Keywords: Interface Cerveau-Machine, Electrocorticographie, Traitement du signal, Multi-voies analyse, Modélisation adaptative, Ondelettes Brain-Computer Interface with cortical electrical activity recording 7 Brain-Computer Interface with cortical electrical activity recording 8 Brain-Computer Interface, Electrocorticography, Signal Processing, Multi-Way Analysis, Adaptive Modeling, Wavelets Brain-Computer Interface with cortical electrical activity recording

Brain-Computer Interface with cortical electrical activity recording 6 méthodes proposées représentent une approche prospective pour de futurs développements de systèmes ICM humains.

Résumé

Une Interface Cerveau-Machine (ICM) est un système permettant de transformer l'activité neurale du cerveau en une commande d'effecteurs externes. Cette étude correspond à une étape vers une ICM totalement autonome fonctionnant dans un environnement naturel ce qui est d'une importance cruciale pour les futures applications cliniques d'une ICM. Pour représenter l'environnement naturel, des expériences avec une ICM binaire asynchrone ont été réalisées avec des animaux libres de se mouvoir. En comparaison avec les études précédentes, des expériences sur le long terme ont été réalisées, ce qui est plus conforme aux exigences des applications de la vie réelle.

L'objectif principal de cette étude est de différencier le modèle spécifique neuronal lié à l'intention d'action de l'activité de fond du cerveau chez des animaux libres de tous mouvements. Pour atteindre le niveau nécessaire de sélectivité, l'analyse Multi-Voies PLS a été choisie sachant qu'elle fournit simultanément un traitement du signal dans plusieurs domaines, à savoir, temporel, fréquentiel et spatial. Pour améliorer la capacité de l'approche générique Multi-Voies PLS pour le traitement de données à grandes dimensions, l'algorithme « Itérative NPLS » est introduit dans notre travail. En ayant des besoins plus faibles en mémoire, cet algorithme fournit des traitements de grands ensembles de données, permet une résolution élevée, préserve l'exactitude de Chapter 1.

FUNCTIONAL BRAIN COMPUTER INTERFACE

Introduction

Brain Computer Interface (BCI), also known as Brain Interface (BI), and Brain Machine Interface (BMI), is a system for translating the brain neural activity into commands to external devices [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. In other words, BCI aims to provide an alternative communication pathway for subjects to interact with their environment without using muscles. Such a system could be helpful for people suffering from severe motor disabilities to control wheelchairs, prostheses, etc. High-level general scheme of BCI system is shown in Figure 1.1.

Indeed, the operation cycle of a BCI system contains two stages: 1) calibration stage; 2) execution stage (also called the close-loop stage). In the close-loop mode, the system controls the external effector by means of the control signal ( ) in the real-time on the basis of the recordings of the brain electrical activity ( ). To create a control Effector ( ) ( ) ( )

Brain

Control Block model used at the second stage, the first stage, namely, calibration is applied. During the calibration stage, the BCI system receives the signal ( ) from the brain as well as information about the effector's state ( ) (the dotted arrow on the figure). This data is used for the BCI system parameterization, and identification of the model, representing the relation between the brain's signals and effector's state. Once all parameters of the model have been defined, the system can be switched to the next stage. In general, the feedback signal ( ) is not available for the control block at the second stage. At the same time, the brain has access to the feedback information (e.g., video, audio, tactile, etc.) on both stages.

Among different BCI systems, the movement-related BCIs are of great importance. Last decades several approaches were developed to face the problem of movement-related signal decoding. Promising results were obtained both in animal [START_REF] Chapin | Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex[END_REF][START_REF] Wessberg | Real-time prediction of hand trajectory by ensembles of cortical neurons in primates[END_REF] and in human [START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF][START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF] studies. To record neural activity, systems with scalp electrodes for electroencephalography (EEG) [START_REF] Wolpaw | An EEG-based brain-computer interface for cursor control[END_REF][START_REF] Birbaumer | A spelling device for the paralysed[END_REF]Kubler et al., 1999), surface electrodes against the dura mater (epidural electrodes) [START_REF] Rouse | Neural adaptation of epidural electrocorticographic (EECoG) signals during closed-loop brain computer interface (BCI) tasks[END_REF][START_REF] Torres Valderrama | Gain of the human dura in vivo and its effects on invasive brain signal feature detection[END_REF], directly against the cortex (subdural electrodes) [START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF], or microelectrode array for deep brain recording of single units [START_REF] Chapin | Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex[END_REF] have been used. Neural activity was analyzed using a variety of methods. Mainly the approaches were aiming at classifying of event-related electrical patterns (for instance, the case of the ‗eventnon-event' classification). The detection of patterns allows triggering the effector, e.g., a cursor on a screen [START_REF] Wolpaw | An EEG-based brain-computer interface for cursor control[END_REF], a motorized device [START_REF] Chapin | Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex[END_REF][START_REF] Wessberg | Real-time prediction of hand trajectory by ensembles of cortical neurons in primates[END_REF], etc. Experiments were carried out either in rodents [START_REF] Chapin | Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex[END_REF]Jensen et Rouse, 2006), or in non-human primates [START_REF] Wessberg | Real-time prediction of hand trajectory by ensembles of cortical neurons in primates[END_REF][START_REF] Serruya | Instant neural control of a movement signal[END_REF], or in human patients [START_REF] Scherer | Frequency component selection for an ECoG-based brain-computer interface[END_REF][START_REF] Leuthardt | Electrocorticography-based brain computer interface--the Seattle experience[END_REF]. Nevertheless, an effective solution of the problem still does not exist. Moreover, a set of simplifications is applied in these BCIs. For instance, the duration of the experiments usually does not exceed several minutes (for instance, see [START_REF] Müller-Putz | Fast set-up asynchronous brain-switch based on detection of foot motor imagery in 1-channel EEG[END_REF]. In addition, the subject is being given a cue to start performing a task [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. And finally, experiment conditions significantly differ from the natural environment (motion restriction, absence of significant external disturbances, and so on. See [START_REF] Zhao | EEG-based asynchronous BCI control of a car in 3D virtual reality environments[END_REF].

Thus, bringing a BCI out of the laboratory to real-life clinical application represents a challenging task. The development of a BCI system that meets the requirements of real applications is the main goal of the CLINATEC/LETI/CEA BCI project. The project includes development and application of implants as well as data acquisition, transferring and processing systems (Figure 1.2). The purpose of the presented work, in the frame of preclinical studying of the BCI project in Clinatec, consists in the development and implementation of methods and algorithms for the signal processing block of the BCI system. Namely, the study is dedicated to the major problems of the clinical applications of BCI: fully autonomous functioning of the system during long term in the real-life environment.

BCI for real-life applications

BCI system for real-life clinical applications must meet a set of requirements, such as health safety, real-life conditions reliability, as well as long-term stability. The main requirements imposed on the system can be summarized as follows:

 Safety

As the system is interacting with the brain, the safety questions, connected with risks of surgical implantation, infection, etc., have a paramount value. The use of deeper brain layers for data acquisition by means of invasive techniques increases risks for health such as brain damage or infection. Besides the risks of surgery intervention, errors in signal interpretation by the BCI system could be dangerous for the user's health and life (e.g., spontaneous motion of wheelchair and so on).

 Signal quality

The question of the signal's quality is of great importance for BCI systems. Broad frequency band, good spatial resolution, as well as the signal-to-noise ratio of the recordings allow increase system reliability (e.g., decreasing the number of the false activations, etc.).

 System's long-time stability

The BCI system should operate for a long period (years). Over this time, the signal can change significantly due to numerous reasons, such as the brain plasticity, data acquisition data transferring signal processing effector  feedback degradation of electrodes, etc. All these effects have to be properly treated by the system.

 Natural environment

As opposed to the laboratory conditions when a subject is concentrated on the task, when external noises are minimized, and computational resources could be rather considerable, it is almost impossible to provide these circumstances in real-life environment. Thereby, the system must be robust enough to guarantee functionality of the BCI in presence of noise and spurious signals.

 Real-time conditions

Operation in real-time is essential for the majority of BCI applications. This requirement imposes a significant restriction on both algorithms and computational facilities. Examples of real-time BCIs could be found in [START_REF] Chapin | Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex[END_REF]; [START_REF] Schalk | Twodimensional movement control using electrocorticographic signals in humans[END_REF]; [START_REF] Zhao | EEG-based asynchronous BCI control of a car in 3D virtual reality environments[END_REF].

 Usability

The system should work without everyday recalibration or this calibration should be acceptably simple and easy (for example, it takes less than several minutes per day and could be made in any conditions).

 Autonomy

The system should be used without any external cue, whenever the user intends.

Thus, the informative component should be detected on the background of the arbitrary user's activity.

 Equipment complexity and expensiveness

Technical problems of the functional BCI system mainly consist in hardware, which should be suitable for the everyday usage in rather rigorous conditions. The BCI system must be fast enough to process all necessary information without significant time delays. From the other hand, it must be small and lightweight not to burden a person using it. Also, the questions of the long-term power supply as well as everyday stable work guarantee are of great importance. Moreover, mass production imposes limitations on the complexity and the cost of the system.

Every block of the BCI system must be optimized according to the points mentioned above. The choice of the method for the brain signals recording is a compromise of the patient's safety, signal's quality, reliability and applicability in natural environment. The methods of data acquisition and treatment must be fast and reliable to function properly in the real-time mode in real-life conditions, etc.

In the following section, the methods and approaches of the signal recording, as well as neurological phenomena and BCI control paradigms, are compared and evaluated according to the requirements of BCI systems.

Neural activity recording

During the last decades, several methods were developed to measure the brain's activity. The methods range from recording activity of single neurons to large-scale brain processes analysis.

A neuron is an electrically excitable cell, which processes and transmits In current BCIs there are different techniques for recording of the brain's neural activity: non-invasive, partially invasive and invasive.

Non-invasive methods

The non-invasive methods are widespread due to their usability and safety. At the same time, the non-invasive signal recording systems produce poor signal resolution because the skull dampens signals, dispersing and blurring the electromagnetic waves created by the neurons.

 Electroencephalography

Electroencephalography (EEG) is one of the most widespread non-invasive techniques for neural activity recording firstly proposed by H. Berger in 1929. Now it is widely used as a diagnostic tool in the clinical purpose [START_REF] Birbaumer | A spelling device for the paralysed[END_REF][START_REF] Wolpaw | The Wadsworth center brain-computer interface (BCI) research and development program[END_REF][START_REF] Blankertz | The Berlin Brain-Computer Interface: EEG-based communication without subject training[END_REF]. Popularity of the EEG could be explained also by the simplicity of the measurements process and cheapness of the necessary equipment.

EEG measurements are made by means of special electrodes with radius about 5 mm.

Electrodes are mounted in the cap which can be easily worn on a subject practically without any preliminary preparation. The EEG approach consists in measurement of the small potentials (up to 100 μV) between the electrodes placed in the different locations on the scalp. Figure 1.4 shows the electrode placement in the EEG cap according to the 10-20 international system and demonstrates an example of the recorded signals. On the physical level, the signal measured with the EEG is caused by the electrical activity at pyramidal neurons in the cerebral cortex [START_REF] Martin | The collective electrical behavior of cortical neurons: The electroencephalogram and the mechanisms of epilepsy[END_REF].

Summation of the currents from neurons can be detected in the EEG [START_REF] Sanei | EEG Signal Processing[END_REF]. For the reason of the tissue barrier, which is between the neurons and the electrodes, it is practically impossible to register a low-energy brain activity, as well as the frequencies higher than 100 Hz. The low limit of the EEG spectrum is 0.1 Hz.

Generally the range from 0.3 Hz to 70 Hz is used in practice [START_REF] Nazarpour | Brain Signal Analysis in Space-Time-Frequency Domain; An Application to Brain Computer Interfacing[END_REF].

Moreover, artifacts from eye blinks, movements and other muscle activities complicate the analysis of EEG data. The amplitude of the artifacts sometimes significantly exceeds the amplitude of the signal of interest. Hence, often artifact filtering must be applied to the signal before data will be used.

Spectrum of the EEG is divided on several bands, which have special names [START_REF] Nazarpour | Brain Signal Analysis in Space-Time-Frequency Domain; An Application to Brain Computer Interfacing[END_REF]:

 Delta (δ): 3.5 Hz  Theta (θ): 4-7 Hz  Alpha (α): 8-13 Hz. An amplitude ranges generally from 20 to 100 μV, mostly below 50 μV. The α rhythm has usually the sinusoidal waveform, it is rhythmic and regular.

 Rolandic Mu (μ): 8-13 Hz. It has the same frequency band as the α rhythm but it is registered in the central and the parietal areas (see Figure 1.4). The μ rhythm has a sharp negative peak and rounded positive phase.

 Beta (β): 14-35 Hz. Located mostly in the frontocentral regions.

 Gamma (γ): 35 Hz.

The EEG is one of the most used techniques for measuring the brain electrical activity due to its safety. It is applied, for example, on research of epilepsy, sleep stages, as well as numerous clinical tasks [START_REF] Acar | Multiway analysis of epilepsy tensors[END_REF]. Unfortunately, shortcomings of the EEG approach significantly restrict its application in BCI systems. For instance, position of the electrodes is unstable, which force frequent system recalibration.

Furthermore, training of subjects is relatively long, and some subjects can never achieve appropriate results [START_REF] Blankertz | Neurophysiological predictor of SMR-based BCI performance[END_REF]. In addition, EEG data frequency range 0.3-70 Hz [START_REF] Nazarpour | Brain Signal Analysis in Space-Time-Frequency Domain; An Application to Brain Computer Interfacing[END_REF] is rather low, whereas a spatial resolution (several centimeters) is too rough due to presence of the skull [START_REF] Srinivasan | Methods to improve the spatial resolution of EEG[END_REF], which dampens the signals. Moreover, as far as the recording's amplitude is rather low (≤100 μV), a significant influence of artifacts on the informative data occurs [START_REF] Hoffmann | Bayesian machine learning applied in a brain-computer interface for disabled users[END_REF]. This leads to impossibility of realization of BCIs with many degrees of freedom. Thus, while currently the EEG is widely used for the BCI systems, its shortcomings lead for searching other methods for more reliable recording of neural activity.

 Magnetoencephalogram

Magnetoencephalogram (MEG) is a non-invasive method for measuring the intensity of a magnetic field (Figure 1.5). In comparison with the EEG, the MEG data are less disturbed by scalp. Thus, magnetoencephalography allows obtain a signal with better spatial resolution. In addition, the MEG is more sensitive to superficial cortical activity. It is useful for studying of the neocortical epilepsy. Signal-to-noise ratio coefficient for MEG is better than the EEG one. The difference is especially significant for the high frequency band (γ-band) [START_REF] Kübler | An introduction to brain computer interfacing[END_REF]. The method cannot be used in real-life applications due to huge size of devices for magnetic field measurements. Moreover, the price of the equipment is very high that leads to impossibility of the MEG systems mass-production. However, some example of MEG-BCIs can be found in [START_REF] Hoffmann | Bayesian machine learning applied in a brain-computer interface for disabled users[END_REF]; [START_REF] Kauhanen | EEG and MEG brain-computer interface for tetraplegic patients[END_REF]; [START_REF] Lal | A brain computer interface with online feedback based on magnetoencephalography[END_REF]; [START_REF] Blankertz | The Berlin Brain-Computer Interface: EEG-based communication without subject training[END_REF]; [START_REF] Buch | Think to Move: a Neuromagnetic Brain-Computer Interface (BCI) System for Chronic Stroke[END_REF].

 Functional Magnetic Resonance Imaging

Functional Magnetic Resonance Imaging (fMRI) is a non-invasive method for measuring the blood oxygen level dependent (BOLD) signal. the fact that local concentration of deoxygenated hemoglobin in the brain depends on the neural activity in the cells. In comparison to the EEG, the fMRI allows detecting activity not only in the cortex but also in deeper structures of the brain [START_REF] Weiskopf | Self-regulation of local brain activity using real-time functional magnetic resonance imaging (fMRI)[END_REF]. Unfortunately, the fMRI like the MEG measurements are very complex and expensive. They can only be made in laboratory conditions and cannot be easily applied in practical BCI systems.

 Near Infrared Spectroscopy

Near Infrared Spectroscopy (NIRS) is a non-invasive method for the measurement of hemodynamic activity. NIPS uses near-infrared light, which depends on the amount of oxygenated and deoxygenated hemoglobin in the blood. In contrast to the fMRI, the NIRS is significantly cheaper and requires less technical effort, whereas the spatial resolution is almost the same. The method was applied for the motor imagery and the movement tasks in BCI systems [START_REF] Coyle | On the suitability of near-infrared (NIR) systems for next-generation brain-computer interfaces[END_REF][START_REF] Sitaram | Near infrared spectroscopy based braincomputer interfaces[END_REF][START_REF] Sitaram | Temporal classification of multichannel near-infrared spectroscopy signals of motor imagery for developing a brain-computer interface[END_REF]. However, an application of NIRS in the real-life BCI is still not possible nowadays due to the complexity of the equipment.

Thus, among the non-invasive methods, mainly EEG is applied in the practical BCI systems. The other methods are mainly used for fundamental researches and medical purposes.

Invasive methods



Figure 1.7

The silicon-based microelectrode array: 100 electrodes spaced at 400-micron intervals [START_REF] Normann | A neural interface for a cortical vision prosthesis[END_REF] and an example of the registered signal [START_REF] Williams | Long-term neural recording characteristics of wire microelectrode arrays implanted in cerebral cortex[END_REF].

The main advantage of the MEA application in comparison with other approaches is its high spatial resolution, which gives opportunity to apply the microelectrode arrays for complex tasks, e.g., real-time 3D motion [START_REF] Taylor | Direct cortical control of 3D neuroprosthetic devices[END_REF] and robotic arm control [START_REF] Wessberg | Real-time prediction of hand trajectory by ensembles of cortical neurons in primates[END_REF][START_REF] Schwartz | Cortical neural prosthetics[END_REF]). The action-potential firing rate allows estimation of the subject's intention of movement [START_REF] Serruya | Instant neural control of a movement signal[END_REF]. For instance, 96 electrodes were implanted in a human subject suffering from tetraplegia that makes possible to the subject to operate with a prosthetic hand and to accomplish other tasks [START_REF] Hochberg | Neuronal ensemble control of prosthetic devices by a human with tetraplegia[END_REF].

The main disadvantage of the MEA application is the penetration of the cortex to a depth of several millimetres [START_REF] Nicolelis | Chronic, multisite, multielectrode recordings in macaque monkeys[END_REF]. Moreover, quality of the registering signals decays with individual neuron death over time [START_REF] Biran | Neuronal cell loss accompanies the brain tissue response to chronically implanted silicon microelectrode arrays[END_REF].

Besides that, the astrocytes and microglia adhere to the device begin to accumulate.

They form a sheath surrounding the array. This both increases the space between the electrode probes and insulates the electrodes. In addition, this increases a measurement impedance. Thus, the data can be recorded only for several months. On the other hand, recent results [START_REF] Hochberg | Neuronal ensemble control of prosthetic devices by a human with tetraplegia[END_REF] demonstrate that a stable recording over a long time (year) is possible but at the expense of losing the signal at numbers of electrodes (about 20 of 150 remained functioning).

The MEA is a remarkable tool for the brain signal registration. Unfortunately, due to the safety questions the microelectrode arrays are used mainly in animals.

Additional information about the method can be found in the review of [START_REF] Lebedev | Brain-machine interfaces: Past, present and future[END_REF].

Thus, the method provides a high quality signal with significant spatial and frequency resolutions, as well as a good signal-to-noise ratio. As far as response time of the system is small enough [START_REF] Taylor | The importance of online error correction and feed-forward adjustment in brain-machine interfaces for restoration of movement[END_REF], whereas the system is compact and relatively cheap, it could be effectively applied for BCI tasks. On the other hand, the method is rather dangerous due to presence of significant risks of the brain damages. In addition, the system long-term stability supply is rather laborious. Moreover, only a small region of the brain could be observed.

Partially-invasive methods

 Electrocorticography

The Electrocorticography (ECoG) uses an electrode grid or strip to record the electrical activity of the brain's cortex. The ECoG was pioneered by W. Penfield and H. Jasper in the 1950's and now it is considerably widespread [START_REF] Huggins | Detection of Event-Related Potentials for Development of a Direct Brain Interface[END_REF][START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF][START_REF] Felton | Electrocorticographically controlled brain-computer interfaces using motor and sensory imagery in patients with temporary subdural electrode implants. Report of four cases[END_REF][START_REF] Schalk | Twodimensional movement control using electrocorticographic signals in humans[END_REF]. Like the EEG, the ECoG is based on measuring of the potential activity in the cortical pyramidal neurons. The difference is that the electrodes are placed immediately on the cortex surface. Thus, the electrical signals must not be conducted through the skull, where their potentials diminish due to the low conductivity of the bone. It leads to increasing of the spatial resolution of the ECoG in comparison with the EEG (tens of millimeters versus centimeters), broader frequency bandwidth (above 300 Hz), and higher signal's amplitude (50-100 μV). In addition, the ECoG recordings are less influenced by artifacts [START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF].

Generally, the ECoG experiments are made in the context of the long-term clinical applications which require continuous monitoring and processing of the neural activity for epileptic patients. For reach the cortex, a craniotomy must be made for a patient, i.e., a part of the skull must be removed to expose the brain surface (see Figure 1.8). This procedure is performed under general or local anesthesia. The electrodes arrays can be placed either on the surface of the dura mater (epidural) or beneath the dura mater (subdural). The electrodes are attached to a flexible frame. Standard spacing between the electrodes in the frame is 1 cm; diameter of the electrodes is up to 5 mm [START_REF] Schuh | Intraoperative Electrocorticography and Direct Cortical Electrical Stimulation[END_REF]. The electrodes are designed in the way to eliminate any injury of the brain during its movements.

Figure 1.9 demonstrates electrodes arrangement in the brain. Figure 1.10 represents the cortex with the electrodes placed for the subdural extraoperative ECoG. [START_REF] Kutsy | TI Focal extratemporal epilepsy: clinical features, EEG patterns, and surgical approach[END_REF].

Figure 1.10 Exposed cerebral cortex and electrode placement for the subdural ECoG [START_REF] Miller | Spectral Changes in Cortical Surface Potentials during Motor Movement[END_REF].

