Marianne Quéméneur
email: marianne.quemeneur@ird.fr

Gaël Erauso

Manon Bartoli

Céline Vandecasteele

Laura Wils

Lisa Gil

Christophe Monnin

Bernard Pelletier

Anne Postec

a novel alkaliphilic anaerobic bacterium isolated from 'La Crouen' alkaline thermal spring, New Caledonia

Keywords: Alkalicella caledoniensis gen. nov., sp. nov Firmicutes, anaerobic, alkaliphilic, alkaline thermal spring, Proteinivoraceae, Alkalicella caledoniensis taxa -Firmicutes (Class Clostridia, Order Clostridiales

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

Over the last twenty years, alkaliphilic bacteria have been extensively studied because they are sources of biomolecules and enzymes stable at extreme conditions, precious in biotechnology and bioremediation [START_REF] Preiss | Alkaliphilic bacteria with impact on industrial applications, concepts of early life forms, and bioenergetics of ATP synthesis[END_REF][START_REF] Sarethy | Alkaliphilic bacteria: applications in industrial biotechnology[END_REF][START_REF] Horikoshi | Alkaliphiles: some applications of their products for biotechnology[END_REF]. Alkaliphiles are microorganisms that optimally grow at pH values above 9 but cannot grow or slowly grow at near-neutral pH [START_REF] Horikoshi | Alkaliphiles: some applications of their products for biotechnology[END_REF]. They inhabit various alkaline ecosystems (e.g., soda lakes, thermal springs, oceans, industrial or miningimpacted areas). Some of them can grow under multiple extreme conditions like a combination of high salt concentrations and high temperature [START_REF] Kevbrin | Isolation and cultivation of alkaliphiles[END_REF]. Anaerobic alkaliphiles, especially members of the order Clostridiales (e.g., the genera Alkaliphilus, Anaerobranca, Natronoincola, Serpentinicella, or Tindallia), have been isolated from different sites around the world and include halo-, meso-, and thermo-alkaliphiles [START_REF] Kevbrin | Isolation and cultivation of alkaliphiles[END_REF][START_REF] Wiegel | Anaerobic alkaliphiles and alkaliphilic poly-extremophiles[END_REF].

In New Caledonia (Southwest Pacific), several thermal springs are located in the southeastern half of the main island [START_REF] Cox | Geochemistry of low temperature geothermal systems in New Caledonia[END_REF] (see the map in [START_REF] Maurizot | Chapter 9: Mineral resources and prospectivity of non-ultramafic rocks of New-Caledonia[END_REF]). Such springs are found down south in the Prony Bay area and the area close to the city of Canala [START_REF] Maurizot | Chapter 9: Mineral resources and prospectivity of non-ultramafic rocks of New-Caledonia[END_REF]. A sampling campaign was conducted in November 2014 during which water and sediment samples were collected at one of the springs of the Canala region, namely the spring feeding the abandoned spa facility of 'La Crouen' (165° 53 ' 20.6" E 21° 32' 06.6" S). On-site measurements showed that its waters are characterized by high pH (9.0), low Eh value (-174 mV, ref H2), low dissolved O2 (0.71 mg/L), low conductivity (314 µS/cm), and moderate temperature (42°C) (data from our study). Deville and Prinzhofer [START_REF] Deville | The origin of N2-H2-CH4-rich natural gas seepages in ophiolitic context: a major and noble gases study of fluid seepages in New Caledonia[END_REF] reported that the gas emitted at the springs is dinitrogen (N2) with traces of methane (CH4 <3%), but we did not observe degassing in the spring during our visit. Our dissolved gases analyses showed that N2 is dominant, followed by O2, but no CH4 or hydrogen (H2) was detected. Both the geological setting and geochemistry differ from those of the well-known hyperalkaline springs located in the peridotite nappe in the Prony Bay Hydrothermal Field (PBHF). These on-shore or submarine springs discharge a warm (<40°C), highly reduced, high-pH (up to 11.2) fluid enriched in N2, H2, and CH4 [START_REF] Monnin | Fluid chemistry of the low temperature hyperalkaline hydrothermal system of the Prony Bay (New Caledonia)[END_REF], produced by the serpentinization process (i.e., a hydrothermal alteration that transforms ferromagnesian minerals contained in ultramafic rocks into serpentine minerals [10]). Prokaryotic communities of PBHF have been studied [START_REF] Quéméneur | Spatial distribution of microbial communities in the shallow submarine alkaline hydrothermal field of the Prony Bay, New Caledonia[END_REF][START_REF] Postec | Microbial diversity in a submarine carbonate edifice from the serpentinizing hydrothermal system of the Prony Bay (New Caledonia) over a 6-year period[END_REF][START_REF] Pisapia | Mineralizing filamentous bacteria from the Prony Bay Hydrothermal Field give new insights into the functioning of serpentinization-based subseafloor ecosystems[END_REF][START_REF] Frouin | Diversity of rare and abundant prokaryotic phylotypes in the Prony Hydrothermal Field and comparison with other serpentinite-hosted ecosystems[END_REF][START_REF] Mei | Metagenomic and PCR-based diversity surveys of [FeFe]-hydrogenases combined with isolation of alkaliphilic hydrogen-producing bacteria from the serpentinite-hosted Prony Hydrothermal Field, New Caledonia[END_REF] and several anaerobic Clostridiales strains have been isolated and described as new taxa [START_REF] Mei | Metagenomic and PCR-based diversity surveys of [FeFe]-hydrogenases combined with isolation of alkaliphilic hydrogen-producing bacteria from the serpentinite-hosted Prony Hydrothermal Field, New Caledonia[END_REF][START_REF] Aissa | Vallitalea pronyensis sp. nov., isolated from a marine alkaline hydrothermal chimney[END_REF][START_REF] Aissa | Characterization of Alkaliphilus hydrothermalis sp. nov., a novel alkaliphilic anaerobic bacterium, isolated from a carbonaceous chimney of the Prony hydrothermal field, New Caledonia[END_REF][START_REF] Bes | Acetoanaerobium pronyense sp. nov., an anaerobic mesophilic bacterium isolated from the Prony alkaline Hydrothermal Field, New Caledonia[END_REF][START_REF] Mei | Fermentative hydrogen production by a new alkaliphilic Clostridium sp. (strain PROH2) isolated from a shallow submarine hydrothermal chimney in Prony Bay, New Caledonia[END_REF][START_REF] Mei | Serpentinicella alkaliphila gen. nov., sp. nov., a novel alkaliphilic anaerobic bacterium isolated from the serpentinitehosted Prony hydrothermal field, New Caledonia[END_REF][START_REF] Postec | Alkaliphilus serpentinus sp. nov. and Alkaliphilus pronyensis sp. nov., two novel anaerobic alkaliphilic species isolated from the serpentinite-hosted Prony Bay Hydrothermal Field (New Caledonia)[END_REF]. On the contrary, microbial communities of 'La Crouen' spring had not been studied previously. Here we report the isolation and characterization of a novel bacterium isolated from 'La Crouen' spring, which original features allow the proposition of a novel species within a novel genus belonging to the family Proteinivoraceae, phylum Firmicutes.

For culturing experiments, a sterile glass bottle was filled with sediment slurry sample, then hermetically sealed to preserve anoxic conditions, and stored at 4°C. In the laboratory, anaerobic enrichment cultures were carried out at 37°C using 0.5 mL of the slurry sample (corresponding to 10% inoculum) in Hungate tubes containing 5 mL of nutrient-rich LB (Luria-Bertani) medium (i.e., 10g/L Tryptone, 5 g/L yeast extract, and 5 g/L NaCl), supplemented with 25 mM Tris-Base. Before inoculation, the culture medium was adjusted to pH 9.5, boiled for 5 min, and then cooled to room temperature under a flow of O2-free N2 gas. The medium was dispensed into Hungate tubes, degassed under a flow of N2, and subsequently autoclaved (45 min, 120°C). The following sterile solutions were injected in each tube: 0.1 mL of 2% Na2S.9H2O (reducing agent) and 0.5 mL of 8% Na2CO3 (to adjust and buffer the final pH to 9.5). The Hungate technique for anaerobic cultivation was used throughout this study [START_REF] Hungate | A roll tube method for cultivation of strict anaerobes[END_REF].

Temperature, pH, and NaCl concentration ranges (respectively tested: 20-70°C, 6.0-12.0, 0-10%) for growth of the isolate were determined in duplicate using Hungate tubes containing LB medium, supplemented with 25 mM Tris-Base (except for testing the optimal pH for growth, which was determined using different buffer solutions: 0 mM Tris-Base for pH 6.0-7.0, 25 mM Tris-Base for pH 7.0-9.0, or 20 mM CAPS for pH 9.0-12.0). The pH of the culture media was adjusted with anaerobic sterile stock solutions of 8% Na2CO3 or 1M NaOH. The pH was always checked both after inoculation and at the end of incubation. Water baths were used for incubation at temperatures up to 70°C. For studies of NaCl requirements, NaCl was weighed and added directly into the tubes before culture medium addition. Each test culture was subcultured at least once under the same experimental conditions before determining growth rates.

A range of inorganic and organic substrates was tested as substrate for the isolated growth in the basal medium containing (per liter of distilled water): 0.3 g KH2PO4, 0.3 g K2HPO4, 1 g NH4Cl, 1 g NaCl, 0.1 g KCl, 0.5 g cysteine.HCl, and 10 mL of trace elements solution [START_REF] Balch | Methanogens: reevaluation of a unique biological group[END_REF].

