Marc Long

Aurélie Lelong

Eva Bucciarelli

Fabienne Le Grand

Hélène Hégaret

Philippe Soudant

Long Marc

Lelong Aurélie

Bucciarelli Eva

Hégaret Hélène

Soudant Philippe

Physiological adaptation of the diatom Pseudo-nitzschia delicatissima under copper starvation

Keywords: bacillariophyta, copper starvation, physiology, membranes, lipid metabolism, energy storage, photosynthesis, photosystem II

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

Introduction

Copper (Cu) is an essential redox-active element with two oxidation states Cu+ and Cu2+ that is essential for phytoplankton growth. When this element is bound to enzymes (e.g. in plastocyanin (Pcn) and cytochrome oxidase) it can be used in electron-transfer reactions from various metabolic functions such as respiration, and photosynthesis [START_REF] Hänsch | Physiological functions of mineral micronutrients (Cu, Zn, Mn[END_REF]. Two studies in oceanic waters reported an increase in phytoplankton biomass following Cu addition [START_REF] Coale | Effects of iron, manganese, copper, and zinc enrichments on productivity and biomass in the subarctic Pacific[END_REF][START_REF] Peers | Copper requirements for iron acquisition and growth of coastal and oceanic diatoms[END_REF].

Coale [START_REF] Coale | Effects of iron, manganese, copper, and zinc enrichments on productivity and biomass in the subarctic Pacific[END_REF] hypothesized that this increase was due to a decrease in microzooplankton grazing due to Cu toxicity, but Peers et al. [START_REF] Peers | Copper requirements for iron acquisition and growth of coastal and oceanic diatoms[END_REF] reported no impact of Cu addition on the zooplankton grazing rate. Two alternative hypotheses were postulated to explain the Cu-induced increase in phytoplankton growth in the subarctic Pacific Ocean. First, the phytoplanktonic Cu requirements of phytoplankton cells in these waters could be linked to their Fe status [START_REF] Peers | Copper requirements for iron acquisition and growth of coastal and oceanic diatoms[END_REF][START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF]. Fe limitation, which occurs in 50% of the global ocean, is pervasive in this oceanic region [START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF][START_REF] Tagliabue | The integral role of iron in ocean biogeochemistry[END_REF].

Fe-limited phytoplankton may need Cu for its role in a multicopper oxidase, that is part of an inducible Fe transport system [START_REF] Wells | Domoic acid: The synergy of iron, copper, and the toxicity of diatoms[END_REF][START_REF] Maldonado | Copper-dependent iron transport in coastal and oceanic diatoms[END_REF]. Due to this enzyme, the Cu demand of certain species under Fe limitation is therefore largely increased [START_REF] Peers | Copper requirements for iron acquisition and growth of coastal and oceanic diatoms[END_REF][START_REF] Annett | The effects of Cu and Fe availability on the growth and Cu:C ratios of marine diatoms[END_REF][START_REF] Guo | The effects of iron and copper availability on the copper stoichiometry of marine phytoplankton1 : Cu quotas of marine phytoplankton[END_REF], and addition of Cu could thus stimulate Fe acquisition and, indirectly, phytoplankton growth. For some diatoms, however, the Cu demand of the high-affinity Fe transport system may be low relative to other cellular Cu pools, such as Cu-containing plastocyanin and superoxide dismutase, and cytochrome c oxidase [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF][START_REF] Annett | The effects of Cu and Fe availability on the growth and Cu:C ratios of marine diatoms[END_REF]. In particular, some oceanic species that needed to adapt to Fe limitation constitutively replaced cytochrome c6, which carries electrons between the cytochrome b6f and the photosystem I (PSI), by its Cu-containing analog plastocyanin (PCn [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF]). In these cases, Cu could stimulate phytoplankton growth independently of Fe [START_REF] Peers | Copper requirements for iron acquisition and growth of coastal and oceanic diatoms[END_REF] especially at surface where Cu concentrations can be limiting due to consumption linked with the biological activity and to the presence of strong organic ligands [START_REF] Takeda | Bioavailability and biogeochemical processes of trace metals in the surface ocean[END_REF][START_REF] Kong | Identification of copper-regulated proteins in an oceanic diatom, Thalassiosira oceanica 1005[END_REF][START_REF] Kong | Transcriptomes of an oceanic diatom reveal the initial and final stages of acclimation to copper deficiency[END_REF]. Experimental studies confirmed that diatom growth was stimulated with the addition of Cu as compared to Cu organic levels, even without Fe limitation [START_REF] Peers | Copper requirements for iron acquisition and growth of coastal and oceanic diatoms[END_REF][START_REF] Annett | The effects of Cu and Fe availability on the growth and Cu:C ratios of marine diatoms[END_REF][START_REF] Kong | Identification of copper-regulated proteins in an oceanic diatom, Thalassiosira oceanica 1005[END_REF].