Besides the clinical applications, the ECoG is considered as promising technique for the BCI systems. It does not penetrate the brain significantly, but has higher spatial and frequency resolutions in comparison with the EEG. Moreover, it has better signalto-noise ratio, which is the very important characteristic for BCIs. The BCI experiments are carried out in subjects with implanted matrix of electrodes. Therefore, the tests cannot be made in healthy humans. Commonly, ECoG is used to localize seizure activity in patients with epilepsy before they undergo surgery [START_REF] Kübler | An introduction to brain computer interfacing[END_REF]. The ECoG experiments have demonstrated that the users can learn to control the ECoG signals through motor imagery [START_REF] Graimann | Toward a direct brain interface based on human subdural recordings and wavelet-packet analysis[END_REF][START_REF] Hill | Classifying EEGand ECoG signals without subject training for fast BCI References Brain-Computer Interface with cortical electrical activity recording 169 implementation: Comparison of nonparalyzed and completely paralyzed subjects[END_REF], speech imagery (Leuthardt et el., 2006), auditory imagery [START_REF] Wilson | ECoG factors underlying multimodal control of a brain-computer interface[END_REF], as well as mental calculation [START_REF] Ramsey | Towards human BCI applications based on cognitive brain systems: An investigation of neural signals recorded from the dorsolateral prefrontal cortex[END_REF]. Most of ECoG BCI studies were carried out

offline and performed open-loop analysis of data [START_REF] Kübler | An introduction to brain computer interfacing[END_REF]Huggins et al., 1999;[START_REF] Levine | Identification of Electrocorticogram Patterns as the Basis for a Direct Brain Interface[END_REF]. [START_REF] Scherer | Frequency component selection for an ECoG-based brain-computer interface[END_REF] reported accuracy between 85 and 91 percent for offline classification of data recorded during self-paced middle finger extension. [START_REF] Hill | Classifying EEGand ECoG signals without subject training for fast BCI References Brain-Computer Interface with cortical electrical activity recording 169 implementation: Comparison of nonparalyzed and completely paralyzed subjects[END_REF] achieved accuracies around 75 percent for motor imagery tasks. However, only in few studies close-loop was realized and feedback of ECoG was provided to the participants [START_REF] Felton | Electrocorticographically controlled brain-computer interfaces using motor and sensory imagery in patients with temporary subdural electrode implants. Report of four cases[END_REF][START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF][START_REF] Leuthardt | Electrocorticography-based brain computer interface--the Seattle experience[END_REF][START_REF] Wilson | ECoG factors underlying multimodal control of a brain-computer interface[END_REF]. [START_REF] Birbaumer | Breaking the silence: brain-computer interfaces (BCI) for communication and motor control[END_REF] implanted electrodes on the cortex of a completely paralyzed patient to classify motor imagery signals. Unfortunately, classification results were at the chance level. Nevertheless, the carried out experiments demonstrated that even more than one year after implantation 50 percent of the electrodes provide stable signal recording. The promising results from new designs of the ECoG matrix electrodes for the long-term registration of the neuronal activity were reported recently [START_REF] Charvet | WIMAGINE: A Wireless, Low Power, 64-Channel ECoG Recording Platform for Implantable BCI Applications[END_REF][START_REF] Ejserholm | A Polymer Based Electrode Array for Recordings in the Cerebellum[END_REF][START_REF] Hirata | An integrative BMI approach for functional restoration using human electrocorticograms[END_REF].

Contrary to the EEG-, ECoG-based BCIs avoid problems with muscular artifacts [START_REF] Sutter | The brain response interface: communication through visuallyinduced electrical brain responses[END_REF][START_REF] Zaveri | Time-frequency representation of electrocorticograms in temporal lobe epilepsy[END_REF], offers better localization of origin of the signals [START_REF] Salanova | Comparison of scalp electroencephalogram with subdural electrocorticogram recordings and functional mapping in frontal lobe epilepsy[END_REF], a wider frequency range [START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF], and higher signal-to-noise ratio [START_REF] Margalit | Visual and electrical evoked response recorded from subdural electrodes implanted above the visual cortex in normal dogs under two methods of anesthesia[END_REF]. In addition, there is a benefit of shorter training times for the ECoG-based BCIs [START_REF] Leuthardt | A braincomputer interface using electrocorticographic signals in humans[END_REF][START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF][START_REF] Huggins | Electrocorticogram as a Brain Computer Interface Signal Source[END_REF].

Thus, the quality of the signal recorded with the ECoG is appropriate to be used in the BCI project. The frequency band and the spatial resolution, as well as the signal- to-noise ratio surpass correspondent parameters of the non-invasive methods. The risk of implantation is not as high as for the microelectrode arrays. Furthermore, since the recording does not depend on a single or several neurons, the system provides better long-term stability than the invasive methods. Response time and size of the system allows its utilization in the real environment in the real-time mode. Hence, as a tradeoff between registered signal quality and safety, the ECoG was selected as the method for the brain activity recording in the Clinatec project.

Nanostructured electrodes for the ECoG recordings

One perspective direction for ECoG recording is application of nanostructured electrodes. The advantages of this approach (in particular, carbon nanotubes (CNT) on the surface of the electrodes) were reported recently by [START_REF] Sauter-Starace | ECoG recordings of a non-human primate using carbon nanotubes electrodes on a flexible polyimide implant[END_REF]. The electrodes covered by CNT allow registration of higher current in comparison to the electrodes without CNT. Their signal-to-noise ratio is better [START_REF] Keefer | Carbon nanotubes coating improves neuronal recordings[END_REF].

A set of experiments were carried out to compare the quality of the signals The experiments demonstrated that during several months the implanted CNT electrodes were able to record neuronal signals. Any epileptic activity, which could be linked to brain irritation, was not observed. Nanostructuration with CNTs is reported as an effective way to improve signal quality by increasing the signal-to-noise ratio in ECoG.

Thus, carbon nanotubes improve quality of the registration of the brain neuron electrical activity. Long-term biocompatibility tests were carried out in vitro by [START_REF] Sauter-Starace | Projet NEUROLINK: vers une interface cerveau machine[END_REF]. At the same time, some long-term experiments in vivo were made to date. However, to have more reliable results the additional tests are required.

Electrophysiological sources of control in BCIs

Numerous neurological phenomena could be used to control the BCI system.

These phenomena are characterized by voltage amplitude, latency, frequency and space distribution. Some of them are described below.

 Slow Cortical Potentials (SCPs)

SCPs are the slow potential changes generated by a user. Usually they have a non-movement source and reflect the changes in the cortical polarization lasting from 300 ms up to several seconds [START_REF] Birbaumer | Slow potentials of the cerebral cortex and behavior[END_REF]. The CSP based BCI systems require users to achieve voluntary regulation of its brain activity. They typically use the traditional S1-S2 paradigm which was proposed in the work of [START_REF] Walter | Contingent negative variation: An electric sign of sensorimotor association and expectancy in the human brain[END_REF]. A negative SCP shift seen after a warning stimulus (S1) two to ten seconds before an imperative stimulus (S2) that requires participants to perform a task (e.g., a button press or cursor movement). An example of the SCP application for the BCI system could be found in [START_REF] Birbaumer | A spelling device for the paralysed[END_REF]. In this work patients suffering from amyotrophic lateral sclerosis (ALS) use the BCI to control a spelling device.

 P300

Auditory, visual and somatosensory stimuli can evoke a positive peak at about 300 ms after the stimulus is received. This positive signal deflection is called P300 [START_REF] Squires | The effect of stimulus sequence on the waveform of the cortical event-related potential[END_REF]. The P300 amplitude varies. It depends on discriminability of standard and target stimuli, overall probability of the target stimuli, and the preceding stimulus sequence, as well as the electrodes position [START_REF] Squires | The effect of stimulus sequence on the waveform of the cortical event-related potential[END_REF]. Mostly, the P300 is observed in the central and the parietal regions.

Farwell and Donchin had shown that the event-related potential could be used to select items displayed on a computer monitor [START_REF] Farwell | Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials[END_REF]. From users it was required that they were able to focus attention and gaze on the target letter for a considerable amount of time.

For more information see [START_REF] Nieuwenhuis | Decision making, the P3, and the locus coeruleus-norepinephrine system[END_REF].

 Visual Evoked Potentials (VEP) and Steady-State Visual Evoked Potentials (SSVEP)

Small changes in the brain signal generated in response to a visual stimulus are called VEPs. Their characteristics depend on the type of the visual stimulus. If a visual stimulus is presented repetitively at a rate of 5-6 Hz or greater a response is termed SSVEP. Its amplitude and phase depend on such stimulus parameters as repetition rate and contrast. Like the P300, the SSVEP requires attention and intact gaze but no user training.

The SSVEP-BCI was used, for instance, in the work of [START_REF] Wang | A practical VEP-based braincomputer interface[END_REF].

Three targets with different flickering frequencies were presented on a monitor.

On general, up to 90% classification accuracy is reported. However, a shortcoming of all SSVEP approaches to the BCI control is their dependence on intact gaze. Hence, the method is unsuitable for patients with restricted eye movement. Moreover, it cannot be used in long-term experiments.

Description of the BCI systems using the SSVEP could be found in [START_REF] Herrmann | Human EEGresponses to 1-100 Hz flicker: Resonance phenomena in visual cortex and their potential correlation to cognitive phenomena[END_REF], [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][START_REF] Lalor | Steady-state VEP-based brain-computer interface control in an immersive 3D gaming environment[END_REF].

 Sensorimotor rhythms (SMR)

The sensorimotor rhythm is brain wave rhythm which appears over the sensorimotor cortex. SMRs are characterized by a decrease (desynchronizing)

and increase (synchronizing) of the signal's energy in specific bands [START_REF] Pfurtscheller | Event-Related Desynchronisation[END_REF]. Moreover, SMRs are evoked by the real movements and by the motor imagery (when no actual movement is required, which is the only possible solution for paralyzed patients). Voluntary modulation of SMR could be achieved by the subject after the first session [START_REF] Blankertz | The Berlin Brain-Computer Interface: EEG-based communication without subject training[END_REF]. Moreover, it was demonstrated by [START_REF] Kübler | Patients with ALS can use sensorimotor rhythms to operate a brain-computer interface[END_REF] that SMR regulation is possible even in the case of significant degeneration of cortical and spinal motor neurons. However, the amplitude of SMR is much higher in healthy individuals [START_REF] Kübler | An introduction to brain computer interfacing[END_REF]. Some examples the BCI systems made on the base of the SMRs could be found in [START_REF] Blankertz | The Berlin Brain-Computer Interface: EEG-based communication without subject training[END_REF], [START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF], [START_REF] Wolpaw | Control of a two-dimensional movement signal by a noninvasive brain-computer interface in humans[END_REF].

The objective of our BCI system is autonomous functioning in the real-time mode. Therefore, among all described above phenomena only sensorimotor rhythms could be utilized, whereas others occur with significant time lag and/or cannot be used in the real environment without external stimulation (SCP, P300, VEP, and SSVEP).

Robustness and stability of BCI system

Robustness and stability are important questions for real-life application of the BCI systems. Essential restrictions are imposed on the stability of the system due to a considerable variability of the registered signal. There are three main types of the signal's variability [START_REF] Tangermann | Contributions from mathematics: Applying machine learning algorithms to BCI[END_REF]:

 subject-to-subject variability  session-to-session variability for the same subject  trial-to-trial variability for the same session and the same subject

The detailed study of the neurological signal variability is presented in the work of [START_REF] Tangermann | Contributions from mathematics: Applying machine learning algorithms to BCI[END_REF]. An example of the subject-to-subject variability is shown in [START_REF] Tangermann | Contributions from mathematics: Applying machine learning algorithms to BCI[END_REF].

Represents the trial-averaged brain patterns made for the 6 subjects for the case of rightand left-hand finger typing. The functional BCI system has to operate in the real-life. Therefore, the signal's variability should be taken into consideration. To take into account the effects of the session-to-session instability of the signal, all currently existing BCI systems require an everyday pre-calibration of the parameters. Moreover, online adaptation of the algorithm could allow adjustment of the method to the data variations during one session. Taking into account different types of the signal's instability, generally, the algorithm used in the BCI system must be sufficiently robust, i.e., its decisions should be made on the base of characteristics which are minimally exposed to the variability of the data recordings.

BCI Control paradigms. Self-paced BCI

BCI systems can be classified according to the way of the no control (NC) periods supporting, when the BCI system is expected to remain in the neutral state, in the other words, to be idle. Moreover, in the real BCI, the NC states happen more frequently than the intentional control (IC) states. Currently, several control paradigms are used in the design of the BCI systems. According to [START_REF] Mason | Temporal control paradigms for direct brain interfaces rethinking the definition of asynchronous and synchronous[END_REF], they can be classified as:

 self-paced (asynchronous)

Figure 1.17 Schematic representation of the different control paradigms [START_REF] Mason | Temporal control paradigms for direct brain interfaces rethinking the definition of asynchronous and synchronous[END_REF].

 system-paced  synchronized (cue-based)

A scheme of the different control paradigms is represented in Figure 1.17.

Only the self-paced and the system-paced paradigms support the NC states. On the other hand, the synchronized and the system paced control strategies use external cues for driving an interaction between a subject and the BCI system. As a result, users are supposed to generate commands only during specific periods. The signals outside the predefined time windows are ignored. However, in the real-life environment this restriction seems to be very burdensome as well as causes user frustration and fatigue.

As opposed to the cue-paced systems, no stimulus is used by the self-paced BCIs, which is continuously available. Users control them at their own intention. However, the performance of the self-paced BCIs reported in the articles is still not suitable for practical application, in particular, because of high level of false system activations [START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF]. It also causes frustration of the user and limits the applications of the system.

Different control paradigms are characterized by various neurological phenomena on which the BCI system is based. The self-paced paradigm can utilize the sensorimotor rhythm phenomenon so long as its nature is not connected with presence of any predefined cue. At the same time, the system-paced BCI can be based on the 2010)).

To concentrate on the main problem, in this work we are considering the simplest case of the BCI system, i.e., the case of the binary self-paced BCI, also known as the brain-switch BCI. The brain-switch BCI was introduced by [START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF], as the system intended to detect only one brain state (brain pattern) in the ongoing brain activity and does not provide any output when the user does not intend to communicate (Pfurtscheller et al., 2010a). Mason and Birch system was able to distinguish voluntary motor-related potentials from ongoing EEG activity in asynchronous mode, by analyzing a low-frequency band. Last time single channel EEGbased brain switch were realized by [START_REF] Pfurtscheller | Beta rebound after different types of motor imagery in man[END_REF]; Pfurtscheller and Solis-Escalante, (2009); [START_REF] Solis-Escalante | Analysis of sensorimotor rhythms for the implementation of a brain switch for healthy subjects[END_REF]. A multiple-electrode brain-switch system was created recently by [START_REF] Barachant | A Brain-Switch using Riemannian Geometry[END_REF]. Except EEG, NIRS [START_REF] Coyle | Brain-computer interface using a simplified functional near-infrared spectroscopy system[END_REF] also can be used for BCI. [START_REF] Pfurtscheller | Brain-computer communication based on the dynamics of brain oscillations[END_REF] demonstrated an ECoGbased brain switch BCI system. Whereas all previously mentioned BCIs were exploiting SMR, brain switches were also realized on the basis of SSVEP [START_REF] Cheng | Design and implementation of a braincomputer interface with high transfer rates[END_REF].

Taking into account advantages of the brain-switch approach, it was chosen for the first version of our BCI system. At the same time, applied algorithms should be easily generalizing to the case of multi-states and continuous-states BCI system, which is the future goal of our project.

In the next section criteria for the self-paced BCI performance evaluation and comparative analysis are discussed.

Evaluation of the self-paced BCI performance

Criteria

The binary BCIs classify incoming data epochs on two types, based on presence or absence of some specific activity. Epochs are classified as ‗event' or ‗non-event'.

For evaluation of the binary BCI, several approaches are used, for example, the error rate or the classification accuracy [START_REF] Blankertz | The BCI competition 2003: progress and perspectives in detection and discrimination of EEG single trials[END_REF], mutual information and the information transfer rate [START_REF] Kronegg | Analysis of bit-rate definitions for Brain-Computer Interfaces[END_REF], the receiver-operator-characteristics (ROC) and the area-under-curve (AUC) [START_REF] Lal | A brain computer interface with online feedback based on magnetoencephalography[END_REF], the correlation coefficient [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] and the mean square error (MSE) [START_REF] Wu | Bayesian Population Decoding of Motor Cortical Activity Using a Kalman Filter[END_REF]. However, these criteria were mainly applied for the synchronous type of the BCI control. The specificity of the self-paced systems consists in significantly different probabilities of ‗event' and ‗non-event' classes. Generally, long periods of the NC are interspersed with brief instances of the IC. It means that the NC states occur with a much higher probability than the IC ones, which contradicts the underlying assumption of equal a priori probability for a variety of the traditional performance estimation methods.

Another important characteristic of the BCI performance, which should be taken into account, is the decision rate, i.e., a number of decisions produced per unit of time.

It could be identical to the sampling rate of the input signal (e.g., [START_REF] Huggins | Detection of Event-Related Potentials for Development of a Direct Brain Interface[END_REF] or could be decimated to some extent (e.g., [START_REF] Kübler | Patients with ALS can use sensorimotor rhythms to operate a brain-computer interface[END_REF]. It is clear that the greater the decision rate is the more the absolute number of errors could be made if the probability of the error remains the same. It should be taken into account during interpretation of the performance properties.

Following characteristics are used to evaluate classification:

 True Positive (TP), which is equal to the number of correctly detected ‗events';

 True Negative (TN), which is equal to the number of correctly detected ‗non-events';

 False Positive (FP), which corresponds to the number of ‗non-event'

situations detected as ‗events';

 False Negative (FN), which is the number of the missed ‗events'.

These four characteristics are summarized in the confusion matrix (Table 1.1). 'Event' rejection FN (event missing) TN (correct rejection)

Figure 1.18 gives a graphical representation of these characteristics for a case when the decision rate coincides with the signal sampling rate. The ECoG signal could not be perfectly synchronized with the moments of the events (for instance, due to the response delay of the system). Thus, online quality evaluation of the self-paced (binary) BCI system should be made. In the current study it is realized in the following way. The event is considered to be detected correctly (TP) if

the system has detected it during the certain (predefined) time interval from the real event (see Figure 1.19 (A)). After every generated detection the system is blocked and cannot produce any other detection during some predefined period of time (see Figure 1.19 (B)). It is made to prevent the multiple system actuation for the same event. On the basis of TP, TN, FP, and FN values, criteria for the system performance evaluation are introduced. Among the most frequently used are [START_REF] Huggins | Detection of Event-Related Potentials for Development of a Direct Brain Interface[END_REF][START_REF] Rijsbergen | Information retrieval[END_REF][START_REF] Schlögl | Evaluation criteria in BCI research[END_REF][START_REF] Fatourechi | A Self-paced Brain Interface System that Uses Movement Related Potentials and Changes in the Power of Brain Rhythms[END_REF]: [START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF], the Positive Predictive Value ( ) [START_REF] Müller-Putz | Fast set-up asynchronous brain-switch based on detection of foot motor imagery in 1-channel EEG[END_REF], etc. While


shows the percentage of successfully detected events, corresponds to the percentage of correct detection.

The confusion matrix and/or a set of criteria, for instance, and , allow characterizing the BCI performance. However, simultaneous comparison of several criteria is not convenient for the calibration purpose. Since standard fails to evaluate the performance of the self-paced BCIs, numerous attempts were made to introduce a single metrics: the weighted [START_REF] Zhu | An evaluation of statistical spam filtering techniques[END_REF], HF-difference [START_REF] Huggins | Detection of Event-Related Potentials for Development of a Direct Brain Interface[END_REF], F1-criterion [START_REF] Rijsbergen | Information retrieval[END_REF], at a fixed false positive rate [START_REF] Borisoff | Brain-computer interface design for asynchronous control applications: improvements to the LF-ASD asynchronous brain switch[END_REF], ratio [START_REF] Fatourechi | A Self-paced Brain Interface System that Uses Movement Related Potentials and Changes in the Power of Brain Rhythms[END_REF] and others. In our study, the average value of and (referred as the Overall Performance, ( ) ) was chosen for the calibration purpose. We apply it to optimize the detection threshold in the BCI system (‗event' -‗non-event'). OP gives a single value for the BCI performance, balancing the False Positives and False Negatives types of errors. The choice of the criterion for the BCI system calibration reflects the goals and priorities of the given BCI study.

Thus, by means of the Overall Performance characteristic, the quality of the BCI system could be measured by a single value.

Comparison

One of the most significant problems of the self-paced BCI systems is high value of the False Positive Rate [START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF], i.e., the number of the false activation during the NC periods. While the current BCI realizations have rather good detection rate (the True Positive Rate or, in other words, the rate of the correct detections of the IC states, see [START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF], the False Positive Rate is still too high for practical applications. For example, if the system makes a decision every 100 ms and has the FP rate equal to 1%, it will generate an error on average every 10 seconds that is unacceptable in the numbers of the real applications due to user's frustration or other reasons.

Low performance of the current self-paced BCI systems [START_REF] Müller-Putz | Fast set-up asynchronous brain-switch based on detection of foot motor imagery in 1-channel EEG[END_REF][START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF][START_REF] Yom-Tov | Detection of movement-related potentials from the electro-encephalogram for possible use in a brain-computer interface[END_REF][START_REF] Townsend | Continuous EEG classification during motor imagery-simulation of an asynchronous BCI[END_REF][START_REF] Bashashati | An improved asynchronous brain interface: making use of the temporal history of the LF-ASD feature vectors[END_REF][START_REF] Fatourechi | Automatic user customization for improving the performance of a self-paced brain interface system[END_REF]Bashashati et al., 2007a;[START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF] is the main restriction for the practical BCI applications. Most of the reported BCIs achieve less than 60% of the control intention detection (TP) with several false positives (FP) activations per minute [START_REF] Mason | A brain-controlled switch for asynchronous control applications[END_REF][START_REF] Yom-Tov | Detection of movement-related potentials from the electro-encephalogram for possible use in a brain-computer interface[END_REF][START_REF] Townsend | Continuous EEG classification during motor imagery-simulation of an asynchronous BCI[END_REF][START_REF] Bashashati | An improved asynchronous brain interface: making use of the temporal history of the LF-ASD feature vectors[END_REF]Bashashati et al., 2007a). The BCI system reported by [START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF] achieved 56% of correct detection, with 0.7 FP per minute: this was obtained by offline analysis of short recordings of only several

(2-5) minutes length and of non-control periods when a subject was resting. These results were obtained from the scalp EEG recordings in highly restricted conditions and time

( )
are not satisfactory in the context of the needs of the long-term clinical applications which require continuous monitoring and processing of the neural activity.