The pH was adjusted to 9.5 with a 1M NaOH solution. The following compounds were tested as potential substrates (at a final concentration of 20 mM for simple substrates and 2g/L for complex substrates): adenine, adenosine 5'-diphosphate, albumin, arabinose, casamino-acids, casein, cellobiose, fructose, galactose, gelatin, glucose, glycerol, guanine, lactate, lactose, mannose, peptone, pyruvate, ribose, starch, sucrose, tryptone, uracil, xylose, yeast extract, H2/CO2 (80/20, v/v) and H2/CO2 (80/20, v/v) in the presence of acetate (2 mM) as carbon source. Elemental sulfur (1% w/v), sulfate (20 mM), thiosulfate (20 mM), sulfite (2 mM), nitrate (10 mM), nitrite (2 mM), fumarate (20 mM), FeIII-citrate (10 mM), and DMSO (10 mM) were tested as terminal electron acceptors using pyruvate as electron donor. Sensitivity to antibiotics (ampicillin, bacitracin, chloramphenicol, cycloserine, kanamycin, metronidazole, novobiocin, penicillin, rifampicin, streptomycin, tetracycline, vancomycin) was tested by adding them separately to the LB medium (supplemented with 25 mM Tris-Base) at a final concentration of 25 µg/mL or 50 µg/mL. Each test and control, carried out in parallel, was run in duplicate Hungate tubes. Controls were (i) inoculated tubes without tested compounds and (ii) uninoculated tubes with tested compounds. Bacterial growth was monitored by measuring the turbidity increase at 600 nm by inserting Hungate tubes into the tube holder of a spectrophotometer (Cary 50, Varian). H2S production was measured photometrically as colloidal CuS, as described by Cord-Ruwisch [START_REF] Cord-Ruwisch | A quick method for the determination of dissolved and precipitated sulfides in cultures of sulfate-reducing bacteria[END_REF]. End-products of metabolism were measured by high-performance liquid chromatography (HPLC) and gas chromatography (GC) after 1 day of incubation at 37°C [START_REF] Mei | Fermentative hydrogen production by a new alkaliphilic Clostridium sp. (strain PROH2) isolated from a shallow submarine hydrothermal chimney in Prony Bay, New Caledonia[END_REF].

Cellular morphology and purity of the strain and the presence of spores were assessed by observation using a phase-contrast microscope (Optiphot; Nikon). For transmission electron microscopy (using a MET FEI Tecnai G2), exponentially grown cells were negatively stained with sodium phosphotungstate (to observe flagella) or fixed with glutaraldehyde and osmium tetraoxide, epon-embedded and stained with uranyl acetate and lead citrate (to watch the cell wall), as described by Quéméneur et al. [START_REF] Quéméneur | Hydrostatic pressure helps to cultivate an original anaerobic bacterium from the Atlantis Massif subseafloor (IODP Expedition 357): Petrocella atlantisensis gen. nov. sp. nov[END_REF].

Cellular fatty acids were determined in the late exponential growth phase (after 24-hour incubation at 37°C using LB-Tris medium at pH 9.5) by the Identification Service of DSMZ with modified Miller extraction protocol [START_REF] Kuykendall | Fatty acids, antibiotic resistance, and deoxyribonucleic acid homology groups of Bradyrhizobium japonicum[END_REF] using the MIDI Microbial Identification system (version 4.0, MIS operating manual March 2001) [START_REF] Sasser | Identification of bacteria by gas chromatography of cellular fatty acids[END_REF]. Near-complete 16S rRNA genes were amplified from genomic DNA extracted as described by Quéméneur et al. [START_REF] Quéméneur | Endolithic microbial communities in carbonate precipitates from serpentinite-hosted hyperalkaline springs of the Voltri massif (Ligurian Alps, Northern Italy)[END_REF], using the primers 27F and 1492R [START_REF] Lane | Nucleic acid techniques in bacterial systematics[END_REF], then sequenced by GATC-Biotech (Konstanz, Germany). The 16S rRNA gene sequence was deposited and aligned with available sequences from NCBI database using the MUSCLE program [START_REF] Edgar | MUSCLE: multiple sequence alignment with high accuracy and high throughput[END_REF] implemented in MEGA7 software [START_REF] Kumar | MEGA7: molecular evolutionary genetics analysis version 7.0 for bigger datasets[END_REF]. Evolutionary analyses were conducted from 16 nucleotide sequences using three different methods: maximum-likelihood (ML) [START_REF] Felsenstein | Evolutionary trees from DNA sequences: a maximum likelihood approach[END_REF], maximum-parsimony (MP) [START_REF] Fitch | Toward defining the course of evolution: minimum change for a specified tree topology[END_REF], and neighbor-joining (NJ) [START_REF] Saitou | The neighbor-joining method: a new method for reconstructing phylogenetic trees[END_REF].

Whole-genome analysis was performed from high molecular weight DNA of the isolate, extracted using a phenol-chloroform method, then sequenced at the GeT-PlaGe platform (INRAe Toulouse, France) by combining long reads (Oxford Nanopore) and short-reads (Illumina MiSeq) technologies, as described by Quéméneur et al. [START_REF] Quéméneur | Hydrostatic pressure helps to cultivate an original anaerobic bacterium from the Atlantis Massif subseafloor (IODP Expedition 357): Petrocella atlantisensis gen. nov. sp. nov[END_REF] with some modifications because it was sequenced by multiplex sequencing according to the manufacturer's instructions "1D Native barcoding genomic DNA" (EXP-NBD104 and SQK-LSK109)". Base-calling was carried out by MinKNOW-Live-Base-calling software version 3.6.0 (Oxford Nanopore Technologies, Oxford, UK). Read demultiplexing and adapter trimming were performed using qcat v1.1.0 (https://github.com/nanoporetech/qcat) and genome assembly with Unicycler v0.4.6 [START_REF] Wick | Unicycler: resolving bacterial genome assemblies from short and long sequencing reads[END_REF]. The genome sequence was deposited in the NCBI database and annotated using the MaGe (Microscope Genome Annotation) [START_REF] Vallenet | MicroScope -an integrated microbial resource for the curation and comparative analysis of genomic and metabolic data[END_REF] and the Integrated Microbial Genomes (IMG) systems [START_REF] Chen | v. 5.0: an integrated data management and comparative analysis system for microbial genomes and microbiomes[END_REF]. The value of average nucleotide identity (ANI) was calculated using OrthoANI with usearch (http://www.ezbiocloud.net/tools/ani). Digital (or in silico) DNA-DNA hybridization (dDDH) was calculated using the Genome-to-Genome Distance Calculator (GGDC) 2.1 (http://ggdc.dsmz.de/) [START_REF] Meier-Kolthoff | Genome sequence-based species delimitation with confidence intervals and improved distance functions[END_REF]. A whole genome-based taxonomic analysis was performed using the bioinformatics platform Type (Strain) Genome Server (TYGS) (https://tygs.dsmz.de) [START_REF] Meier-Kolthoff | TYGS is an automated high-throughput platform for stateof-the-art genome-based taxonomy[END_REF].

In brief, the TYGS analysis included a determination of the closest type strain genomes by (i) comparison against all type strain genomes available in the TYGS database via the MASH algorithm [START_REF] Ondov | Mash: Fast genome and metagenome distance estimation using MinHash[END_REF] and (ii) the closely related type strains determined via the 16S rDNA gene sequences. For the phylogenomic inference, all pairwise comparisons among the genomes were conducted using Genome BLAST Distance Phylogeny (GBDP) and accurate intergenomic distances inferred under the algorithm 'trimming' and distance formula d5. The resulting intergenomic distances were used to infer a balanced minimum evolution tree with branch support via FASTME 2.1.6.1 [START_REF] Lefort | FastME 2.0: A comprehensive, accurate, and fast distance-based phylogeny inference program[END_REF]. Branch support was inferred from 100 pseudo-bootstrap replicates each. The trees were rooted at the midpoint [START_REF] Farris | Estimating phylogenetic trees from distance matrices[END_REF].

The G+C content of the DNA was also determined at DSMZ (Deutsche Sammlung von Mikroorganismen und Zellkulturen, Braunschweig, Germany) using HPLC, as described by Mesbah et al. [START_REF] Mesbah | Precise measurement of the G+C content of deoxyribonucleic acid by high-performance liquid chromatography[END_REF].

The enrichment culture of the 'La Crouen' slurry sample showed significant growth on LB-Tris-Base medium after two days of incubation at 37°C and at pH 9.5. This enrichment was then subcultured several times under the same growth conditions before isolation. The Roll-tube method described originally by Hungate [START_REF] Hungate | A roll tube method for cultivation of strict anaerobes[END_REF] was used for strain isolation. The culture was serially diluted tenth in roll tubes containing molten 1.6% agar medium. Single colonies were transferred into the liquid medium and immediately serially diluted tenth in roll tubes a second time. The colonies grown in roll tubes were white and circular (1-2 mm in diameter after oneweek incubation at 30°C). After the third round of roll tubes isolation, a single colony was finally transferred into the liquid medium, resulting in the strain designated LB2 T that was further physiologically and metabolically characterized.