Cu limitation can therefore induce deleterious effects on microalgal cells, such as inhibiting growth rate [START_REF] Annett | The effects of Cu and Fe availability on the growth and Cu:C ratios of marine diatoms[END_REF][START_REF] Guo | The effects of iron and copper availability on the copper stoichiometry of marine phytoplankton1 : Cu quotas of marine phytoplankton[END_REF], photosynthesis, carbon fixation, respiration [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF][START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF][START_REF] Hippmann | Proteomic analysis of metabolic pathways supports chloroplast-mitochondria cross-talk in a Cu-limited diatom[END_REF]. While some strains can mitigate Cu limitation stress, the research on the physiological adaptations of microalgae to Cu limitation remains scarce. Photophysiology was especially studied as compared to other metabolic pathways. The main physiological response to Cu deficiency is the switch between the Cu-containing Pcn to the Fe-containing cytochrome c6 [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF][START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF][START_REF] Merchant | Dynamic interplay between two copper-titrating components in the transcriptional regulation of cyt c6[END_REF] to mitigate the bottleneck in the electron transfer. Modifications in the electron carrier can be accompanied by larger modifications of the photosynthetic chains. Some strains of the diatom Thalassiosira oceanica can modify their light harvesting complexes to favor photoprotection instead of photochemistry. An overall decrease in the protein content per cell is another potential strategy to decrease the overall Cu requirement of the cells [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF]. Cu limitation modified the transcriptome of the chlorophyte Chlamydomonas reinhardtii with specific modification in the lipid pathways [START_REF] Castruita | Systems biology approach in Chlamydomonas reveals connections between copper nutrition and multiple metabolic steps[END_REF]. The authors from this study highlighted a desaturation of fatty acids associated to thykakoid membranes. To our knowledge, this was not assessed in diatoms despite their significance in global primary production and biogeochemical cycles [START_REF] Tréguer | Reviews and syntheses: The biogeochemical cycle of silicon in the modern ocean[END_REF].

The diatom Pseudo-nitzschia is distributed worldwide in coastal and oceanic areas including high nutrients low chlorophyll (HNLC) areas that are limited by Fe such as the subarctic pacific ocean [START_REF] Marchetti | Identification and assessment of domoic acid production in oceanic Pseudo-nitzschia (Bacillariophyceae) from iron-limited waters in the northeast subarctic Pacific[END_REF]. Therefore, in these environments, Pseudo-nitzschia spp. might also be facing Cu limitations. However, P. delicatissima might have developed physiological strategies to face Cu limitation: under Cu limitation, P. delicatissima cells were able to maintain photosynthesis at the PSII step, unlike under Fe-limitation [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF]. Although the rest of the photosynthetic chain was not studied and might have been partially impaired, the potential physiological adaptation remains uncharacterized. The adaptation could occur at several levels, such as the Pcn/cytochrome c6 level but also at other photosynthetic or nonphotosynthetic levels. Lelong and collaborators [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF] observed significant variations in the overall lipid content and pigment fluorescence in Cu depleted cells but did not thoroughly studied these modifications that might reveal physiological adaptations. Overall, the strategies to cope with Cu limitations in Pseudo-nitzschia spp. and other marine microalgae remain largely unknown.