Functional BCI in natural environment

Stable functioning in the natural environment conditions is required for the applications of the BCI systems. Previous BCI experiments were carried out in the laboratory conditions, which distinguish significantly from the natural environment where subjects are not concentrated properly, can be disturbed by external noises, etc.

As an approximation to the natural conditions, the BCI experiments in freely moving animals (rats and non-human primates) were carried out. The animals continue their general behavior during the experiments. The self-paced BCI system in freely moving animals gives an opportunity to avoid any external control cues, i.e., the animal could control the BCI system barely based on its own intention. Thus, the BCI experiments in freely moving animal were carried out since they model the real-environment conditions.

BCI cognitive tasks. Motor imagery

It is known that paralyzed patients have difficulties in operating with the BCI because of absence of sensory feedback. Nevertheless, it is still possible to use a phantom movement as a basis of the BCI system (BCI cognitive task). Imagination of the movements produces similar to the real-movement activity in the brain; however specificity of the patterns is smaller [START_REF] Anderson | Signal Classification with Different Signal Representations[END_REF][START_REF] Gu | Offline identification of imaged speed of wrist movements in paralyzed ALS patients from single-trial EEG[END_REF][START_REF] Pfurtscheller | Event-Related Desynchronisation[END_REF][START_REF] Beisteiner | Mental representations of movements. Brain potentials associated with imagination of hand movements[END_REF][START_REF] Wolpaw | The Wadsworth center brain-computer interface (BCI) research and development program[END_REF]. An example is given in Figure 1.21. In the current study, we concentrate on the case of real motions, because of difficulty to organize the motor imagery experiments in animals. However, simplicity of the SMR responses for the real and imagery motions allows us to hope for future successful use of the developed methods for the cognitive BCI tasks.

Conclusions

In the frame of the preclinical study of the human BCI system in the context of the Clinatec project, the goal of this thesis is development of algorithms for the signal processing block for the long-term experiments with ECoG-based self-paced binary BCI in freely moving animals.

The ECoG recording of the neural electrical activity is made directly from the surface of the brain cortex. Contrary to other invasive or partially invasive methods, it guarantees high level of safety for subject. The risk of brain damage or infecting is sufficiently moderate. At the same time, the recorded electrical signals are not conducted through the skull. Thus, high spatial and frequency resolutions of the recorded data are provided. Moreover, influence of the artifacts considerably diminishes while the informative signal's amplitude increases. The response time of this registration method is small enough to use the electrocorticography in the real-time mode. In addition, the ECoG system is rather convenient for the everyday use. It is cheap and can be mass-produced relatively easy. Therefore, it was chosen as method of the data acquisition in the Clinatec project.

The self-paced control paradigm is best suited to the experiment when no driving cues are used and a subject takes actions at its own intention. This paradigm provides to the user more freedom and command flexibility in comparison to the other control paradigms. Even taking into account difficulties in development and implementation of the self-paced methods, this approach was chosen for the project as the most promising and the best appropriate for the concept of the planned experiments.

While this work is making as the preclinical prototype of the system operating with patients under the natural conditions, it is crucial to create the system able to function adequately in the environment close to the real-life conditions. The absolute majority of the previously carried out experiments have been essentially limited to the laboratory conditions, at the same time, the whole experiments as well as each their session were relatively short. Contrary to this, our goal is the BCI system which treats properly different forms of animal's behavior, such as eating, sleeping, resting and so on during the long-term experiments. Thus, it could be taken as a satisfactory model of the human long-term real-life BCIs.

For the clinical purpose, significant number of BCI systems should continuously handle an external effector (e.g., prosthesis, wheelchair, etc.). At the same time, the discrete control can approximate the continuous control with any level of precision. We have chosen the binary BCI, which utilizes the simplest case of the discrete control.

Thus, we can concentrate on other problems.

Therefore, the long-term functional ECoG-based self-paced binary BCI in freely moving animal unites applicability and large practical importance. The problem statement lets us focus on the most significant issues of the task of development of the preclinical prototype of the BCI system.

Chapter 2. BCI experiment setup

Installation setup for rodents

The BCI experiments are carrying out by the neurophysiology group of Clinatec. The ECoG signals from the brain of the rat were simultaneously recorded through 14 transcranial screws. Three additional electrodes were used as references, i.e., to distinguish and eliminate a common source noise. Data were continuously recorded by Micromed® system (Micromed SD64, Micromed Italy) as well as by BioMEA system (modular 256-channel Micro-Electrode Array, Biologic, France) compatible with Matlab® based signal processing. The acquisition scheme was developed and implemented by CEA/LETI/DTBS/LE2S, Grenoble, France [START_REF] Charvet | WIMAGINE: A Wireless, Low Power, 64-Channel ECoG Recording Platform for Implantable BCI Applications[END_REF].

Preliminary visual inspection of the recorded signals was made in Spike2® software (Cambridge Electronic Design Limited, UK).

Positions of the electrodes implanted in the brain of the rat are demonstrated in

Installation setup for non-human primates

The BCI experiments in monkeys are carrying out by the neurophysiology group of Clinatec (the author was participating in definition of the experiments' protocol).

Experimental setup for rodents

The BCI experiment in rat consists of two stages (see Figure 2.8). The first one is the Training Stage. During this stage the freely moving rat can voluntary press the pedal without any external cues. The food dispenser is activated by pressing the pedal. Thus, after every pushing the rat obtains a reward, namely one food pellet. During the experiment session the ECoG data from the rat's brain (denoted as in Figure 2.8) is recorded simultaneously with information about state of the pedal ( ( ) equals to 1 if the pedal is pressed at the moment and 0, otherwise). Those experiments, which contain less than 50 pushes, are neglected. One of the others forms a training set which is used for the offline BCI system calibration. Identified model is applied for the ‗pushing-event' prediction.

At the second stage, namely the Close Loop Real-Time BCI, the predictor is directly applied to the signal from the rat's brain in the real-time. On this stage, the pedal is disconnected from the reward distributor. Thus, the rat presses the pedal but it obtains a reward only in the case when the predictor detects a ‗pushing-event' (whatever the position of the pedal is) and generates required control signal ( ).

Experimental setup for non-human primates

Since experiments in monkeys were carried out to verify the approaches applied in rats, the schemes of the experiments are the same (Figure 2.9). During the first stage, namely, the Training Stage, the monkey can voluntary activate a fruit juice dispenser by pressing a pedal. After identification of the model, it is applied on the second stage to activate a reward distributor, whereas the pedal is disconnected. for internal representation of the signal in the system, and ( ) is a command signal generated by the system to control the juice dispenser.

Conclusion

The BCI experiments were carried out in two animals, namely, the rat and the monkey. Neural activity of the animals' brains was recorded by means of the sets of ECoG electrodes which were preliminary implanted on the surfaces of their cortexes.

To modulate the natural environment, the animal was either freely moving (the case of the rat), or minimally restrained during the experiments (the case of the monkey).

During the experimental the animal had different types of activity to model the real-life conditions.

Chapter 3. BCI CONTROL SYSTEM

A general block-scheme of the BCI system consists in two main parts: the

Mathematical model of BCI system

Taking into account that the system is observed in the discrete moments of time * +, let us denote ( ), ( ), and ( ), for all .

Mathematical model of the BCI system can be described as:

{ ( ) ( ) (
{ ̂ ̂( ) ( ̂ ) (3.2)
Finally, ̂ , which is the estimation of the Effector state , is used in the Controller BCI (the predefined function ( )) to produce the control signal (3.2).

Control Block

The Control Block is a part of the BCI system (Figure 1.1). The goal of the Control Block is to translate the Subject's brain activity ( ) into a control command for the Effector ( ) according to a predefined set of rules and previous states of the system. The Control Block consists in two sub-blocks: the Detector BCI and the Controller BCI (Figure 3.1).

Detector BCI

The Detector consists in the Signal Preprocessing, the Feature Extraction, the Model Calibration, and the Model Application (Figure 3.2). In addition, the Model Adaptation block could be included in the Detector.

Preprocessing Sub-Block

In general, the main purpose of the preprocessing is amplification of the internal signal, increasing of the signal-to-noise ratio, as well as artifacts processing (the Block either can remove artifacts from the signal or simply marks some epochs as artifactcontaminated). Several methods and approaches are frequently applied on this stage for ECoG/EEG data preprocessing:

 Spectral Filtering
Finite and Infinite Impulse Response Filters transform the source signal into the filtered one :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),
for all . Here, two sequences and with length and , respectively, can be calculated in several ways according to desired frequency band restrictions [START_REF] Antoniou | Digital Filters: Analysis, Design, and Application[END_REF][START_REF] Shenoi | Introduction to digital signal processing and filter design[END_REF]. This filter is called the Infinite Impulse Response Filter (IIR). In the special case of , the filter is called Finite Impulse

Response Filter (FIR). IIR filters can produce steeper slopes between pass-and stop-bands, however, they can become unstable. FIR filters are always stable.

Impulse Response Filters are widely used for the signal preprocessing in the BCIs [START_REF] Dornhege | General Signal Processing and Machine Learning Tools for BCI Analysis[END_REF][START_REF] Zhang | A maximum mutual information approach for constructing a 1D continuous control signal at a self-paced brain-computer interface[END_REF][START_REF] Pistohl | Prediction of arm movement trajectories from ECoG-recordings in humans[END_REF].

Another method used for the BCI's signal preprocessing is the Fourier-Base Filter [START_REF] Dornhege | General Signal Processing and Machine Learning Tools for BCI Analysis[END_REF]. The method is based on the switching of the signal from the temporal to the spectral domain by calculating the short-time Fourier transform (STFT) [START_REF] Oppenheim | Discrete-time signal processing[END_REF]). In the spectral domain the relevant frequencies are weighted and the Inverse Fourier transform (IFT) is applied.

 Spatial Filtering

Together with the Spectral Filtering, the Spatial Filtering is one of the fundamental approaches for the BSI signals preprocessing. Bipolar Filtering, Common Average Reference, and Laplace Filtering are the most common methods for the Spatial Filters.

To make the Bipolar Filtering all channels are recorded as voltage difference between electrode pairs [START_REF] Mcfarland | Spatial filter selection for EEG-based communication[END_REF][START_REF] Essl | EEG coherence and reference signals: experimental results and mathematical explanations[END_REF][START_REF] Ramoser | Optimal spatial filtering of single trial EEG during imagined hand movement[END_REF]. Thus, the local voltage gradient is calculated, which emphasis a local activity while activity of the distant sources is suppressed.

In the Common Average Reference (CAR) method the mean of all channels is subtracted from each individual channel [START_REF] Mcfarland | Spatial filter selection for EEG-based communication[END_REF][START_REF] Essl | EEG coherence and reference signals: experimental results and mathematical explanations[END_REF][START_REF] Ramoser | Optimal spatial filtering of single trial EEG during imagined hand movement[END_REF]. CAR reduces influence of far sources, but at the same time artifacts from one channel could be spread over all other channels.

Laplace Filtering consists in subtraction of the average of the surroundings signals from each individual channel [START_REF] Mcfarland | Spatial filter selection for EEG-based communication[END_REF][START_REF] Essl | EEG coherence and reference signals: experimental results and mathematical explanations[END_REF][START_REF] Ramoser | Optimal spatial filtering of single trial EEG during imagined hand movement[END_REF]. The choice of the set of the surroundings electrodes defines the properties of the filtering.

In this study we have applied a band-stop filter (IIR notch filter) to cut off the 50 Hz noise (the frequency coming from the power supply) and a CAR filter (to reduce influence of the background noise). Applied methods provide satisfactory quality of the output data by means of significant elimination of irrelevant components of the signal.

At the same time they are computationally efficient and fast enough to satisfy the severe restrictions imposed on the real-time BCI systems. Since quality of the ECoG signal is better than EEG one, artifact removal is absent on the preprocessing stage.

Feature Extraction Sub-Block

The purpose of the Feature Extraction sub-block is the mapping of the input signal to the feature space: ( ) ( ). This feature space should be related to the underlying neurological phenomena correlated with the BCI task (e.g., user's intention of control). The most widespread methods for feature extraction in the ECoG based BCI studies are: time and/or frequency signal representation and parametric modeling (Bashashati et al., 2007b).

 Time and/or frequency methods

Signal Preprocessing

Feature Extraction

Detector BCI

̃( ) ( ) ( ) Model Calibration ( ) (A) ̂( ) ̂( ) ( ) Signal Preprocessing Feature Extraction Model Application Detector BCI ̃( ) ( ) Model Adaptation ( ) (B)
First stage

Second stage

The most common method to estimate power-spectral density (PSD) features of the signal is the Fast Fourier Transform (FFT) [START_REF] Sanei | EEG Signal Processing[END_REF], which has good computational efficiency and provides direct interpretation of the results. A drawback of the method is that it does not provide any time-domain information and, therefore, could not be efficiently applied for non-stationary signals. A windowed version of FFT (short-time Fourier transform, STFT) analyzes the timevarying spectral components of the data. The one-dimensional signal is mapped into a two-dimensional space of time and frequency. Unfortunately, frequency resolution is decreased, and estimated results do not reliably converge to the true PSD peaks [START_REF] Tangermann | Feature Selection for Brain-Computer Interfaces[END_REF]. At the same time, the fixed time-window leads to limitation of the temporal resolution in the higher frequencies. Examples of the Fourier Transform applications in the BCI systems could be found in [START_REF] Kellis | Decoding spoken words using local field potentials recorded from the cortical surface[END_REF]Pfurtscheller et al., 2010b;[START_REF] Galán | Continuous Brain-Actuated Control of an Intelligent Wheelchair by Human EEG[END_REF] To overcome the temporal-resolution limitation of STFT, Wavelet Transform (WT) is often applied. The method maps the signal into a two-dimensional space of time and scales by means of the particular function, called the ‗mother wavelet'. The choice the ‗mother wavelet' is a crucial factor for WT. Like Fourier Transform, WT allows analyzing of both amplitude and phase components of the signal. For more information, see Section 5.2. Wavelet Transform is widely applied in BCI systems, for example see [START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkeys[END_REF][START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF][START_REF] Zhao | EEG-based asynchronous BCI control of a car in 3D virtual reality environments[END_REF][START_REF] Hinterberger | A braincomputer interface (BCI) for the locked-in: comparison of different EEG classifications for the thought translation device[END_REF][START_REF] Sherwood | On Classifiability of Wavelet Features for EEG-Based Brain-Computer Interfaces[END_REF][START_REF] Cabrera | Comparison of feature selection and classification methods for a brain-computer interface driven by non-motor imagery[END_REF] The Correlative Time-Frequency Representation (CTFR) method provides information about the time-frequency interactions between the components of the input signal (Bashashati et al., 2007b) in addition to the spectral information. For instance, in the case of EEG, data samples are not analyzed independently, like in the case of FT or WT, but their relationship is also taken into account. However, the method is sensitive to noise (Garcia et al., 2003a;Garcia et al., 2003b).

 Parametric modeling

The method assumes that the analyzed time series to be the output of a given linear model. The structure and order of the model should be chosen a priori [START_REF] Weitkunat | Digital Biosignal Processing[END_REF]. For short signals parametric modeling results in better frequency resolution in comparison with FT. However, estimation of the model's parameters is very sensitive to artifacts [START_REF] Birch | Single trial EEG signal analysis using outlier information[END_REF]

Model Calibration Sub-Block

The ) in such a way to minimize a functional (e.g., approximation error):

({ ̂ } ) ̂ (3.3)
where, ̂ ̂( ).

Dimensionality reduction and/or feature selection (for instance, narrowing of the frequency range, estimation of the most informative channels, etc.) can be made preliminary (independently) or directly included in the optimization problem ((3.3).

In our case, as well as in the majority of other BCIs, ( ) is a discrete variable.

Mostly, different classifiers were applied to solve optimization problem (3.3), for instance, see a review of [START_REF] Lotte | A review of classification algorithms for eeg-based brain-computer interfaces[END_REF].

Linear Classifiers

Linear classifiers use linear functions to distinguish classes. The most used in the BCI tasks are Linear Discriminant Analysis (LDA, or Fisher's LDA) and Support

Vector Machine (SVM).

 Linear Discriminant Analysis

LDA assumes normal distribution of the data and uses hyperplanes to separate the data from the different classes [START_REF] Fukunaga | Statistical Pattern Recognition[END_REF][START_REF] Duda | Pattern Recognition[END_REF]. The method has low computational requirements, thus it could be effectively used for the online BCIs [START_REF] Pfurtscheller | EEG event-related desynchronization (ERD) and event-related synchronization (ERS) Electroencephalography: Basic Principles, Clinical Applications and Related Fields[END_REF][START_REF] Bostanov | BCI competition 2003-data sets ib and iib: feature extraction from event-related brain potentials with the continuous wavelet transform and the t-value scalogram[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF][START_REF] Scherer | An asynchronously controlled EEG-based virtual keyboard: improvement of the spelling rate[END_REF]. At the same time, on the data of complex non-linear structure the method has demonstrated poor results (Garcia et al., 2003b).

To additionally penalize outliers and improve generalization capabilities of LDA regularization parameters could be introduced. A regularized Fisher's LDA (RFLDA) was used by [START_REF] Blankertz | Classifying single trial EEG: towards brain computer interfacing[END_REF] and [START_REF] Müller | Machine learning techniques for brain-computer interfaces[END_REF].

 Support Vector Machine

To identify classes SVM uses a discriminate hyperplane, selected in the way to maximize the distance between the nearest training points from the different classes [START_REF] Cortes | Support-Vector Networks[END_REF][START_REF] Burges | A tutorial on support vector machines for pattern recognition[END_REF][START_REF] Bennett | Support vector machines: hype or hallelujah?[END_REF]. SVM is known to have good generalization properties and to be steady to over-fitting effect [START_REF] Jain | Statistical pattern recognition: a review[END_REF]. The method is widely applied in BCI [START_REF] Rakotomamonjy | Ensemble of SVMs for improving brain computer interface P300 speller performances[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF][START_REF] Blankertz | Classifying single trial EEG: towards brain computer interfacing[END_REF]. At the same time, SVM allows natural generalization to the non-linear case by using the ‗kernel trick' [START_REF] Aizerman | Theoretical foundations of the potential function method in pattern recognition learning[END_REF][START_REF] Burges | A tutorial on support vector machines for pattern recognition[END_REF]. Kernel SVM has also demonstrated good results in BCI applications [START_REF] Kaper | BCI competition 2003-data set iib: support vector machines for the p300 speller paradigm[END_REF][START_REF] Garrett | Comparison of linear, nonlinear, and feature selection methods for EEG signal classification[END_REF].

Neural Networks

Neural Networks (NN) were used in BCIs for a long time (e.g., [START_REF] Hiraiwa | EEG topography recognition by neural networks[END_REF]. Among different NN the most widespread is the Multilayer Perceptron (MLP).

 Multilayer Perceptron

MLP is an assembly of several layers of artificial neurons: an input layer, hidden layers, and an output layer. When composed of enough neurons MLP can Chapter 3. BCI CONTROL SYSTEM Brain-Computer Interface with cortical electrical activity recording 64 approximate any continuous function [START_REF] Bishop | Neural Networks for Pattern Recognition[END_REF]. The negative side of this property is its sensitivity to overtraining, especially for such noisy and instable data like brain's neuronal activity [START_REF] Balakrishnan | Multilayer perceptrons for the classification of brain computer interface data[END_REF]. A MLP without any hidden layer is called a perceptron and it could be shown that it is equivalent to LDA.

MLPs were applied in variety of BCI tasks: binary [START_REF] Palaniappan | Brain computer interface design using band powers extracted during mental tasks[END_REF], multiclass [START_REF] Anderson | Classification of EEG signals from four subjects during five mental tasks Solving Engineering Problems with Neural Networks[END_REF], synchronous [START_REF] Haselsteiner | Using time-dependant neural networks for EEG classification[END_REF], and asynchronous [START_REF] Chiappa | HMM and IOHMM modeling of EEG rhythms for asynchronous BCI systems[END_REF]. A perceptron without hidden layers were used by [START_REF] Wang | BCI competition 2003data set iv: an algorithm based on CSSD and FDA for classifying single-trial EEG[END_REF] and [START_REF] Congedo | Classification of movement intention by spatially filtered electromagnetic inverse solutions[END_REF].

 Other Neural Networks Architectures

Other Neural Networks used in BCI systems are: Gaussian classifier NN [START_REF] Millán | Local neural classifier for EEG-basedrecognition of mental tasks[END_REF][START_REF] Millán | Noninvasive brain-actuated control of a mobile robot by human[END_REF][START_REF] Solhjoo | Mental task recognition: a comparison between some of classification methods[END_REF][START_REF] Cincotti | Comparison of different feature classifiers for brain computer interfaces[END_REF],

Learning Vector Quantization NN [START_REF] Kohonen | The self-organizing maps[END_REF][START_REF] Pfurtscheller | Brain-computer interface-a new communication device for handicapped persons[END_REF], fuzzy ARTMAP NN [START_REF] Carpenter | Fuzzy artmap: a neural network architecture for incrementalsupervised learning of analog multidimensional maps[END_REF][START_REF] Palaniappan | A new brain-computer interface design using fuzzy artmap[END_REF], dynamic NN [START_REF] Haselsteiner | Using time-dependant neural networks for EEG classification[END_REF][START_REF] Barreto | Classification of spatio-temporal EEG readiness potentials towards the development of a brain-computer interface[END_REF], RBF NN [START_REF] Duda | Pattern Recognition[END_REF][START_REF] Hoya | Classification of single trial EEG signals by a combined principal + independent component analysis and probabilistic neural network approach[END_REF], Bayesian Logistic Regression NN (BLRNN) [START_REF] Penny | EEG-based communication: a pattern recognition approach[END_REF], Adaptive Logic Network (ALN) [START_REF] Kostov | Parallel man-machine training in development of EEGbased cursor control[END_REF], Probability

Estimating Guarded Neural Classifier (PeGNC) [START_REF] Felzer | Analyzing EEG signals using the probability estimating guarded neural classifier[END_REF].