Cells of strain LB2 T stain Gram-positive are motile and endospore-forming bacterium (Fig. 1A). Endospores are terminal and oval (0.8-1 x 1.0-1.5 µm). Rods are approximately 6.0-10.0 µm long and 0.5-0.6 µm wide. They occur singly or in pairs and possessed peritrichous flagella (Fig. 1B). They are pleiomorphic, i.e., rod-shaped during the exponential growth phase (Fig. S1A), they turn coccoid in the stationary phase (after 24h, Fig. S1B). Strain LB2 T is mesophilic and grows only under anaerobic conditions, ranging from 25°C to 50°C (Table 1). The optimum temperature for growth is 37°C. The pH range for growth is 8.2-10.8, with an optimum at an initial pH of 9.5; the final pH is 9.2. NaCl is not required for growth, and the NaCl range for growth is 0-7% (optimum at 0-1%) but is tolerated up to 7%, indicating that strain LB2 T is halotolerant and adapted to the low-salinity and high-pH fluid emitted by the alkaline spring of 'La Crouen' (New Caledonia).

Peptone, pyruvate, xylose, and yeast extract were the only substrates used for growth (Table 1). Yeast extract was degraded to acetate, formate, H2, and propionate. Growth on yeast extract was improved by the addition of thiosulfate used as a terminal electron acceptor, reduced to sulfide. End products detected from pyruvate fermentation were acetate and formate. Both ferric citrate and thiosulfate were used as electron acceptors using pyruvate at pH 9.5. Elemental sulfur, sulfate, nitrate, nitrite, fumarate, and DMSO were not used as terminal electron acceptors, using pyruvate as an electron donor. The maximal growth rate found was 0.07 h -1 using pyruvate and thiosulfate as electron donor and acceptor, respectively. Strain LB2 T was not sensitive to the presence of antibiotics cycloserine and novobiocin added at 25 µg/mL and 50 µg/mL in the culture medium. It also grew in the presence of 25 µg/mL of chloramphenicol and kanamycin, but it was sensitive to these antibiotics at 50 µg/mL. Ampicillin, bacitracin, metronidazole, penicillin, rifampicin, streptomycin, tetracycline, and vancomycin at 25 µg/mL completely inhibited cell growth.

Phylogenetic analysis based on the almost full-length 16S rRNA gene sequence (1245 nt) indicated that strain LB2 T is a member of the family Proteinivoraceae [START_REF] Kevbrin | Proteinivorax tanatarense gen. nov., sp. nov., an anaerobic, haloalkaliphilic, proteolytic bacterium isolated from a decaying algal bloom, and proposal of Proteinivoraceae fam. nov[END_REF], within the order Clostridiales, phylum Firmicutes (Fig. 2). It forms a distinct branch among alkaliphilic clostridia of the family Proteinivoraceae, including the genera Anaerobranca and Proteinivorax (91.2-93.2% 16S rRNA gene sequence identity) (Fig. 2). The phylogenetic position of strain LB2 T was confirmed by using three different methods: maximum-likelihood identity) isolated from a hot lake inlet (Lake Bogoria, Kenya) [START_REF] Prowe | Anaerobranca gottschalkii sp. nov., a novel thermoalkaliphilic bacterium that grows anaerobically at high pH and temperature[END_REF], Anaerobranca horikoshii JWIYL-l3S T (92.6% identity) isolated from Yellowstone National Park (USA) [START_REF] Engle | Isolation and characterization of a novel alkalitolerant thermophile, Anaerobranca horikoshii gen. nov., sp. nov[END_REF], Anaerobranca californiensis PAOHA-1 T (91.7% identity) isolated from alkaline hot spring sediment (Mono Lake, CA, USA) [START_REF] Gorlenko | Anaerobranca californiensis sp. nov., an anaerobic, alkalithermophilic, fermentative bacterium isolated from a hot spring on Mono Lake[END_REF], Anaerobranca zavarzinii JW/VK-KS5Y T (92.2% identity) isolated from a small shallow geothermally heated pool (Kamchatka, Russia) [START_REF] Kevbrin | Anaerobranca zavarzinii sp. nov., an anaerobic, alkalithermophilic bacterium isolated from Kamchatka thermal fields[END_REF],

(
Proteinivorax hydrogeniformans Z-710 T (91.8% identity) and Proteinivorax tanatarense Z-910 T (91.5% identity) isolated from a decaying algal bloom in the alkaline lake Tanatar (Russia) [START_REF] Kevbrin | Proteinivorax tanatarense gen. nov., sp. nov., an anaerobic, haloalkaliphilic, proteolytic bacterium isolated from a decaying algal bloom, and proposal of Proteinivoraceae fam. nov[END_REF][START_REF] Boltyanskaya | Proteinivorax hydrogeniformans sp. nov., an anaerobic, haloalkaliphilic bacterium fermenting proteinaceous compounds with high hydrogen production[END_REF].

The 16S rRNA gene sequence of strain LB2 T was closely related (>97% identity) to few environmental sequences retrieved in August 2020 by a NCBI web BLAST search [START_REF] Altschul | Basic local alignment search tool[END_REF] and originated from low-salt alkaline ecosystems: anaerobic fermentation reactor with waste activated sludge under mesophilic condition (pH 9.0; GU455162) [START_REF] Zhang | Understanding short-chain fatty acids accumulation enhanced in waste activated sludge alkaline fermentation: kinetics and microbiology[END_REF] and surface rock/sediment of Lonar Crater Lake (pH 8.9-10.0; JQ738949) [START_REF] Antony | Molecular characterization of prokaryotic communities associated with lonar crater basalts[END_REF]. 16S rRNA genes affiliated to the family Proteinivoraceae (formerly known as the family Anaerobrancaceae) were also detected in the subsurface hyperalkaline fluids of the Semail Ophiolite (Oman) [START_REF] Rempfert | Geological and geochemical controls on subsurface microbial life in the Samail Ophiolite, Oman[END_REF] and other oligotrophic, rock-hosted ecosystems, such as deep crystalline aquifer of Outokumpu (Finland) [START_REF] Purkamo | Microbial cooccurrence patterns in deep Precambrian bedrock fracture fluids[END_REF]. Still, their potential functional roles in the subsurface remain unclear.

The complete genome sequence of strain LB2 T is a unique circular chromosome of 3.7 Mb long with a 36.2 mol% G+C content (Table 2), close to the G+C content measured by HPLC (36.4 mol%, Table 2). The G+C content of strain LB2 T is higher than that of its closest Anaerobranca relatives (A. californiensis, 30.0-33.4 mol%; A. gottschalkii, 30.0-32.9 mol%; A. horikoshii, 32.0-34.0 mol%; A. zavarzinii, 32.5 mol%), and P. tanatarense (31.9-32.2 mol%) within the family Proteinivoraceae. The genome size of strain LB2 T is much larger than that of its closest phylogenetic neighbor for which genomes are available, A. californiensis (2.0 Mb) and A. gottschalkii (2.3 Mb), but is similar to those of the mesophilic and alkaliphilic genus Alkaliphilus (>3 Mb). The ANI and dDDH values between LB2 T and A. californiensis PAOHA-1 T were 70.61 % and 13.50%. The ANI and dDDH values between LB2 T and A. gottschalkii LBS3 T were 70.78 % and 13.40% (Supplementary Table S1), i.e., far below the species boundaries of 95~96% and 70%, respectively, recommended by Chun et al. [START_REF] Chun | Proposed minimal standards for the use of genome data for the taxonomy of prokaryotes[END_REF]. The LB2 T strain genome comprises a single chromosome and 3,777 genes, including 3,626 proteinencoding genes and 94 RNA encoding genes. Amino acids, carbohydrates, and inorganic ion transport and metabolisms represented 7.4%, 5.7%, and 4.9% of the function-predicted genes.

The distribution of genes into COGs functional categories is shown in Supplementary Table S2. In a phylogenomic analysis (Fig S3), strain LB2 clustered with other alkaliphilic Firmicutes.

It positions in a branch containing members of the genera Alkalibacterium (class Bacilli) and

Serpentinicella (class Clostridia) distantly related to the Anaerobranca species.

The predominant cellular fatty acids (>5% of total fatty acids) were iso-C15:0 (15.3%), C16:0 (10.2%), C14:0 (8.0%), cyclo-C17:0 (6.9%), iso-C17:1 (8.8%), and C18:0 (6.1%). The proportion of iso-C15:0 and C16:0 was lower than that of A. gottschalkii and P. tanatarense, while the iso-C15:0 content was higher than that of A. horikoshii (Supplementary Table S3).

Table 1 summarizes the physiological and biochemical differences between strain LB2 T and the type species of all genera of the family Proteinivoraceae, including A. horikoshii and P. tanatarense. Strain LB2 T shares similar physiological and biochemical properties with this latter (e.g., proteinaceous substrate utilization and high-pH optimal growth ranging between 8.5 and 9.5) but differs by growing optimally higher pH (up to 10.8) compared to other Proteinivoraceae members. Unlike Anaerobranca and Proteinivorax genera, strain LB2 T displays optimal growth in the absence of NaCl addition, similarly to another alkaliphilic anaerobe of the family Clostridiaceae Serpentinicella alkaliphila 3b T , isolated from the serpentinizing Prony Bay Hydrothermal Field of New Caledonia [START_REF] Mei | Serpentinicella alkaliphila gen. nov., sp. nov., a novel alkaliphilic anaerobic bacterium isolated from the serpentinitehosted Prony hydrothermal field, New Caledonia[END_REF]. In contrast to the alkalithermophilic Anaerobranca spp., displaying optimal growth temperatures between 50-60°C, strain LB2 T was mesophilic (the optimum temperature at 37°C) similarly to P. tanatarense and P. hydrogeniformans that grew optimally at 32-38°C and 30-32°C, respectively [START_REF] Kevbrin | Proteinivorax tanatarense gen. nov., sp. nov., an anaerobic, haloalkaliphilic, proteolytic bacterium isolated from a decaying algal bloom, and proposal of Proteinivoraceae fam. nov[END_REF][START_REF] Boltyanskaya | Proteinivorax hydrogeniformans sp. nov., an anaerobic, haloalkaliphilic bacterium fermenting proteinaceous compounds with high hydrogen production[END_REF]. Like other Proteinivoraceae members, strain LB2 T utilizes proteinaceous substrates for growth (i.e., yeast extract). It can also use pyruvate as the sole carbon source, similarly to A. californiensis (or A. zavarzinii that co-metabolizes pyruvate with yeast extract).