In this study, we aimed to further understand the physiological adaptation of the diatom P. delicatissima to Cu limitation and understand the specific roles of lipids. This research focused on the lipid composition (lipid classes and fatty acids) of the diatom, but also integrated the overall physiology (i.e. growth, cell size, C and N contents), especially the photophysiology (pigment composition and fluorescence of PSII) to better interpret the lipidomic responses.

Material and methods

Culture conditions

Batch cultures of the marine pennate diatom Pseudo-nitzschia delicatissima (strain Pd08RB, solitary species isolated in 2008 by Beatriz Beker in the Bay of Brest, France) were grown at 16°C in polycarbonate bottles. Species was determined after sequencing of the ITS-1 fragment, using PnAllR and PnAllF primers [START_REF] Hubbard | Inter-and intraspecific community structure within the genus Pseudo-nitzschia (Bacillariophyceae) 1[END_REF] and obtained sequences were aligned using GenBank. Cultures were grown under cool-white light (OSRAM) over a dark:light cycle of 12:12 h with an irradiance of 130 µmol photons m -2 s -1 . Although the cultures were naturally xenic and grown without antibiotics, the culture media (see below) were sterilized using a microwave [START_REF] Keller | Microwave treatment for sterilization of phytoplankton culture media[END_REF]. Diatoms were pre-acclimated to each culture condition until their growth rate remained stable, and at least 20 generations were grown in the same conditions. Cultures were sampled in triplicates or quadriplicates (see below "Culture media") at mid-exponential phase of growth to maintain a pH lower than 8.5 and avoid CO2 limitation, and at the same time of the day to avoid diel cycle variations.

Culture media

The culture medium consisted of artificial AQUIL seawater enriched with 300 µM nitrate, 10 µM phosphate, 100 µM silicate, 0.55 µg l -1 vitamin B12, 0.5 µg l -1 biotin, 100 µg l -1 thiamin, 10 nM selenite and 100 nM molybdate [START_REF] Price | Preparation and chemistry of the artificial algal culture medium aquil[END_REF]. The synthetic ocean water component (including all salts) and the macronutrients were passed through a Chelex 100 ion exchange resin to remove metal contaminants present in the chemicals [START_REF] Price | Preparation and chemistry of the artificial algal culture medium aquil[END_REF]. The medium also contained a trace metal ion buffer system consisting of 100 µM ethylene diamine tetra acetic acid (EDTA), 50.3 nM Co, 79.7 nM Zn,121 nM Mn and 500 nM Fe. The buffer system generated free ion concentrations of Co, Zn, Mn and Fe of 10 -10.88 , 10 -10.88 , 10 -8.27 and 10 -19.2 M, respectively, at pH 8.1 according to the chemical equilibrium program MINEQL+ (version 4.62.3) for nonilluminated medium. In the replete medium ("Control"), 19.6 nM Cu were added, generating a free Cu concentration of 10 -13.9 M. In the Cu starvation treatment ("Cu-starved"), no Cu was added to the medium.

All bottles and apparatus were acid cleaned, and all manipulations were conducted within a sterile laminar flow hood equipped with a Teflon ® bench using sterile and trace-metal clean techniques [START_REF] Bucciarelli | Effects of an iron-light co-limitation on the elemental composition (Si, C, N) of the marine diatoms Thalassiosira oceanica and Ditylum brightwellii[END_REF]. Cultures were run in triplicates for the Control and in quadriplicates for the Cu-starved condition.

Flow cytometric measurements

A flow cytometer FACScalibur (BD Biosciences, San Jose, CA USA) with an argon blue laser (488 nm) was used, with the same settings for all the duration of the experiment to allow comparison between treatments. Cell concentrations, morphological and physiological measurements of P. delicatissima, quantification of free-living bacteria associated to P. delicatissima and percentage of dead bacteria in the culture were assessed using flow cytometry [START_REF] Lelong | Cell-based measurements to assess physiological status of Pseudo-nitzschia multiseries, a toxic diatom[END_REF]. As Pseudo-nitzschia spp. can form chains, cell concentrations were counted under microscope: each culture bottle was sampled in triplicate and the mean cell concentration is provided as an average.