Non-Linear Classifiers

Non-linear classifiers uses non-linear decision boundaries therefore they can more efficiently reject uncertain samples. At the same time, they are more sensitive to overtraining, more resource consumption, and therefore they are less often applied in BCI systems.

 Bayes Quadratic

In this approach the Bayes rule is used to compute a posteriori probability that a future vector belongs to a given class [START_REF] Duda | Pattern Recognition[END_REF][START_REF] Fukunaga | Statistical Pattern Recognition[END_REF]. The class of the feature vector is estimated by the Maximum a posteriori probability (MAP) rule. It is assumed that the data in the different classes have different normal distribution, which leads to quadratic decision boundary. The method was applied by [START_REF] Lemm | BCI competition 2003-data set iii: probabilistic modeling of sensorimotor mu rhythms for classification of imaginary hand movements[END_REF], [START_REF] Solhjoo | Mental task recognition: a comparison between some of classification methods[END_REF], [START_REF] Keirn | A new mode of communication between man and his surroundings[END_REF], [START_REF] Barreto | On the classification of mental tasks: a performance comparison of neural and statistical approaches[END_REF].

 Hidden Markov Model

Hidden Markov Model (HMM) is a statistical model in which the system is assumed to be a Markov process with unobserved (hidden) states [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF].

HMM are efficiently used for classification of the time series. HMM were realized in BCI systems by [START_REF] Obermeier | Hidden Markov models for online classification of single trial EEG[END_REF], [START_REF] Cincotti | Comparison of different feature classifiers for brain computer interfaces[END_REF], [START_REF] Obermaier | Information transfer rate in a five-classes brain-computer interface[END_REF], [START_REF] Solhjoo | Classification of chaotic signals using HMM classifiers: EEG-based mental task classification[END_REF], [START_REF] Chiappa | HMM and IOHMM modeling of EEG rhythms for asynchronous BCI systems[END_REF].

Nearest neighbor Classifiers

Classifiers of this type assign a feature vector to a class according to its nearest neighbors. These classifiers are non-linear.

 k-Nearest Neighbors

A feature vector is assigned to the dominant class among the k nearest neighbors in the training set [START_REF] Duda | Pattern Recognition[END_REF]. The algorithm was applied in a set of BCIs [START_REF] Blankertz | Classifying single trial EEG: towards brain computer interfacing[END_REF][START_REF] Müller | Machine learning techniques for brain-computer interfaces[END_REF][START_REF] Schlögl | Characterization of four-class motor imagery EEG data for the BCI-competition 2005[END_REF], however it is too sensitive to the curse-of-dimensionality [START_REF] Friedman | On bias, variance, 0/1-loss, and the curse-of-dimensionality[END_REF]. At the same time, in the case of BCI with low dimensional feature vectors, kNN was efficient [START_REF] Borisoff | Brain-computer interface design for asynchronous control applications: improvements to the LF-ASD asynchronous brain switch[END_REF].

 Mahalanobis Distance

The classifier assumes a Gaussian distribution ( ) for each prototype of the class . A feature vector is assigned to the class that corresponds to the nearest prototype according to the Mahalanobis distance ( ) [START_REF] Cincotti | Comparison of different feature classifiers for brain computer interfaces[END_REF]:

( ) √( ) (
) . The method gives simple and robust classifier which was used for the BCI systems by [START_REF] Schlögl | Characterization of four-class motor imagery EEG data for the BCI-competition 2005[END_REF] and [START_REF] Cincotti | Comparison of different feature classifiers for brain computer interfaces[END_REF].

Combinations of Classifiers

A recent trend in the BCI systems consists in combination of different classifiers.

There are several strategies of combination. Here, we are referring to the most popular.

 Boosting

Several classifiers are applied in cascade, each classifier is focusing on the errors committed by the previous ones [START_REF] Duda | Pattern Recognition[END_REF]. Thus, from the several weak classifiers, a powerful one could be constructed. At the same time, it will not be sensitive to overtraining. From the other hand, the method is sensitive to mislabels [START_REF] Jain | Statistical pattern recognition: a review[END_REF]. Boosting was applied in BCI by [START_REF] Boostani | A new approach in the BCI research based on fractal dimension as feature and Adaboost as classifier[END_REF]; [START_REF] Hoffmann | A boosting approach to p300 detection with application to brain-computer interfaces[END_REF].

 Voting

Several classifiers are applied independently and final decision will be that of majority [START_REF] Jain | Statistical pattern recognition: a review[END_REF]. Voting is very popular way of classifiers combination [START_REF] Pfurtscheller | Brain-computer interface-a new communication device for handicapped persons[END_REF][START_REF] Rakotomamonjy | Ensemble of SVMs for improving brain computer interface P300 speller performances[END_REF][START_REF] Qin | ICA and committee machine-based algorithm for cursor control in a BCI system[END_REF].

 Stacking

Two levels of classifier are used: level-0 and level-1.The outputs of a set of independent level-0 classifiers are used as an input for a level-1 (meta) classifier, which makes a final decision [START_REF] Wolpert | Stacked generalization[END_REF]. The method was efficiently applied in the BCI research by [START_REF] Lee | PCA+HMM+SVM for EEG pattern classification[END_REF].

Regression

Although classifiers are used in the majority of the BCIs, regression methods are also applied to solve optimization problem (3.3) [START_REF] Mcfarland | Brain-computer interface signal processing at the Wadsworth Center: mu and sensorimotor beta rhythms[END_REF][START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF][START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkeys[END_REF]. Generally regression approaches are used for continuous values of ( ). For instance, ( ) can describe motion of object in the space (see [START_REF] Kubánek | Decoding flexion of individual fingers using electrocorticographic signals in humans[END_REF][START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkeys[END_REF]. However, the regression methods can be applied also for the discrete case [START_REF] Mcfarland | Brain-computer interface signal processing at the Wadsworth Center: mu and sensorimotor beta rhythms[END_REF]. Regression and classification algorithms for the discrete BCI control demonstrate comparative results [START_REF] Mcfarland | Brain-computer interface signal processing at the Wadsworth Center: mu and sensorimotor beta rhythms[END_REF]. In our study we utilize a linear regression approach with simultaneous projection to a low dimensional feature space and further binarization. The regression was chosen because it allows natural generalization from the case of the binary response variable ( ) * + to the discrete ( ( ) * + , ) or continuous vector ( ) , , which correspond to the discrete multiclass or continuous control.

They are the perspectives of our study. In addition, time-lag information can be easily introduced in both dependent and independent variables.

More precisely the proposed method will be discussed in the next chapter.

Model Application Sub-Block

Founded during the first stage of the experiment estimation ̂( ) is applied to the signal in the Model Application sub-block during the second stage of the BCI experiment. As a result, the estimation of the system state ̂( ) is generated and sent as an output of the Detector BCI to the Controller BCI.

Model Adaptation Sub-Block

The Model Adaptation sub-block is intended for adaptation of the BCI system to

Controller BCI

The Controller BCI translates the features into the control signals by means of the preliminary established function ( ) (see Equation (3.1)). In the case of the binary BCI, the generated control signal ( ) is logical: 0 denotes the NC states (the ‗nonevent' situations when no food is given) and 1 indicates the IC states (the ‗event' detection situations when food is given). The sub-block includes a post-processing of the output control signal to reduce the number of activation of the system (for instance, it could block the system for some time after activation).

Chapter 4. MULTI-WAY ANALYSIS FOR BCI SYSTEMS

Introduction

A common approach for the brain signal processing, intended for event detection/prediction, consists in the extraction of the event-related features of the neuronal activity. Information from spatial [START_REF] Rakotomamonjy | Ensemble of SVMs for improving brain computer interface P300 speller performances[END_REF], frequency [START_REF] Schlögl | Characterization of four-class motor imagery EEG data for the BCI-competition 2005[END_REF], and temporal [START_REF] Vidaurre | Time domain parameters as a feature for EEG-based brain computer interfaces[END_REF] domains could be analyzed by means of the Principal Component Analysis (PCA) [START_REF] Kayser | Event-related brain potentials during auditory and visual word recognition memory tasks[END_REF], the Independent Component Analysis (ICA) [START_REF] Makeig | Functionally independent components of early event-related potentials in a visual spatial attention task[END_REF], the Linear Discriminant Analysis (LDA) [START_REF] Scherer | An asynchronously controlled EEG-based virtual keyboard: improvement of the spelling rate[END_REF], the Common Spatial Patterns (CSP) [START_REF] Zhao | EEG-based asynchronous BCI control of a car in 3D virtual reality environments[END_REF], the Partial Least Squares (PLS) [START_REF] Chao | Long-term asynchronous decoding of 3D hand trajectories using electrocorticographic signals in primates[END_REF], etc. Let us note that the standard methods are designed mainly for vector input variables which present either one domain of analysis or several domains unfolded in one line. Using only one domain usually does not provide satisfactory results. Combination of several domains is thus necessary. In most cases, two or three ways of analysis are applied sequentially. For example, see [START_REF] Galán | Continuous Brain-Actuated Control of an Intelligent Wheelchair by Human EEG[END_REF], where first stable frequency components are determined and second the best electrodes are chosen. A multi-way analysis allows simultaneous treatment of several domains, by means of a tensor-based data representation. In recent years, it was applied in several BCI studies and demonstrated promising results (e.g., tensor factorization with PARAFAC [START_REF] Nazarpour | Parallel space-timefrequency decomposition of eeg signals for brain computer interfacing[END_REF], Tucker [START_REF] Zhao | Slice Oriented Tensor Decomposition of EEG Data for Feature Extraction in Space, Frequency and Time Domains[END_REF], Non-negative Tensor Factorization [START_REF] Mørup | Algorithms for Sparse Nonnegative Tucker Decomposition[END_REF],

Multi-way Partial Least Squares (NPLS) [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF], General Tensor Discriminant Analysis [START_REF] Li | A prior neurophysiologic knowledge free tensor-based scheme for single trial EEG classification[END_REF], Regularized Tensor Discriminant Analysis [START_REF] Li | Regularized tensor discriminant analysis for single trial EEG classification in BCI Pattern Recognition[END_REF]). Therefore, it is chosen for the ECoG data representation in the current study.

The signals from the different electrodes are mapped by a continuous wavelet transform to the temporal-frequency-spatial space. As a result, every time epoch is represented by a cube (a third-order tensor). All cubes are stored in a tensor of observations (a fourth-order tensor).

The Multi-way Partial Least Squares (N-way Partial Least Squares, NPLS) [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF] presents an effective approach for tensor data analysis. Namely, NPLS is a statistical method for a linear regression identification in the case of tensor input or/and output variables. It is particularly suited for high dimensional data. While NPLS works without any prior knowledge, it can be efficiently applied for automatic identification of the model aimed at prediction of the BCI events from the measurement of the brain neuronal activity. In comparison to the ordinary Partial Least Squares or other vector oriented algorithms that can be applied to the tensor data after unfolding, NPLS preserves the multimodality of the data in a robust way [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF]. In addition, it allows identifying how much each feature contributes to the decision rule (for instance, which electrode is the most informative). NPLS is based on tensor factorization and data projection to a low dimensional feature space. In comparison to other tensor-based methods that were previously applied to BCI studies recently, NPLS involves supervised tensor decomposition which significantly increases the efficiency of modeling. At the same time, whereas others methods are restricted to discriminative task, NPLS can be applied for both classification and regression. Besides that, NPLS was successfully used for epilepsy seizure recognition [START_REF] Acar | Multiway analysis of epilepsy tensors[END_REF]. That is why it was chosen as the basic approach in our work. However, the NPLS algorithm has a set of drawbacks connected with possible excessive complexity of the identified model, method's computational efficiency and resources consumption. To overcome these drawbacks, penalized, iterative and recursive versions of the algorithm are proposed, which allow informative subsets selection, huge datasets processing, and online adaptation, correspondently. Detailed description of the NPLS approach as well as of its modifications are given in this chapter.

Tensor notations and preliminaries

In this section, some background information about tensors is given. For more details see, for example, [START_REF] Kolda | Tensor Decompositions and applications[END_REF] or [START_REF] Comon | Tensor Decompositions, State of the Art and Applications[END_REF].

Tensors (multi-way arrays) are higher-order generalization of vectors and matrices (this notion of tensors must be distinguished from tensors in physics (e.g., is referred to as 1 2 . Here, is the order of the tensor, i.e., its number of dimensions, also known as ways or modes. Vectors and matrices are tensors of order one and two, respectively. The number of the variables in the -th mode shows dimensionality of this mode (indices typically range from 1 to their capital version, e.g.,

). An example of a third-order tensor is given in Figure 4.1.

Rank of tensor. Rank-one tensors

A tensor

1 × 2 × ×
always admits a decomposition into a sum of outer products as [START_REF] Comon | Generic and Typical Ranks of Three-Way Arrays[END_REF]:

∑ (4.1)
where for each the vector , and -‖ denotes the outer product. For more information see [START_REF] Kolda | Tensor Decompositions and applications[END_REF].

Each tensor's element of Equation (4.1) could be written as:

1 2 ∑ 1 2 (4.2)
where, is the -element of the vector .

The rank of the given tensor is the minimal integer number such that this decomposition (4.1) is exactly satisfied [START_REF] Comon | Generic and Typical Ranks of Three-Way Arrays[END_REF]. For example, the tensor : . /, . / is the rank-one tensor:

. / . / . /.

Tensor operations

 Unfolding (tensor → matrix)

Tensor unfolding, also known as matricization or flattening, is the mapping of a tensor into a matrix. The mode-unfolding ( ) of a tensor

1 × 2 × ×
arranges the mode-fibers (see Figure 4.4) to be the columns of the matrix. Namely, a tensor element 1 2 is mapped to a matrix element :

∑ ( ) ,
where

∏ ( if , if and
). The unfolding operation is illustrated for a third-order tensor in Figure 4.3.  Vectorization (matrix → vector)

Vectorization of a matrix , denoted as ( ), is a column-wise unfolding of into a column vector. For example, if , then

( ) [ ].

 Norm

The norm of a tensor

1 × 2 × ×
is generalization of the Frobenius norm:

‖ ‖ √ ∑ ∑ | 1 | 1 1 (4.4)  -mode (vector) product
The -mode (vector) product of a tensor

1 × 2 × ×
and a vector is denoted as × . The result tensor has ( ) order and its size is × × × × × . Elementwise,

( × ) 1 1 1 ∑ 1 (4.5)

Multi-way representation of ECoG data

To form the BCI feature tensor, the ECoG signal (see Figure 2.

Statement of problem of BCI system calibration from experimental data

The goal of the Calibration stage in BCI experiments consists in the determination of dependency between the brain neural activity signal (an observed multivariate variable) and the state of the system (a response binary variable). Let us denote the explanatory variable as ( ) and the response variable as ( ). A model of as a function of is estimated during the calibration stage, by solving the optimization problem (3.3). To this purpose training data { } are formed from observed data samples. In the case of regression, the minimization functional is often taken as sum of squares of residual norms:

∑ ‖ ̂ ‖ ̂ ̂ ̂( ).
(4.6)

In the present study, specificity of optimization task (4.6) consists in tensor representation of the data and high dimensionality (more variables than observations).

Moreover, the variables are highly correlated. The methods, applied for the model identification, should take into account these peculiarities. In particular, to reduce the dimensionality a set of projections, saving tensor data structure, should be applied. The regression is determined either simultaneously with projecting or after it.

Two-way modeling

Before the description of regression approaches for tensor data, we consider the matrix methods in the case of large dimension and highly correlated variables. Let matrix of residuals. In addition, prior to application of the PCA decomposition, the variables in must be column mean-centered.

Graphical representation of the PCA approach is shown in Figure 4.6.

Each consecutive principal component is chosen to maximize the decrease of variation of the matrix . It can be shown that in PCA the loading vectors , are the sorted eigenvectors of the matrix , whereas the scores , are the sorted eigenvectors of the matrix [START_REF] Geladi | Partial least-squares regression: a tutorial[END_REF].

Thus, PCA is theoretically the optimal linear scheme, in terms of the least mean square error, for compressing a set of high dimensional vectors into a set of lower dimensional vectors. The solution defined by PCA is not unique, there is a rotational freedom. Therefore, it should be made some type of post-processing to select the most appropriate decomposition for the task.

After the principal components for the matrix have been found, a regression on can be identified. For example, it could be a linear regression (PCR) ̂ ̂ . In this case, the least square solution gives ̂ ( ) .

Partial Least Squares Regression (PLS)

Partial Least Squares (PLS) regression is a statistical method for vector-based data analyses, particularly appropriate to the case of high dimension [START_REF] Geladi | Partial least-squares regression: a tutorial[END_REF]. As opposed to other widely used projection-based methods, like PCA, PLS uses for factorization not only the matrix of the independent variables but also Finally, PLS properly handles situations when the matrix of observations contains more variables than observations and the variables are highly correlated [START_REF] Geladi | Partial least-squares regression: a tutorial[END_REF].

Recursive Partial Least Squares Regression (RPLS)

Several Recursive PLS algorithms were invented to take into account timedependent changes of data as well as to be able to handle large data sets. [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF] and [START_REF] Dayal | Improved PLS algorithm[END_REF] introduced the most known approaches. The method introduced by Dayal and MacGregor has better performance and does not suffer from problems concerned with short data windows (for more information, see [START_REF] Dayal | Improved PLS algorithm[END_REF]. However, Dayal's algorithm stores in the active memory the covariance matrix , which dimension is equal to square of . This requirement is excessively difficult for high dimensionality tasks. In our study, data dimensionality is extremely large ( ). Therefore, we focus ourselves on Qin's algorithms [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF].

According to Qin's algorithm, the matrices and are decomposed by the batch-wise PLS algorithm with orthonormal latent variables. In general, the PLS model is constructed as: , , (4.9) where * +. The matrix of latent variables is orthonormal (see [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF]:

( ( )) ( ( ))
, (4.10)

where is an by identity matrix. In addition, is orthogonal to both residuals matrices: , (4.11) , (4.12)

In his work, Qin has shown that if the number of factors is large enough to provide the condition , then and . This yields that for the new data pair * +, the regressions on the next data sets will be equivalent:

{[ ] [ ]} {[ ] [ ]}. (4.13)
Thus, the old data and are captured by the loading matrices and , as well as by the coefficient matrix , whereas the new data are concatenated to their combination. As the result, the algorithm always keeps the size of the stored in the active memory datasets. In addition, this approach represents an effective tool for the adaptive learning, by introducing weights for the old and new data (see [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF].

Generic multi-way approaches

PCA and PLS methods were originally designed for the vector input variables.

In practice for a wide variety of tasks several domains must be analyzed simultaneously to obtain proper results (e.g., time-series analysis, chromatography, spectral data, etc.).

The methods described in this section, namely PARAFAC and NPLS, are generalization of the PCA and PLS approaches, discussed in Section 4.5. They provide the multi-modal data treatment. Unlike the vector oriented algorithms that can be applied to the multi-way (tensor) data after unfolding, the multi-way models preserve the structure of the data, improve robustness of the results as well as allows identifying relative impact of each feature.

In this section explanatory and response variables are represented by the and -order tensors, respectively:

1 × × and 1 × × . After observation of data samples of both types of variables, tensors are formed by means of concatenation of the data samples along the first modality:

× 1 × ×
and × 1 × × .

PARAFAC

PARAFAC (Parallel Factor analysis) is a generalization of the PCA approach to data structures of high orders [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis[END_REF]. It is used for decomposition of the tensor into sets of score and loading vectors, that describes the data in a compressed way.

The PARAFAC decomposition of the observation tensor

× 1 × ×
is expressed in the form of outer product [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF]:

∑ ‖ ‖ (4.14)
where, , is a matrix of the extracted score vectors, , represents a matrix of the extracted loadings for -th modality (

), and is a tensor of residuals. The tensor assumed to be mean-centered along its first modality. As in the case of PCA, the loadings are chosen in a such way to minimize the norm of the residual tensor ‖ ‖ . Thus, PARAFAC provides the best approximation of the tensor by the sum of rank-one tensors.

To find the solution of the problem (4.14) the Alternating Least Squares (ALS) algorithm [START_REF] Yates | The analysis of replicated experiments when the field results are incomplete[END_REF]) could be applied [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF]. ALS fixes all parameters except one, which is estimated in a least square sense. The procedure is repeated for all parameters until convergence. Unfortunately, the ALS algorithm, as well as its simple modifications, suffers from unstable convergence properties, moreover, the solution is quite sensible with respect to noise [START_REF] Bro | Multi-way analysis in the food industry: models, algorithms, and applications[END_REF][START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation[END_REF][START_REF] Albright | Algorithms, initializations, and convergence for the nonnegative matrix factorization[END_REF]. However, the algorithm is often utilized because of its programming simplicity [START_REF] Comon | Tensor Decompositions, State of the Art and Applications[END_REF].

To explain PARAFAC and ALS algorithms in details, let us consider the case of third-order tensor is the Moore-Penrose pseudoinverse of [START_REF] Golub | Matrix Computations[END_REF].

Let us note that the PARAFAC solution is unique up to an arbitrary factor [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF]. Thus, PARAFAC has no rotation freedom unlike PCA, and, therefore, no postprocessing is needed to detect the most appropriate solution.

Graphical representation of the PARAFAC approach is shown in Figure 4.8. After decomposition of the tensor , a regression of the response tensor on the matrix is build. For simplicity, we will consider the vector case of response variable: . Then, for example, a linear regression could be constructed ̂ ̂, ̂ ( ) .

Multi-linear PLS Regression (NPLS)

Multi-linear or N-way PLS Regression (NPLS) was introduced as generalization of the two-way PLS to data of higher orders [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF]. The goal of this method is the model the relation between the explanatory tensor

× 1 × ×
and the tensor of

responses × 1 × × .
The NPLS decomposition of the observation tensor 3 + , related to the second, the third, and the fourth modes of , respectively , is formed in such a way that projection of the tensor on these vectors results in .

