑜𝑜𝑘𝑚𝑎𝑘𝑒𝑟 =	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	𝑠∈𝑆 𝑝𝑟𝑜𝑑 ∑ 𝑅𝑒𝑐𝑎𝑙𝑙 𝑠 + 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 𝑠	-1,
	𝐽𝑎𝑐𝑐𝑎𝑟𝑑 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 =	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	∑ 𝑠∈𝑆 𝑝𝑟𝑜𝑑	𝑡𝑝 𝑠 𝑡𝑝 𝑠 + 𝑓𝑛 𝑠 + 𝑓𝑝 𝑠	,
	𝐻𝐹 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	𝑠∈𝑆 𝑝𝑟𝑜𝑑 ∑ 𝑅𝑒𝑐𝑎𝑙𝑙 𝑠 + 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑠 -1	,
	𝐶𝑙𝑎𝑠𝑠 𝐵𝑎𝑙𝑎𝑛𝑐𝑒 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 =	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	∑ 𝑠∈𝑆 𝑝𝑟𝑜𝑑	𝑡𝑝 𝑠 𝑡𝑝 𝑠 + max (𝑓𝑛 𝑠 , 𝑓𝑝 𝑠 ) ,
	𝑀𝑎𝑡𝑡ℎ𝑒𝑤𝑠 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
	=	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	∑ 𝑡𝑝 𝑠 × 𝑠∈𝑆 𝑝𝑟𝑜𝑑	,
	𝐾𝑎𝑝𝑝𝑎 =	1 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )	∑ 𝑠∈𝑆 𝑝𝑟𝑜𝑑	𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 𝑠 -𝐶ℎ𝑎𝑛𝑐𝑒𝐿𝑒𝑣𝑒𝑙 𝑠 1 -𝐶ℎ𝑎𝑛𝑐𝑒𝐿𝑒𝑣𝑒𝑙 𝑠	𝑤𝑖𝑡ℎ
	𝐶ℎ𝑎𝑛𝑐𝑒𝐿𝑒𝑣𝑒𝑙 𝑘 =	(𝑡𝑝 𝑠 + 𝑓𝑛 𝑠 )(𝑡𝑝 𝑠 + 𝑓𝑝 𝑠 ) + (𝑡𝑛 𝑠 + 𝑓𝑝 𝑠 )(𝑡𝑛 𝑠 + 𝑓𝑛 𝑠 ) (𝑡𝑝 𝑠 + 𝑓𝑛 𝑠 + 𝑡𝑛 𝑠 +𝑓𝑝 𝑠 ) 2 ,
			𝐺𝑚𝑒𝑎𝑛 =	√ ∏ 𝑅𝑒𝑐𝑎𝑙𝑙 𝑠 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 )
						𝑠∈𝑆 𝑝𝑟𝑜𝑑

Table 2 :

 2 Linear approximation of the performance indicators across time estimated for the 8D virtual avatar experiments. In order to evaluate the evolution of the performance indicators across time, linear approximations of the performance across time were computed. The estimated slope and associated bias are shown in the table. The p-values indicating the significance of the results are also presented.

	𝑎𝑥 + 𝑏
	𝑎	𝑏	𝑝 -𝑣𝑎𝑙𝑢𝑒

Table 3 :

 3 Linear approximation of the performance indicators across time estimated for the 8D exoskeleton experiments. In order to evaluate the evolution of the performance indicators across time, linear regression fitting of the performance across time was achieved. The estimated slope and associated bias are shown in the table. The p-values indicating the significance and reliability of the results are also presented.

	𝑎𝑥 + 𝑏
	𝑎	𝑏	𝑝 -𝑣𝑎𝑙𝑢𝑒

  Table 4 and in the Table5for the right hand translation decoding. Numerous models highlighted statistical performance difference with the REW-NPLS model. As examples of statistical performance difference among the previously mentioned models, the L0-PREW-NPLS performance highlighted statistical differences with 𝜆 ∈ {0.01,0.026,0.04} for the left hand translation study and 𝜆 ∈ {0.01,0.018,0.24} for the right hand translation decoding study. For L0.5-PREW-NPLS differences were stressed for 𝜆 ∈ {0.22,0.03} and 𝜆 ∈ {0.1,0.16,0.04} for the left and right hand translation study. Finally, L1-PREW-NPLS showed statistical performance differences with REW-NPLS algorithm for the left hand translation tasks with 𝜆 ∈ {0.12,0.34,0.36} as well as for the right hand translation decoding study with 𝜆 ∈ {0.04,0.06,0.1, 0.22, 0.26, 0.38}.