Physiological measurements

Specific growth rate (μ, d -1 ) was determined by linear regression of the natural log (cell concentration) versus time. Mortality of P. delicatissima was assessed by staining cultures with 0.1 µM of SYTOX Green (Molecular probes, Invitrogen, Eugene, Oregon, USA) for 30 minutes. More than 95 % of living cells could be observed in all cultures, which ensures that the physiological measurements were performed on living cells.

Bio-dilution

When cells divide, the cytoplasm and its content are divided between the two daughter cells.

To assess the bio-dilution effect at steady state, the production rate (in arbitrary units per cell per day (AU cell -1 d -1 ) of lipids can be calculated by multiplying their content (in AU cell -1 ) by the acclimated specific growth rate (in d -1 ) [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF].

Bacteria

To estimate free-living bacteria concentration and viability, bacteria were analyzed after 15 min incubation with a final concentration of 1/10000 of the commercial solution of SYBR Green I (Molecular probes, Invitrogen, Eugene, Oregon, USA) and propidium iodide (PI, Sigma, St. Louis, MO, USA) at 10 µg ml -1 [START_REF] Lelong | Cell-based measurements to assess physiological status of Pseudo-nitzschia multiseries, a toxic diatom[END_REF].

Cell volume

For each replicate, a minimum of 50 cells were photographed using a Leica microscope and the Axiovision imaging software. For each cell, cell length and cell width were measured.

Cell volume was estimated using the following formulae of an ellipsoid [START_REF] Hillebrand | Biovolume calculation for pelagic and benthic microalgae[END_REF]: Cell volume=1/6π * length * width * width.

Pulse-amplitude modulated (PAM) fluorimetry

Maximum quantum yield (ΦPSII = (Fm-F0)/Fm=Fv/Fm), which is a measurement of the efficiency of the photosynthesis at the photosystem II (PSII) level, was measured using the AquaPen-C AP-C 100 fluorometer (Photo Systems Instruments, Czech Republic), where F0 and Fm are respectively the minimum and maximum fluorescence of cells at 455 nm. The measurement of ΦPSII was performed after 20 min of dark adaptation of the cells at 16°C. Chl a fluorescence induction transient (OJIP) curves were also performed on cultures to determine if Cu starvation modified the photosynthetic electron transport of P. delicatissima.

Measurements were performed applying the internal protocol with blue light (455 nm).

Complete dataset of fluorescent variables acquired during this study are available in the supplementary files.

The relative electron transport rate through PSII (rETRPSII) was calculated as: rETR = ΦPSII * I with I being the light intensity (comprised between 0 and 500 µmol photons m -2 ).

Pigment composition and chlorophyll content

Duplicate samples from each culture bottle were filtered onto glass fiber filters (GF/F) for Chlorophyll a and pigments, and rinsed with artificial seawater previously sterilized by microwave. Filters for Chl a were immediately stored in glass tubes at -80 °C. Chl a was measured by fluorometry after extraction into 90% acetone [START_REF] Lorenzen | Determination of chlorophyll and pheo-pigments: spectrophotometric equations[END_REF].

Qualitative pigment composition was analyzed on pigments extracted from frozen cells (-80°C) by methanol using high-performance liquid chromatography (HPLC) according to the method described in [START_REF] Ras | Spatial variability of phytoplankton pigment distributions in the Subtropical South Pacific Ocean: comparison between in situ and predicted data[END_REF], adapted from [START_REF] Van Heukelem | Computer-assisted high-performance liquid chromatography method development with applications to the isolation and analysis of phytoplankton pigments[END_REF]. All the pigment standards were purchased from DHI (HØRSHOLM, Denmark). Duplicate cultures were used for pigment composition.

Cellular nitrogen and carbon

To determine the cellular C and N, culture samples were filtered as duplicates on a precombusted Whatman GF/F filter (450 °C for 4 hours) and rinsed with sterile artificial seawater containing no nutrient. The filters were dried at 60 °C overnight in pre-combusted glassware and stored until analysis using an elemental analyzer Thermo Fisher NA 2100 CN.

