Francesco Miceli

Charissa Millevert

Maria Virginia Soldovieri

Ilaria Mosca

Paolo Ambrosino

Lidia Carotenuto

Dewi Schrader

Hyun Kyung Lee

James Riviello

William Hong

Sarah Risen

Lisa Emrick

Hitha Amin

Doroth Ee Ville

Patrick Edery

Julitta De Bellescize

Vincent Michaud

Julien Van-Gils

Cyril Goizet

Marjolein H Willemsen

Tjitske Kleefstra

Rikke S Møller

Allan Bayat

Orrin Devinsky

Tristan Sands

G Christoph Korenke

Gerhard Kluger

Heather C Mefford

Eva Brilstra

Gaetan Lesca

Mathieu Milh

Edward C Cooper

Maurizio Taglialatela

Sarah Weckhuysen
email: sarah.weckhuysen@uantwerpen.vib.be

KCNQ2 R144 variants cause neurodevelopmental disability with language impairment and autistic features without neonatal seizures through a gain-of-function mechanism

Keywords: KCNQ2, Gain-of-function, Amitriptyline, Autism, Developmental and epileptic encephalopathy

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Evidence before this study Pathogenic variants in KCNQ2, encoding for Kv7.2 voltage-gated potassium channel subunits, represent one of the most common causes of childhood-onset genetic epilepsies, with an incidence in young population of 1 per 17,000 live births. Most disease-causing KCNQ2 variants cause loss-of-function (LoF) when studied in vitro, and have been associated with a broad spectrum of neonatal-onset epilepsy phenotypes. In addition to LoF, few gain-of-function (GoF) variants have been described; these affect residues within the voltage-sensing domain (VSD) and are associated with a distinct phenotype, as evidenced by the absence of neonatal seizures. GoF and LoF variants may need different therapeutic approaches, but, unlike for KCNQ2 LoF, targeted treatment for KCNQ2/3 GoF patients have not emerged to date.

Added value of this study

Twenty patients (14 novel and 6 published), all carrying pathogenic variants at position R144 in the VSD, were identified. Detailed clinical information was collected on 9 R144W and 6 R144Q individuals for analysis of clinical features. All patients had developmental delay with prominent language impairment. R144Q patients were generally more severely affected than R144W patients; in fact, patients carrying the R144W variant showed a mild to severe degree of intellectual disability (ID), whereas all patients with the R144Q variant had severe ID. Infantile to childhood onset epilepsy was seen in 40%, while 67% of sleep-EEGs showed sleep-activated epileptiform activity. None of the patients had neonatal-onset epilepsy. Ten patients (67%) presented autistic features. In vitro, Kv7.2 channels incorporating R144 pathogenic variants displayed a leftward shift in activation gating, suggesting GoF effects; the extent of this shift was larger for R144G (»25 mV) and R144Q (»18 mV) than for R144W (»4 mV) channels, and Introduction KCNQ2 and KCNQ3 genes encoding Kv7.2 and Kv7.3 voltage-gated potassium (K + ) channel subunits are widely expressed in the nervous system. [START_REF] Dirkx | The role of Kv7.2 in neurodevelopment: insights and gaps in our understanding[END_REF] In neurons, Kv7.2 subunits form homotetrameric or heterotetrameric channels with Kv7.3 subunits to generate the Mcurrent (IK M ), a non-inactivating K + current with slow activation and deactivation kinetics that regulates the resting membrane potential and suppresses repetitive neuronal firing. [START_REF] Dirkx | The role of Kv7.2 in neurodevelopment: insights and gaps in our understanding[END_REF] Each Kv7 subunit is formed by six transmembrane segments (S 1 -S 6 ); the voltage-sensing domain (VSD) is encompassed by S 1 -S 4 , whereas S 5 , S 6 and the S 5 -S 6 intervening linker form the pore region. [START_REF] Soldovieri | Driving with no brakes: molecular pathophysiology of Kv7 potassium channels[END_REF] In addition to membrane voltage, Kv7 channel opening requires phosphatidylinositol 4,5-bisphosphate (PIP 2 ), a membrane lipid which binds to distinct channel regions at each VSD gating state. [START_REF] Zaydman | PIP2 regulation of KCNQ channels: biophysical and molecular mechanisms for lipid modulation of voltagedependent gating[END_REF] KCNQ2/3 pathogenic variants exerting loss-of-function (LoF) effects are associated with a spectrum of neonatal-onset phenotypes ranging from self-limited familial neonatal epilepsy (SLFNE, MIM:121200) to KCNQ2 developmental and epileptic encephalopathy (KCNQ2-DEE, MIM:613720) with therapy-resistant seizures and intellectual disability (ID). [START_REF] Weckhuysen | KCNQ2 encephalopathy: emerging phenotype of a neonatal epileptic encephalopathy[END_REF] Most KCNQ2-DEE pathogenic variants exert dominant-negative LoF effects when studied in vitro. [START_REF] Miceli | Genotype-phenotype correlations in neonatal epilepsies caused by mutations in the voltage sensor of K(v)7.2 potassium channel subunits[END_REF] Seizures in these children tend to respond well to sodium channel blockers, now recommended as first-line treatment, [START_REF] Pisano | Early and effective treatment of KCNQ2 encephalopathy[END_REF] and a randomized, double-blind, placebo-controlled trial with the potassium channel opener retigabine has recently been initiated (ClinicalTrials.gov Identifier:NCT04639310). [START_REF] Millichap | KCNQ2 encephalopathy: Features, mutational hot spots, and ezogabine treatment of 11 patients[END_REF] Few heterozygous KCNQ2 missense variants enhance channel function (gain-of-function, GoF) by causing a hyperpolarizing shift in voltage-dependent activation. [START_REF] Mulkey | Neonatal nonepileptic myoclonus is a prominent clinical feature of KCNQ2 gain-of-function variants R201C and R201[END_REF][START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF][START_REF] Millichap | Infantile spasms and encephalopathy without preceding neonatal seizures caused by KCNQ2 R198Q, a gain-of-function variant[END_REF][START_REF] Devaux | A Kv7.2 mutation associated with early onset epileptic encephalopathy with suppression-burst enhances Kv7/M channel activity[END_REF] Phenotypes associated with GoF variants differ from the classical KCNQ2-DEE phenotype. They range from profound ID without neonatal seizures but with a characteristic nonepileptic myoclonus and poor prognosis for the R201C variant with strong GoF effect, [START_REF] Mulkey | Neonatal nonepileptic myoclonus is a prominent clinical feature of KCNQ2 gain-of-function variants R201C and R201[END_REF] to epileptic spasms with hypsarrhythmia and severe developmental delay (DD) for the R198Q variant with milder GoF effect. [START_REF] Millichap | Infantile spasms and encephalopathy without preceding neonatal seizures caused by KCNQ2 R198Q, a gain-of-function variant[END_REF] Interestingly, KCNQ3 variants at R230, the homologous residue of KCNQ2 R201, also have strong GoF in vitro effects; patients carrying these variants display global developmental delay within the first 2 years of life, sleep-activated near-continuous multifocal spikes, and autistic features, but no neonatal seizures. [START_REF] Sands | Autism and developmental disability caused by KCNQ3 gain-of-function variants[END_REF] Prognosis is often poor since most patients with KCNQ2/3 GoF variants do not respond well to anti-seizure medications. [START_REF] Mulkey | Neonatal nonepileptic myoclonus is a prominent clinical feature of KCNQ2 gain-of-function variants R201C and R201[END_REF] GoF and LoF variants may need different therapeutic approaches, but, unlike for KCNQ2 LoF, targeted treatment for KCNQ2/3 GoF patients have not emerged to date.

We have previously shown that the KCNQ2 R144Q variant, identified in a single patient, results in a GoF effect. [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF] Here, we investigated the phenotypic spectrum associated with KCNQ2 variants at the R144 position by describing detailed clinical features of 15 patients, and summarizing published clinical data on 5 additional cases. To highlight potential genotype-phenotype correlations, we compared the functional properties of Kv7.2 R144W and R144G mutant subunits with those of the previously described R144Q. [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF] At last, we demonstrate that amitriptyline inhibits channels carrying both wildtype and mutant Kv7.2 R144Q subunits, suggesting a potential targeted treatment for patients with KCNQ2 R144 variants. correlated with severity of neurodevelopmental problems, highlighting novel genotype-phenotype correlations and supporting a role for in vitro functional assays among predictors of neurodevelopmental trajectories in these children. Amitriptyline (1À10 mM) dose-dependently blocked channels containing both Kv7.2 and Kv7.2 R144Q subunits.

Implications of all the available evidence

This study confirms the existence of a distinct phenotype associated with KCNQ2 GoF variants characterized by developmental delay (DD) with prominent language impairment and autistic features in the absence of neonatal seizures. In some (but not all), DD is accompanied by infantile-to childhood-onset epilepsy and EEG sleepactivated epileptiform activity. These phenotypic characteristics are clearly distinct from those of patients carrying KCNQ2 loss-of-function variants, where early neonatal-onset epilepsy is the unifying clinical feature across the severity spectrum. The present results provide novel genotype-phenotype correlations in KCNQ2related disorders, and underline the importance of including the "epilepsy gene" KCNQ2 in diagnostic intellectual disability (ID) gene panels. The absence of neonatal seizures has emerged as a robust differentiator between the (rarer) KCNQ2 GoF and (more common) LoF variants. This distinction is clinically very important, as the entities present in very different clinical settings and require development of different therapeutic approaches. Raising awareness about these distinct phenotypes has become especially important now that an international randomized clinical trial with a pediatric formulation of the potassium channel opener retigabine (ClinicalTrials.gov Identifier: NCT04639310; https:// clinicaltrials.gov/ct2/show/NCT04639310) for patients with KCNQ2-encephalopathy has recently been initiated. In this trial, only patients with clear LoF variants can be included, and the "presence of a known GoF variant in the KCNQ2 gene, or clinical characteristics consistent with previously reported pathogenic GoF variants in the KCNQ2 gene" are explicitly listed among exclusion criteria for enrolment in the trial. Finally, the Kv7.2/7.3 channel blocking effect of amitriptyline holds promise for its use as a potential targeted treatment for patients with KCNQ2 GoF variants and deserves further study.

Methods

Patient series collection

We collected detailed clinical information of patients carrying a KCNQ2 R144W (NM_172107.3:c.430C>T), R144G (NM_172107.3:c.430C>G), or R144Q (NM_172107.3:c.431G>A) variant. Patients were recruited through the Rational Intervention of KCNQ2 Epileptic Encephalopathy (RIKEE) database (http:// www.RIKEE.org), a curated database aggregating (un) published patient data provided by physicians or families after parental informed consent. The site is hosted at Baylor College of Medicine under an institutional review boardÀ approved research protocol. Additional patients were directly referred for inclusion in the study by collaborating paediatric neurologists and medical geneticists. A standardized phenotyping sheet, including data fields on technical exams as EEG and MRI of the brain, was sent to the referring physician of all unpublished patients (n=14) and of Patient 11, who was previously reported with minimal clinical details as part of an Epi4K epileptic encephalopathy cohort. [START_REF] Allen | De novo mutations in epileptic encephalopathies[END_REF] Degree of ID was defined based on the level of adaptive functioning across multiple domains (cognitive, social, practical), based on local multidisciplinary clinical assessment, as proposed by the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5). [START_REF]American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders (DSM-5)[END_REF]

Mutagenesis and heterologous expression of KCNQ2 and KCNQ3 cDNAs

Variants were engineered in KCNQ2 human cDNA cloned into pcDNA3.1 by QuikChange site-directed mutagenesis (Agilent Technologies, Milan, Italy), as described. [START_REF] Miceli | Genotype-phenotype correlations in neonatal epilepsies caused by mutations in the voltage sensor of K(v)7.2 potassium channel subunits[END_REF] Channel subunits were expressed in Chinese Hamster Ovary (CHO) cells by transient transfection using Lipofectamine 2000 (Invitrogen, Milan, Italy). Total cDNA in the transfection mixture was kept constant at 4 mg.