Coefficients of the linear regression of , depending on the current set of latent variables * + , are calculated. This regression is used for deflation on each iteration. Besides that, the tensor is also deflated according to the decomposition. Let us note that there are modifications of NPLS (e.g., [START_REF] De | Regression coefficients in multilinear PLS[END_REF], where deflation of the tensor is not applied. The procedure is repeated a pre-defined number of times.

In details, the pseudo-code of the method is presented by Algorithm 4.3.

Algorithm 4.3 NPLS

Input:

, , number of factors .

( ( )) ( ( ) )
Outputs: { } , { } . 1. Centering of { } 2.
3.

4. for to 5. ×

6. { } = ALS( ) 7. ‖ ‖ 8. × × × 9. [ ] 10. ( )
11.
12. // deflation 13. end for a

The -mode vector product of a tensor and vector is denoted as

× (see Section 0). b
The vector outer product is denoted as (see Section 4.2.1). In details, the prediction steps are shown in Algorithm 4.4.

Sets of the regression coefficients { } and the projection vectors

Algorithm 4.4 NPLS Prediction

Input: The vector outer product is denoted as (see Section 4.2.1).

× 1 × 2 × 3 , { } , { } . Outputs: ̂ . 1.
Both PARAFAC and NPLS are widespread projective methods for the case of highly correlated tensor data of high dimensionality. However, contrary to PARAFAC, NPLS constructs a set of projectors taking into account information about the response variables. Thus, it was chosen as the basis approach for our BCI experiments.

Modality Influence Analysis

The elements of the input data have an implicit impact on the NPLS regression model through the latent variables. The Modality Influence (MI) analysis [START_REF] Cook | Residuals and Influence in Regression[END_REF][START_REF] Martens | Maltivariate Calibration[END_REF][START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF][START_REF] Nilsson | Multiway calibration in 3D QSAR[END_REF] allows estimating the relative importance of the elements of each mode for the final model. For example, in BCI experiments MI analysis can be applied to estimate the importance of electrodes, frequency bands, and time intervals (Eliseyev et al., 2011a).

Estimation ̂ of the dependent variable according to the model ̂ ∑ can be represented as:

̂ ∑ ∑ ∑( ‖ ‖) ‖ ‖ ∑ ‖ ‖
For the given mode , which is characterized by the set of the projectors { } , the matrix [ ] is formed. The vector of leverages for theth modality is defined as . ( ) /. All the elements of the vector are non-negative. Moreover, the elements of the leverage which are close to zero have not affected the model very much, while the elements with high values are more important.

This information could be useful for reduction of the modality dimension. Those components of the modality which have relatively small weights in the vector of leverages can be excluded from the further analysis.

Shortcomings of the generic NPLS approach

Despite many positive properties, the generic NPLS method has drawbacks. For instance, the identified model is of excessive complexity. Other shortcoming of the generic approach consists in its significant consumption of memory resources. The method continuously keeps in the active memory the tensor , size of which could be too great. Finally, generic NPLS cannot be used online for adaptive adjustment of the identified model to changings in the data flow.

In this study three modifications of the generic NPLS approach are proposed. A penalized version of the generic algorithm, providing an opportunity for selection of informative features, is represented in Section 4.7. An iterative version of the generic algorithm, which allows storing only part of the tensor in the active memory, is described in Section 4.8. Presented in Section 4.9 a recursive NPLS algorithm provides an opportunity of online treatment of the data.

L1-Penalized N-PLS algorithm (PNPLS)

In this study we propose generalization of the NPLS algorithm which gives us an opportunity to impose the additional constrains on feature selection. For this purpose, the Alternating Least Squares (Algorithm 4.2) can be substituted for its penalized version and used for decomposition of the tensor × in the NPLS algorithm (see Algorithm 4.3; [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF]. In the general case, the optimization problem for the Penalized ALS algorithm has the form:

* ̂ ̂ ̂ + 1 2 3 (‖ ‖ ( )) (4.16)
here, , respectively (see, [START_REF] Hoerl | Ridge Regression: biased estimation for nonorthogonal problem[END_REF]). However, to have a sparse solution ( ) could be chosen as a non-convex function with a singularity at the origin. For example, -norm penalty can be often used: ‖ ‖ ∑ | | . Some particular cases of the function ( ) are:

 ( ) ‖ ‖ , the Least Absolute Shrinkage Selection Operator (LASSO) [START_REF] Tibshirani | Regression shrinkage and variable selection via the lasso[END_REF];  ( ) ‖ ‖ , the -Fusion Lasso‖ [START_REF] Land | Variable fusion: a new method of adaptive signal regression[END_REF]. Here, is the first order difference operator;

 the Smooth Clipped Absolute Deviation (SCAD) [START_REF] Fan | Comments on -Wavelet in Statistics: A Review[END_REF]; [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF].

 ( ) ‖ ‖ ‖ ‖ , the Elastic Net (Enet)
To obtain a sparse solution, the -norm penalty (LASSO) is often used. The LASSO can be easily implemented providing a sufficient level of selectivity. In the present study, the -penalty was integrated into the ALS (Algorithm 4.2). At each step of the ALS algorithm, all the projectors are fixed except one leading to the optimization:

̂ .‖ ‖ ‖ ‖ / (4.17)
For the particular case :

̂ 1 .‖ ‖ ‖ ‖ / (4.18)
The optimization problem (4.18) can be rewritten in the matrix notation:

̂ 1 .‖ ‖ ‖ ‖ / (4.19)
where ( ) is unfolding of the tensor , and ( ) (see Section 0).

A possible approach to solve this optimization problem with the -penalization is the Gauss-Seidel algorithm [START_REF] Shevade | A simple and efficient algorithm for gene selection using sparse logistic regression[END_REF][START_REF] Schmidt | Least Squares Optimization with L1-Norm Regularization[END_REF]. The advantages of this algorithm are its simplicity and low computational cost, as well as low memory consumption which is of great importance for BCI tasks. We have applied this approach to solve the optimization task (4.19). Namely, the anti-gradient of the To automatically select the optimal value of different approaches can be used:

Cross-validation [START_REF] Devijver | Pattern Recognition: A Statistical Approach[END_REF], generalized cross-validation [START_REF] Golub | Generalization cross-validation as a method for choosing a good ridge parameter[END_REF]), Akaike's Information Criterion [START_REF] Akaike | A new look at the statistical model identification[END_REF], or Schwartz's Bayesian Information Criterion [START_REF] Schwartz | Estimating the dimension of a model[END_REF].

Penalized decomposition of the tensor × results in a set of sparse factor * + * ̂ ̂ ̂ + (see Algorithm 4.3).

Thus, in the work the task of Penalized NPLS algorithm is formulated. For its solving the Alternative Penalized Least Squares is proposed.

The L1-Penalized NPLS algorithm combines computational simplicity and moderate memory consumption with sufficient selectivity. Thus, it could be applied for BCI system calibration and for electrode subset selection.

Iterative N-PLS algorithm (INPLS)

In spite of efficiency, the curse-of-dimensionality has significant influence on the application of NPLS to the BCI systems. The amount of data increases exponentially with dimensionality of the feature tensor/vector. Thus, restrictions on the data set size considerably limits frequency/temporal resolution of the signal decomposition, as well as the number of simultaneously analyzed electrodes, due to huge storage consumption for saving the tensor in the active memory. To overcome the problem, we proposed the Iterative Multi-way Partial Least Squares (INPLS) algorithms (Eliseyev et al., 2011a), which properly treat the data set of huge dimension.

It is based on fragmentation of the initial dataset on several subsets and their sequential treatment. Thus, at each instant only a small part of the data is stored in the active The -mode vector product of a tensor and vector is denoted as

× (see Section 0). ** b
The vector outer product is denoted as (see Section 4.2.1).

Algorithm 4.6 Tensor Factorization

Input:

1 × 2 × 3 , * +, .

Outputs: * +.

1. { } = ALS( * +) 2. ( (
The random noise × was drawn from a multivariate normal distribution ( ), which is a generalization of Gaussian distribution for tensors [START_REF] Hamedani | On the determination of the bivariate normal distribution from distributions of linear combinations of the variables[END_REF]. Parameter is introduced to control the signal-to-noise ratio (SNR). The noise has the same amplitude as the signal ( ) in the case of .

{ ( ) × × ( ) * + } , { ( ) × × ( ) * + } , { ( ) × × ( ) * + } , { ( ) × × ( ) * + } .
Here, corresponds to traditional NPLS. The number of factors equal to 2, 5, and 10 was considered. The noise amplitude varied from 1 to 4, i.e., from 100% to 400% of noise. Resulted regression models were validated on the test dataset.

General performance of the algorithm was estimated by comparison of predicted ̂ with using Euclidian distance: (∑ ̂ ) .

In all the tests, INPLS either demonstrated comparable results or surpassed traditional NPLS. Moreover, INPLS noticeably outperformed NPLS in the case of large factor number and high noise level( , ) (Figure 4.12). This advantage can be explained by overfitting effect suppression in INPLS. Proposed iterative algorithm demonstrated good accuracy and was applied for the BCI system calibration.

Recursive N-PLS algorithm (RNPLS)

One of the major problems in BCI studies is the variability of the neuronal signals, in particular, due to the brain plasticity. These changes in the neuronal activity require recalibration of the BCI systems. The full system recalibration is a time and labor consuming procedure. Adaptive calibration aims to provide a fast adjustment of the BCI system to mild changes of the signal. Although INPLS allows treating data arrays of huge dimension, this method cannot be applied for adaptive learning. In this section the Recursive NPLS (RNPLS) algorithm is proposed (Eliseyev et al., 2011b). It allows online processing of the multi-modal data. Moreover, weighted RNPLS can be applied for adaptive learning to treat time-dependent recordings.

Algorithm description

The RNPLS algorithm unites the RPLS scheme of recursive calculation with multimodal data representation of NPLS. To apply the recursive approach to the NPLS algorithm described above, orthonormality of the latent variables as well as orthogonality of the latent variables to the residuals should be provided (see conditions (4.10), (4.11), and (4.12)).

Let us consider a fourth-order tensor × 1 × 2 × 3 of observations and a vector of observations . At the first step, the tensor is represented by a set of factors

{ } : (4.20)
constructed by the NPLS algorithm (Algorithm 4.3). For simplicity of notation, let us unfold the tensor along the first mode into the matrix ×( 1 2 3 ) [START_REF] Kolda | Tensor Decompositions and applications[END_REF]. At the same time let us denote as 1 2 3 vectorization of a tensor

. The score vector is a projection of the observation matrix on :

. In this matrix notation, Equation (4.20) can be rewritten in the form (on iteration :

).

In general, the NPLS latent vector variables are not orthogonal, but this restriction is necessary for the recursive scheme of calculation (see [START_REF] Qin | Recursive PLS algorithms for adaptive data modeling[END_REF]. Let us apply orthonormalization of the latent variables: , ( ) . Here, is a matrix of orthonormalization. It could be obtained from the QR decomposition procedure [START_REF] Golub | Matrix Computations[END_REF], where any rectangular matrix × could be represented as a product of two matrices (

× ,
× is an upper triangular matrix). There are several methods for computing the QR decomposition, such as the Gram-Schmidt process [START_REF] Golub | Matrix Computations[END_REF], Householder transformations [START_REF] Householder | Unitary Triangularization of a Nonsymmetric Matrix[END_REF], or Givens rotations [START_REF] Givens | Computation of plane unitary rotations transforming a general matrix to triangular form[END_REF], each has a number of advantages and disadvantages. For the new orthonormal latent variables the matrix could be decomposed as , where

. The Ordinary Least Squares (OLS) coefficients of regression of on the orthonormal latent variable equal to . After iterations:

, .

At this step conditions (4.10) and (4.12) are satisfied: ( ) and ( ) .

Then to provide orthogonality of the matrix and the residual matrix

{[ ] 0 1} {[ ̃ ] [ ]}.
The tensor ̃ is obtained from the matrix ̃, with as dimensionality of the first mode.

Dimensions of the other modes are equal to the dimensions of the other modes of the tensor . Thus, the RNPLS algorithm inherits the tensor representation of data from NPLS and allows effective adaptive learning, which is the property of the recursive PLS.

In addition to identification of the tensor ̃ and the vector , which contain information about the training sets on the previous iterations, the algorithm generates also sets of the vectors { } and the projection vectors { } , which are used on the prediction stage for determination of an estimation of the depended variable ̂ in the same way as in the traditional NPLS approach (Algorithm 4.3).

The graphical representation of the RNPLS algorithm is shown in Moreover, it showed better robustness. Variation of the prediction errors was essentially smaller for the recursive algorithm for small number of factors for noise level up to 500%. Advantages of RNPLS can be explained by overfitting effect suppression.

Outputs: { } , { } , ̃ , ( ) , ̅ , ̅ . 1. ̅ ( ) ̅ ( ) 2. ̅ 3. ̅ ( ) ̅ ( ) 4. ̅ 5. [ ̃ ] [ ] 6

Convergence of RNPLS

To study convergence properties of the RNPLS method, a set of computational experiments were carried out with artificial data described in Section 4.9.2.

At each iteration the RNPLS algorithm generates the current sets of coefficients { } as well as projection vectors { } and { } (see Figure 4.13), which can be used to predict of an estimation ̂( ) of the dependent variable ( ) from the independent variable ( ). The model can be represented as linear in the original variables ( ) and ( ). For the case of scalar ( ) and matrix ( ) 1 × 2 (see the dataset from Section 4.9.2) linear relations between them can be represented as:

In our computational experiments Convergence of the matrixes is represented in Figure 4.16. As it could be seen from Figure 4.17, the changes in the regression coefficients matrix become insignificant after the 10-th iteration. data realizations). In parallel, the means and the standard deviations of ( ) computed for the data generated by the NPLS algorithm is represented for every level of noise. As it could be seen, the mean value of the distances between the RNPLS and PLS regression coefficients already after 1-2 iterations (20-40 points) are significantly less than the ones between the NPLS and PLS regression coefficients computed for the whole training sets (800 points). In addition, the standard deviations of the RNPLS results are generally less than those obtained by NPLS. In conclusion, as it was shown in the set of computational experiments, both the RNPLS and the NPLS algorithms approximate the PLS solution obtained in the case when the noise was absent. RNPLS approach demonstrates faster convergence (20-40 points instead of 800 for NPLS) and is considerably noise-steady. For all tested levels of noise the disturbance in the RNPLS solution was appreciably less relatively to NPLS.

Conclusions

Multi-way approach is promising for BCI tasks. It allows simultaneous analysis of different modalities, preserves the structure of the data, improves robustness of results, as well as allows identifying relative impact of each feature on the final model.

PARAFAC and NPLS are widespread project methods for the case of highly correlated multi-way (tensor) data of high dimensionality. However, NPLS constructs the projectors by taking into account information about the response variables. Thus, it was chosen as the basis approach in this study. The shortcomings of the NPLS algorithm are excessive complexity of the identified model, method's computational efficiency and resources consumption. A set of modifications of NPLS were proposed in this chapter:

L1-Penalized NPLS (providing an opportunity for selection of informative features),

Iterative NPLS (allows storing only part of the processing data in the active memory)

and Recursive NPLS (provides an opportunity for online treatment of the data).

Suggested algorithms were tested in the computational experiments on artificial datasets. Experiments demonstrated promising results. Thus, the methods could be efficiently applied for the BCI tasks. However, the proposed approaches have some drawbacks. For instance, the INPLS algorithm cannot treat data flow, L1-Penalized NPLS gives a sparse solution that could be not appropriate for some modalities (e.g., frequency), and RNPLS could bring to some deterioration of the prediction quality. During the second stage, the Close Loop Real-Time BCI, the predictor is applied immediately to the signal from the animal's brain in real-time. The animal presses the pedal but it obtains the reward only in the case when the predictor detects the pushing event. More precise description of the experimental setup is given in Chapter 2.

Signal acquisition and pre-processing

A set of electrodes was implanted on the surface of the rat's and the monkey's cortex for the neural activity recording. The ECoG signals were registered at 6.5 kHz, or 13 kHz sampling rate either with the Biomea® system (Biologic, Grenoble, France), or at 1 kHz by the Micromed® system (Micromed SD64, Micromed Italy). In parallel, the monkey was sitting in a custom made primate chair minimally restrained and its neck was collar hooked to the chair.

The signal was downsampled to 1.3 kHz or 1 kHz. The Common Average Reference (CAR) filter was applied to the signal of the rat, i.e., average signal among all the electrodes was subtracted to eliminate a ‗common source' [START_REF] Ludwig | Using a Common Average Reference to Improve Cortical Neuron Recordings From Microelectrode Arrays[END_REF].

Application of this filter is made due to the presence of the strong noise signal sources whereas the signal of interest is relatively weak in the case of the rat.

Feature extraction

Continuous Wavelet Transform (CWT) was applied for feature extraction:

( ) * +( ), * +( ) ( ) ∫ ( ) ( ) ( ) ( ) (5.1)
where and represent the scale and the shift in the time-frequency domain, respectively. ( ) is a wavelet function, which effectively limits duration and has an average value of zero. For more information, see [START_REF] Teolis | Computational Signal Processing with Wavelets[END_REF].

In the present study, Meyer wavelet (Figure 5.1) was chosen as a mother wavelet taking into account its computational efficiency [START_REF] Sherwood | On Classifiability of Wavelet Features for EEG-Based Brain-Computer Interfaces[END_REF].

Analytical representation of the wavelet in the frequency domain is given by equation

(5.2). ̂( ) { ( . /) ( . /) 0 1 , ( ) ( ), , - (5.2)

Tensor representation of the ECoG data

To form the BCI feature tensor, each ECoG epoch sample was mapped to the temporal-frequency-spatial space by CWT. For each epoch (determined by its final moment ), electrode , frequency and time shift , elements of the tensor were calculated as the norm of CWT coefficients of the ECoG signal (see Figure 5.2).

Frequency band , -Hz with step Hz was chosen for both monkey and rat. The analyzed time interval was equal to s (with resolution s) for the rat and s (with resolution s) for the monkey (according to the biological point of view). For the case of the rat , and for the case of the monkey. The resulting dimensions are ( × × ) and ( × × ),

respectively. The binary dependent variable was set to one, , if the pedal was pressed at the moment , and , otherwise.

BCI system calibration

The tensor and the vector , representing the training dataset, were used for the BCI system calibration. Because of huge dimension of the feature tensor , an iterative method was employed. In order to apply the INPLS algorithm, the training dataset was split into subsets in such a way that each one consisted of 30 points. Total number of the points in the training dataset was , including all event-related points and randomly selected non-event ones. The number of factors was chosen by tenfold cross validation ( for the rat and for the monkey). On the basis of the given data INPLS identified a predictive regression model between the dependent variable and the tensor of independent variables. In addition a set of projection vectors for each modality was generated. Let us note that each factor could be considered as a spatial-frequency-temporal filter, formed as outer product [START_REF] Kolda | Tensor Decompositions and applications[END_REF] of particular filters related to each modality: spatial, frequency, and temporal.

The overall goal of our study was to calibrate the BCI system using the signal of a single electrode. For this purpose, in the case of experiments in rats, the most informative electrode was chosen according to the Modality Influence Analysis, as the one with the highest weight across all the electrodes (see Section 4.6.3). In the case of monkey, the Recursive NPLS algorithm was applied to select the most informative electrode, since RNPLS has demonstrated better efficiency in comparison with MI mapping analysis when number of electrodes was great. The training tensors related to the selected electrodes were analyzed with INPLS to construct the single-electrode predictors.

Let us note that we estimate the dependent variable ̂( ) as a linear combination of the components of the ECoG signal decomposition at the time moment : ̂( ) ∑ ∑ ( ) . Here, ( ) is the absolute value of the continuous wavelet decomposition coefficient for the frequency at the moment . Thus, the resulting model presets a dynamical filter.

For binarization of the predicted output variable ̂( ), a scalar threshold was chosen maximizing the overall performance ( ( ) ⁄ , see Section 1.2.5) of the BCI algorithm on the training data set running in the online mode simulation.

Calibration was made on one recording. The resulting model was tested in all recorded datasets as well as in the online experiments.

Simulation of BCI experiments

Offline emulations of BCI experiment, were fully reproducing online sessions and were carried out to study a generalization ability of the predictive model. For this purpose, the set of recordings of behavioral experiments were played back. The decisions (‗event' or ‗non-event') were made sequentially for each half-a-second-length buffer of the recordings. This corresponds to the real-time data acquisition in the CLINATEC/LETI/CEA BCI experimental platform. The predictors were calculated every 0.125 s, i.e., 4 times per buffer. The buffer was considered as containing the ‗event' if at least one of these predictors surpassed the threshold of binarization. After each detection, the system was blocked for 5 seconds in the case of the rat and 1.5 seconds in the case of the monkey to prevent multiple activations. Following to [START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF] the real event was considered as detected (True Positive, see Section 1.2.5) if the time interval between the real event and its detection did not exceed 1.5 s (Figure 1.19).

Proof and comparison of solutions

To confirm the selected approaches and implementations, a set of comparative experiments were carried out.

The present section consists of three parts. In the first one, several mother wavelets were compared by means of correlation between the wavelet coefficients and the signal of the pedal. For this purpose, we used the series of recordings, made during the long-term experiments in the rat freely moving in the natural-like environment. The second part is a comparison of the unsupervised (PARAFAC) and supervised (INPLS)

approaches for classification of the real BCI data recorded in the experiment in monkey.

The last part is devoted to the study of the efficiency of different classifiers in the space of the INPLS latent variables. Simulations of the online experiments were made on the basis of 9 recordings of the experiments in the rat.

Comparison of different mother wavelets

To project recorded data to the feature space continuous wavelet transform is applied. To found the most effective function for this projection several mother wavelets were compared, namely, Meyer, Morlet, Symlet ‗7' and ‗8', 2 nd and 10 th orders Debauchies, Coiflets ‗5', and Haar (Figure 5.3).

2 .| ( )| ( )/3 ( ( ) ( )) ∑ ( ( ) ̅)( ( ) ̅ ) √∑ ( ( ) ̅) ∑ ( ( ) ̅ ) ̅ ∑ ( ) , ̅ ∑ ( ) ,
where corresponds to the frequencies of the band , -Hz and ( ) * + represents the position of the pedal at the moment .