Table 4 :

 4 Significance of the differences between the REW-NPLS decoder and the L0, L0.5 or L1 PREW-NPLS algorithm in the pseudo-online left hand translation decoding study using the non-parametric paired Wilcoxon signed rank test with and without the Bonferroni correction (𝛼 = 0.05 and 𝛼 𝐵𝑜𝑛𝑓𝑒𝑟𝑟𝑜𝑛𝑖 = 0.00161).The bolded P-values represent the statistical differences with 𝛼 = 0.05 whereas the bolded underlined P-values highlight the statistical difference with the Bonferroni correction.

	𝜆 0.02 0.04 0.06 0.08 0.10 0.12	0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.30 0.32	0.34	0.36	0.38	0.40
	𝐿 1 0.73 0.15 𝟎. 𝟎𝟐𝟏 𝟎. 𝟎𝟏𝟖 0.53 𝟎. 𝟎𝟎𝟏𝟔 𝟎. 𝟎𝟎𝟎𝟓 0.24 𝟎. 𝟎𝟎𝟏𝟐 0.5	0.15	𝟎. 𝟎𝟐𝟕 0.24 𝟎. 𝟎𝟎𝟐𝟑 0.24	0.21	0.24	0.21	0.21	0.21

Table 5 :

 5 Significance of the differences between the REW-NPLS decoder and the L0, L0.5 or L1 PREW-NPLS algorithm in the pseudo-online right hand translation decoding study using the non-parametric paired Wilcoxon signed rank test with and without the Bonferroni correction (𝛼 = 0.05 and 𝛼 𝐵𝑜𝑛𝑓𝑒𝑟𝑟𝑜𝑛𝑖 = 0.00161).The bolded P-values represent the statistical differences with 𝛼 = 0.05 whereas the bolded underlined P-values highlight the statistical difference with the Bonferroni correction. 𝟓 𝟎. 𝟎𝟑𝟖 𝟎. 𝟎𝟎𝟎𝟒 𝟏𝟎 -𝟓 𝟎. 𝟎𝟎𝟏𝟔 𝟏𝟎 -𝟔 𝟎. 𝟎𝟎𝟎𝟒 𝟏𝟎 -𝟔 0.22 𝟎. 𝟎𝟏 𝟎. 𝟎𝟓 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟓 𝟎. 𝟎𝟎𝟑 𝟎. 𝟎𝟎𝟑

	𝜆	0.02 0.04 0.06	0.08	0.10	0.12 0.14	0.16	0.18	0.2	0.22	0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40
	𝐿 1 𝟎. 𝟎𝟎𝟓 𝟏𝟎 -𝟓 𝟎. 𝟎𝟎𝟎𝟐 0.25	𝟏𝟎 -𝟔	𝟏𝟎 -