Whole-cell electrophysiology

Currents from CHO cells were recorded at room temperature (20À22 °C) 1-2 days after transfection, using the whole-cell configuration of the patch-clamp technique with glass micropipettes of 3À5 MV resistance. The extracellular solution contained (in millimolar): 138 NaCl, 2 CaCl 2 , 5.4 KCl, 1 MgCl 2 , 10 glucose and 10 HEPES, pH 7.4 with NaOH. The pipette (intracellular) solution contained (in millimolar): 140 KCl, 2 MgCl 2 , 10 EGTA, 10 HEPES, 5 Mg-ATP, pH 7.3À7.4 with KOH. The pCLAMP software (version 10.0.2) was used for data acquisition and analysis. Current densities, conductance-voltage curves, and current activation and deactivation kinetics were obtained and analysed as previously described. [START_REF] Soldovieri | Atypical gating of M-type potassium channels conferred by mutations in uncharged residues in the S4 region of KCNQ2 causing benign familial neonatal convulsions[END_REF] A stock solution (10 mM) of amitriptyline (Sigma-Aldrich, Germany) was prepared in dimethyl sulfoxide (DMSO); extracellular solution was used for subsequent drug dilutions. In the experiments with amitriptyline, currents were activated by 3s voltage ramps from À80 to +20 mV applied every 15 s. Currents at +20 mV were measured before and after drug application, and drug's effects expressed as % of blockade.

Structural modelling

Three-dimensional models of Kv7.2 subunits in activated and resting gating states were generated by multistate modelling, as previously described. [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF]

Statistics

Values are expressed as the mean §SEM of cells recorded in at least three independent experimental sessions. Statistically significant differences between experimental groups were evaluated with the Student's t-test (p<0.05).

Role of the funding source

The funders had no role in study design, data collection, data analyses, interpretation, or writing of report.

Results

Cohort description and genotypes

Fourteen previously unpublished KCNQ2-DEE patients were included in our study: 9 carrying the p.R144W variant, and 5 the p.R144Q variant (eTable 1 and 2). We also collected clinical data on a sixth p.R144Q patient (patient 11) who was previously published with limited clinical information. [START_REF] Allen | De novo mutations in epileptic encephalopathies[END_REF] We defined this cohort of 15 wellphenotyped patients as our "study cohort". Eight of the 15 were male. Age at last follow-up of R144W patients and R144Q patients ranged from 3.5 to 27 years (median 7.5 years), and from 3 years 9 months to 26 years (median 9 years), respectively. In 14 patients, the pathogenic variant occurred de novo. For patient 13, the variant was shown to be absent in mother, but paternal DNA was not available.

We also summarized published phenotypical data of 5 patients: 3 carrying the p.R144W, [START_REF] Mary | Pathogenic variants in KCNQ2 cause intellectual deficiency without epilepsy: Broadening the phenotypic spectrum of a potassium channelopathy[END_REF]17 1 the p.R144Q [START_REF] Mary | Pathogenic variants in KCNQ2 cause intellectual deficiency without epilepsy: Broadening the phenotypic spectrum of a potassium channelopathy[END_REF] and 1 the p.R144G 17 variant (eTable 3). Due to the limited clinical data available, these previously published patients were not included in the analysis of specific clinical features. R144W, R144Q, and R144G are all considered pathogenic according to ACMG/AMP criteria (PS2, PS3, PM1, PM2, PM5, PP2, PP3). [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: a joint consensus recommendation of the American college of medical genetics and genomics and the association for molecular pathology[END_REF] Neurodevelopmental and behavioural phenotype of patients in the study cohort Detailed clinical information can be found in eTable 1, a summary is provided in Table 1. Figure 1 gives a graphical overview of the main clinical features shared by patients with the R144W and R144Q variant.

All patients had DD noted during the first year of life. Patients carrying the R144W variant showed a mild to severe degree of ID; in contrast, all patients with the R144Q variant had severe ID.

Except for patient 2, all patients had marked language impairment. At last follow-up, six patients (40%, aged 3 years 9 months to 27 years) were non-verbal (2/9 R144W, 22%; and 4/6 R144Q, 67%). Five patients (33%, aged 4.5 to 26 years) spoke a few words (3/9 R144W, 33%; and 2/6 R144Q, 33%), and 4 patients (27%, aged 4 to 21 years) spoke (short) sentences (4/9 R144W, 44%).

All patients had delayed motor milestones. 8/9 R144W (89%) patients walked independently at last follow-up, whereas only 2/6 R144Q (33%) patients did so. Age of independent walking in the R144W patients ranged from 16 months to 5.5 years, and all but one walked independently before the end of 2 years (median 1 year 9 months). Age of independent walking in the 2 R144Q patients was 3 years and 3.5 years, respectively (median 3 years 3 months). Age at last follow-up of the patients that could not walk ranged from 3.5 to 14 years.

Developmental regression was present in 3 patients and correlated with time of seizures onset in 2 (patient 11 and 15).

Ten patients (67%; 7/9 R144W and 3/6 R144Q) were reported by their child neurologist to have autistic features (described as difficulties in social interaction or in understanding emotions, appearing "in their own world", and/or engaging in repetitive, stereotypic movements). Episodes of aggressive behaviour were reported for 3 individuals (20%, 1 R144W and 2 R144Q). Features of attention deficit hyperactivity disorder (ADHD; e.g., hyperactivity, poor attention, and impulsivity) were reported in 7 (47%; 4/9 R144W and 3/6 R144Q).

Neurological exam of patients in the study cohort

Five R144W patients (56%) and 5 R144Q patients (83%) showed signs of hypotonia; 1 R144W and 1 R144Q patient showed axial hypotonia combined with hypertonia in extremities. In 1 R144Q patient only peripheral hypertonia was described. Ataxia was present in 2/9 R144W (22%) and 2/6 R144Q (33%) patients. Head circumference was available for five patients (33%). All had a head circumference smaller than average, two of whom (1 R144W (À2.5 SD) and 1 R144Q (À3.6 SD)) had true microcephaly.

Hyperopia plus strabismus were reported in 3 patients (1 R144W and 2 R144Q). Two R144Q patients showed strabismus or hyperopia alone. Hyperlaxity or marfanoid habitus (moderate hyperlaxity, arachnodactyly, high arche in the palate, scoliosis, tall stature, and dolichostenomelia) was described in 4/6 R144Q cases, but not in R144W cases.

Epilepsy phenotype of patients in the study cohort

In total, epilepsy was diagnosed in 6 of 15 patients (Table 1, Figure 1, eTable 2). At last follow-up, epilepsy had been diagnosed in 2/9 (22%) and was suspected in 1/9 patients with the R144W variant. In patients with the R144Q variant, epilepsy was diagnosed in 4/6 (67%) cases, and suspected in 1 other. Two R144W cases and 1 R144Q patient experienced one or two simple febrile seizures only, and were not included in the group of patients with a diagnosis of epilepsy.

Among those diagnosed with epilepsy, seizure onset ages ranged from 4 months to 5 years (median 13 months). Seizure types varied, and included epileptic spasms (1/9 R144W patients, 2/6 R144Q patients), myoclonic jerks (3/6 R144Q patients) and atonic seizures (1 R144W). In 1 R144W patient, and 4 R144Q patients, episodes with impaired awareness were reported. None of the latter events have been captured on video-EEG, and the distinction between an epileptic or behavioural non-epileptic event was often unclear. Epileptic spasms were controlled with vigabatrin and/or corticosteroids. Where seizure types other than epileptic spasms occurred, valproic acid was the most frequently used agent, and led to seizure freedom in all individuals in whom epileptic seizures were diagnosed.

EEG was performed in 14/15 patients and showed (multi)focal epileptiform abnormalities in 13/14 (92%) patients, most prominent in posterior (7/13) and centrotemporal (4/13) leads in the majority of patients. Hypsarrhythmia was seen in all 3 patients with epileptic spasms. Sleep-EEG was performed in 12 patients, and in 8 patients (67%) activation or augmentation of epileptic activity during sleep was present, without fulfilling criteria of continuous spike and wave during slow wave sleep (CSWS). [START_REF] Singhal | Continuous spike-wave during slow wave sleep and related conditions[END_REF] Brain MRI was normal in 10/14 (71%) patients. Of the scans with abnormalities, three (1 R144W and 2 R144Qpatients) showed delayed myelination. Abbreviations: ADHD = attention deficit hyperactivity disorder; ASD = autism spectrum disorder; EEG = electroencephalogram; FU = follow-up; ID = intellectual disability; M = month; Y= year.

Comparison with previously published cases

Although 5 individuals carrying KCNQ2 R144W, 3 R144Q, 1 and R144G 1 variants have been identified in two earlier studies, [START_REF] Mary | Pathogenic variants in KCNQ2 cause intellectual deficiency without epilepsy: Broadening the phenotypic spectrum of a potassium channelopathy[END_REF]17 phenotypic details were available for only 4 of the patients (eTable 3). All 5 patients had DD, and all reported (4/4) had language impairment. Focal seizures were reported in 1 patient. EEG was reported for 3 patients and showed multifocal abnormalities in all, with activation of epileptic activity during sleep described in two. Two patients were reported to have behavioural problems (1 with hyperactivity, and 1 with autistic traits).