Comparison of the mother wavelets shows that 2 nd order Daubechies and Haar lead to a relatively low level of correlation, whereas the performance of all other wavelets is comparable. Meyer wavelet was chosen for the present study as the mother function, due to its computational efficiency [START_REF] Sherwood | On Classifiability of Wavelet Features for EEG-Based Brain-Computer Interfaces[END_REF].

Results are shown in Figure 5.4.

Comparison of unsupervised and supervised classifiers

Unsupervised methods, like PARAFAC, Tucker Decomposition, or Nonnegative Tensor Factorization are often used to form a feature space in the tasks with multimodal data [START_REF] Nazarpour | Parallel space-timefrequency decomposition of eeg signals for brain computer interfacing[END_REF][START_REF] Zhao | Slice Oriented Tensor Decomposition of EEG Data for Feature Extraction in Space, Frequency and Time Domains[END_REF][START_REF] Mørup | Algorithms for Sparse Nonnegative Tucker Decomposition[END_REF]. These formation (e.g., General Tensor Discriminant Analysis [START_REF] Li | A prior neurophysiologic knowledge free tensor-based scheme for single trial EEG classification[END_REF], Regularized

Tensor Discriminant Analysis [START_REF] Li | Regularized tensor discriminant analysis for single trial EEG classification in BCI Pattern Recognition[END_REF], N-way PLS [START_REF] Bro | Multiway calibration. Multi-linear PLS[END_REF]) are preferable for the BCI tasks. In this section we compare effectiveness of PARAFAC combined with a linear regression and the NPLS algorithm on the real data recorded in the experiment in monkey. The main difference in the applied approaches is that whereas the NPLS algorithm simultaneously finds the space of latent variables and builds a linear regression, PARAFAC independently projects data into the space of latent variables (identified without any information about the classes) and after we perform a linear multiple regression on the obtained projections.

During it the monkey was pressing the pedals which were located in different This experiment also has demonstrated better efficiency of the NPLS algorithm for classification tasks in comparison with PARAFAC approach. performed by the NPLS and the PARAFAC approaches for different number of factors.

Comparison of different classifiers

The NPLS algorithm is based on the data projection to the low dimensional feature space (the space of latent variables), with simultaneous construction of a linear regression. The binary BCI leads to the problem of two-classes discrimination.

Although the PLS algorithm (as well as NPLS and its derivations) was not inherently designed for classification, it is widely applied to solve this problem [START_REF] Barker | Partial least squares for discrimination[END_REF]. In particular in the studies (Eliseyev et al., 2011a;[START_REF] Chao | Long-term asynchronous decoding of arm motion using electrocorticographic signals in monkeys[END_REF]Acar et al., 2010) the NPLS linear regression was applied for classification using binary output variables. Otherwise PLS was used as a dimensionality reduction tool and coupled with different classifiers in the space of latent variables (for more details see [START_REF] Barker | Partial least squares for discrimination[END_REF]. The goal of the present section is to study the efficiency of different classifiers in the space of the INPLS latent variables in the context of the binary self-paced BCI. For the comparison of classifiers, we used the recordings from the series of 9 long-term experiments in the rat freely moving in the nature-like environment.

Different types of classification methods have been applied in the BCI tasks [START_REF] Lotte | A review of classification algorithms for eeg-based brain-computer interfaces[END_REF]. Several linear and non-linear classifiers widely used in the BCI research were chosen in this study. They were compared using a given set of the INPLS features.

Linear classifiers:

 Linear Discriminant Analysis (Fisher's LDA): in the space of the latent variables it constructs a hyperplane to separate two classes [START_REF] Duda | Pattern Recognition[END_REF][START_REF] Fukunaga | Statistical Pattern Recognition[END_REF].

The method assumes normal distribution of the data, as well as equal covariance matrix for both classes. The separation hyperplane is seeking to maximize the distance between the classes and minimize the interclass variance. Non-linear classifiers:

 Quadratic Discriminant Analysis: the method assumes different normal distributions of data in the different classes [START_REF] Duda | Pattern Recognition[END_REF][START_REF] Fukunaga | Statistical Pattern Recognition[END_REF]. The Bayes rule is used to compute a posteriori probability that point belongs to a given class. The task leads to a quadratic decision surface, which explains the name of the classifier.



Logistic Regression: the method assumes binomial distribution of the data [START_REF] Hilbe | Logistic Regression Models[END_REF]. The model supposes that probability that the point belongs to the first class can be expressed as ( ) ( ( )), where is a linear function of . The unknown parameters of this linear function are usually estimated on the training set by maximum likelihood procedure.

 Kernel Support Vector Machine: the method uses a discriminant hyperplane to identify the classes [START_REF] Cortes | Support-Vector Networks[END_REF], which is found in the way to maximize the margin, i.e., the distance between the nearest training points belonging to the different classes. By using a ‗kernel trick' [START_REF] Aizerman | Theoretical foundations of the potential function method in pattern recognition learning[END_REF] the method could be applied as non-linear classifier, by mapping of the data points to higher dimension

Conclusion

In this chapter, structural parameters of the applied method were compared in the simulation of the online BCI experiments.

Comparison of different mother wavelets used for mapping of the ECoG signals has demonstrated similar performance of all bases of decomposition. Thus, taking into account its computational efficiency, Meyer wavelet was chosen.

A set of test experiments has confirmed that supervised classifier (INPLS), which took into account information about classes, was preferable than unsupervised classifier (based on PARAFAC). Thus, the initial choice of the supervised method was justified.

Different classifiers in the space of the latent variables were compared in a set of simulations of the online BCI experiments. The quadratic classifier has demonstrated the best overall performance. However, it did not significantly outperform the linear regression with binarization threshold resulted from the INPLS algorithm. Thus, application of INPLS without any additional classifier is preferable.

Chapter 6. BCI EXPERIMENTS RESULTS

Experiments in rats

Experimental data were collected during the behavioral experiments in a rat based on a simple reward-oriented task (see Chapter 2). Three series of behavioral experiments, 34 experiments in total (see Table 6.1), were collected over more than 8 months. The experiments lasted from 5 min up to 1 h (22 min in average). A part of one recording (#1 in Table 6.1) was used for calibration to identify the predictive model.

The training data set included all event-related epochs and randomly selected ‗nonevent' epochs. The threshold of binarization was adjusted to every recording. 3). These coefficients were (Figure 6.2). The MI analysis revealed the leverages of each element for each modality (Figure 6.3). In particular, applied to the spatial modality, the MI analysis indicates occipital electrode number 15

Calibration results

For

as having the highest impact on the decision rule (~57% of the extracted information).

High frequencies [100, 300] Hz provide the main contribution to the decision in the frequency modality (~86%). In the time domain the interval [-0.5, 0] s before the event is the most significant (~68%).

To verify different frequency bands, the calibration procedure was applied

Offline and open-loop validation of the BCI system

For offline validation of the BCI system, the predictive model identified at the calibration stage is applied to the previously recorded data. This type of verification is preferable at the initial stages of the experiments when the animal is not trained enough and false activations of the distributor as well as misses of the pedal pushing significantly confuse the rat.

In the simulation mode, a decision about event/non-event state of the system was made every half a second buffer: the predictors were calculated each 0.125 s, i.e., 4 times per buffer. The buffer was considered as containing the ‗event' if at least one of these predictors surpassed the threshold of binarization. After each detection, the system was blocked for 5 s to prevent multiple activations. Following to [START_REF] Fatourechi | A self-paced brain-computer interface system with a low false positive rate[END_REF] the real event was considered as detected (True Positive) if the time interval between the real event and its detection did not exceed 1.5 s.

For the first two groups of the BCI experiments (carried out in July and October, 2009; see from #1 to #12 in Table 6.1, Figure 6. , 2010;Table 6.1, Figure 6.11). In these experiments the rat obtained rewards after every pressing regardless decision of the predictive model. The answers generated by the system were analyzed independently afterwards.

Like in the case of the offline validation, this type of the experiment was made to prevent the animal to be confused by false activation and pushing omissions.

Experiments have demonstrated rather considerable decreasing of the quality: mean overall performance OP ol = 50.34% was achieved with 1.46 false activations per minute.

However, further examination of the rat's brain and the implanted electrodes, made after death of the animal, showed that this decreasing of the BCI performance was significantly caused by the electrodes degradation. Like in the simulation mode and open-loop mode, the decision about event/nonevent state of the system was made every half a second buffer (4 predictors per buffer, the buffer contained the ‗event' if at least one of the predictors surpassed the threshold of binarization). The real event was considered as detected if distance between the real event and its detection did not exceed 1.5 s; the system was blocked for 5 s after the detection.

The

Validation of the RNPLS algorithm with real data

The RNPLS algorithm was tested with the real data sets which were collected during the BCI experiments in freely moving rat. The data were extracted from the record #1 (see Table 6.1). 1000 randomly selected points (700 correspond to -non-events‖ and 300 to -events‖) formed the training set, whereas 400 points (300+100)

were used as the test set. The intersection of the training and the test data sets was empty. The NPLS algorithm was trained on the whole training set. For the RNPLS the training set was split into disjoint subsets with 10 and 100 points. Then the projectors and the predictive models were identified. Figure 6.16 represents the first two factors (the total number of factors was taken equal to 5 by cross-validation procedure) calculated by the RNPLS (10). The relative weights of the factors in the final decomposition are demonstrated in Figure 6.17. Figure 6.18 shows the percentage of the resulting prediction errors. With respect to NPLS, the RNPLS algorithm demonstrates minimal deterioration in the prediction quality: for RNPLS (100) it is about 0.1%, whereas for RNPLS (10) it is about 0.2%.

Thus, the proposed recursive algorithm demonstrated the excellent performance in comparison to the NPLS approach in terms of accuracy and convergence rate on the real data. At the same time, taking into account that requirements for computation resources (memory) are low and do not depend on size of the processed data, the method is favorable for the BCI application.

Experiments in monkeys

Experimental data were collected during the behavioral experiments based on a simple reward-oriented task (see Chapter 2). The model was applied to the data recorded in 24 experiments (see Table 6.2, Figure 6.25). The experiments lasted from 4 min up to 20 min, 8 min in average. Parts of four recordings were used for the calibration that is to identify the predictive models. The training data sets included all event-related epochs and randomly selected ‗non-event' epochs. The threshold of binarization was adjusted to each recording.

Results of the calibration

The monkey's brain activity signal of the training data set was mapped to the temporal-frequency-spatial space for the calibration purposes. Then, five factors (the number is defined by the cross-validation procedure) and the corresponding latent All modalities of the first factor are shown in Figure 6.20 for each position of the pedal.

The MI analysis revealed the leverages of elements of each modality (Figure 6.21).

Applied to the spatial modality, the MI analysis indicates that the electrode #22 located in the motor primary cortex has the highest impact on the decision rule (84%, 97%, 89%, and 75% of extracted information for ‗left', ‗right', ‗up', and ‗down' positions of the pedal, respectively). High frequencies ( Hz) make significant contribution to the decision in the frequency modality, however, contrary to the rat, the influence of the low frequencies ( Hz) is also rather considerable, especially, in the case of the ‗left' position of the pedal. In the time domain the interval [-0.2, 0] s before the event is the most significant for all positions of the lever. The MI analysis revealed the leverages of elements of each modality for the best electrode case (Figure 6.24). Like in the case of the rat, taking into account the computational cost, the single-electrode predictive model was chosen for the BCI system.

The calibration procedure results in the predictor of the pressing events and the threshold-based decision rule. Calibration-based model was applied for prediction of the animal's control intention in the series of offline simulation BCI experiments.

Offline validation of the BCI system

The event prediction procedure was validated in a set of simulations of the realtime experiments. A decision was made every 0.5 s buffer: the predictors were calculated each 0.125 sec. The buffer was considered as containing an ‗event' if at least one of these predictors surpassed the threshold of binarization. After each detection, the system was blocked for 1.5 s to prevent multiple activations, the real event was considered as detected if the time interval between the real event and its detection did not exceed 1.5 s.

To make calibration of the BCI system and assess performance of the predictive models 24 experiments (from 4 to 20 min, 8 min in average) were carried out (see Table 6.2, Figure 6.25). Four recordings were used for calibration. Identified models were tested in all 24 recordings. A fragment of one simulated experiment as well as its time-delay histogram are shown in Figure 6.26. In average, the event prediction was made 0.04 s before the event. Figure 6.27 demonstrates a set of photos of the real-time experiment.

Validation of the LPNPLS algorithm with real data

The L1-Penalized NPLS (LPNPLS) method was tested with real data collected during the BCI experiments in monkey. For validation of the LPNPLS algorithm it was compared with generic NPLS. Four files were used for calibration (#2 for ‗left', #11 for ‗right', #13 for ‗up', and #21 for ‗down' pedal positions, see Table 6.2, Section 6.2.1).

The factors number in the predictive models ranges from 1 to 5. Identified models were applied to the test files (one file for each pedal position).

The computational experiments demonstrated that results of L1-Penalized NPLS are comparable with NPLS ones or outperform them in terms of root mean squared error (RMSE). The comparison was carried out for all positions of the pedal and number of factors ranging from 1 to 5 (Figure 6.28). At the same time, it should be emphasized that the LPNPLS algorithm identifies the sparse predictive model. Contrary to the NPLS approach, only few electrodes are used for decision making (6, 6, 7, and 9 for ‗left', ‗right', ‗up', and ‗down' positions of the pedal, respectively). Results of the MI analysis are represented for each pedal position in Figure 6.21 for LPNPLS, and in Figure 6.29

for NPLS (number of factors ). Thus, the Penalized NPLS algorithm effectively reduces the number of utilized electrodes without loss of prediction quality. Moreover, due to the reduction of the feature space, a considerable prediction improvement can be achieved.

Discussion

Experiments in rodent

The experiments in rodents have demonstrated that the most posterior electrodes have the strongest influence on the resulting model. These electrodes are located in the region of cerebellum. Additional experiments in several animals will allow better studying the location of zone of interest, its stability, evolution over time, neuroplasticity, etc.

The identified factors can be interpreted taking into account their influence on the final model. The first factor is responsible for almost 57% of extracted information (see Section 6.1.1). For this factor the weights in the frequency-modality are positive for the low and high frequencies ( -and -bands: 10-15 and 55-300 Hz), and negative for -band Hz. The signs of weights can be resulted from the changes of the signal energy in the corresponding frequency bands. The decrease of signal intensity in theband could be associated with a motor related potential (MRCP) (e.g., [START_REF] Nascimento | Movement-related parameters modulate cortical activity during imaginary isometric plantar-flexions[END_REF][START_REF] Boye | Identification of movement-related cortical potentials with optimized spatial filtering and principal component analysis[END_REF][START_REF] Gu | Offline identification of imaged speed of wrist movements in paralyzed ALS patients from single-trial EEG[END_REF][START_REF] Cabrera | Comparison of feature selection and classification methods for a brain-computer interface driven by non-motor imagery[END_REF]. At the same time, the MI analysis demonstrates that high frequencies (>100 Hz) have the most significant influence on the decisions. However, in the low frequency (~10.5 Hz) the event-related activity appears approximately 100 ms before the activity in -band. In parallel, the MI analysis determined one electrode in the occipital cortex which has the greatest influence on the decision. It was demonstrated that the detection model, using only this electrode, has comparable prediction quality with the one, using all electrodes. Thus, taking into account the question of computational efficiency, the one-electrode predictive model seems preferable for the future implementation in the real-time BCI the recording chain (connectors and so on) and can be eliminated using wireless data transferring [START_REF] Charvet | WIMAGINE: A Wireless, Low Power, 64-Channel ECoG Recording Platform for Implantable BCI Applications[END_REF]. However, additional experiments in several animals are necessary to make more reliable conclusions. Finally, carried out close-loop realtime experiments have demonstrated efficiency of implementation of the proposed method in the real BCI systems. Moreover, data collecting and processing time (for a 0.5 s buffer of data: 0.25 s and 0.06 s, respectively) allows application of the proposed approach for several degrees of freedom.

Experiments in monkey

The experiments in the monkey were carried out to verify the proposed approaches in other animals. The experiments have shown that contrary to the rat, the strongest influence on the resulting model have electrodes located in the motor primary cortex. However, to determine a long-term stability, time evolution, neuroplasticity, etc., additional experiments should be carried out.

The one-electrode model calibration was made on the basis of data from the primary motor cortex electrode, determined by the L1-Penalized NPLS algorithm. The MI analysis applied to the identified model has shown that, similar to the case of the rat, the low frequency band (<45 Hz) had relatively small influence on decision, whereas the high frequencies (>100 Hz) contain the most significant data. Thus, although the informative electrodes are located in the different parts of the rat's and monkey's brains Chapter 7. CONCLUSIONS

Summary

This study was undertaken as a step toward the fully autonomous (self-paced) BCI functioning in natural environment which is of crucial importance for efficient BCI clinical applications. The self-paced BCI in noisy natural environment requires a high level of selectivity for identification and discrimination of the specific neuronal activity against the background brain functioning during continuous monitoring. To approach this goal, we have studied the binary self-paced BCI in animals (rats freely moving in the cage and monkeys sitting on chair). Namely, we have analyzed the recordings of the series of the behavioral experiments in rodents and primates controlling a food dispenser by pushing a lever. Contrary to experiments carried out previously, the longterm tests were realized in the present study. The short-term experiments allow the subject to be concentrated on the task that significantly simplifies data analysis.

However, the long-term experiments are required for the real-life applications.

Durations of our tests varied from 5 minutes up to 1 hour. Let us stress that in average feeding was taking about 40% of the experimental session in the case of the rats and 35% in the case of the monkeys, the rest of the time was spent by the animals in spontaneous various activities. While the main goal of the study was to discriminate the specific neuronal pattern related to the control action, the additional goals were to make the decision using the single electrode recordings and to predict the control actions rather than detect them. Single electrode classification is desirable for the following self-paced BCI applications with multiple degrees of freedom. Early detection (prediction) allows avoiding the delays of execution.

To achieve the necessary level of selectivity the Multi-way Analysis was chosen since it provides a simultaneous signal processing in several domains. In the present study, we have applied the mapping of the ECoG signal to the temporal-frequency-spatial space. However, additional modalities, like the phase information, can be easily incorporated. To extract knowledge from the experimental data, a variety of machine learning methods was applied in the BCI research. We have chosen NPLS as a basic approach of multi-modal analysis for extraction the BCI features and for the BCI system calibration. Specially developed for high dimensional and highly correlated observations, this method requires neither exhaustive search of the model nor regularization of the task. It does not depend on the metrics of the BCI system comparison. In our study we have only applied the criterion of the BCI performance to determine the threshold of binarization.

To improve the capacity of the NPLS approach for treatment of highdimensional data, the Iterative NPLS algorithm (Eliseyev et al., 2011a;patent received) is introduced in the current study. Having lower memory requirements it allows huge datasets treatment, allows higher resolution of analyses, and preserves the accuracy of the generic algorithm. Moreover, INPLS demonstrates better robustness against noise which can be explained by the overfitting effect suppression. The method was implemented in the BCI system, which demonstrated successful results in series of the close-loop real-time experiments in rats.

Selecting an effective subset of features could significantly optimize efficiency of the model. Proposed L1-Penalized NPLS algorithm [START_REF] Eliseyev | L1-Penalized N-way PLS for subset of electrodes selection in BCI experiments[END_REF]patent received) directly includes feature selection in the modeling process. Contrary to other approaches which lead to a linear combination of all features, the L1-PNPLS provides a sparse solution. Applied in the BCI experiments with non-human primates, the method demonstrated efficient selection of the optimal subset of electrodes.

One of the major problems of the BCI studies is significant variability of the neuronal signals, in particular, due to the brain plasticity. These changes in the recorded neuronal activity require recalibration of the BCI systems. The full system recalibration is a time and labor consuming procedure. The adaptive calibration aims to provide a fast adjustment of the BCI system to handle the time changes of the signal. Although the INPLS algorithm allows treating data arrays of huge dimension, this method cannot be applied for the adaptive learning. To overcome the problem, the Recursive NPLS algorithm (Eliseyev et al., 2011b) is proposed in the study. It allows online processing of the multi-modal data. The method can be efficiency applied for the adaptive learning to treat the time-dependent recordings. Moreover, the proposed recursive algorithm demonstrated an excellent performance in comparison with NPLS in terms of accuracy (the difference in prediction quality in the real-data experiments did not surpass 0.2%) and convergence rate (in the model experiments the method converged after the first or the second iteration). The requirements for computation resources (memory) are low and do not depend on size of processed data, since the data is treated in the flow. In the real-data experiments, the same predictive quality (less than 0.2% prediction deterioration) was achieved with storing in the active memory only 10 points instead of the whole training set of 1000 points. Thus, RNPLS can be efficiently applied in the BCI systems, which is the perspective of the current study.

The proposed approaches were tested in a set of long-term binary self-paced BCI experiments in animals (rats and non-human primates) and demonstrated promising results for the offline as well as for the open-and close-loop real-time modes (about 80% of correct detections for the rat and up to 70% for the monkey, with acceptably low level of false activation: 1% and 3%, respectively). High frequencies are the most significant for decision making in the cases of both animals. Event-prediction was achieved ~100 ms before the moment of event for the rat, and ~20 ms, for the monkey.

Whereas experimental conditions are more complicated than in experiments of other groups, performance characteristics of our methods either surpass or are comparable with others. Detection of event-intention instead of event-detection was achieved.

Moreover, only one electrode was used for decision. Computational efficiency of the algorithms allowed their application in the real-time BCI systems. Thus, the suggested approaches provide the essential basis for further development of a human BCI system.

At the same time, additional experiments will allow following studying of the long-term robustness or/and instability of the predictive models, effects of the brain plasticity, etc. Moreover, the experiments in different animals will make it possible to evaluate the discriminative efficiency of the methods for different BCI tasks.

Perspectives

In the current study a set of methods have been proposed for the binary closeloop real-time BCI. However, the binary BCI represents a particular case of the applications of the suggested approaches. The same methods can be applied for the multi-class BCI with minimal modifications. The computational efficiency of the proposed algorithms is sufficient enough for the BCI system with multiple degrees of freedom and using a standard portable computer. Moreover, it is possible to apply these methods for the continuous control ( ).

The Recursive NPLS algorithm was tested with real data but in the simulation mode. The method showed itself to good advantage and should be applied for the adaptive calibration of the BCI system.