As reported for A. zavarzinii, both formate and acetate were the major end products detected. It suggests that pyruvate is mainly metabolized via pyruvate: formate lyase and not via the pyruvate: ferredoxin oxidoreductase as commonly reported in Firmicutes. In the latter case, acetate plus H2 would be the major metabolic end-products. Genomic data supported this hypothesis. The gene encoding the formate lyase, also known as formate C-acetyltransferase (EC 2.3.1.54), is found in the genome of strain LB2 T and not the gene of the pyruvate: ferredoxin oxidoreductase (PFOR, EC 1.2.7.1). The ability of strain LB2 T to use both ferric citrate and thiosulfate as electron acceptors were also shared with A. californiensis. Sugars (i.e., xylose) was used by strain LB2 T , similarly to A. gottschalkii, as well as A. californiensis, A. zavarzinii, 'P. hydrogeniformans' and P. tanatarense ferment some sugars, but in the presence of tryptone or yeast extract.

In conclusion, the strain LB2 T differs from other species of the closest genera Anaerobranca and Proteinivorax (belonging to the family Proteinivoraceae) by several features: (i) it forms a distinct branch among the family Proteinivoraceae in the 16S rRNA gene phylogenetic trees, (ii) it is only distantly related to Anaerobranca spp. in phylogenomic analyses, (iii) it displays a larger genome size and G+C content than other Proteinivoraceae members, (iv) it grows optimally at higher pH (up to 10.8) than other Proteinivoraceae members, (v) it grows optimally in the absence of NaCl addition, (vi) it has a much lower optimal growth temperature (37°C) than the closest Anaerobranca members, (vii) it is the only one in the family Proteinivoraceae (including pleomorphic bacteria) occurring as rods or cocci depending on the growth phase, as observed for the actinobacterial genus Arthrobacter [START_REF] Luscombe | Characteristics of Arthrobacter grown in continuous culture[END_REF]. In contrast, cells of the genus Anaerobranca are rod-shaped, sometimes showing branching [START_REF] Engle | Isolation and characterization of a novel alkalitolerant thermophile, Anaerobranca horikoshii gen. nov., sp. nov[END_REF][START_REF] Kevbrin | Anaerobranca zavarzinii sp. nov., an anaerobic, alkalithermophilic bacterium isolated from Kamchatka thermal fields[END_REF]. Based on morphological, phenotypic, phylogenetic, and genomic, strain LB2 T is proposed to represent a novel species of the new genus Alkalicella, A. caledoniensis gen. nov., sp. nov. Alkalicella caledoniensis (ca.le.do.ni.en′sis. N.L. fem. adj. caledoniensis pertaining to New Caledonia, where strain LB2 T was first isolated).

Displays the following characteristics in addition to those listed in the genus description. Cells are approximately 6.0-10.0 µm long and 0.5-0.6 µm wide, occurring singly or in pairs. Cells are motile and possess multiple flagella. They are pleiomorphic depending on the growth phases (from rod to cocci). Colonies are white and circular and about 1-2 mm in diameter after seven days of incubation at 30°C. Growth occurs at 25-50°C (optimum 37°C), at pH 8.2-10.8 (optimum pH 9.5) with 0-7% NaCl (optimum 0-1%). Added NaCl was not required for growth. Pyruvate, yeast extract, xylose, and peptone were used as electron donors. Still, adenine, adenosine 5'-diphosphate, albumin, arabinose, casamino-acids, casein, cellobiose, fructose, galactose, gelatin, glucose, glycerol, guanine, lactate, lactose, mannose, ribose, starch, sucrose, tryptone, uracil, and H2/CO2 were not. End products detected from pyruvate fermentation were acetate and formate. Both ferric citrate and thiosulfate were used as electron acceptors using pyruvate and reduced to ferrous iron and sulfide. Yeast extract was degraded into acetate, formate, H2, and propionate. Elemental sulfur, nitrate, nitrite, fumarate, sulfate, sulfite, and DMSO were not used as terminal electron acceptors. The predominant fatty acids were iso-C15:0 (15.3%), C16:0 (10.2%), C14:0 (8.0%), cyclo-C17:0 (6.9%), iso-C17:1 (8.8%), and C18:0 (6.1%). The GenBank accession number for the 16S rRNA gene sequence of strain LB2 T is KR349724, and that for the whole genome sequence is CP058559.

The type strain LB2 T (= DSM 100588 T = JCM 30958 T ) was isolated from a sediment slurry of 'La Crouen' thermal spring near Canala, New Caledonia.

Figures legends

  ML, Fig 2), maximum-parsimony (MP, Fig S2A), and neighbor-joining (NJ, Fig S2B). The closest relatives of strain LB2 T are type species: Anaerobranca gottschalkii LBS3 T (93.2%

  Description of Alkalicella gen. nov. Alkalicella (Al.ka.li.cel'la. N.L. n. alkali from Arabic al-qaliy ashes of saltwort; L. fem. n. cella a storeroom, chamber, and in biology a cell; N.L. fem. n. Alkalicella, a cell living in an alkaline environment).Gram-stain-positive (terminal and oval) endospore-forming bacterium with anaerobic chemoorganotrophic metabolism. Occurs as rods or cocci under different conditions. Able to utilize sugars and proteinaceous substrates. Use ferric citrate and thiosulfate as electron acceptors. Alkaliphilic and mesophilic. Displays optimal growth in the absence of NaCl addition. Belongs to the family Proteinivoraceae, order Clostridiales, class Clostridia within the phylum Firmicutes. Most closely related to Anaerobranca gottschalkii LBS3 T (93.2% 16S rRNA gene sequence identity). The G+C content of the genomic DNA is 36.2%. The type species is Alkalicella caledoniensis, and the genus description is based on that of the type species, currently the only species ascribed to this genus. Description of Alkalicella caledoniensis sp. nov.

Fig 1 :

 1 Fig 1: Phase-contrast micrographs of cells of strain LB2 T with terminal and oval endospore formation (A; bar 10 µm), and transmission electron micrograph showing cells of strain LB2 T

Fig. 2 :

 2 Fig. 2: Maximum-likelihood (ML) phylogenetic tree based on 1245 aligned bp of 16S rRNA gene sequences showing the position of strain LB2 T among the other family members

Table 1 .

 1 Differential characteristics between strain LB2 T (obtained in this study) and data obtained from previous studies for all type species of all genera of the family Proteinivoraceae. Strains: 1, Strain LB2 T ; 2, Anaerobranca horikoshii JWIYL-l3S T[START_REF] Engle | Isolation and characterization of a novel alkalitolerant thermophile, Anaerobranca horikoshii gen. nov., sp. nov[END_REF]; 3, Anaerobranca gottschalkii LBS3 T[START_REF] Prowe | Anaerobranca gottschalkii sp. nov., a novel thermoalkaliphilic bacterium that grows anaerobically at high pH and temperature[END_REF]; 4, Anaerobranca californiensis PAOHA-1 T [47]; 5, Anaerobranca zavarzinii JW/VK-KS5Y T[START_REF] Kevbrin | Anaerobranca zavarzinii sp. nov., an anaerobic, alkalithermophilic bacterium isolated from Kamchatka thermal fields[END_REF]; 6, Proteinivorax tanatarense Z-910 T[START_REF] Kevbrin | Proteinivorax tanatarense gen. nov., sp. nov., an anaerobic, haloalkaliphilic, proteolytic bacterium isolated from a decaying algal bloom, and proposal of Proteinivoraceae fam. nov[END_REF]; 7, Proteinivorax hydrogeniformans Z-710 T[START_REF] Boltyanskaya | Proteinivorax hydrogeniformans sp. nov., an anaerobic, haloalkaliphilic bacterium fermenting proteinaceous compounds with high hydrogen production[END_REF].