Functional properties of channels containing Kv7.2 R144W/Q/G subunits

The R144 residue, highly conserved among Kv channels, [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF] is located at the bottom of the S 2 segment within the VSD (Figure 2A). CHO cells expressing homomeric Kv7.2 channels generated voltage-dependent, K + -selective currents characterized by a slow time course of activation/deactivation and an activation threshold around À50 mV (Figure 2B,C). When compared to wild-type (wt) Kv7.2 channels, homomeric Kv7.2 R144W, R144Q or R144G channels displayed similar maximal current densities at +20 mV but a significant hyperpolarizing shift in voltage-dependent activation gating by about 4, 18, and 25 mV, respectively (Figure 2B,C; Table 2).

wtKv7.2 channels showed activation kinetics which were adequately fitted by a biexponential function with a fast and a slow time constant (t f and t s , respectively). [START_REF] Soldovieri | Atypical gating of M-type potassium channels conferred by mutations in uncharged residues in the S4 region of KCNQ2 causing benign familial neonatal convulsions[END_REF] The relative amplitude of the fast component (A f ) is dominant over that of the slow component (A s ); indeed, the A f /A f +A s was close to unity at each potential investigated (Figure 2D). By contrast, while no change in activation t f and t s was observed (data not shown), the A f / A f +A s ratio for the currents carried by K7.2 R144W/Q/ G channels was significantly decreased at all potentials investigated (Figure 2D). In addition, when compared to wtKv7.2 channels, currents carried by Kv7.2 W/Q/G channels all showed slower deactivation kinetics, with Kv7.2 R144G channels having the strongest effect; indeed, the time constant of deactivation was 21.1 § 2.1 ms, 37.6 §4.3 ms, 30.1 §3.0 ms and 52.4 §3.7 ms for wtKv7.2, Kv7.2 R144Q, Kv7.2 R144W and Kv7.2 R144G channels, respectively (p<0.05 for wtKv7.2 channels vs Kv7.2 R144W/Q/G channels). Despite such changes in gating, selectivity for K + over Na + ions as indicated by the current reversal potential in our recording conditions, as well as sensitivity to blockade by tetraethylammonium (TEA) was unaffected in mutant channels (Table 2), confirming that these pathogenic variants did not affect pore function. Expression of Kv7.2 precedes that of Kv7.3 at early developmental stages 1 whereas in adult superior sympathetic ganglion neurons, I KM is mainly formed by heteromeric assembly of Kv7.2 and Kv7.3 subunits. [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF][START_REF] Wang | KCNQ2 and KCNQ3 potassium channel subunits: molecular correlates of the M-channel[END_REF] Therefore, the effect of all the three variant subunits in heteromeric configuration with wtKv7.2 and/or wtKv7.2/Kv7.3 subunits were also evaluated. Currents from wtKv7.2+Kv7.2R144W-, wtKv7.2+Kv7.2R144Q-, and wtKv7.2+Kv7.2R144Gtransfected cells (1:1 cDNA ratio) displayed a significant leftward shift in activation voltage-dependence when compared to Kv7.2 channels, although the magnitude of this effect was smaller than that observed in homomeric configuration. When compared to wtKv7.2/Kv7.3, wtKv7.2+Kv7.2R144G+Kv7.3, but not wtKv7.2++Kv7.2R144Q+Kv7.3 and wtKv7.2 +Kv7.2R144W+Kv7.3 heteromeric channels, displayed statistically-significant differences in activation gating (Table 2).

Structural basis for the GoF effect of the Kv7.2 R144W/ Q/G variants

To gain insight into the structural consequences of R144W, Q and G substitutions, we investigated a previously-described homology model of a Kv7.2 subunit in both resting and activated states, [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF][START_REF] Jensen | Mechanism of voltage gating in potassium channels[END_REF] and an atomic structure of Kv7.2 bound to retigabine/ezogabine. [START_REF] Li | Molecular basis for ligand activation of the human KCNQ2 channel[END_REF] The latter structure is likely akin to the VSD-activated, conducting state, although bound PIP 2 could not be visualized. In the homology model resting state, the R144 side chain in S 2 forms an electrostatic interaction with the side chain of D172 in S 3 (Figure 3A). In the activated states of both the homology model and the cryoEM structure, this direct interaction is absent, suggesting its prominent role in resting-state VSD stabilization. When the R144 side chain is replaced with W, Q or G, the ionized hydrogen bond between the 144 position and D172 is lost (Figure 3 B,C,D), potentially destabilizing the VSD resting state and favoring transition to the activated state. Notably, substitution of R144 with W, but not Q or G, introduces two novel hydrophobic interactions with the C169 and the I173 residues in the S 3 segment (Figure 3B), partially compensating for the variant-induced loss of the electrostatic interaction with D172.

Possible involvement of PIP 2 in the gating changes triggered by the Kv7.2 R144Q variant

The anionic phospholipid PIP 2 acts as a critical Kv7 gating modulator by promoting the electromechanical coupling between VSD movement and pore opening. [START_REF] Zaydman | PIP2 regulation of KCNQ channels: biophysical and molecular mechanisms for lipid modulation of voltagedependent gating[END_REF] To investigate whether the GoF effects observed in channels carrying R144 neutralizations were due to a facilitation of PIP 2 -dependent transitions to the open state, resting PIP 2 levels were either increased by coexpression of the PIP 2 -synthesizing enzyme PIP5K, or decreased by co-expressing a voltage-sensitive phosphatase (DrVSP) that activates at very depolarized voltages. [START_REF] Hossain | Enzyme domain affects the movement of the voltage sensor in ascidian and zebrafish voltage-sensing phosphatases[END_REF] The R144Q variant was selected for these experiments as this substitution is associated with a lesser degree of a-helical distortion when compared to the R144G, and with stronger functional changes when compared to the R144W.

PIP5K co-expression enhanced wtKv7.2 peak currents, also causing a negative shift in their half-activation potential (DV1/ 2 = -14.5 mV) (Figure 2A,B; Table 1). 23À25 When PIP5K was co-expressed with Kv7.2 R144Q channels, a similar 3-fold increase in maximal currents was observed, but the leftward shift in activation gating was larger than that observed in wtKv7.2 channels (DV1/ 2 =-36.2 mV) (Figure 4A,B; Table 2).

In CHO cells expressing DrVSP, depolarization to +100 mV time-dependently reduced wtKv7.2 currents to 29.5 §2.6% of their starting value after 2 s (n=23; Figure 3C); by contrast, Kv7.2 R144Q currents were reduced to only 49.0 §2.0% (n=10; p<0.05 vs Kv7.2). Notably, this functional difference in DrVSP sensitivity between wtKv7.2 and Kv7.2 R144Q channels was abrogated when PIP 2 levels were increased upon simultaneous co-expression of both DrVSP and PIP5K enzymes (Figure 3C).

Amitriptyline blockade of Kv7.2-and Kv7.2 R144Qcontaining channels

The tricyclic antidepressant drug amitriptyline has been described as a Kv7.2/3 blocker (26); given the GoF effects triggered by the R144 variants, we investigated whether Kv7.2 channels containing R144Q mutant subunits could also be blocked by amitriptyline. Amitriptyline (1 and 10 µM) dose-dependently and reversibly inhibited both heteromeric wtKv7.2+Kv7.3 and wtKv7.2 +Kv7.2R144Q+Kv7.3 currents by the same extent (Figure 5A,B).

Discussion

Pathogenic KCNQ2 variants represent one of the most common causes of childhood-onset genetic epilepsy, with an estimated incidence within the first three years of life of 1 per 17,000. [START_REF] Symonds | Incidence and phenotypes of childhood-onset genetic epilepsies: a prospective population-based national cohort[END_REF] Most individual diagnosed with pathogenic KCNQ2 variants experience neonatal-onset epilepsy; the responsible variants include deletions, truncating variants and missense variants exhibiting LoF in vitro.

In addition to LoF, a few GoF missense variants have been described; to date these have been located within the VSD and associated with a different phenotypic spectrum, lacking neonatal seizures (Figure 6). 8- 11,28 Children carrying KCNQ2 GoF variants at the R201 position displayed a very severe neonatal phenotype characterized by encephalopathy, burst suppression EEG, bouts of non-epileptic myoclonus, hypoventilation, and evidence of hypomyelination on MRI. Evolution to infantile epileptic spasms syndrome is frequent, profound DD persists, and early mortality is frequent. [START_REF] Mulkey | Neonatal nonepileptic myoclonus is a prominent clinical feature of KCNQ2 gain-of-function variants R201C and R201[END_REF] Children with the KCNQ2 R198Q variant showed epileptic spasms without preceding neonatal seizures or noted neonatal encephalopathy, suggesting that KCNQ2 is a rare cause of infantile epileptic spasms syndrome. [START_REF] Millichap | Infantile spasms and encephalopathy without preceding neonatal seizures caused by KCNQ2 R198Q, a gain-of-function variant[END_REF][START_REF] Liu | Gene and phenotype expansion of unexplained early infantile epileptic encephalopathy[END_REF] In the present work, we describe the phenotypes of 20 patients (14 novel and 6 published) carrying KCNQ2 R144 variants (12 R144W, 7 R144Q, and 1 R144G). New clinical information was gathered on 9 R144W and 6 R144Q individuals.

Although little information was available on the one (previously published) R144G patient, the more numerous patients with R144W or R144Q variants showed several phenotypic commonalities. First, none had neonatal seizures. Instead, the first signs of medical concern were neurodevelopmental problems. All were diagnosed with DD. Language delay was particularly prominent. Neuropsychiatric problems were common: autistic features were reported by treating physicians in more than 65%, and aggressive behaviour and/or hyperactivity in more than half of the patients. Seizures ultimately occurred in 40%, always with an age of onset after the neonatal period. Seizure types included epileptic spasms (with hypsarrhythmia on EEG), and myoclonic, atonic and focal seizures. Both myoclonic seizures and epileptic spasms have previously been associated with KCNQ2 GoF variants. [START_REF] Mulkey | Neonatal nonepileptic myoclonus is a prominent clinical feature of KCNQ2 gain-of-function variants R201C and R201[END_REF][START_REF] Millichap | Infantile spasms and encephalopathy without preceding neonatal seizures caused by KCNQ2 R198Q, a gain-of-function variant[END_REF][START_REF] Devaux | A Kv7.2 mutation associated with early onset epileptic encephalopathy with suppression-burst enhances Kv7/M channel activity[END_REF][START_REF] Samanta | Myoclonic epilepsy evolved into west syndrome: a patient with a novel de novo KCNQ2 mutation[END_REF][START_REF] Weckhuysen | Extending the KCNQ2 encephalopathy spectrum: clinical and neuroimaging findings in 17 patients[END_REF][START_REF] Hortig€ Uela | Clinical and genetic features of 13 Spanish patients with KCNQ2 mutations[END_REF][START_REF] Olson | Genetics and genotypephenotype correlations in early onset epileptic encephalopathy with burst suppression[END_REF][START_REF] Kojima | A patient with early myoclonic encephalopathy (EME) with a de novo KCNQ2 mutation[END_REF] In a third of the children, episodes of reduced responsiveness were reported, but none were captured by video-EEG. Thus, their possible epileptic nature is uncertain. Interestingly, unresponsive staring episodes of unclear origin were also seen in patients with KCNQ3 R230 variants and DD. [START_REF] Sands | Autism and developmental disability caused by KCNQ3 gain-of-function variants[END_REF] More extensive video-EEG may reveal the exact aetiology of these episodes. Interictal EEGs typically showed (multi) focal epileptiform abnormalities, even in 7 of the 8 individuals who did not have clinical seizures but had EEGs (in 1 patient, EEG was not performed).