  Les interfaces cerveau-Machine (ICM) sont des systèmes permettant de traduire l'activité cérébrale d'un individu via un ordinateur afin d'effectuer des tâches nécessitant normalement une action des nerfs périphériques et/ou musculaires. En se basant sur l'essai clinique de CEA/LETI/CLINATEC, nommé « BCI et Tétraplégie », les recherches de thèse présentées dans ce manuscrit répondent aux challenges de l'intégration ICM dans la vie quotidienne, à savoir le contrôle complexe asynchrone d'effecteurs multi-membres et l'adaptation en temps réel des algorithmes de décodage durant des expériences en closed-loop. Pour répondre aux challenges de l'essai clinique, l'algorithme incrémental adapatif en temps réel Recursive exponentially weighted Markov switching multi-linear model (REW-MSLM) a été créé. L'algorithme REW-MSLM repose sur une architecture du type Mixture d'Experts (ME). Les MEs combinent plusieurs décodeurs continus dit «experts » dont les prédictions sont pondérées par un modèle discret nommé «gate » . Les modèles des experts sont estimés via l'algorithme REW-NPLS tandis que le modèle de gate est un modèle de markov caché (HMM). L'algorithme REW-MSLM a permis à un patient tétraplégique de contrôler un exosquelette 4 membres avec 8 degrés de liberté via le décodage de signaux électrocorticographiques (ECoG) enregistrés avec deux implants épiduraux sans fil nommé WIMAGINE. Dans une même expérience, avec un même modèle, le patient a réalisé des tâches de contrôle alternatif des mouvements du bras gauche et du bras droit dans l'espace 3D et de rotation 1D des poignets gauche et droit. Le contrôle du patient durant ces tâches est resté très stable, même durant des expériences 6 mois après la fin de la mise à jour du modèle. Par exemple, les performances de contrôle réalisées 0 à 37 jours après la dernière mise à jour du modèle ont montré un score de réussite de 71 ± 12% et 99 ± 2% pour les tâches de contrôle 3D des mouvements des mains et les tâches de contrôle 1D de rotation des poignets. Les performances de contrôle réalisées de 0 à 167 jours après la dernière mise à jour du modèle ont montré un score de réussite de 67 ± 21% et 93 ± 12% pour les tâches de contrôle 3D des mouvements des mains et les tâches de contrôle 1D de rotation des poignets. Dans le but d'être intégré dans REW-MSLM, de nouveaux algorithmes, à savoir le (Automatic) Penalized Recursive Exponentially Weighted N-way Partial Least Squares (PREW-NPLS et APREW-NPLS) et une version du Modèle de Markov Caché Hiérarchique (H2M2), ont été imaginés et testés en offline.

Figure 10-14: Parameter weights of the Lp REW-NPLS and REW-NPLS models estimated offline in the left arm decoding study. Model parameter weights of the tested algorithms for 3D left hand translation movements decoding from the D dataset according to the spatial, frequency or temporal modalities. The parameter weights related to the 𝑦 1 , 𝑦 2 and 𝑦 3 axis are represented using blue, orange and yellow lines respectively.

Figure 10-22: Parameter weights of the REW-NPLS and APREW-NPLS models estimated offline in the right hand translation decoding study. Model parameter weights of the tested algorithms for 3D right hand translation decoding from the D dataset according to the spatial, frequency or temporal modalities. The parameter weights related to the 𝑦 1 , 𝑦 2 and 𝑦 3 axis are represented using blue, orange and yellow lines respectively.

Acknowledgements

Premièrement, je tiens sincèrement à remercier le Professeur François CABESTAING, le Professeur Alim Louis BENABID et Monsieur Solaiman SHOKUR pour avoir accepté de faire partie des membres du jury et de participer à l'évaluation de mes travaux de recherche. Plus particulièrement, j'aimerai faire part de ma gratitude la plus sincère au Docteur Fabien LOTTE et au Professeur Saïd MOUSSAOUI pour avoir endossé les rôles de rapporteurs de mon manuscrit de thèse.

Chapter 3

Transducers in Brain Computer Interfaces

Chapter 7

Experimental setup and data

Chapter 2

Scientific context: BCI and Tetraplegia clinical trial procedure in considered in [START_REF] Moly | Online adaptive group-wise sparse NPLS for ECoG neural signal decodin[END_REF] and an efficient integration to REW-NPLS algorithm is proposed in the manuscript.

Unlike the non-regularized ALS algorithm (5.2.3)- (5.2.5), the norms of projectors are not arbitrary parameters anymore due to penalization terms. Therefore, the normalization of current estimate is added into ALS optimization cycle [START_REF] Moly | Online adaptive group-wise sparse NPLS for ECoG neural signal decodin[END_REF].