Lipids

About 9.10 6 cells from algal cultures were filtered on a GF/F filter. Boiling water was immediately added on filter to prevent lipid degradation through lipase activity. Total lipid extraction was performed placing the filter with algae in glass vials containing 6mL of a chloroform-methanol mixture (2/1, v/v). Lipid extract vials were then sealed under N2 and stored at -20°C before further analyses. Lipid extracts were stored at -20°C under nitrogen (N2(g)) until analysis.

Lipid class composition analyses were performed by high-performance thin layer chromatography (HPTLC) using a CAMAG auto-sampler to spot the sample on HPTLC glass plates pre-coated with silica gel (Merck & Co., Ltd., Darmstadt, Germany). Neutral and polar lipid classes were analyzed according to [START_REF] Moutel | Development of a screening procedure for the characterization of Botryococcus braunii strains for biofuel application[END_REF]. For polar and neutral lipid fatty acid (FA) analysis, an aliquot of the chloroform: methanol (2:1, v:v) extract was dried under N2(g) and then resuspended in chloroform:methanol (98:2, v:v) prior to neutral and polar lipid separation. Separation of neutral and polar lipids was realized by solid phase extraction [START_REF] Le Grand | Disseminated neoplasia in the soft-shell clam Mya arenaria: membrane lipid composition and functional parameters of circulating cells[END_REF].

Polar and neutral fractions were transesterified and the resulting fatty acid methyl esters (FAME) were analyzed and quantified by gas chromatography (GC-FID) according to the method from [START_REF] Le Grand | Disseminated neoplasia in the soft-shell clam Mya arenaria: membrane lipid composition and functional parameters of circulating cells[END_REF]. Lipid class and fatty acid standards were the same standards used in [START_REF] Le Grand | Disseminated neoplasia in the soft-shell clam Mya arenaria: membrane lipid composition and functional parameters of circulating cells[END_REF].

FA were expressed as percentage of total FA in polar and neutral lipids. The level of unsaturation was calculated from the % FA derived from the gas chromatographic data according to the equation:

𝑈𝑛𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑙𝑒𝑣𝑒𝑙 = ∑[% 𝑜𝑓 𝑓𝑎𝑡𝑡𝑦 𝑎𝑐𝑖𝑑 × 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑜𝑢𝑏𝑙𝑒 𝑏𝑜𝑛𝑑]

Statistics

Effects of Cu starvation on the physiological parameters of P. delicatissima were tested using T-tests with the software StatGraphics Plus (Manugistics, Inc, Rockville, MD, USA). The test of rank used was the Tukey test (variance homogeneity was first tested and confirmed for all the parameters). For all statistical results, a probability of p < 0.05 was considered significant.

Results

Flow cytometry and microscopy parameters

Specific growth rate

Control cultures grew at µmax = 1.01 ± 0.01 d -1 (n=3), while Cu-starved cultures grew significantly slower (p<0.01), with a growth rate of 0.66 ± 0.01 d -1 (n=4) (Table 1). Custarved cells had a significantly higher length and width (p<0.01); leading to a cell volume 24% higher than control cells (p<0.01) (Table 1). Results are expressed as mean ± standard deviation. Statistical differences between the two treatments are indicated as followed "NS" not significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01.

Control

Cu-starved p-value µ (day

Dead and active cells

Cu starvation did not have any effect on the mortality of P. delicatissima. Less than 3% of the cells were considered dead (i.e. cells stained by SYTOX green) in the control and in Cu-starved conditions (Table 1), which showed no significant difference.

Bacteria

There were significantly (p<0.01) more bacteria in the Cu-starved cultures per P.

delicatissima cell (Table 1). The biovolume of bacteria was estimated to represent less than 4% of microalgal biomass, therefore, the contribution of bacteria to the biomass was considered as negligible in our biochemical analyses (C, N and lipid contents).

Photosynthetic parameters

Photosystem II activity

The maximum PSII quantum yield (Figure 1A) was significantly lower (p<0.01) in Custarved cells (0.58 ± 0.01) than in control cells (0.62 ± 0.00). Light curves (Figure 1B) show that Cu-starved cells exhibited lower rETRPSII than control cells, regardless of the value of the light pulse (20 to 500 µmol photons.m -2 ).