The L1-Penalized NPLS algorithm was used for selection of the effective subset of electrodes. The next step would be the application of this method for other modalities (e.g., frequency).

In the present study, we have applied the INPLS/RNPLS/L1-PNPLS regressions, which are combinations of projection techniques with linear regression.

However, the NPLS projections can be coupled with other regression and/or classification techniques that can provide further improvement.

Finally, additional computational optimization should be provided in real-time BCI system. Internal structure of the proposed methods allows an effective code parallelization which will be included in further implementation of our BCI.

Acknowledgement

Working toward this thesis during three years I should say it was an interesting and exciting experience. At the same time it would be impossible to finish this study without help, encouragement and support of people around me.

First of all I would like to express my gratitude to Dr. Tatiana Aksenova who was my supervisor and gave me the possibility to do a PhD. Her patience, knowledge and advices had a crucial importance for me.

I would like to thank all the people from the CEA/LETI/CLINATEC team with whom I was working during these years. My special thanks to Prof. Alim-Louis

Benabid who is the heart of the project.

In addition, I thank to the CEA/LETI/LE2S laboratory, people from which are productively participating in our project.

Last but not least, my special thanks to my family and all close to me people in Ukraine, France, Russia, Switzerland, and USA who were supporting and encouraging me during the last three years.

The study was supported by Foundation of Nanoscience in frame of the project CE ICoBI.

  Figure 1.1 General scheme of BCI system. Here 𝐬(𝑡) is a vector reflecting brain neural activity, 𝐮(𝑡) is a vector of control block command, and 𝐲(𝑡) represents the state of the effector at the moment 𝑡. The dotted arrow in the figure represents the connection which is active only at a certain period of the BCI's operation cycle.

Figure 1 . 2

 12 Figure 1.2 CLINATEC/LETI/CEA project: real-life human BCI.

  information by means of electrical and chemical signals. The neuron consists in a cell body (soma), an axon, and dendrites (Figure 1.3). Different neurons are connected to each other to form a neural network. In this network, signals are sent from the axon of one neuron to dendrites of others by means of electrochemical pulse (action potential).

Figure 1 . 3

 13 Figure 1.3 Basic structure of a neuron.

Figure 1 . 4

 14 Figure 1.4 EEG cap with 10-20 international system of electrode placement on the scalp and general view of the recorded signals.

Figure 1 . 5

 15 Figure 1.5 The UUMSI MEG system from Elekta Neuromag Oy in Helsinki (Finland).

  Figure 1.6 demonstrates an example of the fMRI scanner as well as examples of the obtained brain images.

Figure 1 . 6

 16 Figure 1.6 Berkeley's 4T fMRI scanner and brain images example.

  Microelectrode arrays Microelectrode arrays (MEA) are a technique for registering activity of small groups of neurons (local field potentials) or even a single neuron (single-unit action potentials). To record signals, needle-like electrodes are placed into the brain cortex during surgery. First arrays were implanted by C. Thomas in 1972 (for more information, see Pine, 2006). Electrode diameter ranges between 10 and 30 μm. The method allows recording of the signals with frequency up to 5000 Hz. Examples of the MEA electrodes and signal are given in Figure 1.7. On average, the microelectrode array is covering a surface up to about 50 mm 2 .

Figure 1 . 8

 18 Figure 1.8 Tissue layers covering the cortex.

Figure 1 . 9

 19 Figure 1.9 Spatial electrode arrangement shown through MRI image[START_REF] Kutsy | TI Focal extratemporal epilepsy: clinical features, EEG patterns, and surgical approach[END_REF].

Figure 1 .

 1 Figure 1.11 CNT electrodes on the flexible structure (Sauter-Starace et al., 2009).

  Figure 1.12 (B). It could be seen from the figure that signals from CNTs electrodes are richer than ones from TiN electrodes. This observation was consistent over the course of the study, irrespective to variation in amplitude (between 50 μV and 200 μV) and in the level of noise (between 20 μV and 50 μV). Power spectra represented in Figure 1.13 also demonstrates advantage of CNTs electrodes.

Figure 1 .Figure 1 .

 11 Figure 1.12 (A) Scanning electronic microscopy (SEM) picture of carbon nanotubes (Sauter-Starace et al., 2009). (B) Typical recorded ECoG signal (Sauter-Starace et al., 2009). Top and bottom pictures are sonograms of TiN and CNT electrodes, respectively (here, x: time in seconds, y: frequency in Hz).

Figure 1 .

 1 Figure 1.14. The averaged brain patterns exhibit a great diversity for 6 subjects for the case of right and left hand finger typing.

Figure 1 .

 1 Figure 1.14 Example of the subject-to-subject variability[START_REF] Tangermann | Contributions from mathematics: Applying machine learning algorithms to BCI[END_REF].

Figure 1 .

 1 Figure 1.15 Example of the session-to-session variability (Tangermann, 2008). Represents an average across 140 trials made for the same subject (different sessions are shown on the different columns) for the case of right-and left-hand finger typing.

Figure 1 .

 1 Figure 1.16 Example of the trial-to-trial variability[START_REF] Tangermann | Contributions from mathematics: Applying machine learning algorithms to BCI[END_REF]. It represents the alpha-band power for twelve 3.5 seconds-length trials for the case of right-and lefthand finger typing.

Figure 1 .

 1 Figure 1.18 Graphical representation of the decision characteristics.

  Figure 1.19 (A) After each detection the system cannot generate other detections during the predefined time interval. (B) Detection is treated as True Positive (TP) if it is made during the certain time interval from the real event.

  Whereas the BCI system has to work in the real-time conditions, the response time (RT) of the system generally is of great importance. It could be characterized by the probability of the different values of the response lags. An example of the responsetime histogram is represented in Figure 1.20.

Figure 1 .

 1 Figure 1.20 An example of the Response-Time Histogram: represents the probability of the different response delays.

Figure 1 .

 1 Figure 1.21 Topographies for mu-and beta-band activity. (A) Left-hand movement versus rest and right-hand movement versus rest. (B) Left-hand imagery versus rest. (C) Movement versus rest and imagery versus rest. (D) Right-versus left-hand movement and imagery. Taken from McFarland et al., (2000).

  The author was participating in definition of the experiments' protocol and assisted in the experiments. The experiments are based on a simple reward-oriented task. Freely moving in the ABETT® behavioral cage (Abett II Starter kit, Campden Instruments; Lafayette Instrument Co., Leicestershire, UK) rat has an opportunity to push a pedal, mounted in a wall of the cage, see Figure2.1. Every pushing event activates a food dispenser and the rat obtains a reward (a food pellet). The rat is trained to press the pedal without any cue or conditioning stimulus.

Figure 2 . 1

 21 Figure 2.1 Photo of the installation for the experiments in rats (CLINATEC/LETI/CEA).

Figure 2

 2 Figure 2.2, whereas Figure 2.3 gives an example of the recorded signals (visualization made by Spike2® software). The lines ##1-11 of Figure 2.3 represent the ECoG recordings of the rat. The specificity of the data consists in plenty of chewing artifacts, which are produced by the rat's jaw muscles. These artifacts mainly are located after pressing event and have different duration. The 12th line demonstrates the state of the pedal (‗pressed' or ‗released').

Figure 2 . 2 Figure 2 . 3

 2223 Figure 2.2 Position of the electrodes implanted in the rat's brain. Anatomical distribution of the electrodes on atlas (L. Swanson plates) coronal sections (A) and on sagittal view (C). (B) from back to front, the electrodes are situated over the cerebellum (retro-lambdatic: 8, 15), the occipital (visual area: 6, 13), the postcentral (4, 12), the precentral (3, 10), and the prefrontal (2, 9) cortices. 4 additional electrodes are temporal, left and right: anterior (5,12), and posterior(7,1). (D) Skull distribution. (E) and (F) represent X-Ray images of the implanted electrodes (CLINATEC/LETI/CEA).

  Data was collected from behavioral experiments in non-human primates (Rhesus Macaque) based on a simple reward-oriented task. During the experiment the monkey is sitting in a custom made primate chair minimally restrained, its neck collar hooked to the chair, see Figure 2.4. The monkey has to push a pedal which can be mounted in for different positions (‗left', ‗right', ‗up', and ‗down') on a vertical panel facing the monkey (Figure 2.5). Every correct push event activates a food dispenser and the monkey obtains a reward (fruit juice). No cue or conditioning stimulus were used. A set of ECoG recordings was collected from 32 surface electrodes chronically implanted on the surface of monkey's cortex (see Figure 2.6). The electrode matrix was produced by the Ad-Tech Medical Instrument Corporation (Racine, USA). Diameter of the electrodes is 2 mm.

Figure 2 . 4

 24 Figure 2.4 Photo of the installation for the experiments in monkeys.

Figure 2 . 5

 25 Figure 2.5 Four possible positions of the pedal on the panel (not to scale).

Figure 2 Figure 2

 22 Figure 2.6 (A) Position of the electrodes implanted in the monkey's brain. (B) The photo of the implant. (C) X-Ray images of the implanted electrodes (CLINATEC/LETI/CEA).

Figure 2 . 8

 28 Figure 2.8 General scheme of the BCI experiments in rats. The training stage aims at the BCI system calibration resulting in a decision rule, which is applied immediately to the signal from the rat's brain during the Close Loop Real-Time experiments to activate the reward distributor. Here, ( ) characterizes the state of the pedal at the moment , contains the signal from the brain recording during the experiment, is used for internal representation of the signal in the system, and ( ) is a command signal generated by the system to control the food dispenser.

Figure 2 . 9

 29 Figure 2.9 General scheme of the monkey's BCI experiments. The training stage aims at the BCI system calibration resulted in a decision rule, which is applied immediately to the signal from the monkey's brain during the Close Loop Real-Time experiments to activate the reward distributor. Here, ( ) characterizes the position of the pedal at the moment , contains signal from the brain recording during the experiment, is used

Control

  Block and the Object (Figure 3.1). The Control Block contains the Detector and the Controller BCI sub-blocks; the Object consists in the Effector (the pedal and the reward distributor in the Clinatec BCI experiment) and the Subject (a rat or a monkey). The Effector, characterized by its state (position of the pedal) ( ) * + at the moment . It can affect the Subject according to the control signal ( ) (the reward distributor gives a food pellet if ( ) , otherwise ( ) ). The signals from the Subject ( ) (the recordings of the rat's or the monkey's brain activity) as well as information about the state of the Effector ( ) are received by the Detector. During the first stage of the BCI experiment these data are used to estimate an unknown relation between the signals ( ) and ( ). During the second stage, the Detector controls the Effector by means of the Controller BCI according to the estimated Effector's state ̂( ).

Figure 3 . 1

 31 Figure 3.1 Block-scheme of the BCI Control Block. (A) corresponds to the training stage, (B) represents the second stage of the BCI system (the Close-Loop Real-Time Mode). 𝑠(𝑡) is the signal from the Subject at the moment 𝑡, 𝑦(𝑡) and 𝑦 ̂(𝑡) characterize the real-and estimated state of the Effector, correspondently. 𝑢(𝑡) is the signal to control the Effector. The Subject interacts with the Effector: arrows i (influence) and r (response/feedback). In our experiments, the influence of the Subject on the Effector is a pressing event, whereas the response of the Effector to the Subject is reward distribution. On the second stage of the experiment (B), the influence i is not taken into account (dotted arrow) and the response is determined by the Controller BCI, at the same time the Detector BCI no longer has an access to the state of the Effector 𝑦(𝑡) (dotted arrow).

  Figure 3.1 (B)).

Figure 3 . 2

 32 Figure 3.2 The general scheme the Detector Block. (A) During the first stage the signal from the Subject is used for the calibration of the model. (B) During the second stage the estimated model is used to evaluate the state of the Effector.

  goal of the Model Calibration sub-block is the estimation of the unknown function ( ) (see Equation (3.1)). For this purpose, during the first stage of the experiment (the calibration stage) the system records the signals ( ) and ( ) to form the training set { } . On the base of the function ̂( ) is found in a class of functions (e.g., linear, quadratic, polynomial functions, etc.

  the different types of changes in the signal over time due to the brain plasticity or other external reasons. During the adaptation stage the sub-block has a temporal access to ( ), which describes the real state of the Effector (in Figure 3.2 (B) shown as a dashed arrow). The Model Adaptation sub-block activates occasionally and adjusts the function ̂( ) to minimize a misalignment between the Effector state estimation ̂( ) and its real state ( ).

Figure 4 . 1

 41 Figure 4.1 A third-order tensor 𝐗 𝐼 1 ×𝐼 2 ×𝐼 3 .

Figure 4 . 2

 42 Figure 4.2 The third-order rank-one tensor 𝐗 𝐼 1 ×𝐼 2 ×𝐼 3 : 𝐗 𝐚 𝟏 𝐚 𝟐 𝐚 𝟑 .

Figure 4 . 3

 43 Figure 4.3 Different ways to unfold a third-order tensor 𝐗 𝐼 1 ×𝐼 2 ×𝐼 3 .

Figure 4 . 4

 44 Figure 4.4 Fibers for a third-order tensor 1 × 2 × 3 .

  3 and Figure 2.7) is mapped to the temporal-frequency-spatial space by Continuous Wavelet Transform (CWT) (Figure 4.5). The observation tensor is formed from a set of epochs, such as for each epoch (determined by its final moment ), electrode , frequency and time shift , elements of the tensor were calculated as absolute value of CWT coefficients of the ECoG signal.

Figure 4 . 5

 45 Figure 4.5 The multi-channel ECoG recording is mapped to the temporal-frequencyspatial feature space.

  Figure 4.6 Graphical representation of the PCA decomposition of a matrix 𝐗.

  the matrix of the dependent variables , this increases the efficiency in selection of the most relevant to features. A PLS model tries to find such principal components to variables observations explain the maximum variance of and simultaneously. Algebraically it could be shown as: of the extracted loadings, and , are matrices of residuals. In addition, like in the case of PCA, prior to application of the PLS decomposition, the variables in and must be column meancentered.Graphical representation of the PLS approach is shown in Figure4.7.

Figure 4 . 7

 47 Figure 4.7 Graphical representation of the PLS decomposition of the matrices 𝐗 and 𝐘.

Figure 4 . 8

 48 Figure 4.8 Graphical representation of the PARAFAC decomposition of a tensor 𝐗.

Figure 4 . 9

 49 Figure 4.9 Graphical representation of the NPLS decomposition of the tensors 𝐗 and 𝐘.

  4.3 are used for prediction of the output variable ̂ for the new data.

  regularization parameter, which quantifies the relative importance of the penalization. Depending on the penalization operator ( ), several formulations of the optimization task can be considered. Detailed description of different optimization problems could be found in the work of[START_REF] Martínez-Montes | Penalized PARAFAC analysis of spontaneous EEEG recordings[END_REF]. One particular case is the well-known Ridge regression, where the penalty function is ( that operate on the columns and rows of

Chapter 4 .

 4 MULTI-WAY ANALYSIS FOR BCI SYSTEMS Brain-Computer Interface with cortical electrical activity recording 90 memory. The algorithm was tested with artificial and real datasets and demonstrated efficiency and robustness (see Sections 4.8.2 for artificial and Section 6to the number of modalities) is constructed by successive refinement of their values on every subset { ( ) ( ) }. For this purpose the iterative Tensor Factorization is used (see Algorithm 4.6). It is again based on the ALS algorithm (see Algorithm 4.2) in which previously calculated tensor decomposition is taken as initial estimation. The current approximation of the projection vectors is obtained as a weighted sum of the weights generated by ALS on the current iteration and the vectors from the previous iteration. After the set of projectors * + is identified, the generic NPLS algorithm is applied for determination of the regression coefficients in the space of the latent variables (see Algorithm 4.3: steps 8-12). This procedure is repeated a pre-defined number of times. The set of coefficients vectors { } and projection vectors { } are used for prediction in the same way as in traditional NPLS (Algorithm 4.4). Graphical representation of the algorithm is shown in Figure 4.10.

Figure 4 .

 4 Figure 4.10 Graphical representation of the INPLS algorithm applied to the tensors and (the general case: and are the tensors of the orders and , respectively).

  Figure 4.11 shows the examples of graphical representation of tensors for different values of .

Figure 4 .

 4 Figure 4.11 Example of points , for with different noise levels: presenting the artificial dataset used for INPLS performance study.

Figure 4 .

 4 Figure 4.12 Comparison of INPLS vs. NPLS for the different noise level , the factor number and data fragmentation manner , , , and . Here, corresponds to generic NPLS.

  (4.11), let us subtract from the residual matrix its projection on all latent variables { } : 11) and (4.12) are satisfied, we get ̃ ̃ and ̃ . Similar to RPLS for a new data pair { } regressions on the next data sets will be equivalent:

  Figure 4.13 (see details in Algorithm 4.7).

Figure 4 .

 4 Figure 4.13 The RNPLS scheme. Information used for decomposition of the tensors of observation and is captured by their loading tensors and as well as by the coefficient matrix (the general case: and are the tensors of the orders and , respectively). In addition, on every iteration the algorithm generates the current sets of the coefficient vectors { } as well as of the projection vectors { } and

  an average of the tensor along the first mode. b The -mode vector product of a tensor and vector is denoted as × (see Section 0). c ( ) is the mode-unfolding of the tensor (see Section 0). d ( ) is vectorization of the tensor (see Section 0To study performance and prediction accuracy of RNPLS, we compared it with the traditional NPLS algorithm on simulated data sets with different noise level and for different number of factors. Taking into account specificity of the binary-BCI task (‗event'-‗non-event' discrimination), the tests were performed only for binary output variables. An artificial data set 2 It was added to the templates with a parameter introducing the signal-to-noise ratio: * +. The noise has the same amplitude as the signal ( ) in the case when . The entire data set was split into the training and the test data sets of equal size.

Figure 4 .

 4 Figure 4.14 Example of points { } from the artificial dataset, with different levels of noise.

Figure 4 .

 4 Figure 4.15 Comparison of prediction errors (root mean squared error, RMSE) for the NPLS and the RNPLS algorithms on the test sets for different levels of noise  and different number of used factors F.

.

  To analyze the convergence of the set of matrixes * + we introduce the distance between two matrices and as: where, ‖ ‖ is the Frobenius norm (see Equation (4.4)). To study the convergence of the RNPLS algorithm computational experiments were carried out. A training set consisting of 800 points was split on 40 disjoint subsets Interface with cortical electrical activity recording 101 (each one contains 20 points). The level of noise was taken equal to 1000% of the signal amplitude ( ). The number of factors was chosen in a way to minimize the average RMSE on the test set over 10 realizations of the training set (see Figure 4.15).

Figure 4 .

 4 Figure 4.16 Convergence of the coefficient matrices versus the iterations of the RNPLS algorithm. The distance between two successive coefficient matrices decreases significantly after the first 10 iterations.

Figure 4 .

 4 Figure 4.17 The regression coefficients depending on the iteration number of the RNPLS algorithm.

Figure 4 .

 4 Figure 4.18 The regression coefficients defined by the RNPLS and NPLS algorithms in the 10 realization of the training set ().

Figure 4 .

 4 Figure 4.19 Comparison of the regression coefficients defined by the RNPLS and the NPLS algorithms (average over 10 realizations, ) with the ones generated by PLS for undisturbed classes (noise is absent).

Figure 4 .

 4 Figure 4.20 Red lines: the mean values and the standard deviations (for 10 experiments) of the distances between the regression coefficients generated by the RNPLS and the ordinary PLS ( ) for different noise levels over the RNPLS algorithms recursive iterations. Blue lines: the mean values and the standard deviations (for 10 experiments) of the distances between the regression coefficients generated by the NPLS (applied for whole training sets) and the ordinary PLS ( ) for different levels of noise.

  see Chapter 2), based on a simple reward oriented task, were carrying out by the Clinatec neurological team. Each animal has the opportunity to push a wall-mounted pedal. Every pushing event activates the food dispenser and the animal obtains a reward (food pellet or juice). The animal was trained to press the pedal without any cue or conditioning stimulus. According to Chapter 2, BCI experiment consists of two stages. The first one is the Training Stage. During this stage, the training dataset is collected for the following BCI system calibration. Namely, the ECoG signals and the pedal signal are recorded simultaneously during the behavior experiments described in Chapter 2. During the first stage, the food dispenser is activated by the pedal. Those training sets which contain less than 50 pushes are neglected. The offline calibration procedure based on the collected recordings results in the event predictor computation.

  SD32 system (Micromed, Italy) records and displays the signals. During the experiments the rat was freely moving in the Abet II behavioral cage whereas the Chapter 5. APPLICATION OF MULTIMODAL ANALYSIS TO BCI EXPERIMENTS Brain-Computer Interface with cortical electrical activity recording 108

Figure 5 . 1

 51 Figure 5.1 Meyer wavelet.

Figure 5 . 2

 52 Figure 5.2 Time epoch of the multi-channel ECoG recording mapped by continuous wavelet transform to the temporal-frequency-spatial feature space.

Figure 5 . 3

 53 Figure 5.3 -functions of the examined mother wavelets.

Figure 5 . 4

 54 Figure 5.4 Maximum of correlation between the wavelet coefficients and the signal of the pedal for different types of the mother wavelets.

  approaches could be effectively applied for dimensionality reduction by means of selection of a set of directions with largest variation of the data. At the same time, these Chapter 5. APPLICATION OF MULTIMODAL ANALYSIS TO BCI EXPERIMENTS Brain-Computer Interface with cortical electrical activity recording 114 methods do not take into account distribution of the classes along the considered directions. Thus founded projectors could be not effective enough for the purposes of separation of different classes. For this reason, supervised methods of the feature space

  positions. The recorded data for two positions of the pedal (‗left' and ‗up') were split on the training and the test datasets of the same size. Comparison of the methods consists of two parts. The first one is separation of the background monkey's activity from the pressings of the ‗left' pedal. The second one is discrimination of the ‗left' pressings from the ‗up' ones. The number of factors was ranging from 1 to 20.

Figure 5 .

 5 Figure 5.5 demonstrates the root mean squared error (RMSE) calculated on the test set by both methods for separation of the ‗non-event' and the ‗left-event' epochs.The training and the test sets contain 350 points (50 ‗events' and 300 ‗non-events'. The optimum achieved by the NPLS approach outperforms the one of PARAFAC in terms of the RMSE, at the same time NPLS requires less number of factors to achieve the extremum.