min 𝐰 ̃1 (‖𝐕 (1) -𝐰 ̃1(𝐰 3 ⊗ 𝐰 2 ) T ‖ 2 + 𝜆 1 ‖𝐰 ̃1‖ 𝑞,ℒ 1 ) 𝑤𝑖𝑡ℎ 𝐰 1 = 𝐰 ̃1 ‖𝐰 ̃1‖ , (5.3.8) min 𝐰 ̃2 (‖𝐕 (2) -𝐰 ̃2(𝐰 3 ⊗ 𝐰 1 ) T ‖ 2 + 𝜆 2 ‖𝐰 ̃2‖ 𝑞,ℒ 2 ) 𝑤𝑖𝑡ℎ 𝐰 2 = 𝐰 ̃2 ‖𝐰 ̃2‖ , (5.3.9) min 𝐰 ̃3 (‖𝐕 (3) -𝐰 ̃3(𝐰 2 ⊗ 𝐰 1 ) T ‖ 2 + 𝜆 3 ‖𝐰 ̃3‖ 𝑞,ℒ 3 ) 𝑤𝑖𝑡ℎ 𝐰 3 = 𝐰 ̃3 ‖𝐰 ̃3‖

. (5.3.10) Next, for faster computing, it can be noted that all considered regularization functions are decomposed as a sum of element-wise functions. Consequently, similarly to (5.3.2)- (5.3.4) optimization tasks (5.3.8)- (5.3.10) are split into element-wise optimization: . (5.3.14) In the next subsections, the particular cases of L0, L1, L0.5 penalizations are presented. Details of the demonstration are available in [START_REF] Moly | Online adaptive group-wise sparse NPLS for ECoG neural signal decodin[END_REF].

In the case of 𝑳 𝟎 penalization which penalized the parameter weights depending on the number of non-zero coefficients, and considering one of the optimization step, e.g. (5.3.11) of ALS optimization [START_REF] Moly | Online adaptive group-wise sparse NPLS for ECoG neural signal decodin[END_REF] The solution turns out to be an elementwise hard thresholding of the least square solution (w 𝑗 1 ) 𝐿𝑆 𝑗 = 1, … , 𝐼 1 leading to [START_REF] Moly | Online adaptive group-wise sparse NPLS for ECoG neural signal decodin[END_REF]: (6.3.1) To simplify the notation and to be closer to the BCI application presented in future chapters, in the next section, the H2M2 is considered in the specific case where 𝑧 𝑡 ∈ 𝑆 𝑝𝑟𝑜𝑑 with 𝑐𝑎𝑟𝑑(𝑆 𝑝𝑟𝑜𝑑 ) = 5 production states. The architecture of the considered H2M2 algorithm is represented in the Figure 6-2. Nevertheless, all the presented methods, algorithms and strategies can be generalized to any H2M2 architecture.

In this particular case, the H2M2 architecture is split in 𝐷 = 2 layers composed of 3 sub-HMMs with the number of sub-HMM at the layer 𝑑 = 2 is 𝐻 𝑑 = 2. This example is made of five production states 𝑆 𝑝𝑟𝑜𝑑 = {𝑠 1,1,1 , 𝑠 1,1,2 , 𝑠 2,1,2 , 𝑠 1,2,2 , 𝑠 2,2,2 } and two internal states 𝑆 𝑖𝑛𝑡 = {𝑠 2,1,1 , 𝑠 3,1,1 }. The transition matrix 𝐀 1,1 ∈ ℝ 3×3 describes the transition between the states 𝑠 1,1,1 , 𝑠 2,1,1 , 𝑠 3,1,1 whereas 𝐀 1,2 ∈ ℝ 2×2 and 𝐀 2,2 ∈ ℝ 2×2 evaluate the transitions between 𝑠 1,1,2 , 𝑠 2,1,2 and 𝑠 1,2,2 , 𝑠 2,2,2 respectively. A schematic of the considered HHMM in represented in Figure 6-2. trained and tested using the online closed-loop adaptive decoder REW-MSLM during 10 experiments (titled series F of experiments). REW-MSLM was calibrated during the first five experiments before to be tested. Model calibration lasted in total 81 min whereas all the test dataset represented 105min of experiments. In the opposite to the previously reported online experiments, during the calibration and the test phase, all the possible state transitions were experimented (including AS to AS transitions). Chapter 9

Performance evaluation and analysis

Chapter 9 : Performance evaluation and analysis

Secondly, the gate model is based on dynamic decoding using HMM. Therefore, we demonstrated that HMM dynamic gating enhanced classification performance compared to the classic sample-wise gating.