Sleep-activated epileptic activity was seen in 8 of the 12 individuals that had sleep-EEGs. Similar findings have been shown for KCNQ3 GoF variants. [START_REF] Sands | Autism and developmental disability caused by KCNQ3 gain-of-function variants[END_REF] It is currently unclear whether treatment of abundant epileptiform activity during sleep with antiseizure medication can improve developmental outcome. Sands et al. could not consistently demonstrate clinical improvement in patients carrying KCNQ3 GoF variants when sleep-activated spikes were reduced by high-dose oral diazepam or corticosteroids. Nevertheless, the study population was too small to draw conclusions, and prospective studies with standardized sleep-EEG analysis and neuropsychological assessment are warranted to address this question. [START_REF] Sands | Autism and developmental disability caused by KCNQ3 gain-of-function variants[END_REF] Overall, patients carrying KCNQ2 R144Q variants had a more severe neurodevelopmental outcome compared to R144W patients: first, all R144Q patients had severe ID, whereas the degree of ID of R144W patients ranged from mild to severe; second, 5/6 (83%) R144Q patients were non-verbal and/or non-ambulatory, compared to 2/9 (22%) R144W patients; and third, seizures were more frequently seen in R144Q cases compared to R144W cases (4/6 (67%) versus 2/9 (22%)). No difference in prevalence of behavioral problems was observed.

Patch clamp electrophysiology revealed GoF effects for all three Kv7.2 R144 variants, arising from negative shifts in their activation gating. This effect was largest (»27 mV) in R144G, intermediate (»18 mV) in R144Q and smallest (»4 mV) in R144W channels. Together with the slowing of deactivation observed at negative potentials, these functional changes suggest a destabilization of Kv7.2 VSD resting state. Structural modelling revealed that the R144 residue is involved in an Remarkably, phenotypic severity appears correlated with the magnitude of the V1/2 shift, but even small shifts (R144W, R214W) can be pathogenic. These correlations are not interpreted as a necessarily complete pathogenic mechanism; and for LoF variants, additional disease pathomechanisms have been described (reviewed in 40).

electrostatic interaction with the D172 residue in S 3 when the VSD occupies the resting configuration; by contrast, no such interaction occurs when the VSD is in the activated state, consistent with the hypothesis that substitutions neutralizing R144 selectively destabilize the resting VSD configuration, thereby favoring the occupancy of the activated state. Moreover, replacement of R144 with W, but not Q or G residues introduces two novel hydrophobic interactions that could partially compensate for the loss of the electrostatic interaction with D172. This mechanism provides a plausible structural explanation for the gating changes observed under voltage-clamp, and for the smaller shift in V 1/2 of Kv7.2 R144W channels compared to Kv7.2 R144Q or R144G channels.

Kv7 currents are characterized by their absolute functional requirement for PIP 2 . [START_REF] Zaydman | PIP2 regulation of KCNQ channels: biophysical and molecular mechanisms for lipid modulation of voltagedependent gating[END_REF] Multiple and distinct PIP 2 binding sites have been described in each closed and open configuration [START_REF] Zaydman | PIP2 regulation of KCNQ channels: biophysical and molecular mechanisms for lipid modulation of voltagedependent gating[END_REF][START_REF] Salzer | Phosphorylation regulates the sensitivity of voltage-gated Kv7.2 channels towards phosphatidylinositol-4,5-bisphosphate[END_REF] ; in particular, the linker region between S 2 and S 3 has been proposed as a critical site for VSD coupling to pore opening in Kv7.1, 35 Kv7.2 [START_REF] Zaydman | Kv7.1 ion channels require a lipid to couple voltage sensing to pore opening[END_REF][START_REF] Chen | Migration of PIP2 lipids on voltagegated potassium channel surface influences channel deactivation[END_REF] and Kv7.3 37 channels; in particular, PIP 2 migration from a peripheral S 2 -S 3 linker site to a more central S 4 -S 5 /pre-S 6 binding region appears required for pore opening. [START_REF] Zhang | Dynamic PIP2 interactions with voltage sensor elements contribute to KCNQ2 channel gating[END_REF] The R144 residue is located in S 2 , immediately above this PIP 2 binding site, suggesting its possible role in PIP 2 -dependent modulation of Kv7.2 channels gating. When compared to Kv7.2 channels, Kv7.2 R144Q channels display a greater voltage-shift at increased PIP 2 levels and decreased sensitivity to PIP 2 depletion, suggesting that an increased PIP 2 apparent affinity might contribute to the GoF effects triggered by this variant. Given that PIP 2 is a negatively charged molecule, and that the Kv7.2 variants herein investigated neutralize a positively-charged arginine residue, the present hypothesis seems consistent with an indirect (allosteric) effect of the variant on PIP 2 affinity and, potentially, PIP 2 -dependent VSD-pore coupling.

While our paper was in preparation, Mary et al. described 2 R144W and 1 R144Q patients among a series of four KCNQ2 patients with DD without epilepsy and considered the potential role of unknown second genetic or environmental factors, or female gender as possible explanations for the absence of seizures. [START_REF] Mary | Pathogenic variants in KCNQ2 cause intellectual deficiency without epilepsy: Broadening the phenotypic spectrum of a potassium channelopathy[END_REF] Here, through identification of a larger study cohort and inclusion of functional analysis, we provide evidence supportive of a single gene with phenotypic heterogeneity model. The R144W/Q alleles are recurrently associated with a neurodevelopmental disorder without neonatal seizures. This phenotype clearly differs from the phenotype seen in individuals with KCNQ2 variants with LoF effects, and it now strongly predicts a GoF effect of KCNQ2 (and KCNQ3) missense variants (Figure 6). [START_REF] Mulkey | Neonatal nonepileptic myoclonus is a prominent clinical feature of KCNQ2 gain-of-function variants R201C and R201[END_REF][START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF][START_REF] Millichap | Infantile spasms and encephalopathy without preceding neonatal seizures caused by KCNQ2 R198Q, a gain-of-function variant[END_REF][START_REF] Devaux | A Kv7.2 mutation associated with early onset epileptic encephalopathy with suppression-burst enhances Kv7/M channel activity[END_REF][START_REF] Sands | Autism and developmental disability caused by KCNQ3 gain-of-function variants[END_REF][START_REF] Samanta | Myoclonic epilepsy evolved into west syndrome: a patient with a novel de novo KCNQ2 mutation[END_REF] Recognizing the clinical characteristics of pathogenic KCNQ2 GoF variants, with the absence of neonatal seizures as robust differentiator between KCNQ2 GoF and LoF variants, is of clinical importance since individuals with KCNQ2 GoF variants are likely to need different therapeutic approaches compared to individuals with KCNQ2 LoF variants. Recently, an international randomized clinical trial with a pediatric formulation of the neuronal KCNQ channel opener retigabine (Clinical-Trials.gov Identifier: NCT04639310; https://clinical trials.gov/ct2/show/NCT04639310) has been initiated. In this trial, eligibility is restricted to patients with pathogenic KCNQ2 LoF variants. Individuals with known GoF KCNQ2 variant or individuals showing clinical characteristics consistent with previously reported pathogenic GoF KCNQ2 variants, are excluded. This highlights the continued need for description of rare patients such as performed by Mary et al. and in the current study.

The present data also reveal novel genotype-phenotype correlations in the GoF spectrum of KCNQ2-DEE, as the severity of the phenotype seems to correlate with the electrophysiological effect of the variant in vitro, with larger functional changes of R144Q-carrying channels paralleling the more severe neurodevelopmental outcome (Figure 6). Patients carrying the KCNQ2 R144 GoF variants are also less severely affected than those carrying the strong KCNQ2 GoF variants at R201 in S 4 . [START_REF] Miceli | Early-onset epileptic encephalopathy caused by gain-of-function mutations in the voltage sensor of Kv7.2 and Kv7.3 potassium channel subunits[END_REF] The KCNQ2 R198Q variant appears more similar to R144, both in the patch clamp results and phenotypes, namely the emergence of DD after a neonatal period without developmental concerns or seizures. [START_REF] Millichap | Infantile spasms and encephalopathy without preceding neonatal seizures caused by KCNQ2 R198Q, a gain-of-function variant[END_REF][START_REF] Liu | Gene and phenotype expansion of unexplained early infantile epileptic encephalopathy[END_REF] Interestingly, the phenotype of patients with KCNQ2 R144W is similar to that of patients with KCNQ3 R230C, even though KCNQ3 R230C has much larger effects on voltage-dependence in vitro. [START_REF] Sands | Autism and developmental disability caused by KCNQ3 gain-of-function variants[END_REF] This discrepancy may potentially be due to the higher expression levels of KCNQ2 compared to KCNQ3 during early neurodevelopment. [START_REF] Kanaumi | Developmental changes in KCNQ2 and KCNQ3 expression in human brain: possible contribution to the age-dependent etiology of benign familial neonatal convulsions[END_REF] Genotype-phenotype correlations have been also described in the LoF spectrum of KCNQ2-related disorders (reviewed in). [START_REF] Nappi | Epileptic channelopathies caused by neuronal Kv7 (KCNQ) channel dysfunction[END_REF] As illustrated in Figure 6, VSD missense variants responsible for self-limited familial neonatal epilepsy (SLFNE) caused milder effects on activation gating (e.g. R214W), [START_REF] Del Giudice | Benign familial neonatal convulsions (BFNC) resulting from mutation of the KCNQ2 voltage sensor[END_REF][START_REF] Castaldo | Benign familial neonatal convulsions caused by altered gating of KCNQ2/KCNQ3 potassium channels[END_REF] whereas more severe functional defects are prompted by DEE-causing variants in the same region (e. g. R213Q). [START_REF] Weckhuysen | KCNQ2 encephalopathy: emerging phenotype of a neonatal epileptic encephalopathy[END_REF][START_REF] Miceli | Genotype-phenotype correlations in neonatal epilepsies caused by mutations in the voltage sensor of K(v)7.2 potassium channel subunits[END_REF][START_REF] Millichap | KCNQ2 encephalopathy: Features, mutational hot spots, and ezogabine treatment of 11 patients[END_REF][START_REF] Orhan | Dominant-negative effects of KCNQ2 mutations are associated with epileptic encephalopathy[END_REF][START_REF] Trump | Improving diagnosis and broadening the phenotypes in early-onset seizure and severe developmental delay disorders through gene panel analysis[END_REF][START_REF] Na | Targeted gene panel sequencing in early infantile onset developmental and epileptic encephalopathy[END_REF][START_REF] Kim | Clinical characteristics of KCNQ2 encephalopathy[END_REF][START_REF] Xu | Early initial video-electro-encephalography combined with variant location predict prognosis of KCNQ2related disorder[END_REF] Finally, although patients with clinically proven seizures in our cohort responded well to treatment with valproic acid, vigabatrin, or corticosteroids, they all show significant neurodevelopmental problems. This highlights the need for better treatment, ideally reversing the mutational effect of pathogenic GoF variants and improving neurodevelopment. Amitriptyline is a tricyclic antidepressant with analgesic properties due to the serotonin and norepinephrine reuptake inhibition, which also targets some voltage-gated ion channels and receptors. [START_REF] Punke | Amitriptyline is a potent blocker of human Kv1.1 and Kv7.2/7.3 channels[END_REF] It has been widely used for depression, migraine, and neuropathic pain, and has been trialed in children (6 years of age) for nocturnal enuresis. [START_REF] Caldwell | Tricyclic and related drugs for nocturnal enuresis in children[END_REF] Based on pharmacokinetic analysis in experimental animals, it has been suggested that clinically-effective steady-state plasma concentrations of amitriptyline in humans (150À250 ng/ml) correspond to brain drug concentrations of 5À7 mM. [START_REF] Glotzbach | Brain concentrations of tricyclic antidepressants: single-dose kinetics and relationship to plasma concentrations in chronically dosed rats[END_REF][START_REF] Atkin | A comprehensive approach to identifying repurposed drugs to treat SCN8A epilepsy[END_REF] These low mM concentrations of amitriptyline similarly block Kv7.2/7.3 and Kv7.2/Kv7.2R144Q/Kv7.3 channels in vitro, [START_REF] Punke | Amitriptyline is a potent blocker of human Kv1.1 and Kv7.2/7.3 channels[END_REF] highlighting the potential ability of this drug to counteract in vivo the GoF effect caused by the R144Q variant.