	Characteristic	1	2	3	4	5	6	7
	Origin	La Crouen spring, New Caledonia	Yellowstone National Park, USA	Lake Bogoria, Kenya	Mono Lake, USA	Kamchatka, Russia	Lake Tanatar VI, Russia	Lake Tanatar III, Russia
	Cell size (µm)	0.5-0.6 x 6-10	0.5-0.6 x 11-20	0.3-0.5 x 3-5	0.26-0.31 x 2.4-5.0	0.36-0.45 x 3.6-7.7	0.5 x 1.6-4	0.5-0.6 x 2.0-4.5
	Spore formation	+	-	-	-	-	+	-
	Motility	+	+	+	+	-	+/-	+
	Temperature for growth (°C)
	Range	25-50	34-64	30-65	45-70	34-64	16-52	12-45
	Optimum	37	57	50-55	58	54-60	32-38	30-32
	pH for growth
	Range	8.2-10.8*	6.7-10.3	6.0-10.5	8.6-10.4	7.7-9.9	7.6-10.5	7.7-10.4
	Optimum	9.5	8.5	9.5	9.0-9.5	8.5-9.0	8.8	8.3-8.9
	NaCl for growth (%, w/v)
	Range	0-7	ND	0-4	0-6	ND	0.4-3.6	0.7-3.0
	Optimum	0*-1	ND	1	1-2.5	ND	2-3	1.0-1.4
	DNA G+C content (mol%) 1	36.2 1*	34 2	30.9 2	30 2	32.5 2	32.2 2	33.9 2
	Substrates	Peptone, pyruvate, xylose, yeast extract	Peptone, tryptone, yeast extract	Peptone, tryptone, yeast extract, mono-and polysaccharides	Peptone, soytone, malt extract, Casamino acids	Soytone and yeast extract	Albumine, casein, gelatin, peptone, yeast extract soytone, tryptone,	Peptone, soytone, tryptone, yeast extract
							From tryptone:	From tryptone:
		From yeast extract:					acetate, formate, NH4,	acetate, hydrogen,
	Fermentation products	acetate and propionate; from pyruvate: acetate and	From yeast extract: acetate, H2, CO2	From glucose or starch: Acetate, ethanol (traces)	ND	From YE: Formate, acetate, H2 (major), propionate (minor)	H2, propionate, succinate, n-butyrate, iso-butyrate, 2-	ammonium (major), formate, propionate, n-butyrate, iso-
		formate					methylbutyrate, iso-	butyrate, iso-valerate
							valerate	and succinate (minor).
	Electron acceptors	Ferric citrate, thiosulfate	Fumarate	Thiosulfate	Sulfur, polysulfide, thiosulfate, ferric citrate	Fumarate, thiosulfate, sulfur	Fumarate, DMSO, thiosulfate, sulfur	Fumarate, thiosulfate, sulfur
	+, Positive; -, negative; ND, not data available.

1

G+C content determined from genomic data.

2

G+C content determined by using HPLC. * Characteristics distinguishing the strain LB2 T from others presented.

Table 2 .

 2 Genome statistics of strain LB2 T , Anaerobranca californiensis PAOHA-1 T[START_REF] Gorlenko | Anaerobranca californiensis sp. nov., an anaerobic, alkalithermophilic, fermentative bacterium isolated from a hot spring on Mono Lake[END_REF], and Anaerobranca gottschalkii LBS3 T[START_REF] Prowe | Anaerobranca gottschalkii sp. nov., a novel thermoalkaliphilic bacterium that grows anaerobically at high pH and temperature[END_REF].

	Genome Name /	LB2 T	Anaerobranca	Anaerobranca
	Sample Name		californiensis	gottschalkii LBS3 T
			PAOHA-1 T
	GenBank accession	CP058559	NZ_FRAI00000000 NZ_FOIF01000000
	number
	IMG Genome ID	2888992949	2585428191	2602042032
	Genome Size (bp)	3,695,371	2,034,462	2,300,172
	Contig count	NA	51	115
	Scaffold count	1	NA	110
	GC (%)	36.2	33.4	32.9
	Gene Count	3,777	2,132	2,382
	Protein count	3,626	2,006	2,246
	rRNA Count	27	22	16
	tRNA Count	62	52	55
	Other RNA	5	5	4

NA, data not applicable.

Funding Information

This project was financially supported by the French Research Institute for Development (IRD), the INTERVIE CNRS INSU program, the H2NAT project funded by the Midi-Pyrénées Observatory (OMP, Toulouse) and ANR MICROPRONY (N°19-CE02-0020-02). We thank Céline Roques at the GeT core facility, Toulouse, France (http://get.genotoul.fr) for the genome sequencing of LB2 strain, supported by France génomique. The LABGeM (CEA/Genoscope & CNRS UMR8030) is acknowledged for support within the MicroScope annotation platform.

Conflicts of interest

The authors declare that there are no conflicts of interest.


Toufik Bakir

Bernard Bonnard

Sandrine Gayrard

Jérémy Rouot

Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation

Keywords: Biomechanics, Force-fatigue models, Sampled-data control, Nonlinear input-output approximation, Predictive-correction methods in optimization

Recent dynamical models, based on the seminal work of V. Hill, allow to predict the muscular response to functional electrostimulation (FES), in the isometric and non-isometric cases. The physical controls are modeled as Dirac pulses and lead to a sampled-data control system, sampling corresponding to times of the stimulation, where the output is the muscular force response. Such a dynamics is suitable to compute optimized controls aiming to produce a constant force or force strengthening, but is complex for real time applications. The objective of this article is to construct a finite dimensional approximation of this response to provide fast optimizing schemes for the design of a smart electrostimulator for muscular reinforcement or rehabilitation. It is an on-going industrial project based on force-fatigue models, validated by experiments. Moreover it opens the road to application of optimal control to track a reference trajectory in the joint angular variable to produce movement in the non-isometric models.

Introduction

Based on the seminal work of V. Hill depicted in [START_REF] Gesztelyi | The Hill equation and the origin of quantitative pharmacology[END_REF], recent mathematical models (experimentally validated) allow to predict the force response to external stimulation. They are presented and discussed in details in [START_REF] Wilson | Force response of locust skeletal muscle[END_REF] in the non-fatigue This paper was not presented at any IFAC meeting. It benefited from the support of the FMJH Program PGMO and from the support of EDF; Thales, Orange and the authors are partially supported by the Labex AMIES. Corresponding author J. Rouot. Email. jeremy.rouot@univbrest.fr.

Email addresses: toufik.bakir@u-bourgogne.fr (Toufik Bakir), bernard.bonnard@u-bourgogne.fr (Bernard Bonnard), sandrine.gayrard@grenoble-inp.org (Sandrine Gayrard), jeremy.rouot@univ-brest.fr (Jérémy Rouot). isometric case. They were extended in particular by [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF]Ding et al. ( , 2002a,b) ,b) to take into account the muscular fatigue due to a long stimulation period and later in [START_REF] Marion | Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration[END_REF] to analyze the joint angular variable response in the non-isometric case aiming to produce movements. Such models contain two basic nonlinearities and constitute the intricate part of the dynamics. First of all, the ionic conduction and the nonlinear effect of successive pulses on the Ca 2+ -concentration. Second, the nonlinear dynamics relating the muscular force response to such concentration, modeled by the Michaelis-Menten-Hill functions, see [START_REF] Michaelis | Die Kinetik der Intertinwerkung[END_REF].

For each train of pulses, due to digital constraints, only a finite number of pulses can be applied and from the optimal control point of view, the problem fits into the frame of optimal sampled-data control problems, studied in particular in [START_REF] Bourdin | Optimal sampled-data control, and generalizations on time scales[END_REF] to derive Pontryagin necessary conditions. They can be analyzed to determine optimized train pulses and compared with direct optimizing schemes. A previous series of articles described the optimal control problems related to track a reference force or force strengthening, the control being either the interpulse I i = t i -t i-1 between two successive pulses or the amplitude of each pulse. In particular, a model predictive control (MPC) method is presented in [START_REF] Bakir | A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model[END_REF] and is based on online optimized closed loop control using a force-fatigue model, as suggested in [START_REF] Doll | Optimization of a Stimulation Train based on a Predictive Model of Muscle Force and Fatigue[END_REF]. Direct methods vs indirect methods based on Pontryagin type necessary conditions are discussed and numerically implemented in [START_REF] Bakir | Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations[END_REF] for the isometric case and in [START_REF] Bakir | Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations[END_REF] for the non-isometric case.

The conclusion of aforementioned articles is that the nonlinear dynamics is computationally expensive due to the numerical integration procedure and a challenging task is to reduce this time for real time computation in the applications. This article is motivated by the design of a smart electrostimulator, where the Ding et al. model is used to adjust automatically the frequency and the amplitude of the stimulations and to compute the sequence of stimulations and rest periods adapted to the task of the training program, e.g. endurance program or force strengthening program. The objective of this article being to bypass the computational difficulty by constructing a finite dimensional approximation of the force response, depending upon the parameters of each individual, which can be online estimated, aiming a real time computation of the optimized amplitudes and impulse times. Note that this approximation has been coded and the application scheme to the smart electrostimulator is briefly presented in the final section.

The article is organized as follows. In section 2, the mathematical model called the Ding et al. model presented by [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF]Ding et al. ( , 2002a,b) ,b) is introduced and the main properties of the dynamics are described, reflecting the features of the muscular activity. Hence our analysis can be applied to different models discussed in [START_REF] Wilson | Force response of locust skeletal muscle[END_REF]. The section 3 presents the optimization problems, in relation with muscular dynamics and oriented towards the design of a smart electrostimulator, where each training program must be translated into an optimization problem. Section 4 is the technical contribution of this article, that is the construction of the approximation for real time computation. In section 5, we present some numerical simulations to validate the approximation and the optimizing scheme. In the final section 6, we outline the application to the design of the smart electrostimulator. It is based on a nonlinear output tracking, see [START_REF] Hirschorn | Output tracking for nonlinear systems with singular points[END_REF][START_REF] Hirschorn | Global output tracking for nonlinear systems[END_REF]; [START_REF] Isidori | Nonlinear Control Systems[END_REF] as a general theoretical frame, and is applied to produce a constant force in our situation, but it can be extended to the non-isometric case to obtain any reference force. The conclusion indicates directions to complete our analysis, related to online parameters estimation of the problems [START_REF] Stein | Ecological Modeling from Time-Series Inference: Insight into Dynamics and Stability of Intestinal Microbiota[END_REF]; [START_REF] Wilson | Force response of locust skeletal muscle[END_REF] and MPC-methods presented in [START_REF] Richalet | Industrial applications of model based predictive control[END_REF]; [START_REF] Wang | Fast Model Predictive Control using Online Optimization[END_REF] suitable for practical applications.