Finally, the REW-MSLM algorithm benefits from the ME structure which splits the neural space into state related subsets associated to independent expert decoders (left arm translation expert, right arm translation expert, etc.). As a result, the training data are divided into subsets associated with particular experts allowing independent expert learning. However, the continuous experts were trained on a smaller specific subset of the training dataset. This may affect regression performance. The expert-specific subset training strategy was evaluated.

To evaluate the first key feature presented above, the REW-MSLM gating (Figure 9-1C) is compared to the continuous REW-NPLS model [START_REF] Eliseyev | Recursive Exponentially Weighted N-way Partial Least Squares Regression with Recursive-Validation of Hyper-Parameters in Brain-Computer Interface Applications[END_REF] thresholded in post-processing to label the continuous decoding results as discrete IS and AS states. Such a comparison stresses the importance of the ME structure which dedicates a specific model to discrete state decoding (Figure 9-1A). The REW-NPLS with discretized output evaluated thanks to the post-processing threshold is named REW-NPLST.

Next, to determine the benefits of the dynamic HMM gating, the REW-MSLM gating model was compared to its own variant without dynamic HMM (Figure 9-1B). The version of the REW-MSLM algorithm with a static gating model referred as the REW-SLM algorithm (see section 4.5.2) was evaluated.

A similar pseudo-online study was performed to decode the right hand translation movements. The results are presented in the Figure 10 L0 REW-NPLS algorithm (Figure 10-13A) shows performance improvements with sparse solutions for different penalization hyperparameter 𝜆. For 𝜆 = 0.01, the 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 4.68% corresponding to only 3 electrode parameter weights set to zero value but the dot product (cosine similarity) was evaluated at 0.157 (0.1018 -0.203. These performance represent a cosine similarity enhancement of 24%, 117% and 30% for the median, the Q 1 and Q 3 metrics respectively. For 𝜆 = 0.018, the cosine similarity was estimated at 0.157 (0.0989 -0.185) with a 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 37.5%. For 𝜆 = 0.024, sparser solution was obtained with a sparsity index of 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 45.31% and a cosine similarity estimated at 0.153 (0.0786 -0.198). For 𝜆 > 0.04 L0 PREW-NPLS models converged to sparse solution with 48 electrodes over 64 (𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 75%) removed from the model which highlighted decoding performance similar to the state of the art REW-NPLS. The best performance of the models with 𝜆 > 0.04 was estimated at 0.128 (0.058 -0.168).

L0.5 REW-NPLS algorithm (Figure 10-13B), similarly to L0 REW-NPLS decoder, highlighted better decoding performance than REW-NPLS with sparser solutions for some penalization parameter 𝜆. For 𝜆 = 0.1, 23 electrodes were removed from the model (𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 35.94%) and the cosined similarity was estimated at 0.136 (0.100 -0.177). With higher penalization parameter 𝜆 = 0.16 sparser model was computed with 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 54.69% without decreasing the decoding performance 𝐷𝑜𝑡𝑃 = 0.150 (0.0881 -0.176). The sparsest models were obtained for 𝜆 = 0.26 and 𝜆 = 0.28 showing a sparsity index of 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 79.69% and 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 78.13% respectively. Finally, the models converged to the same solutions for 𝜆 > 0.36 with a sparsity of 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 68.75% (44 electrodes removed from the final solution)

and a cosine similarity of 𝐷𝑜𝑡𝑃 = 0.131 (0.0835 -0.186).