Limitations

This study has limitations that are intrinsic to our approach: data were collected retrospectively, and as a result not all desired clinical information was available for every patient (e.g., not all had sleep-EEG recordings capturing sleep), and some reports were impossible to further validate (e.g., the etiology of reduced responsiveness episodes). Sampling bias may be arise due to recruitment via a voluntary family registry and referral to specialized centres. In addition, the small size of the two subgroups within our cohort (R144W and R144Q), limits the strength of conclusions drawn about phenotypic differences between these subgroups. Despite these limitations, this study strengthens available evidence that KCNQ2 GoF variants cause DD in the absence of neonatal seizures and that disease severity is correlated with the extent of the V 1/2 shift in vitro. Thus, the KCNQ2 GoF spectrum is distinct from that associated with KCNQ2 LoF variants. Owing to rarity, the known set of KCNQ2 missense variants causing GoF and DD likely remains incomplete. To accelerate variant discovery, KCNQ2 should be included in diagnostic ID/DD/ASD multigene panels. Finally, agents withKv7.2/ 7.3 channel blocking effects, such as amitriptyline, represent candidate as targeted treatments for patients with KCNQ2 GoF variants and deserve further investigation.

Figure 1 .

 1 Figure 1. Bar graphs showing the main clinical features shared by 9 patients with Kv7.2 R144W variants (age at last follow up between 3.5 and 27 years) and 6 patients with R144Q variants (age at last follow up between 3 years 9 months and 26 years). Distribution of (a) grade of intellectual disability, (b) language outcome, (c) highest level of motor functioning reached at 3 years of age, (d) behavioral phenotype, and (e) seizure phenotype. Persons presenting only paroxysmal episodes (such as staring) of unclear origin are not included in the epilepsy group.

Figure 2 .

 2 Figure 2. Topological location of the R144 residue and functional characterization of the Kv7.2 R144 W/Q/G variants. a. Homology model of a wtKv7.2 channel, built upon the Kv1.2/2.1 chimera crystal structure (PDB code: 2R9R), showing the localization of the R144 position (side chain colored in red using a space-filling model). For clarity only two of the four subunits are shown. b. Macroscopic currents from wtKv7.2 (Kv7.2), Kv7.2 R144Q, Kv7.2 R144W, and Kv7.2 R144G homomeric channels, in response to the indicated voltage protocol. Current scale, 200 pA; time scale, 0.2 s. c. Conductance/voltage curves for the indicated channels. Continuous lines are Boltzmann fits to the experimental data. Each data point is the Mean §S.E.M. of 13À21 cells recorded in at least 3 separate experimental sessions. d. Relative amplitudes of the fast and slow current activation components (expressed as A f /A f + As ), for wtKv7.2, and Kv7.2 R144Q/W/G homomeric channels as a function of membrane voltage.

Figure 3 .

 3 Figure 3. An enlarged view of the resting state configuration of the region where R144 is located in wtKv7.2 (a), Kv7.2 R144W (b), Kv7.2 R144Q (c) and Kv7.2 R144G (d) channel.

Figure 4 .

 4 Figure 4. Effect of changes in PIP 2 availability on channels incorporating Kv7.2 R144Q mutant subunits. a. Representative current traces from cells expressing wtKv7.2 and Kv7.2 R144Q homomeric channels in the absence or in the presence of PIP5K in response to the voltage protocols shown in each respective panel. Current scale, 100 pA; time scale, 0.2 s. b. Conductance/voltage curves for the indicated channels with or without PIP5K co-expression. Continuous lines are Boltzmann fits to the experimental data. c. Current inhibition upon time-dependent activation of Dr-VSP. Asterisks (*) indicate values significantly different from each respective control (p<0.05).

Figure 5 .

 5 Figure 5. Effect of amitriptyline on heteromeric channels incorporating Kv7.2 R144Q mutant subunits. a. Representative current traces recorded from cells transfected with the indicated cDNA combinations in response to the voltage ramp protocol reported, both before and after amitriptyline (AMI) exposure (1 and 10 µM). Current density scale, 20 pA/pF; time scale 0.2 s. b. Percent of current inhibition by amitriptyline. Asterisks (*) indicate values significantly different from each respective control recorded before drug exposure (p<0.05).

Figure 6 .

 6 Figure 6. Correlation between the shift in voltage-dependent activation midpoint (V1/2) and phenotypes of pathogenic KCNQ2 variants in the VSD. Listed are 6 variants (at 5 codons) found recurrently, the shift observed under three informative and physiologically relevant subunit expression ratios (indicated schematically at top), and the associated phenotypes. The voltage shifts are taken from the publications listed. Remarkably, phenotypic severity appears correlated with the magnitude of the V1/2 shift, but even small shifts (R144W, R214W) can be pathogenic. These correlations are not interpreted as a necessarily complete pathogenic mechanism; and for LoF variants, additional disease pathomechanisms have been described (reviewed in 40).

Contributors

  Conceptualization: E.C., S.W., M.T.; Data curation (access and verification of the data): S.W., M.T., F.M., C. M.; Formal analysis: E.C., S.W., M.T., F.M., C.M.; Wri-tingÀoriginal draft: E.C., S.W., M.T., F.M., C.M.; Wri-tingÀreview & editing: All authors. All authors read and approved the final version of the manuscript.

Table 1 :

 1 Overview of clinical features of patients with KCNQ2 R144 gain-of-function variants.

	Patient	Age at last FU	KCNQ2	Degree of ID (DSM-V)	Behavioural	Current level of functioning	Onset-age	Episodes with	EEG
			variant		phenotype		of epilepsy	reduced	sleep-activated
								responsiveness	epileptiform
								of unclear origin	activity
		4.5y	p.R144W	Mild	Features of ASD and ADHD	Speaks words, walks independently	/		Yes
		6y	p.R144W	Mild to moderate	Features of ADHD	Speaks sentences, walks independently	/		No
		4.5y	p.R144W	Mild to moderate	Features of ASD and ADHD	Speaks words, walks independently	/		No sleep-EEG
		11y	p.R144W	Mild to moderate	Features of ASD	Speaks short sentences, walks independently	/		Yes
		21y	p.R144W	Moderate	Features of ASD and	Speaks short sentences, walks independently	/	Yes (3-6y)	No sleep-EEG
					aggressive behavior
		8y	p.R144W	Moderate	Features of ASD	Speaks words, walks independently	/		Yes
		7.5y	p.R144W	Moderate	Features of ASD	Speaks short sentences, walks independently	/		No
		27y	p.R144W	Severe	Features of ASD and ADHD	Nonverbal, walks independently	5y		Yes
		3.5y	p.R144W	Severe	Normal	Nonverbal, cannot sit or walk independently	4m		Yes
		4y	p.R144Q	Severe	Aggressive behavior	Nonverbal, cannot sit or walk independently	/	Yes (12m)	No
		14y	p.R144Q	Severe	Features of ASD and ADHD	Nonverbal, walks independently	6m	Yes	Yes
		14y	p.R144Q	Severe	Normal	Nonverbal, sits independently	18m	Yes	No sleep-EEG
		26y	p.R144Q	Severe	Features of ASD and ADHD	Speaks words, walks independently	8m		No
		4y	p.R144Q	Severe	Features of ASD and ADHD,	Speaks words, sits independently	/		Yes
					aggressive behavior
		3y9m	p.R144Q	Severe	Normal	Nonverbal, sits independently	2y	Yes	Yes

Table 2 :

 2 Biophysical and pharmacological properties of channels containing wild-type and R144 mutant Kv7.2 subunits.

					TEA blockade (% of inhibition)
		n	V mid (mV)	K	Maximal current	Current reversal	0.3 mM	3 mM	30 mM
					density (pA/pF)	potential (mV)
	Kv7.2	16	À25.0 § 1.2	14.1 § 0.7	36.5 §6.6	À78 §1	53.5 § 11.4	87.9 § 5.1	93.0 § 7.5
	Kv7.2+ PIP5K	14	À39.5 §0.6*	10.1 §0.5	107.1 §7.3*	À	À	À	À
	Kv7.2 R144Q	19	À43.2 § 1.6*	17.2 § 0.9*	38.2 §5.5	À79 §1	57.5 § 0.4	85.5 § 4.1	93.0 § 4.6
	Kv7.2 R144Q + PIP5K	13	À81.3 §0.7**	10.5 §0.6	100.4 §17.2**	À	À	À	À
	Kv7.2 R144W	15	À29.1 § 1.5*	17.9 § 1.1*	25.6 §3.3	À79 §1	60.0 § 2.8	91.7 § 0.9	97.7 § 0.6
	Kv7.2 R144G	22	À49.9 § 1.3*	12.9 § 0.4	36.0 §5.4	À79 §1	59.7 § 3.6	93.2 § 2.1	97.2 § 1.3
	Kv7.2 + Kv7.2 R144Q	12	À35.7 § 1.9*	15.1 § 0.4	39.4 §4.5	À	À	À	À
	Kv7.2 + Kv7.2 R144W	6	À27.9 §0.9*	15.8 §1.0	36.4 §10.1	À	À	À	À
	Kv7.2 + Kv7.2 R144G	6	À36.4 § 3.4*	14.8 § 0.9	25.7 § 6.6	À	À	À	À
	Kv7.2 + Kv7.3	26	À29.9 § 1.1	12.7 § 0.5	119.7 § 12.2	À	9.2 § 2.2	42.1 § 7.4	73.1 § 6.8
	Kv7.2 + Kv7.2R144Q + Kv7.3	12	À33.5 § 1.8	11.5 § 0.4	94.8 § 18.9	À	19.0 §10.0	49 §10.0	88 §2.0
	Kv7.2 + Kv7.2R144W + Kv7.3	14	À30.5 § 1.9	11.3 § 0.5	111.9 § 16.7	À	7.8 § 6.3	42.0 § 8.2	78.7 § 5.9
	Kv7.2 + Kv7.2R144G + Kv7.3	22	À37.0 § 1.4 $	12.5 § 0.7	87.4 § 10.3	À	8.0 § 5.3	44 .0 § 5.0	78.5 § 5.2
	* p<0.05 versus Kv7.2,
	** p<0.05 versus Kv7.2 R144Q

$ p<0.05 versus Kv7.2+Kv7.3.

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Acknowledgments

The authors thank the families and the children for their participation in this study.