Force-fatigue model

We present the Ding et al. force-fatigue model [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF]Ding et al. ( , 2002a,b),b), extension of the original Hill model [START_REF] Gesztelyi | The Hill equation and the origin of quantitative pharmacology[END_REF].

Ding et al. force fatigue model.

The FES input over a pulse train [0, T ] is the signal

t → n i=0 η i δ(t -t i ), t ∈ [0, T ],
(1) Such physical control provides the FES-signal

where 0 = t 0 < t 1 < • • • < t n < t n+1 = T
E(t) = 1 τ c n i=0 R i e -t-t i τc η i H(t -t i ),
where H is the Heaviside function. E depends upon the time response parameter τ c and the scaling function R i is defined, given a fatigue parameter R, by

R i = 1 if i = 0
1 + (R -1) e -(ti-ti-1)/τc for i = 1, . . . , n, which codes the memory effect of successive muscle contractions and is associated to tetanus [START_REF] Wilson | Force response of locust skeletal muscle[END_REF].

The FES signal drives the evolution of the electrical conduction describing the evolution of Ca 2+ -concentration c N , which is related to the force F and the fatigue variables (A, K m , τ 1 ) through the Michaelis-Menten-Hill functions introduced in [START_REF] Michaelis | Die Kinetik der Intertinwerkung[END_REF] and defined by

m 1 (t) = c N (t) K m (t) + c N (t) , m 2 (t) = 1 τ 1 (t) + τ 2 m 1 (t) . (2)
The time evolution of these variables is governed by the Ding et al. force-fatigue model

ċN (t) = E(t) - c N (t) τ c , (3)
Ḟ (t) = -m 2 (t) F (t) + m 1 (t)A(t), (4)
Ȧ(t) = - A(t) -A 0 τ f at + α A F (t), (5)
Km (t) = - K m (t) -K m,0 τ f at + α Km F (t), (6) τ1 (t) = - τ 1 (t) -τ 1,0 τ f at + α τ1 F (t), (7)
where typical parameters values are reported in Table 1.

2.2 Non-fatigue model.

The variables ξ 1 (t) = α Km (A(t) -A 0 ) -α A (K m (t) - K m,0 ) and ξ 2 (t) = α τ1 (A(t) -A 0 ) -α A (τ 1 (t) -τ 1,0 ) satisfy the linear differential equations ξi (t) = - 1 τ f at ξ i (t), i = 1, 2,
and we obtain an explicit expression of K m (t) and τ 1 (t) as soon as A(t) is explicit. Moreover, the fatigue dynamics ( 5) is linear and can be integrated explicitly once F -or its approximation -is fixed in some appropriate category of functions. This is the purpose of Section 4 to provide such category in which an approximation of F is derived.

Therefore, the intricate part of model ( 3)-( 7) consists of the nonlinear force dynamics (4), which is related to the historical Hill model recollected by [START_REF] Gesztelyi | The Hill equation and the origin of quantitative pharmacology[END_REF] and we shall restrict our mathematical analysis to the case where the fatigue variables (A, K m , τ 1 ) have constant dynamics (non-fatigue model).

The concentration c N (•) can be written explicitly as

c N (t) = 1 τ c n i=0 R i η i (t -t i ) e -t-t i τc H(t -t i ), (8)
and the model considered in this paper consists of the nonlinear non-fatigue model

Ḟ (t) = -m 2 (t) F (t) + m 1 (t)A, (9)
where the fatigue variables 1).

(A(•), K m (•), τ 1 (•)) ≡ (A 0 , K m,0 , τ 1,0 ) are constant (see Table
This restriction to a simpler model is not a limiting aspect of the method presented in Section 4, which can be adapted to various models derived from the Ding et al. model, in particular in the nonisometric case [START_REF] Marion | Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration[END_REF]. More generally, we believe that our method is suitable for models arising from a sampled-data control framework but this generalization is not the purpose of this article.

Sampled-data control system. Writing E as

E(t) = e -t/τc τ c n i=0 R i η i e -ti/τc H(t -t i ) ui(t)
, the non-fatigue model can be formulated as

ẋ(t) = g(x(t)) + e -t/τc τ c n i=0 u i (t) e, ( 10
)
where x = (c N , F ), g is defined from ( 3) and ( 9), e = (1, 0) and u i (t) is the effect of the pulse and is constant over each interval [t i , t i+1 ], i = 0, . . . , n.

Observe that each impulse has an effect on the whole train [0, T ]. We come accross a sampled-data control problem defined below.

Definition 1 Consider a control system of the form:

dx dt = f (x, u) where x ∈ R d , u ∈ U ⊂ R m . It is said permanent if u is a measurable bounded mapping valued in U . It is called a sampled-data control system if the set of controls is restricted to the set of piecewise cons- tant mappings [u 0 , u 1 , . . . , u n ], u i ∈ U over a set of times t 0 = 0 < t 1 < • • • < t n < T = t n+1 ,
where n is a fixed integer.

Input-Output mapping. The variable σ

= (t 1 , . . . , t n , η 0 , η 1 , . . . , η n ) belongs to the finite dimen- sional input-space I ⊂ [0, T ] n × [0, 1] n+1 defined by the constraints 0 ≤ η i ≤ 1, i = 0, . . . , n, t 0 = 0 < t 1 < • • • < t n < t n+1 = T, t i -t i-1 ≥ I min , i = 1, . . . , n ,
where I min is the smallest admissible interpulse.

The control system is observed using the following observation function

y(t; σ) = h(x(t; σ)), (11)
and h : x → F serves as a direct measure of the muscular force, where to emphasize the dependence of a function, say ψ, with respect to some parameters γ, we use the notation ψ(•; γ).

We derive an important regularity property of the inputoutput function, which is the object of study of this article.

Integrating (9), the force with F (0) = 0 can be written for t ∈ [0, T ] as 3 Optimization problems related to the design of the electrostimulator

F (t; σ) = A 0 M (t; σ) t 0 M -1 (s; σ)m 1 (s; σ) ds, (12)

Standard electrostimulators vs smart electrostimulators

The standard commercial electrostimulators apply a sequence of pulses trains and rest periods, where on each train [0, T ] the user only imposes the amplitude of the pulses trains and the frequency is related to the training program, typically low frequency for endurance program and high frequency for force strengthening program. Our aim is to introduce optimization problems related to the design of a smart electrostimulator, which will be discussed in Section 6.

Optimization problems

The punch program

In this case, we wish to optimize the force at the end of the train over each train [0, T ]. This leads to:

OCP1: max σ F (T ).
The amplitudes can be held at the constant maximal values η i = 1, i = 0, . . . , n and the optimization variables are the impulse times:

0 = t 0 < t 1 < • • • < t n < T.
Since one considers a single train, the force model is sufficient.

The train endurance program

We consider a single train [0, T ] on which the model is the force model and the corresponding problem is

OCP2: min σ T 0 |F (t) -F ref | 2 dt.
Here, the amplitudes are appended to the impulse times to form the optimization variables and we use the convexified amplitudes constraints:

η i ∈ [0, 1], i = 0, . . . , n.
The reference force has to be adjusted in relation with the user and can be set to F max /ρ, where ρ is a suitable positive number greater than 1 and F max is deduced from OCP1.

4 Construction of an integrable model for real time application

Mathematical analysis of c N

The response c N to a pulses train σ = (t 1 , . . . , t n , η 0 , . . . , η n ), given by ( 8), is a sum of lobes 13)) reaches its maximum at t = t k +τc and is equal to

R k η k /e. It is zero for t ≤ t k , strictly increasing on [t k , t k + τc] and strictly decreasing [t k + τc, T ]. k is concave on [t k , t k + 2τc] and convex on [t k + 2τc, T ]. For all t ≥ t k + 5τc, | k (t)| ≤ R 5e -5 ≤ 3.4%R. k , k = 0, . . . , n defined as k (t) := R k η k t -t k τ c e -(t-t k )/τc H(t -t k ). ( 13
)
The mathematical properties of a lobe are outlined in Fig. 1.

Notation. Throughout the paper, the notation g k refers to the restriction of a function g : [0, T ] → R on [t k , t k+1 ] and g k refers to the average of

g k on [t k , t k+1 ].
Computing, we have

c k N (t) = k i=0 R i η i t -t i τ c e -(t-ti)/τc . ( 14
)
4.2 Approximation of Michaelis-Menten-Hill functions.

For k ∈ {0, . . . , n}, we have ṁ1 = K m ċN /(K m + c N ) 2 and ṁ2 = τ 2 ṁ1 /(τ 1 + τ 2 m 1 ) 2 , therefore m k 1 (resp. m k 2 ) is maximal (resp. minimal) when c k N is maximal, that is at time t * k := argmax t∈[t k ,t k+1 ] c k N (t)
and, from ( 14), we get

t * k = τ c + k i=0 R i η i t i e ti/τc k i=0 R i η i e ti/τc ≤ t k +τ c . ( 15
)
In the rest of the paper, we assume that (η 0 , . . . , η n ) is a nonzero vector and

I min ≥ τ c to ensure that t k+1 ≥ t k + τ c ≥ t * k . It follows that m k 1 (resp. m k 2 ) is strictly increasing (resp. decreasing) on [t k , t * k ] and strictly decreasing (resp. in- creasing) of [t * k , t k+1 ].
Computing, we have

mk 1 = K m (K m + c k N ) 2 ck N - 2 ċk N 2 K m + c k N and ck N (t) = k i=0 R i η i τ 2 c t -t i τ c -2 e -(t-ti)/τc .
Hence, we draw:

ck N (t) ≤ 0 ⇔ k i=0 R i η i (t -t i -2τ c )e ti/τc ≤ 0 ⇔ t ≤ t * k + τ c and m k 1 is concave on [t k , min(t * k + τ c , t k+1 )]. Likewise, mk 2 = - A 0 τ 2 (τ 1 + τ 2 m k 1 ) 2 mk 1 - 2τ 2 ṁk 1 2 τ 1 + τ 2 m k 1 and m k 2 is convex on [t k , min(t * k + τ c , t k+1 )].
We define the category of functions in which we will construct the approximation of the input-output mapping.