L1 REW-NPLS algorithm (Figure 10-13C) results show better decoding performance than REW-NPLS algorithm for numerous penalization parameter 𝜆. Several models with small penalization parameter 𝜆 = 0.04, 0.06 and 0.1 without setting any electrode to zero (𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 0%) highlighted a cosine similarity of 𝐷𝑜𝑡𝑃 = 0.154 (0.0915 -0.202), 𝐷𝑜𝑡𝑃 = 0.158 (0.0791 -0.184) and 𝐷𝑜𝑡𝑃 = 0.164 (0.0959 -0.191) representing a median improvements of 21%, 24% and 29% respectively. Similar decoding performance were obtained for higher penalization parameter 𝜆 = 0.22 and 𝜆 = 0.26 with a dot product indicator of 𝐷𝑜𝑡𝑃 = 0.154 (0.101 -0.192) and 𝐷𝑜𝑡𝑃 = 0.152 (0.0872 -0.197) but with 33 (51.56%) and 44 (68.75%) electrodes parameters weights set to zero, respectively. Finally, for a penalization parameter 𝜆 > 0.38, the models calibration stabilized to a solution with a sparsity indicator of 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 68.75% with a decoding performance of 𝐷𝑜𝑡𝑃 = 0.131 (0.0835 -0.186).

APREW-NPLS decoding performance

The results of the 3D left and right hand translation decoding studies are displayed in the For each study, the decoding performance and the sparsity indicators were evaluated for the six models obtained using the APREW-NPLS model calibration procedure. In addition to the performance of the models on the test dataset, the rank of each model estimated during the last model calibration iteration is tagged. The performance of the models is compared to the state of the art REW-NPLS algorithm. The REW-NPLS decoder highlighted a cosine similarity of 𝐷𝑜𝑡𝑃 = 0.223 (0.158 -0.266) and 𝐷𝑜𝑡𝑃 = 0.127 (0.0468 -0.155) in the pseudoonline left and right hand translation decoding studies respectively.

For the left hand translation models , the APREW-NPLS model with 𝜆 = 0.1 was ranked last during the model calibration procedure and achieved the worst decoding performance with a cosine similarity of 𝐷𝑜𝑡𝑃 = 0.168 (0.106 -0.245) without setting any parameter weight to zero (𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 0%). For 𝜆 = 0.2, the model was ranked 3 rd . The sparsity index reached 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 60.94%. The decoding performance stressed small improvements compared to REW-NPLS algorithm with a cosine similarity of 0.241 (0.151 -0.300). The models characterized with a penalization coefficient of 𝜆 = 0.3 ranked penultimate models according to APREW-NPLS algorithm highlighted similar results than REW-NPLS algorithms with a dot product indicator of 0.212 (0.160 -0.270) but with multiple electrodes parameter weights set to zero: 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 78.13%. The APREW-NPLS algorithm assigned the best ranked to the model with 𝜆 = 0.4 which highlighted good decoding performance 0.243 (0.160 -0.290) with a sparse model 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 73.44%. The last two models with the penalization coefficient 𝜆 = 0.5 and 0.6 showed a decoding performance of 0.244 (0.142 -0.270) and 0.237 (0.149 -0.290) with a sparsity index of 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 62.5% and 71.88% respectively. These models were rated as fourth and second best models.

For the right hand models , the APREW-NPLS models highlighted better decoding performance than REW-NPLS algorithms for 𝜆 = 0.1, 0.2 and 0.6 which were ranked 6 th ,5 th and 2 nd best models according to APREW-NPLS calibration procedure. These models exhibited a decoding performance of 0.160 (0.0766 -0.194) for 𝜆 = 0.1, 0.155 (0.0898 -0.174) for 𝜆 = 0.2 and 0.145 (0.0841 -0.197) for 𝜆 = 0.6 with a sparsity index of 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 0%, 56.35 and 60.94% respectively. These APREW-NPLS models exhibited a median cosine similarity improvement of 26%, 22% and 14% respectively compared to the median REW-NPLS decoding performance. The APREW-NPLS with a penalization hyperparameter equal to 𝜆 = 0.3, 0.4 and 0.5 presented similar results than REW-NPLS algorithm with a sparsity index between 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 39.06% and 𝑆𝑝𝑎𝑟𝑠𝑒𝐼𝑑𝑥 = 71.88%. These models were classified 3 rd , 1 st , and 4 th best models respectively.