SW received funding from FWO (1861419N and G041821N); Queen Elisabeth Medical Foundation for Neurosciences (GSKE); KCNQ2-Cure; Jack Pribaz Foundation; KCNQ2e.v.; European Joint Programme on Rare Disease JTC 2020 (TreatKCNQ). MT received funding from the Italian Ministry for University and Research (MIUR) (PRIN 2017ALCR7C); the Italian Ministry of Health (Project RF-2019-12370491); the European Commission H2020 (UNI-COM À 875299); European Joint Programme on Rare Disease JTC 2020 (TreatKCNQ). ECC received funding from NINDS NS49119 and NS108874; The Jack Pribaz Foundation; The Chalk Family Foundation and The Miles Family Fund. FM received funding from the Italian Ministry for University and Research (MIUR) (PRIN 2017YH3SXK). CM received funding from the University of Antwerp (BOF DOCPRO4À project ID 42329).

Funding Supported by FWO, GSKE, KCNQ2-Cure, Jack Pribaz Foundation, European Joint Programme on Rare Disease 2020, the Italian Ministry for University and Research, the Italian Ministry of Health, the European Commission, the University of Antwerp, NINDS, and Chalk Family Foundation.

Data availability

All data that support the findings of the study are available from the corresponding author, with the exception of primary patient sequencing data, as they are derived from patient samples with unique variants that are impossible to guarantee anonymity for.

Data sharing statement

De-identified participant data and study protocol can be requested by contacting one of the corresponding authors by email (sarah.weckhuysen@uantwerpen.vib.be).

Ethics

Written informed consent for participation/publication was obtained from all parents or legal guardians of recruited patients. This study was approved by the Human Research Ethics Committees of the University Hospital of Antwerp, Belgium (Ref. 19/20/257).

Declaration of interests

SW received consultancy and speaker fees from UCB, Biocodex, Xenon, Zogenix, Lundbeck, Knopp Biosciences, Encoded MT received consultancy fees from Xenon.

ECC received consultancy fees from Xenon, Knopp, these activities have been reviewed and approved by Baylor College of Medicine according to its policy on disclosure of outside interests.

RSM received consultancy and speaker fees from EISAI and UCB. The remaining authors have no conflicts of interest.

Supplementary materials

Supplementary material associated with this article can be found in the online version at doi:10.1016/j. ebiom.2022.104130.


Lélio Chetot
email: lelio.chetot@insa-lyon.fr

Malcolm Egan
email: malcom.egan@inria.fr

Active User Detection and Channel Estimation for Grant-Free Random Access with Gaussian Correlated Activity

Keywords: active user detection, channel estimation, bayesian compressed sensing, approximate message passing, correlated activity, internet of things Algorithm Correlation matrix K ∈ [-1, 1] N

Industrial IoT (IIoT) is one of the major verticals targeted by the next generations of wireless networks. In order to provide industrial plants with features relying on wireless communications, the grant-free RA (GFRA) protocol appears to be a promising means for supporting massive ultra-reliable connectivity; at the same time, it is a critical bottleneck that requires an access point (AP) to be able to jointly perform active user detection and channel estimation (AUDaCE) to fulfill its main mission of allowing industrial wireless devices to access the core network. This mission is even harder when the GFRA requests are correlated because of event-driven activity triggers. This paper proposes a new tractable gaussian correlated activity (GCA) model for this scenario. The corresponding AUDaCE problem is then studied in the Bayesian compressed sensing (BCS) framework. An hybrid instance of the generalized AMP (GAMP) algorithm is derived and its capability to perform AUDaCE is numerically assessed by extensive Monte-Carlo simulations. The numerical results show gains of 2.5dB in channel estimation gain for twice less detection errors w.r.t. state-of-theart algorithms.

I. INTRODUCTION

One of the major target verticals of sixth generation (6G) is IIoT consisting of a large, densely populated collection of wireless sensors interfaced via wireless links with cloud-based computing systems. IIoT systems are characterized by their large scale, strict reliability and latency constraints. An open challenge is to develop for the facility sensors transmission strategies dealing with all these requirements. Since the sensors cannot stay in a connected mode with dedicated resources because of their number, each time they transmit, they must complete a random access (RA) procedure. To this end, GFRA was introduced in 5G [1], [2], to reduce the latency and control overhead. In order for the AP to reliably acknowledge the RA requests, estimating the channel state in the uplink based on the RA signaling simultaneously to the active user detection is also required. Performing AUDaCE is then at the center of a successful GFRA procedure.

A. Related work

As suggested in [START_REF] Dai | A Survey of Non-Orthogonal Multiple Access for 5G[END_REF], [START_REF] Shahab | Grant-Free Non-Orthogonal Multiple Access for IoT: A Survey[END_REF], leveraging BCS-non-orthogonal multiple access (NOMA) appears to be relevant for GFRA in IIoT by exploiting the sparse sensor activities.

BCS relies on message-passing algorithms based on belief propagation (BP), expectation propagation (EP) or approximate message passing (AMP). In the case of independent sensors activity, a number of methods have been developed [START_REF] Liu | Sparse Signal Processing for Grant-Free Massive Connectivity: A Future Paradigm for Random Access Protocols in the Internet of Things[END_REF]- [START_REF] Ayachi | Massive Unsourced Random Access Based on Bilinear Vector Approximate Message Passing[END_REF]. In [START_REF] Liu | Sparse Signal Processing for Grant-Free Massive Connectivity: A Future Paradigm for Random Access Protocols in the Internet of Things[END_REF], the simplest version of AMP has been used to perform AUDaCE. In [START_REF] Ke | Compressive Sensing-Based Adaptive Active User Detection and Channel Estimation: Massive Access Meets Massive MIMO[END_REF], a GAMP algorithm was suggested, empowered by expectation maximization (EM) to estimate hyper parameters with an ad-hoc active user detection. In [START_REF] Zou | Message Passing Based Joint Channel and User Activity Estimation for Uplink Grant-Free Massive MIMO Systems With Low-Precision ADCs[END_REF], [START_REF] Zou | A Low-Complexity Joint User Activity, Channel and Data Estimation for Grant-Free Massive MIMO Systems[END_REF], hybrid GAMP (HGAMP) is used to perform AUDaCE with low-precision analog-to-digital converter, incorporating independent activity prior information for the active user detection. In [START_REF] Ayachi | Massive Unsourced Random Access Based on Bilinear Vector Approximate Message Passing[END_REF], AUDaCE is accomplished using bilinear vector AMP in the context of massive unsourced RA.

The compressed sensing (CS) algorithms in [START_REF] Liu | Sparse Signal Processing for Grant-Free Massive Connectivity: A Future Paradigm for Random Access Protocols in the Internet of Things[END_REF], [START_REF] Zou | Message Passing Based Joint Channel and User Activity Estimation for Uplink Grant-Free Massive MIMO Systems With Low-Precision ADCs[END_REF], [START_REF] Zou | A Low-Complexity Joint User Activity, Channel and Data Estimation for Grant-Free Massive MIMO Systems[END_REF], [START_REF] Ayachi | Massive Unsourced Random Access Based on Bilinear Vector Approximate Message Passing[END_REF], [START_REF] Ke | Compressive Sensing-Based Adaptive Active User Detection and Channel Estimation: Massive Access Meets Massive MIMO[END_REF] rely on the assumption that the random sensor activities are mutually independent. However, when sensors monitor the same machines, events may be detected by multiple sensors, leading to correlated activity. While CS algorithms for independent activity may be applied, this can lead to significant performance degradations [START_REF] Chetot | Joint Identification and Channel Estimation for Fault Detection in Industrial IoT with Correlated Sensors[END_REF]. To the best of our knowledge, the only work addressing AUDaCE for sensors with correlated activity is [START_REF] Chetot | Joint Identification and Channel Estimation for Fault Detection in Industrial IoT with Correlated Sensors[END_REF]. However, correlation in activity was restricted to be large and homogeneous among groups of users.

B. Main Contributions

Our main contributions are as follows:

• Model of GCA: we propose a new tractable statistical model for user activity exploiting the correlated Gaussian distribution. Existing activity models either assume independence (e.g. [START_REF] Zou | Message Passing Based Joint Channel and User Activity Estimation for Uplink Grant-Free Massive MIMO Systems With Low-Precision ADCs[END_REF]) or high levels of correlation ( [START_REF] Rangan | Hybrid Approximate Message Passing[END_REF], [START_REF] Dai | Multivariate Bernoulli distribution[END_REF]). In contrast, the GCA model can support heterogeneous correlation between devices, ranging from independence to high correlation. RETURN: Desired sample s.

• Numerical Evaluation: we validate our GCA-HGAMP algorithm via a comprehensive numerical study. We show that our GCA-HGAMP algorithm either outperforms or does not suffer from performances losses in all correlation regimes that have been previously considered by state-ofthe-art algorithms with an average gain of 2.5dB in the channel estimation error and twice less detection errors.

C. Notations

The sets R, R M , R M ×N (resp. C, C M , C M ×N ) respectively denote the sets of real (resp. complex) scalars, vectors of dimension M and matrices of dimensions (M, N ). The integer set [N ] is a shorthand notation for {1, . . . , N } (resp. ∅) if N ≥ 1 (resp. N < 1).

We respectively denote by x (or X), x, X a scalar, vector and matrix deterministic variable and by x (or X), x, X their random counterparts. The operators X T , X * , X † denotes the transpose, conjugate and conjugate transpose of a matrix.

The probability density function (pdf), cumulative distribution function (cdf) and probability mass function (pmf) of X are respectively denoted by f X , F X and P X . Expectation and variance are denoted by

E[•] and V[•].
The real (resp. complex) Gaussian distribution with matrix mean M and covariance C is denoted by Norm(M , C) and its pdf is N (•; M , C) (resp. CNorm(M , C) and CN (•; M , C)).

II. SYSTEM MODEL FOR GAUSSIAN CORRELATED ACTIVITY

A. Transmission of the idenfication sequences

We consider a single-carrier orthogonal frequency division multiplexing (OFDM) communication system with N singleantenna sensors and 1 K-antenna AP. The digital baseband signal received at the AP is

Y = XH(s) + W (1)
where X = x 1 . . . x N ∈ C M ×N is the known matrix of the N concatenated sensors identification sequences spanning M OFDM symbols, s ∈ {0, 1} N is the random vector of the N sensor activities,

H(s) = [h nk (s n )] (n,k)∈[N,K]
is the random matrix of the channel coefficients between the N sensors and the K AP's antennas, is the random matrix of the additive noise coefficients and

W = [w mk ] (m,k)∈[M,K] -2 0 2 cn c ∼ Norm(0, K)) -2 0
Y = [y mk ] (m,k)∈[M,K] is the random matrix of the output signal coefficients. For (n, m, k) ∈ [N ] × [M ] × [K], µ h ∈ C and (τ w , τ h ) ∈ R 2
+, * , we assume the noise coefficients to be i.i.d. as

w mk ∼ CNorm(0, τ w ) (2)
and the channel coefficients to be i.i.d. as

h nk (s n ) | s n = s ∼ Dirac(0) if s = 0 CNorm(µ h , τ h ) if s = 1 . (3)
Note that the channel coefficients are only assumed to have conditional independence given we know all the sensors activities s = s.