Figure 5 . 5

 55 Figure 5.5 Comparison of RMSE values for classification ‗non-event' vs. ‗left-event' performed by the NPLS and the PARAFAC approaches for different numbers of factors. NPLS outperforms PARAFAC in terms of RMSE, moreover it requires less number of factors to achieve the optimal classification results.

Figure 5 .

 5 Figure 5.6 represents the RMSE calculated on the test for discrimination of the ‗up-events' from the ‗left-events'. The training consists of 100 points (50 ‗left-events'and 50 ‗up-events'), the test set holds 100 points (50 ‗left-events' and 50 ‗up-events').

Figure 5 . 6

 56 Figure 5.6 Comparison of RMSE values for classification ‗up-event' vs. ‗left-event'



  INPLS regression with binarization: the INPLS algorithm generates a linear regression model in the latent variables space to predict ̂ corresponding to the output variable . For binarization of ̂ a scalar threshold was found using the training set by means of maximization of the Overall Performance (OP).

Figure 5 . 7

 57 Figure 5.7 The Overall Performance (OP) for the series of the simulated self-paced BCI experiments using different classifiers in the space of the INPLS latent variables. Black circles represent the average value over all the experiments.

  calibration purposes, the signal of the training data set was mapped to the temporal-frequency-spatial space. Then, eight factors and the corresponding latent variables , , were extracted by INPLS. All modalities of the first and the second factors are shown in Figure 6.1. Besides that, the predictive model for the pedal's position was constructed: ̂ ∑ , where is a parameter of centering. While INPLS extracts task-related factors, the influence of the factors is different. The coefficients , of the normalized model ̂ ∑ , correspond to the weights of related factors in the final decomposition (see Section 4.6.

  separately for ECoG signal filtered in -(10-15 Hz), - bands. The relative weights of these bands in the final decomposition are: , , (Figure6.3). MI analysis applied to identified models revealed the leverages of elements of each modality and for every frequency band (Figure6.4). Inband the most informative are: frequency ~147 Hz, moment 290 ms before event, and occipital electrode #15. In -band the most informative are: frequency 10.5 Hz, moment 390 ms before event, and occipital electrodes #8 and #15. In -band the most informative are: frequency ~33 Hz, occipital electrodes #8 and #15, the moment is rather defused and random (it could be explained by insignificance for BCI system of information contained in the -band).The temporal-frequency analysis for the best electrode (#15, over the cerebellar cortex) was the next step. Six factors were extracted by INPLS. The first three of them are shown in Figure 6.5. The relative weights of all six factors in the final decomposition are (Figure 6.6). Like in the case of all electrodes, the calibration procedure was applied separately for ECoG signal filtered in -(10-15 Hz), -(15-40 Hz), and -(55-300 Hz) bands for the electrode #15. MI analysis applied to identified models revealed the leverages of elements of each modality and for every frequency band (Figure 6.7). Inband the most informative are: frequency ~110 Hz, moment 290 ms before event. Inband the most informative are: frequency 10.5 Hz, moment 410 ms before event. Inband the most informative are: frequency ~33 Hz, like in the case of all electrodes the time moment is rather defused and random. Comparison of the BCI performances (all electrodes versus single electrode) demonstrates that using of all electrodes does not significantly improve the approximation of the output variable on the training data set (Figure 6.8). The difference in RMSE is less than 2% for the number of factors . Projections of the points of observation into the first and the second factors (Figure 6.9) show good separability of the classes ‗event' and ‗non-event' in the single-electrode case even over almost eight months. From a computational point of view, the single-electrode predictive model was chosen. The calibration procedure resulted in the predictor of the pedal pressing events

Figure 6 . 1

 61 Figure 6.1 The first and the second factors (which are the most contributive out of eight): frequency, temporal and spatial projections; the values of elements of the spatial projector are shown in colors according to the color bar; positions of the electrodes are indicated by numbers.

Figure 6 . 2

 62 Figure 6.2 Factor's weights in the final decomposition.

Figure 6 . 3

 63 Figure 6.3 Impact on the predictive model of different modalities components according to MI analysis; the spatial modality is represented by the graph and the corresponding color map.

Figure 6 . 4

 64 Figure 6.4 Impact on the predictive model of the components of different modalities according to the MI analysis; the spatial modality is represented by the graph and the corresponding color map. The predictive models are identified for three frequency bands: -band (10-15 Hz), -band (15-40 Hz), and -band (55-300 Hz).

Figure 6 . 5

 65 Figure 6.5 The best-electrode-calibration: frequency and temporal projections of the first three factors.

Figure 6 . 6

 66 Figure 6.6 The best-electrode-calibration: weights of the factors in the final decomposition.

Figure 6 . 7

 67 Figure 6.7 Impact on the predictive model of the components of different modalities according to the MI analysis. The predictive models are identified for three frequency bands: -band (10-15 Hz), -band (15-40 Hz), and -band (55-300 Hz).

Figure 6 . 8

 68 Figure 6.8 The best-electrode-calibration: comparison of the relative approximation errors, calculated on the training data set for the different numbers of factors: all electrodes versus the best electrode.

Figure 6 . 9

 69 Figure 6.9 The best-electrode-calibration: projections of the ‗event' and ‗non-event' points of observations from the training recording to the first and the second factors.Even in eight months the quality of the separation remains considerable.

Figure 6 .

 6 Figure 6.10 The best-electrode-calibration: the threshold of event detection is identifying according to the OP = (TPR+PPV)/2 criterion: TPR represents the rate of the detected events; (1-PPV) corresponds to the number of FP related to the number of system activations.

  Figure 6.12. Delay time histograms for all these experiments are shown in Figure 6.13.The open-loop experiments were carried out in the third session (the experiments ofFebruary and March, 2010; Table 6.1, Figure 6.11). In these experiments

Figure 6 .

 6 Figure 6.11 The Overall Performance (OP) and the number of false activation per minute calculated over 8 months (offline validation, open-and close-loop real-time experiments).

Figure 6 .

 6 Figure 6.12 10-minutes-length fragment of the experiment #11 (from 20/10/09).

Figure 6 .

 6 Figure 6.13 Time delay histograms of detection for the experiments from #1 to #12, zero moment corresponds to the real event time.

  mean overall performance of the close-loop experiments OP cl = 53.61% with 1.52 false activations per minute. It should be especially emphasized that performances of the open-loop and the close-loop tests are similar at the same period of time (OP ol = 50.34% and 1.46 false activations per minute, see Section 6.1.2). This conclusion has one significant practical consequence. Carrying out of the close-loop experiments in animals is connected with considerable difficulties: contrary to humans, the animals are very confused by the errors of the BCI system that could bring to impossibility of the experiment continuation. At the same time, the open-loop experiments could be easily implemented in animals without any considerable influence on the model's quality evaluation.Another important requirement imposed on the BCI system is its possibility to function in the real-time conditions. Carried out experiments have confirmed satisfactory computation efficiency of the algorithm for the practical applications. The time for processing 0.5 second-buffer does not surpass 0.1 s (Intel Dual Core, 3.16 GHz; RAM 3.25 Gb). In addition, the evaluation experiments allow estimation of the response time of software that is on average 0.31 s including data collecting and data processing time: 0.25 and 0.06 s, respectively.

Figure 6 .

 6 Figure 6.14 The close-loop real-time stage of the experiment. The rat presses the pedal but decision whether to give a reward is made on the basis of the ECoG signal.

Figure 6 .

 6 Figure 6.15 25 second-length example of the close-loop BCI experiment: two real events were detected by the system.

Figure 6 .

 6 Figure 6.16 Frequency, temporal, and spatial projectors of the first and the second factors identified by RNPLS (10).

Figure 6 .

 6 Figure 6.17 Weights of the factors in the final decomposition (evaluated by the RNPLS (10) algorithm).

Figure 6 .

 6 Figure 6.18 Comparison of the test data prediction error (RMSE). RNPLS (10)the training set is split into 10-points disjoint subsets; RNPLS (100)the training set is split into 100-points disjoint subsets, NPLS (1000)generic NPLS using the whole training set.

  all four positions of the lever. The algorithm was chosen taking into account computation restrictions of the real-time model application to find a subset of electrodes mostly participating in the final model. The coefficients , of the normalized predictive model ̂ ∑ , correspond to the weights of the related factors in the final decomposition (see Section 4.6.3). These coefficients are represented in Figure 6.19 for all positions of the pedal. The values of the coefficients are:

(

  The temporalfrequency analysis for the best electrode (#22) was the next step. Seven factors were extracted by INPLS. The first one of them is shown in Figure 6.22 for every position of the pedal. The relative weights of all seven factors in the final decomposition are

Figure 6 .

 6 Figure 6.19 Factors weights in the final decomposition for every position of the pedal.

Figure 6 .

 6 Figure 6.20 The first factor (according to the influence on the final decision): frequency, temporal and spatial projections for every position of the pedal. The values of elements of the spatial projectors are shown in colors according to the color bar; the electrodes positions are indicated by numbers.

Figure 6 .

 6 Figure 6.21 Impact on the predictive model of the components of different modalities according to the MI analysis of the LPNPLS predictive models for each pedal position; the spatial modalities are represented by the graphs and the corresponding color map.

Figure 6 .

 6 Figure 6.22 The best-electrode-calibration: frequency and temporal projections of the first factors for each position of the pedal.

Figure 6 .

 6 Figure 6.23 The best-electrode-calibration: the factors weights in the final decomposition for each position of the pedal.

Figure 6 .

 6 Figure 6.24 Impact on the predictive model of the components of the different modalities according to the MI analysis for each pedal position.

Figure 6 .

 6 Figure 6.25 The Overall Performance (OP) and the number of false activations per minute obtained in the experiments for each pedal position.

Figure 6 .

 6 Figure 6.26 1.5 min length fragment of the experiment #2 (from 14/03/2011) and the time-delay histogram of detections.

Figure 6 .

 6 Figure 6.27 Photos of the real-time stage of the experiment.

Figure 6 .

 6 Figure 6.28 Comparison of prediction errors (root mean squared error, RMSE (∑ ̂ ) ) for the NPLS and the LPNPLS algorithms on the test set for different number of factors and different pedal positions.

Figure 6 .

 6 Figure 6.29 Impact on the predictive model of the components of different modalities according to the MI analysis of the NPLS predictive models for each pedal position; the spatial modalities are represented by the graphs and the corresponding color map.

  data set in the rat's experiments was composed from 10minutes recording (about 1.3% of all available data), the observations of more than 750 minutes (12.5 hours) were used for test. The same predictive model was applied in all tests. The only adjusted parameter was the detection threshold. It was slightly varying daily, that could be explained by instability of the acquisition chain as well as by variability of the brain activity. This parameter remains stable during the day of experiment. It can be adjusted from the first several minutes (3-5 min) of the experimental session of the real-time BCI. Three groups of experiments were carried out over almost eight months. In the first two groups of experiments (simulation mode, Table6.1) about 80% of the correct event detections were achieved (TPR = 78±13% in average), while the level of the false activations was acceptably low (FPR = 0.44±0.24% in average). In almost eight months after the calibration model was obtained (the last group of the experiments in Table6.1), TPR = 48±13%, FPR = 1.25±0.29% in the open-loop and TPR = 54±10%, FPR = 1.30±0.23% in the close-loop experiments. The deterioration of the system's performance could be explained by significant degradation of electrodes. At the same time, it should be mentioned that both the open-and the close-loop tests gave similar results. Moreover, at least part of the false positives of all the groups of experiments resulted from artifacts produced along

Table 1 .1

 1 Confusion matrix contains information about the number of the correct detections, event missings, false activations, and correct rejections.

	'Event'	'Non-event'

  Accuracy of Classification:

	(Schlögl et al., 2007). The True Positive Rate (	), and the False Positives Rate
	(	(A) ) are widely used in the self-paced BCIs. They show the relative amount of (B)
	silent interval successfully detected events and of false activations. Let us note that TP interval TP interval is influenced
	detection by the decision rate and the ratio of the classes. This complicates the BCI systems detection time time
	moment comparison. That is why additional criteria characterizing false activations of the self-moment
	paced BCIs were proposed: the number of the False Activation per Minute
		 Error Rate:
		 True Positive Rate (Hit Rate, Sensitivity):
		 False Positive Rate:
		 False Detection Rate:
		 False Activation per Minute
		 Positive Predictive Value (Precision):
		 Specificity:
		 Harmonic Mean of Precision (	):
		 Hit/False-difference:
		The accuracy of classification (	) is a commonly used evaluation criterion in
	the BCI research. It shows the percentage of correctly classified samples. However,
	being efficient for the cue-paced BCIs,	, as well as the error rate	, fail to
	characterize the performance of a self-paced BCI due to highly unbalanced classes

  . A widely used example of the parametric model

	is the autoregressive (AR) model: ( )	∑	( ) (	) , here is the order
	of the model, and	( ( )	( )) are AR-coefficients. The vector of
	coefficients is used further as the feature vector. Parametric models were applied
	for BCI, for instance, by Scherer et al., 2007; Kubánek et al., 2009.
	In this study Continuous Wavelet Transform (CWT, see Section 5.2) is chosen,
	since this method allows efficiently revealing of underlying neurological phenomena
	correlated with BCI tasks. Unlike STFT, WT has no fixed time-window that leads to

absence of limitation of the temporal resolution in the high frequencies. At the same time, contrary to discrete WT, continuous WT allows analyzing of the frequency range with better resolution.

Table 6 .1

 6 Performances of both open-loop and close-loop BCI experiments.

	No	Date	No. of events	Time (s)	FP FN TP TN	TPR (%)	PPV (%)	FPR (%)	ERR (%)	OP (%)	FP min -1 Conditions
	1 a 09.07.2009	73	592	2	3	70 1109 95,89 97,22	0,18	0,42 96,56	0,20	s
	2	09.07.2009	74	895	9	13 61 1707 82,43 87,14	0,52	1,23 84,79	0,60	s
	3	09.07.2009	56	895	11 24 32 1723 57,14 74,42	0,63	1,96 65,78	0,74	s
	4	09.07.2009	68	537	6	11 57 1000 83,82 90,48	0,60	1,58 87,15	0,67	s
	5	16.07.2009	37	310	0	7	30 583 81,08 100,00 0,00	1,13 90,54	0,00	s
	6	16.07.2009	33	427	3	6	27 818 81,82 90,00	0,37	1,05 85,91	0,42	s
	7	13.10.2009	49	599	3	5	44 1146 89,80 93,62	0,26	0,67 91,71	0,30	s
	8	15.10.2009	46	964	15 18 28 1867 60,87 65,12	0,80	1,71 62,99	0,93	s
	9	15.10.2009	28	397	6	10 18 760 64,29 75,00	0,78	2,02 69,64	0,91	s
	10 19.10.2009	99	3587 32 40 59 7043 59,60 64,84	0,45	1,00 62,22	0,54	s
	11 20.10.2009	169	3600 18 17 152 7013 89,94 89,41	0,26	0,49 89,68	0,30	s
	12 23.10.2009	114	1808 14 18 96 3488 84,21 87,27	0,40	0,88 85,74	0,46	s
	13 19.02.2010	33	2400 73 17 16 4694 48,48 17,98	1,53	1,88 33,23	1,83	c
	14 19.02.2010	37	940	29 26 11 1814 29,73 27,50	1,57	2,93 28,61	1,85	o
	15 22.02.2010	30	385	10 14 16 730 53,33 61,54	1,35	3,12 57,44	1,56	c
	16 22.02.2010	33	715	17 21 12 1380 36,36 41,38	1,22	2,66 38,87	1,43	o
	17 23.02.2010	45	800	24 16 29 1531 64,44 54,72	1,54	2,50 59,58	1,80	c
	a This recording was used for calibration.
	s -simulation mode, o -open-loop mode, c -close-loop mode.

Brain-Computer Interface with cortical electrical activity recording 133

Table 6 .1

 6 Performances of both open-loop and close-loop BCI experiments.

	No	Date	No. of events	Time (s)	FP FN TP TN	TPR (%)	PPV (%)	FPR (%)	ERR (%)	OP (%)	FP min -1 Conditions
	18 23.02.2010	77	2050 43 23 54 3980 70,13 55,67	1,07	1,61 62,90	1,26	c
	19 23.02.2010	31	535	14 14 17 1025 54,84 54,84	1,35	2,62 54,84	1,57	o
	20 24.02.2010	112	1980 40 59 53 3808 47,32 56,99	1,04	2,50 52,16	1,21	c
	21 24.02.2010	30	490	8	12 18 942 60,00 69,23	0,84	2,04 64,62	0,98	o
	22 25.02.2010	152	2500 66 55 97 4782 63,82 59,51	1,36	2,42 61,66	1,58	c
	23 25.02.2010	31	505	9	12 19 970 61,29 67,86	0,92	2,08 64,57	1,07	o
	24 26.02.2010	28	480	9	15 13 923 46,43 59,09	0,97	2,50 52,76	1,13	c
	25 26.02.2010	115	2170 58 59 56 4167 48,70 49,12	1,37	2,70 48,91	1,60	c
	26 26.02.2010	39	615	17 14 25 1174 64,10 59,52	1,43	2,52 61,81	1,66	o
	27 01.03.2010	180	3000 81 62 118 5739 65,56 59,30	1,39	2,38 62,43	1,62	c
	28 01.03.2010	29	540	16 18 11 1035 37,93 40,74	1,52	3,15 39,34	1,78	o
	29 02.03.2010	163	2350 54 74 89 4483 54,60 62,24	1,19	2,72 58,42	1,38	c
	30 02.03.2010	30	500	15 13 17 955 56,67 53,13	1,55	2,80 54,90	1,80	o
	31 03.03.2010	247	3600 78 159 88 6875 35,63 53,01	1,12	3,29 44,32	1,30	c
	32 03.03.2010	31	700	17 15 16 1352 51,61 48,48	1,24	2,29 50,05	1,46	o
	33 04.03.2010	140	2400 79 68 72 4581 51,43 47,68	1,70	3,06 49,56	1,98	c
	34 04.03.2010	59	800	13 40 19 1528 32,20 59,38	0,84	3,31 45,79	0,98	o
	s -simulation mode, o -open-loop mode, c -close-loop mode.

Table 6 .2

 6 Performance of the BCI experiments.

		No	Date	No. of events	Time (s)	FP FN TP TN	TPR (%)	PPV (%)	FPR (%)	ERR (%)	OP (%)	FP min -1
		1 10.03.2011	85	468,5	29 27 58 823	68,24	66,67	3,40	5,98	67,45	3,71
		2 a 14.03.2011	109	845,5	32 31 78 1550 71,56	70,91	2,02	3,73	71,23	2,27
	Left	3 18.03.2011 4 24.03.2011	83 109	347 506	23 24 59 588 32 39 70 871	71,08 64,22	71,95 68,63	3,76 3,54	6,77 7,02	71,52 66,42	3,98 3,79
		5 14.04.2011	83	515	30 30 53 917	63,86	63,86	3,17	5,83	63,86	3,50
		6 19.04.2011	19	276	5	9	10 528	52,63	66,67	0,94	2,54	59,65	1,09
		7 10.03.2011	82	498,5	36 27 55 879	67,07	60,44	3,93	6,32	63,76	4,33
	Right	8 11.03.2011 9 30.03.2011 10 04.04.2011 11 a 14.04.2011	94 62 85 88	1107 283 304,5 227,5	65 55 39 2055 41,49 12 10 52 492 83,87 32 33 52 492 61,18 9 10 78 358 88,64	37,50 81,25 61,90 89,66	3,07 2,38 6,11 2,45	5,42 3,89 10,67 4,18	39,49 82,56 61,54 89,15	3,52 2,54 6,31 2,37
		12 18.04.2011	78	353,5	26 20 58 603	74,36	69,05	4,13	6,51	71,70	4,41
		13 a 15.03.2011	104	535	27 31 73 939	70,19	73,00	2,80	5,42	71,60	3,03
		14 17.03.2011	69	672	29 35 34 1246 49,28	53,97	2,27	4,76	51,62	2,59
	Up	15 24.03.2011 16 25.03.2011	32 63	336 380,5	14 23 26 37 675 8 24 626	75,00 58,73	63,16 61,67	2,19 3,30	3,27 6,44	69,08 60,20	2,50 3,63
		17 30.03.2011	90	339,5	21 26 64 568	71,11	75,29	3,57	6,92	73,20	3,71
		18 14.04.2011	70	361,5	11 23 47 642	67,14	81,03	1,68	4,70	74,09	1,83
		19 10.03.2011	38	521,5	27 29	9	978	23,68	25,00	2,69	5,37	24,34	3,11
	Down	20 28.03.2011 21 a 30.03.2011 22 14.04.2011	85 53 47	824,5 322,5 341,5	45 42 43 1519 50,59 20 20 33 572 62,26 17 18 29 619 61,70	48,86 62,26 63,04	2,88 3,38 2,67	5,28 6,20 5,12	49,73 62,26 62,37	3,27 3,72 2,99
		23 18.04.2011	52	516	26 29 23 954	44,23	46,94	2,65	5,33	45,58	3,02
		24 19.04.2011	76	390,5	27 34 42 678	55,26	60,87	3,83	7,81	58,07	4,15
	a These recordings are used for calibration.

2.5. ConclusionBrain-Computer Interface with cortical electrical activity recording

6.1. Experiments in ratsBrain-Computer Interface with cortical electrical activity recording

(Figure 6.11, Table 6.1). To clarify the organization of the experiments, some

(cerebellum region in the case of rat, and motor primary cortex in the case of monkey),

the identified activity responsible for the pedal-pressing events is rather similar.

The training data set in the monkey's experiments were composed from four recordings (30-minutes length in total, which is about 17% of all available data), the observations of more than 180 minutes (3 hours) duration were used for testing. The only adjusted parameter was the detection threshold. Like in the case of the rat, it was slightly varying daily. During the tests, up to 70% of the correct event detections were achieved in average ( , ,

, and ), while the average level of the false activations was acceptably low ( , , , and ). Some part of the false positives can be also eliminated by means of the wireless data transferring.

Additional experiments should be carried out to make more reliable conclusions about other possible reasons of the quality deterioration.

List of Publications

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