Cellular nitrogen and carbon

Cu-starved cells contained significantly more C (p<0.01) and N (p<0.05) than the cells in replete conditions (Table 2). When corrected to cell volume, the difference in C and N contents between the two conditions were not significant anymore (p>0.05). However, the C/N ratio was significantly higher by 20% in Cu-starved cells than in cells from the control (p<0.01).

Pigment composition and chlorophyll content

The chlorophyll content was significantly (p<0.01) increased by 54% in Cu-starved cells as compared to control cells (Table 3). When corrected to cell volume, chlorophyll a content of Cu-starved cells was still significantly higher (p<0.05), but only by 23%. The overall pigment composition of P. delicatissima was affected by Cu starvation with an increase in the proportion of three pigments (Chlorophyllide a, Fucoxanthine, Diadinoxanthine) and a decrease in the proportion of two pigments (Chlorophyll C2, Chlorophyll a).

Tableau 3: Pigment composition and chlorophyll content of P. delicatissima in the control and in the Cu-starved conditions.

Results are expressed as mean ± standard deviation. Statistical differences could not be tested for the % of each pigment class (N = 2). Statistical differences in the chlorophyll (Chl) content are indicated as followed "NS" not significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01.

Control

Lipid classes

Copper starvation induced a significant increase in the content and productivity of lipids by P.

delicatissima (Table 4). The composition of lipid classes was also affected with significant decreases in the proportions of sulfoquinovosyl diacylglycerol (SQDG) (P<0.05), phosphatidylglycerine (PG) (P<0.001), free fatty acids (FFA) (P<0.01) and significant increases in the proportions of digalactosyl diacylglycerol (DGDG) (P<0.001) and triglycerides (TAG) (P<0.01).

Fatty acid (FA) composition and phytosterols

Changes in lipid classes occurred alongside significant modifications in polar (Figure 2A-B) and neutral (Figure 2C-D) FA compositions. Cu starvation induced a significant increase in the proportions of polar 16:1n-7, 18:1n-7, 18:1n-5 and neutral 16:1n-7, 16:2n-4 FA, while the proportions of neutral 16:0, 18:0 and 22:6n-3 FA significantly decreased. The proportions of the two phytosterols, were also significantly modified by the unavailability of Cu in the culture media, with a decreased cholesterol, while the 24-methylenecholesterol was increased (Figure 3). Photophysiology of Cu starvation. While literature reports contrasting effects of Culimitation on the physiological/morphological responses, including photophysiology, of microalgae [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF][START_REF] Sandmann | Photosynthetic and respiratory electron transport in Cu 2+ -deficient Dunaliella[END_REF][START_REF] Ayala | Changes in carotenoids and fatty acids in photosystem II of Cudeficient pea plants[END_REF], the photosystem II (PSII) of P. delicatissima was only slightly affected by Cu starvation which corroborates with results from a previous study [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF]. However, the relative electron flow at PSII (rETRPSII) was significantly decreased, as similarly observed in However, the rETRPSII was not totally suppressed, and cells were still able to grow, suggesting that cells were able to adapt to Cu starvation stress. Some algae contain both PCn and cytochrome c6 and/or can shift from Cu-containing PCn to Fe-containing cytochrome c6

when Cu is limiting [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF]. This is for example the case of coastal diatoms that are rarely facing metal limitations and can synthesize both Cu-containing PCn and Fe-containing cytochrome c6 [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF][START_REF] Strzepek | Photosynthetic architecture differs in coastal and oceanic diatoms[END_REF]. It is likely that the coastal strain of P. delicatissima used in our study and in [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF] can synthesize cytochrome c6 in addition to PCn. This would also explain the growth rate and maximum PSII QY inhibitions of P. delicatissima observed under Fe limitation [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF].

Therefore, under Cu-starved, P. delicatissima might have shifted from PCn to more cytochrome c6 electron carriers to counteract the bottleneck in the electron flow and compensate Cu starvation. While in the literature the photosynthetic chain received most of the attention, Cu is a constituent of several proteins and its unavailability might have inhibited other metabolic pathways downstream of the photosynthetic chain that might have impaired the electron transport. Quantification of PCn and Cyt c6 in P. delicatissima is required to fully understand the inhibition of photosynthesis.