B. Gaussian Correlated Activity Model

For n ∈ [N ], denote by q n ∈ [0, 1] the random activity probability of the sensor state s n such that

q n ∼ Beta(α n , β n ) s n | q n = q n ∼ Bern(q n ) (4)
for (α n , β n ) ∈ R 2 + * . The motivation for q n to be Beta distributed is similar to [START_REF] Chetot | Joint Identification and Channel Estimation for Fault Detection in Industrial IoT with Correlated Sensors[END_REF]: this distribution describes a wide range of probability profiles with only two parameters, as suggested by its pdf

f qn (q) = q αn-1 (1 -q) βn-1 B(α n , β n ) (5)
where B(α, β) denotes the beta function evaluated at (α, β).

In order to model the state correlation, we seek to impose a dependence structure at the level of the activity probability random vector q, targeting Beta marginal distributions. Let c ∼ Norm(0 N ×1 , K) be a correlated Gaussian random vector parameterized by K ∈ [-1, 1] N,N , a, potentially degenerated, correlation matrix. Assume that

∀n ∈ [N ], q n = T n (c n ) with T n = F -1 qn • F cn (6)
where F qn and F cn respectively denote the marginal cdfs of q n and c n . Since K is a correlation matrix, the entries of c

fq qn M fq→qn (qn) ∝ [0,1]N-1 fq(q)   n ∈[N ]\{n} M fq←q n (qn )dqn   M fq←qn (qn) ∝ M Ps n|qn →qn (qn)
identically follow the standard normal distribution Norm(0, 1). Hence

∀n ∈ [N ], u n = F (c ) ∼ Unif([0, 1]) q n = F -1 qn (u n ) ∼ n , β n ) (7)
or, in a more compact form,

q = T (c) = [T n (c n )] T n∈[N ] (8)
The element-wise transform T conserves the targeted Beta marginals of q and transports the dependence structure from c to q.

The interest of this model w.r.t. the one of [START_REF] Chetot | Joint Identification and Channel Estimation for Fault Detection in Industrial IoT with Correlated Sensors[END_REF] is its greater flexibility. First, the parameters of the beta distributions are not shared anymore between subgroups of activity probabilities. Second, the correlation matrix K adds freedom in capturing custom pairwise state dependence structures. Together, they provide a tractable model for GCA in sensor networks at the expanse of meaningful additional parameters.

For instance, a system where the N sensors are split into G non-overlapping groups, with S g sensors in group g ∈ [G] with Gaussian correlation ρ g ∈ [0, 1], can be described by a correlation matrix of the form

K = diag(K 1 , . . . , K G ) (9)
with

∀g ∈ [G], K g = (1 -ρ g )I Sg + ρ g 1 Sg×Sg . (10)
If the sensor n has Beta parameters α n = 1 -β n , its average activity probability would be

E[q n ] = α n α n + β n = α n . (11)

III. ACTIVE USER DETECTION AND CHANNEL ESTIMATION

A. Problem formulation

Based on the model of Sec. II, the AUDaCE problem consists in jointly estimating an activity pattern s = s and the corresponding channel matrix H(s) = H . To do so, we want to compute estimates ŝ ≈ s and Ĥ ≈ H given the AP observed the output signal Y = Y with

ŝ = arg max s∈{0,1} N f s|Y (s | Y ), (Active User Detection) (12) Ĥ = E[H | Y = Y ] . (Channel Estimation) (13)
The estimators require the computation of the following marginal posteriors

P s|Y (s | Y ) = C M ×N f s,H(s)|Y (s, H | Y )dH (14)
f H|Y (H | Y ) = s∈{0,1} N f s,H(s)|Y (s, H | Y ) (15)
where each marginalization involves the posterior density

f s,H(s)|Y (s, H | Y ) = [0,1] N f q,s,H(s)|Y (q, s, H | Y )dq (16) = f q,s,H(s),Y (q, s, H, Y ) [0,1] N s∈{0,1} N C M ×N f q,s,H(s),Y (q, s, H, Y )dHdq . ( 17
)
Since the system variables form the following Markov chain

q → s → H → Y (18)
after Bayes' theorem, the full joint density which appears in the numerator and denominator of (17) factorizes as

f q,s,H,Y (q, s, H, Y) = f Y|H (Y | H)f H|s (H | s)P s|q (s | q)f q (q) (19) with f Y|H (Y | H) = (m,k)∈[M ]×[K] f y mk |h k (y mk | h k ), (20)
f H|s (H | s) = (n,k)∈[N ]×[K] f h nk |sn (h nk | s n ), (21)
P s|q (s | q) = n∈[N ] P sn|qn (s n | q n ). (22)
Note that a factorization of f q (q) is not provided since q is assumed to be correlated without any specific structure other Algorithm 2 GCA-HGAMP Description: GCA-HGAMP consists into two parts. The black lines corresponds to the updates of the GAMP variables for estimating the channel and the blue lines corresponds to the updates of the pattern variables with BP. Estimates of the system variables are colored in red.

1 input: Y , X, µh, τh, τw, Imax than [START_REF] Zou | A Low-Complexity Joint User Activity, Channel and Data Estimation for Grant-Free Massive MIMO Systems[END_REF] in the general case. The factor graph corresponding to such a factorization is depicted in Fig. 2. From (17), it is clear that computing ( 14) and ( 15) is intractable because of the high-dimensional sum and integrals. Hence, we will first resort to the BP algorithm to approximate the posterior distributions ( 14) and ( 15) and then derive a HGAMP algorithm to make the approximations and estimations tractable.

2 init: 3 i = 0 4 ∀(n, k) ∈ [N ] × [K] ĥi nk = µh, τ i h,nk = τh 5 ∀(m, k) ∈ [N ] × [K] ûi mk = 0 6 end 7 for i ∈ [Imax] do: 8 for (m, k) ∈ [M ] × [K] do: 9 τ i p,mk = N n=1 |xmn| 2 τ i-1 h,
23 qi n,n = [0,1]N qnfq(q) n ∈[N ]\{n} (1-q n )φ 0,n +q n φ 1,n dqn [0,1]N fq(q) n ∈[N ]\{n} (1-q n )φ 0,n +q n φ 1,n dqn 24 LLRn = log qi n,g (1-q i n,g ) φ1,n φ0,n 25 ŝi n = 1(LLRn > 0) 26 γn = (1 + exp(-LLRn)) -1 27 qi n = [0,1]N qnfq(q) n ∈[N ] (1-q n )φ 0,n +q n φ 1,n dqn [0,1]N fq(q) n ∈[N ] (1-q n )φ 0,n +q n φ 1,

B. HGAMP for AUDACE

The derivation of the HGAMP algorithm to address the AU-DaCE problem needs first the construction of a BP algorithm based on the factor graph in Fig. 2

∀(n, k) ∈ [N ] × [K], B h nk = M f h nk |sn →h nk M f h nk |sn ←h nk (24)
We then build the following posterior approximates as

P s|Y (s | Y ) ≈ n∈[N ] B sn (s n ) (25) f H|Y (H | Y ) ≈ (n,k)∈[N ]×[K] B h nk (h nk ) (26)
Substituting these approximates in ( 12) and ( 13) allows to perform AUDaCE using the works in [START_REF] Rangan | Hybrid Approximate Message Passing[END_REF], [START_REF] Rangan | Generalized approximate message passing for estimation with random linear mixing[END_REF], we can partially approximate the BP algorithm in Tab. I using the framework of HGAMP. For large systems when N/M = N →+∞ O(1), the following approximation of (29) can be made

∀n ∈ [N ], ŝn ≈ arg max s∈{0,1} B sn (s) (27) ∀(n, k) ∈ [N ] × [K], ĥnk ≈ ŝn C hB h nk (h)dh (
M f h nk |sn ←h (h) ≈ CN (h; rnk , τr,nk ). ( 30
)
where (r nk , τr,nk ) are mean and variance variables that are iteratively updated by Algo. 2. Propagating this approximation in the BP messages in Tab. I, we obtain the HGAMP instance in Algo. 2 tailored to the GCA model, namely GCA-HGAMP. From Tab. I and Algo. 2, one can see that the original BP algorithm has been turned into an iterative algorithm where the updated quantities are not functions anymore but scalars. Also, note that the updates at line 23 and 27 in Algo. 2 require numerical integration. It is handled using Monte-Carlo integration with I samples samples {q i } i∈[Isamples] drawn with Algo. 1. As a whole, we can show that the algorithm complexity is

C GCA = O(N M K + N K + N I samples ) (31)
which is comparable to the state-of-the-art algorithm (see [START_REF] Rangan | Hybrid Approximate Message Passing[END_REF] and [START_REF] Chetot | Joint Identification and Channel Estimation for Fault Detection in Industrial IoT with Correlated Sensors[END_REF]).

IV. NUMERICAL RESULTS The performances of the new GCA-HGAMP are numerically studied based on extensive Monte-Carlo simulations. For a collection of I MC ground truth values {(s i , H i )} i∈[IMC] and their respective estimates {(ŝ i , Ĥi )} i∈[IMC] , the capability of GCA-HGAMP to perform active user detection is assessed by the UER

UER({s i , ŝi } i∈[IMC] ) = 1 N I MC i∈[IMC] (N -(s i ) T ŝi ) (32)
and the quality of its channel estimation is assessed by the NMSE

NMSE({H i , Ĥi } i∈[IMC] ) = 1 I MC i∈[IMC] H i -Ĥi 2 2 H i 2 2

. (33)

In order to compare the performances of our GCA-HGAMP, we also provide the performances for the following algorithms. A. Benchmark a) Modified GAMP (based on [START_REF] Rangan | Generalized approximate message passing for estimation with random linear mixing[END_REF]): This modified GAMP corresponds to the algorithm originally described in [START_REF] Rangan | Generalized approximate message passing for estimation with random linear mixing[END_REF] but with a slight modification to allow activity detection. To do so, we have considered, as in [START_REF] Ke | Compressive Sensing-Based Adaptive Active User Detection and Channel Estimation: Massive Access Meets Massive MIMO[END_REF], a Bernoulli-Gaussian prior for the channel coefficients so that

f H (H) = (n,k)∈[N ]×[K] (1 -q n )δ(h nk ) + q n CN (h nk ; µ h , τ h ) (34) where ∀n ∈ [N ], q n = α n /(α n + β n ). (35)
The log-likelihood ratios for this GAMP become

LLR n = log α n α n + β n + log φ 0,n -log β n α n + β n -log φ 1,n (36)
that will be used for activity detection. This modified GAMP does not account for any activity correlation. b) Group-sparse activity (GSA)-HGAMP [START_REF] Rangan | Hybrid Approximate Message Passing[END_REF]: For the simulations, the activity probability of the group g will be identified to

q g = 1 S g n∈Gg α n α n + β n (37)
which consists in computing the average of the ratios as a proxy for the group activity probability. c) Group-homogeneous activity (GHomA)-HGAMP [START_REF] Chetot | Joint Identification and Channel Estimation for Fault Detection in Industrial IoT with Correlated Sensors[END_REF]: The average activity probability of sensors belonging to the same group is assumed to be the same and defined as

B. Relative performances of GCA to the benchmark

The numerical study focuses on the relative performances of GCA-HGAMP against modified GAMP, GSA-HGAMP and GHomA-HGAMP. The simulation parameters are given in Tab. II and the corresponding NMSE and UER have been drawn in Fig. 3. Note that the simulation scenarios (ρ = 0.25 and ρ = 0.75) are tailored to GCA-HGAMP. Hence, the main goal of the simulations is to assess the gains of the proposed GCA-HGAMP algorithm w.r.t. to the benchmark algorithms that do not account for the more general Gaussian correlated activity.