Definition 3 A function f : [0, T ] → R is in the (piecewise smooth) polynomial-exponential category C pe if f can be expressed on each interval [t k , t k+1 ] as sums of products of polynomials P i and exponential terms : for

t ∈ [t k , t k+1 ], f k (t) = i P i (t)e λit , λ i ∈ R.
The category C pe is stable with respect to integration and derivation. The c N function is in C pe but m 1 , m 2 , F do not.

An important question is to determine the "best category" of functions we should use to construct approximations of m 1 , m 2 , F . Since the functions m 1 , m 2 are (piecewise) concave and convex respectively, it is natural to compute (piecewise) polynomial approximations m 1 , m 2 as follows.

We consider a finer partition of (t i ) 1≤i≤n denoted as (t i+j/p ) 0≤i≤n, 0≤j≤p-1 , p ∈ N * , such that it satisfies

t i < t i+1/p < • • • < t i+(p-1)/p < t i+1 .
The approximation of m 1 , m 2 goes as follows. For simplicity, we present the construction on a specific example used for the numerical simulations in Section 5. We choose p = 2 and on [t k+j/2 , t k+(j+1)/2 ], j = 0, 1, the functions m 1 , m 2 , are approximated by the piecewise affine functions:

m k 1 (t) = a j,k (t -t k+j/2 ) + b j,k , m k 2 (t) = m 2 (t k+j/2 ) + m 2 (t k+(j+1)/2 ) 2 ,
for k = 0, . . . , n, where t k+1/2 := t * k is given by ( 15) and a j,k , b j,k are parameters computed with the conditions m 1 (t k+j/2 ) = m 1 (t k+j/2 ) and m 1 (t k+(j+1)/2 ) = m 1 (t k+(j+1)/2 ). It follows

a j,k = m 1 (t k+(j+1)/2 ) -m 1 (t k+j/2 ) t k+(j+1)/2 -t k+j/2 , b j,k = m 1 (t k+j/2 ).
The key point is that the coefficients a j,k , b j,k depend explicitly on t 1 , . . . , t n , T and can be generated using symbolic computation.

A geometric picture of this construction is given in Fig. 2.

Approximation of F

Using (12), we construct an explicit approximation F of F using the approximations m 1 and m 2 as follows.

Proposition 4 Choosing m 1 in C pe and m 2 as a piecewise constant function on [0, T ], the function:

F (t) = A 0 t 0 M (t) M -1 (s) m 1 (s) ds, ( 16
)
where

M (t) = exp - t 0 m 2 (s) ds , has a closed-form expression in C pe .
PROOF. Decomposing the integral ( 16) as a sum of integrals over the partition (t i+j/p ) ij gives the expression:

F (t)/A 0 = kt-1 i=0 p-1 j=0 t i+(j+1)/p t i+j/p M (t) M -1 (s) m 1 (s) ds + jt-1 j=0 t k t +(j+1)/p t k t +j/p M (t) M -1 (s) m 1 (s) ds + t t k t +j t /p M (t) M -1 (s) m 1 (s) ds, ( 17
)
for t ∈ [t kt+jt/p , t kt+(jt+1)/p ], k t = 0, . . . , n, j t = 0, . . . , p -1. The term M (t) M -1 (s) m 1 (s) = exp t s m 2 (u) du m 1 (s) has an explicit expression in C pe since m 2 is piecewise constant and m 1 is in C pe . Therefore F has an explicit expression in C pe .

For practical reason, it is important to obtain upper and lower approximations of the force F . The following proposition goes into this direction and is proved in the Appendix.

Proposition 5 Take k ∈ {0, . . . , n}. Assume m 2 is a piecewise constant approximation of m 2 such that m k 2 (.; ν) = ν m k 2 (.). Then there exists ν + ≥ 1 (resp. ν -≤ 1) such that the approximation F + (.; ν) (resp. F -(.; ν)), defined by ( 16) with m 2 is replaced by m 2 (.; ν), satisfies

F + (t k ; ν) ≤ F (t k ) for all ν ≥ ν + ≥ 1 if 0 ≤ m 1 ≤ m 1 on [0, T ], and
F -(t k ; ν) ≥ F (t k ) for all ν ≤ ν -≤ 1 if 0 ≤ m 1 ≤ m 1 on [0, T ].
Error estimate. The following proposition gives an upper bound on the error between F and F and is proved in the Appendix.

Proposition 6 Assume that m 1 is in C pe satisfying 0 ≤ m 1 (t) ≤ 1 and m 2 is the piecewise constant appro- ximation defined by m k 2 = m k 2 for k = 0, . . . , n.
Then, the error between the force F and its approximation F defined by (16) satisfies for k = 0, . . . , n:

|F (t k ) -F (t k )|/A 0 ≤ t k 0 |m 1 (s) -m 1 (s)| ds + t k t k 0 |m 2 (s) -m 2 (s)| ds.
5 Numerical results

Functional specification for the computation of a pulses train

The aim is to compute a local minimum σ * = (η * 0 , . . . , η * n , t * 1 , . . . , t * n , T ) ∈ R 2n+2 + of a cost function denoted as Θ depending on the training program we have chosen. The free final time T adjusts automatically the optimal frequency of the pulses train. The functional specification of the electrostimulator imposes efficient computation of this minimum (real time computation) and this prevents us (at least when Θ involves the force) from using direct or indirect methods such as those presented in [START_REF] Bakir | Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations[END_REF], mainly because these methods are based on a numerical integration scheme to approximate the variable F .

Our numerical scheme presented in this section first generates using symbolic computations the explicit expression of an approximation of Θ denoted as Θ in terms of the optimization variables η i , t i . This expression can be computed offline and constitutes the objective function of the optimization routine.

Finite dimensional optimization methods

We recall basic facts about finite dimensional optimization, see [START_REF] Boyd | Convex Optimization[END_REF] for details, to emphasize that an optimal sampled-data control problem can be viewed as an instance of such optimization problem.

The optimization problems, associated to the optimal sampled-data control problems OCP1 and OCP2 presented in Section 3.2, can be written in the form:

min σ Θ(σ), (σ) ≤ 0, (18)
where (σ) = (Ξ 1 (σ), . . . , Ξ 3n+5 (σ)) is the vector of constraints defined by:

Ξ i (σ * ) = t * i-1 -t * i + I min , i = 1, . . . , n, Ξ n+1 (σ * ) = t * n -T, Ξ n+2+i (σ * ) = -η * i , i = 0, . . . , n + 1, Ξ 2n+4+i (σ * ) = η * i -1, i = 0, . . . , n + 1.
The cost Θ : σ → Θ(σ) is related to the endurance or the force strengthening program and is smooth with respect to σ (see Proposition 2). Consider the Lagrangian defined for all (σ, µ) ∈ R 2n+1 × R 3n+5 + by:

L(σ, µ) := Θ(σ) + µ • (σ).
The problem (18) can be written as inf

σ∈R 2n+1 sup µ∈R 3n+5 + L(σ, µ)
and its dual as sup µ∈R 3n+5 + inf σ∈R 2n+1 L(σ, µ). First order necessary optimality conditions for σ * to be a local minimizer state that there exists a Lagrange multiplier λ ∈ R 3n+5 such that

∇ σ L(σ * , λ) = 0, λ i Ξ i (σ * ) = 0 λ i ≥ 0, Ξ i (σ * ) ≤ 0, i = 1, . . . , 3n + 5.
We usually do not solve directly these optimality conditions to compute an primal-dual pair (σ * , λ), but a relaxation of these conditions can lead to efficient algorithm, namely the primal-dual interior point method [START_REF] Boyd | Convex Optimization[END_REF].

Force optimization

We consider the endurance and force strengthening optimization problems OCP1 and OCP2. For each problem, we give the approximation Θ of the cost functions Θ based on the approximation F of the variable F described in section 4. We solve the associated problem ( 18) -where Θ is replaced by its approximation Θ -using an interior point method on a standard computer1 . Note that Θ may consist of million of bytes, for that reason we compute an approximation of the gradient of Θ with respect to t i , i = 1, . . . , n using finite differences (although formal computation of the gradient of Θ may be carried out offline). We initialize the pulses train to a regular partition of [0, 1] and the initial amplitudes to 1.

The force approximation F is defined by ( 16) and the piecewise affine functions m 1 , m 2 are taken to be equal on [t k , t k+1 ], k = 0, . . . , n to:

m 1 (t) = m 1 (t k+1/2 ) if t ∈ [t k , t k+1/2 ] a 1j,k (t -t k+1 ) + b 1j,k , if t ∈ [t k+1/2 , t k+1 ] , m 2 (t) = m2(tk)+m2(t k+1/2 ) 2 if t ∈ [t k , t k+1/2 ] m2(t k+1/2 )+m2(tk+1) 2 , if t ∈ [t k+1/2 , t k+1 ] , ( 19
)
where

t k+1/2 = argmax u∈[t k ,t k+1 ] c N (u), a 1j,k = (m 1 (t k+1 ) - m 1 (t k+1/2 ))/(t k+1 -t k+1/2
) and b 1j,k = m 1 (t k+1 ).