Chapter 10 : Results and 89.5% ± 3.2 for the HMMlimited and HMMfull and H2M2 classifier respectively. With the exception of the precision indicator, they all increased with H2M2 classifier compared to HMMlimited and compared to HMMfull. However the performance variation between HMMfull and the H2M2 classifier were small with an average improvements of 1.5% ± 0.83. Compared to the HMMlimited, H2M2 showed an average improvements of 4.9% ± 2.9. The Gmean performance indicator of the H2M2 algorithm was estimated at 73.5% ± 4.4 whereas HMMlimited and HMMfull reached a Gmean classification score of 65.6% ± 5.4 and 70.9% ± 4.8. The highest performance improvements were estimated for Recall, HF difference, Bookmaker and Gmean indicators with a classification enhancement of 6.9%, 8.2%, 7.8% and 7.9% compared to HMMlimited, and 2.4%, 1.5%, 2.7% and 2.6% compared to HMMfull respectively. H2M2 gating algorithm offline comparative study 259

The estimated median latency of the three decoders are represented in Figure 10-26C. H2M2 showed lower decoding latency than HMMlimited and HMMfull. Using the notation: median (Q 1 -Q 3 ), the H2M2 latency was computed at 2.6 (1.9 -4.9) whereas the HMMlimited and HMMfull algorithms showed a median latency of 4.1 (3.0 -7.25) and 3.2 (2.3 -5.9) respectively. The median latency was reduced by 37% and 19% compared to HMMlimited and HMMfull.

H2M2 highlighted an error length block duration of 2.2 (0.9 -5.

PREW-NPLS offline evaluation

The PREW-NPLS was tested offline in a pseudo-online study using dataset recorded during the online closed-loop 8D experiments performed with the REW-MSLM algorithms. In order to stick to the online closed-loop experiments restrictions, the pseudo-online Lp-PREW-NPLS calibration dataset was restricted to the same 6 calibration experiments which were used for the online REW-MSLM experts and gating model estimations. Lp-PREW-NPLS performance were only estimated for the 3D left and right hand translation tasks as the REW-NPLS expert of REW-MSLM already highlighted good decoding performance for the left and right wrist rotation tasks. The model was evaluated with L0, L0,5 and L1 norm regularization and multiple penalization hyperparameters in order to estimate the impact of the sparsity on the decoding performance. In the presented pseudo-online study, the group-wise regularization was focused on the spatial modality.

The PREW-NPLS decoders highlighted equivalent or better decoding performance than the REW-NPLS algorithm for the majority of the penalized hyperparameters. As an example, for the right hand decoding study, the L1-PREW-NPLS model with a penalization hyperparameter of 𝜆 = 0.26, highlighted a significant (𝑝 -𝑣𝑎𝑙𝑢𝑒 = 10 -6 ) cosine similarity improvement of 21%, 116% and 24% of the median, 25 th and 75 th percentiles respectively with less than half of the electrodes maintained to a non-zero value (33 electrodes set to zero). In the case of small penalization hyperparameters such as 𝜆 = 0.1, the model converged to a non-sparse solution with significant cosine similarity differences between REW-NPLS and L1-PREW-NPLS models (𝑝 -𝑣𝑎𝑙𝑢𝑒 = 10 -6 ) leading to a median, 25 th and 75 th percentile enhancement of 24%, 104%

and 23% respectively. The sparsest solution with the L1-PREW-NPLS algorithm removed 75% of the electrodes without decreasing the cosine similarity indicator reducing the features space from 10 × 15 × 64 = 9600 features to 10 × 15 × 16 = 2400. If the sparse models with a limited number of electrodes turns out to be stable, the feature space dimension can be definitively reduced.

Significant indicator differences between REW-NPLS and L1-PREW-NPLS models were more evident for the 3D right hand translation models than for the 3D left hand translation ones. In all the manuscript, the 3D right hand translation models highlighted lower decoding performance than the left hand translation models. Group-wise sparse regularization may be a solution to enhance the neural signal decoding for 3D right hand translation tasks. Although not all the L1-PREW-NPLS models highlighted better cosine similarity than the REW-NPLS model, for most of the left and right L1-PREW-NPLS models, similar results were obtained with only a small number of spatial features compared to REW-NPLS.

Finally, reducing the feature space dimension led to more interpretable models. Several activation patterns were discernable. In the spatial domain, the electrodes closed to the motor and sensory sulci exhibit important parameter weights whereas, in the frequency domains, frequency bands below 30Hz and between 60Hz and 110Hz were dominant.