C. Discussion of the results

The important indicator are the correlation coefficients {ρ g } g∈[G] that fully describe the correlation matrix K of the underlying Gaussian random vector c. For a particular group g ∈ [G], when ρ g is close to 0 the matrix is close to I Sg and when ρ g is close to 1 the matrix is close to 1 Sg×Sg , making K block diagonal. Hence, two correlation regimes, low ρ g = 0.25 and high ρ g = 0.75, have been considered to assess the performances, for which we give a review hereafter.

In all the correlation regimes, GSA-HGAMP always performs the worst and GCA-HGAMP always performs the best.

In the low correlation regime, the activity of the sensors is close to be independent (ρ = 0) so that the modified GAMP algorithm is as good as GCA-HGAMP over the whole SNR range. GAMP and GCA reduce the NMSE by ∼ 2.5dB for the channel estimation w.r.t. GHomA (cf. Fig. 3a) below . The reason is that GHomA-HGAMP still relies on the underlying group structure of the sensors, which is responsible for a higher number of false alarms and missed detections for GHomA-HGAMP, leading to an overall twice worse UER (see Fig. 3a). Indeed, when the activity correlation is low, the activity behavior of sensors within the same group is very unlikely to be the same.

In the high correlation regime, the trend observed in the low correlation regime is reversed. Both GHomA and GCA-HGAMP have the best performance, reducing the NMSE by ∼ 2.5dB w.r.t. the modified GAMP algorithm in Fig. 3b; the UER is again halved in favor of GHomA and GCA-HGAMP. The reason is that GAMP cannot leverage the correlation structure which is favorable to similar activity probabilities within each group of sensors to improve its detection capability.

An additional remark is that GCA-HGAMP is able to perform the best in all scenarios where GHomA-HGAMP and GAMP only perform best when the correlation regime respectively approaches the high and low correlation ones, which are their reference scenarios. As a whole, the results show the ability of GCA-HGAMP to adapt to various correlation structures with significant gains in terms of NMSE and UER.

V. CONCLUSION

This paper addresses the problem of AUDaCE in the context of massive GFRA where the activity of transmitting devices is correlated. A tractable GCA model is proposed leveraging the correlated distribution of the device activity probabilities. Based on this new model, the AUDaCE problem is formulated within the BCS framework. A GCA-HGAMP algorithm with polynomial complexity is then derived to approximate the optimal solutions of the AUDaCE problem. Finally, numerical simulations show significant gains in terms of UER and NMSE for the proposed algorithm to account for more general correlation against state-of-the-art algorithms that account for a limited form of correlated activity, or even no correlation at all. For future works, more general correlated activity models could be considered using the framework of the copula theory.

END 5 Sample 6 7 8

 5678 c ∼ Norm(0 N ×1 , K).Compute the vectoru = [Fc n (cn)] T n∈[N ] .Compute the vector q = [F -1 qn (un; αn, βn)] T n∈[N ] .Sample s = [sn] T where ∀n ∈ [N ], sn ∼ Bern(qn).

9

 9

Fig. 1 :

 1 Fig. 1: Scatter plots of the vectors c, u and q based on (7). The correlation matrix is K = ρ1 2×2 + (1 -ρ)I 2 with ρ = 0.25 for the top row and ρ = 0.75 for the bottom row. The parameters of the Beta distributions are (αn, α n , βn, β n ) = (0.2, 2, 0.8, 5).

Fig. 2 :

 2 Fig. 2: Factor graph of (19).

  . The corresponding BP messages are given in Tab. I and the corresponding beliefs are ∀n ∈ [N ], B sn = M P sn |qn →sn M P sn|qn ←sn (23)

  28) However, for (n, k) ∈ [N ] × [K], the computation of the message M f h nk |sn ←h nk = M m=1 M f h nk |sn ←h nk (29) requires the computation of the messages M f h nk |sn ←h nk which are in general intractable high dimensional integrals.

Fig. 3 :

 3 Fig. 3: Performances in terms of user error rate (UER) and normalized mean squared error (NMSE) (see (32) and (33)).

  α n α n + β n and βg = 1 -αg . (38) Parameter = Value Description IMC = 1000 # of Monte-Carlo iterations Imax = 100 # of GAMP or HGAMP iterations Isamples = 100 # of samples for the integration G = 64 # of groups ∀g ∈ [G], |Gg| = 4 # of sensors per group N = 256 # of sensors M = 128 # of OFDM symbols K = 2 # of antennas at the AP X = [xn/ xn 2]n∈[N ], xn ∼ CNorm(0N×1, IN ) identification sequences ∀n ∈ [N ], (αn, βn) = (0.35, 0.65) Beta distribution's parameters ∀g ∈ [G], ρg = {0.25, 0.75} correlation within each group

TABLE I BELIEF

 I PROPAGATION MESSAGES CORRESPONDING TO FIG. 2.

	Factor Variable Factor → Variable													Factor ← Variable
																	
	fy mk |hk	hnk	M mk |hk →hnk fy	(hnk) ∝	CN-1	fy mk |hk (ymk | h:k)	 n ∈[N ]\{n}	M m k |hk ←h n k fy	(hn k)dhn k		fy	mk |hk ←hnk M	(hnk) ∝	fh nk |sn →hnk M	(hnk)	m ∈[M ]\{m}	fy mk |hk →hnk M	(hnk)
						1													M
	fh nk |sn	hnk	M nk |sn →hnk fh	(hnk) ∝	s=0	fh nk |sn ←sn fh nk |sn (hnk | s) M	(s)				fh nk |sn ←hnk M	(hnk) ∝	m=1	fy	mk |hk →hnk M	(hnk)
	fh nk |sn	sn	fh nk |sn →sn M	(sn) ∝			fh	nk |sn ←hnk	(h)dh					fh nk |sn ←sn M	(sn) ∝	Ps n|qn →sn M	(sn)	k ∈[K]\{k}	fh	nk |sn →sn M	(sn)
						1													K
	Ps n|qn	sn	M Ps n|qn →sn	(sn) ∝	0	Ps n|qn ←qn Ps n |qn(sn | q) M	(q)dq					Ps n|qn ←sn M	(sn) ∝	k=1	fh nk |sn →sn M	(sn)
						1
	Ps n|qn	qn	M Ps n |qn→qn	(qn) ∝	s=0	Ps n |qn←sn Ps n |qn(s | qn) M	(s)							Ps n |qn←qn M	(qn) ∝	M	Ps n |qn→qn	(qn)

C fh nk |sn (h | sn) M fq→qn (qn) M

  It is the algorithm introduced in this paper. Different from GAMP, GSA-HGAMP and GHomA-HGAMP, it can deal with more general dependence and correlation structures, still introduced at the level of the activity probability that affects the state correlation as explained in Sec. II-B.

	TABLE II
	SIMULATION PARAMETERS FOR THE COMPARATIVE STUDY OF GCA
	AGAINST MODIFIED GAMP, GS-HGAMP AND GHOMA-HGAMP
	d) GCA-HGAMP:


R Di Stasi

F Cerrotti

L Rat-Fischer

Rana Esseily

Rana Esseily Positive

Positive Emotional Reactions and Arousal enhance Infants' Learning

Positive Emotional Reactions and Arousal enhance Infants'

Learning.

R. Di Stasi 1 , F. Cerrotti 1 , L. Rat-Fischer 1 & R. Esseily 1

1 Laborotoire Ethologie Cognition Développement (UR LECD), Paris Nanterre University

Material and Methods

A Multidimensional Method

Behavioural Analysis

(using The Observer XT15 by Noldus)

ECBQ Temperament Questionnaire Facial Expression Analysis

(using BabyFace Reader by Noldus)

Physiological Analysis

(using E4 from Empatica)

119 infants from 14 to 22 months old.

Task : Infant (B) has to imitate the use of a rake to retrieve an out of reach toy after a demonstration by an experimenter (L1 or L2 depending on the phase).

Results and Discussion

Two types of demonstration: Neutral or Humorous. andEmotion, 30(4), 817-825

What mechanisms may explain this association: humor per se, surprise, positive emotions or arousal?

To tackle this question, a multidimensional apporach was used.

Infants exposed to Humorous Demonstration learned better than the others (Fig A).

Infants produce more social referencing during the humorous than the neutral demonstration: higher frequency (p< 0.001) and gaze duration (p= 0.015) towards the experimenter. This result was independent of demonstration type and of infants' emotional reaction during the demo.

In both demonstration types

Experiment exemple (Humorous Demonstration)

Humorous demonstrations (probably due to a surprise effect) and infants' positive emotional reactions improved infants' learning performance. A stress reduction effect (i.e. an increase in heart rate variability) was observed but not related to demo type and to infants' laughing or smiling reaction. It is possible that infants expressed other emotions during the demonstration that could explain this HRV dynamic. By associating physiological data with facial expression analysis with a specific algorithm, we hope to understand the link between HRV, emotions and learning in infants during a humorous context. These results are consistent with the literature suggesting a positive effect of stress reduction on learning (1). The decrease of the effect is likely due to infants' habituation to the task.

Conclusion

These results suggest a surprise effect (3) induced by the humorous demonstration in addition to a positive emotions effect (4).

  Banas, J. A., Dunbar, N., Rodriguez, D., & Liu, S. J. (2011). A review of humor in educational settings: Four decades of research. Communication Education, 60(1), 115-144 (2) Esseily, R., Rat-Fischer, L., Somogyi, E., O'Regan, K. J., & Fagard, J. (2015). Humour production may enhance observational learning of a new tool-use action in 18-month-old infants. Cognition

  . (3) Emde, R. N. (1992). Social referencing research: Uncertainty, self, and the search for meaning. Social referencing and the social construction of reality in infancy, 79-94. (4) Stahl, A. E., & Feigenson, L. (2017). Expectancy violations promote learning in young children. Cognition, 163, 1-14. (5) Quintana, D. S., Guastella, A. J., Outhred, T., Hickie, I. B., & Kemp, A. H. (2012). Heart rate variability is associated with emotion recognition: Direct evidence for a relationship between the autonomic nervous system and social cognition. International journal of psychophysiology, 86(2), 168-172. Positive emotions induced in humorous contexts can have a positive influence on learning in adults and school-aged children (1). What about babies? A first study showed that 18-month-old infants who laughed at the humorous demonstration improved their learning in a tool use imitation task (2).

  , infants who smiled and laughed during the demonstration imitated more (Fig B). Infants with the lowest arousal (ΔHRV 1 ) at the beginning of the demo were also the most successful after demo (Fig C). This correlation was only a trend at the middle of the demo (Fig D).

1

  ΔHRV = difference between heart rate variability during the baseline and during the beginning of the demo (Fig C) or the middle of the demo (Fig D)