Problem OCP1.

The objective function Θ(σ) = -F (T ) is approximated by the function Θ(σ) = -F (T ). The optimization varia-bles consist in the impulse times while the amplitudes are fixed to 1.

Numerical result:

The optimal solution σ * , the force response F and its approximation F are depicted in Fig. 3.

Fig. 3. The optimal solution σ * = (t * 1 , . . . , t * n , T ) minimizes the cost Θ(σ) = maxσ F (T ) (T free) (see ( 16) for the definition of F ) under the constraints Ξi ≤ 0, i = 1, . . . , n + 1 (see ( 18)). The continuous curve is the response t → F (t) to σ * . Values of the constants are τc = 20ms, n = 7, Imin = 20ms.

Problem OCP2

.

The cost Θ(σ) = T 0 |F (s) -F ref | 2 ds is approximated by: Θ(σ) = n k=0 F + (t k+1 ; ν) -F ref 2 (t k+1 -t k ),
where the upper approximation F + (.; ν) (introduced in Proposition 5) is defined from m 1 given by ( 19) and m 2 (.; ν) given by

m k 2 (t; ν) = ν m2(t k )+m2(t k+1/2 ) 2 if t ∈ [t k , t k+1/2 ] m2(t k+1/2 )+m2(t k+1 ) 2 , if t ∈ [t k+1/2 , t k+1 ]
.

Numerical result: The optimal solution σ * , the force response F and its upper approximation F + (.; ν = 0.8) are depicted in Fig. 4. We also represent the lower approximation F -(.; ν = 1.1), which is the response force for the input σ * .

6 Isometric case: design of a smart muscular electrostimulator

In this section we apply our study in the isometric case associated to the conception of a smart electrostimulator. In this case, the task is to assign a reference force, but the general frame is suboptimal motion planning, see [START_REF] Hirschorn | Output tracking for nonlinear systems with singular points[END_REF][START_REF] Hirschorn | Global output tracking for nonlinear systems[END_REF] We illustrate the implementation of the endurance program in the smart electrostimulator (see Fig. 5) associated to optimization program OCP2 defined in Section 3.2.2.

(1) First we estimate, using the methods proposed in [START_REF] Stein | Ecological Modeling from Time-Series Inference: Insight into Dynamics and Stability of Intestinal Microbiota[END_REF]; [START_REF] Wilson | Force response of locust skeletal muscle[END_REF], the parameters R, τ 1,0 , K m,0 , τ 2 , τ c , A 0 , α A , τ f at of the reduced force-fatigue model.

Conclusion

In this short article, we have mainly constructed a finite dimensional approximation of the muscular force response to FES-input exploiting the mathematical structure of the model. The construction is based on the Ding et al. model but can be adapted to deal with the different models discussed in [START_REF] Wilson | Force response of locust skeletal muscle[END_REF]. We have presented one application of our study in the isometric case in view to design a smart electrostimulator, which is an ongoing industrial project. Our approximation can be used to parameters estimation [START_REF] Stein | Ecological Modeling from Time-Series Inference: Insight into Dynamics and Stability of Intestinal Microbiota[END_REF]Wil-(2011) to design MPC-optimized sampled-data control schemes, applying standard [START_REF] Richalet | Industrial applications of model based predictive control[END_REF]; [START_REF] Wang | Fast Model Predictive Control using Online Optimization[END_REF] to this situation. Another application of our study is to track in the nonisometric case a path in the joint angle variable and this will be developed in a forthcoming article.

Appendix

This appendix give the proofs of the propositions 5 and 6.

Using ( 12)-( 16), we can compute F (t k ) -F (t k ), k ∈ {0, . . . , n} as follows:

(F (t k ) -F (t k ))/A = With these conditions, we rewrite the last integral term of (20) as

Clearly, if m 1 ≤ m 1 there exists ν + ≥ 1 large enough such that for all ν ≥ ν + , F (t k ) ≥ F (t k ). Likewise, if m 1 ≥ m 1 there exists ν -≤ 1 small enough such that for all ν ≤ ν -, F (t k ) ≤ F (t k ). 2

Proof of Proposition 6. Here we assume that the approximations m 1 , m 2 satisfy 0 ≤ m k 1 ≤ m k 1 ≤ 1 and 0 ≤ m k 2 = m k 2 on [t k , t k+1 ], for all k ∈ {0, . . . , n}.

From (20), we can write

|F (t k ) -F (t k )|/A ≤ t k 0 |m 1 (s) -m 1 (s)| ds +

Fig. 1 .

 1 Fig.1. The lobe k (see (13)) reaches its maximum at t = t k +τc and is equal toR k η k /e. It is zero for t ≤ t k , strictly increasing on [t k , t k + τc] and strictly decreasing [t k + τc, T ]. k is concave on [t k , t k + 2τc] and convex on [t k + 2τc, T ]. For all t ≥ t k + 5τc, | k (t)| ≤ R 5e -5 ≤ 3.4%R.

Fig. 2 .

 2 Fig. 2. Construction of the approximation m1, m2 of the Michaelis-Menten-Hill functions.

  Fig. 4. The optimal solution σ * = (η * 0 , . . . , η * n , t * 1 , . . . , t * n , T ) minimizes the cost minσ

  (2) Once the model is calibrated, we compute the maximal muscle strength F tet , which is the asymptotic force response to a high frequency train over [0, T ].(3) This allows to fix a reference forceF ref = F tet /ρ,where ρ > 1 and we compute the input σ * minimizingT 0 | F (s) -F ref | 2 dsover σ (see Section 5). (4) We apply the train pulses defined by σ * and we come back to instruction (1).

t k 0 MM

 0 (t k )M -1 (s)m 1 (s) -M (t k ) M -1 (s) m 1 (s(t k )M -1 (s)(m 1 (s) -m 1 (s)) ds + t k 0 (M (t k )M -1 (s) -M (t k ) M -1 (s)) m 1 (sProof of Proposition 5. Assume the approximation F is constructed from the piecewise approximations m 1 , m 2 satisfying m 1 ≥ 0 and for all k ∈ {0, . . . , n}: m k 2 = ν m k 2 , where ν ∈ R is a parameter.

  u) -m 2 (u)| du, which yields the expected result. 2

Table 1

 1 List of variables and values of the constant parameters in the Ding et al. model Coefficient for the force-model parameter A in the fatigue model αK m Coefficient for the force-model parameter Km in the fatigue model ατ 1

	Symbol Unit Value	Description
	CN	-	-	Normalized amount of Ca 2+ -troponin complex
	F	kN	-	Force generated by muscle
	ti	s	-	Time of the i th pulse
	n	-	-	Total number of the pulses before time t
	i	-	-	Stimulation pulse index
	τc	s	0.02	Time constant that commands the rise and the decay of CN
	R	-	1.143	Term of the enhancement in CN from successive stimuli
	A τ1	mN s s	--	Scaling factor for the force and the shortening velocity of muscle Force decline time constant when strongly bound cross-bridges
				absent
	τ2	s	0.1244	Force decline time constant due to friction between actin and
				myosin
	Km	-	-	Sensitivity of strongly bound cross-bridges to CN
	A0 Km,0	mN s -	3.009 0.103	Value of the parameter A when muscle is not fatigued Value of the parameter Km when muscle is not fatigued
	τ1,0	s	0.05095	The value of the parameter τ1 when muscle is not fatigued
	αA τ f at	1 s 2 -4.0 10 -1 1 s•mN 1.9 10 -2 1 2.1 10 -2 Coefficient for force-model parameter τ1 in the fatigue model mN s 127 Time constant controlling the recovery of (A, Km, τ1)

where M (t; σ) = exp -t 0 m 2 (s; σ) ds .

Since c N is a (piecewise) sum of products of polynomial and exponential functions with respect to σ (see (

8

)), the function σ → M -1 (s; σ)m 1 (s; σ) is a (piecewise) sum of rational products of polynomial and exponential functions on [0, T ]. Their partial derivatives with respect to η i , t i exist and are continuous on I (bounded) for almost every s ∈ (0, T ). According to the regularity of an integral involving parameters (Ambrosio et al., 2011, Appendix A), σ → F (t; σ) has continuous differential, and we deduce the following proposition.

Proposition 2 The input-output mapping σ → y(t; σ) is differentiable with continuous partial derivatives on I.

  The muscle fatigue effect is taken into account by reducing the reference force F ref during the training period and is related to new estimation of the fatigue.

	Choose the program's algorithm
	Example case: the Endurance Program
			Scan the muscle to estimate the parameters
					(see Table 1)
			Compute
	maximal strength (high frequency train : tetany)
			Define
	0.08
	0.06
	0.04
	0.02
	0	0.9	1.8	2.7
	0.08
	0.06
	0.04
	0.02
	0.2		0.4	0.6	0.8
		response to	solution of
	Fig. 5. Illustration of the endurance program for a smart
	electrostimulator.

4 Intel@Core T M i5 CPU @

2.4Ghz

Acknowledgements

The authors thank Mokrane Abdiche for his involvement in the project, as well as Segula Matra Automotive and French Association Nationale de la Recherche et de la Technologie (ANRT) for their financial support.