Rearrangement of photosynthetic membranes. Cu starvation induced a bottleneck in the photosynthetic electron flow that increases the excitation in the pigment antennae and favors the production of ROS and photodamages. However, under the light conditions of this experiment, the maximum PSII QY of P. delicatissima was only slightly decreased, highlighting some adaptation of the photosynthetic apparatus. Among the mechanisms to cope with the lack of Cu, the oceanic diatom Thalassiosira oceanica rearranges its light harvesting complexes (LHCs) from light harvesting to photoprotection under Cu limitation [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF].

Furthermore the proportion of pigments (fucoxanthin, Chl c,a and Ddx) associated to energy dissipation in fucoxanthin chlorophyll a/c-binding protein (FCP) [START_REF] Wang | Structural basis for blue-green light harvesting and energy dissipation in diatoms[END_REF], the protein forming LHC, was significantly increased in the absence of Cu. Overall, this rearrangement results in an increase in a better dissipation of energy coming from the electron flow bottleneck.

Rearrangement of LHCs is accompanied with biochemical modifications of the membranes.

In our experiments, LHCs were not quantified, but the biochemistry of membranes highlighted a rearrangement of P. delicatissima photosynthetic chain with an increase in lipid classes associated to light harvesting complex II (LHCII) such as digalactosyl diacylglycerol (DGDG) and phosphatidylglycerine (PG) [START_REF] Barros | Crystallisation, structure and function of plant lightharvesting Complex II[END_REF]. The modifications in the proportion of lipid classes as observed with SQDG, DGDG and PG might be responsible for the modification in polar FA composition. The modifications in these FA might also come from modifications of FA within specific lipid classes: in the green algae Chlamydomonas reinharditii, Cu limitation induced a modification of galactolipid composition and a specific desaturation of FA from DGDG [START_REF] Castruita | Systems biology approach in Chlamydomonas reveals connections between copper nutrition and multiple metabolic steps[END_REF]. All these modifications in the lipid and pigment compositions suggests a potential increase in LHCs in P. delicatissima in response to Cu starvation. The photophysiological responses of the strain of P. delicatissima from our study are somewhat similar from the photophysiological responses of T. oceanica [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF]: the rearrangement of the photosynthetic apparatus in T. oceanica leads to the increase of LHCs associated with an increase in NPQ. The biochemical modification of P. delicatissima photosynthetic membranes could support the hypothesis of a restructuration of LHC towards photoprotection, although no significant increase in NPQ was measured, even at saturating light intensities (> 300 µmol photons m -2 ).

Cu-starvation modifies the overall lipid metabolism. Under the light conditions used for culture growth (130 µmol photon m -2 ), while the photosynthesis was only slightly inhibited, the microalgal growth was significantly inhibited by 35%. Therefore, the energy produced by photosynthesis that was not used for growth had to be somehow used or stored.

"Excess" energy storage can be stored under several forms in microalgae [START_REF] Li | Photosynthetic carbon partitioning and lipid production in the oleaginous microalga Pseudochlorococcum sp. (Chlorophyceae) under nitrogen-limited conditions[END_REF][START_REF] Da Costa | Effects of growth phase and nitrogen limitation on biochemical composition of two strains of Tisochrysis lutea[END_REF]. The C/N ratio was higher in Cu-starved cells, and this increase could be linked to the increase in polysaccharides and lipids, more likely than an increase in the production of proteins. When microalgae are not able to divide, they can store their energy in the form of lipids [START_REF] Giordano | Fourrier transform infrared spectroscopy a novel tool to investigate changes in intracellular macromolecular pools in the marine microalga Chaetoceros muelleri (Bacillariophyceae)[END_REF], as observed with the daily production rate of storage lipids that significantly increased in Custarved conditions in this study and a previous one [START_REF] Lelong | Iron and copper limitations differently affect growth rates and photosynthetic and physiological parameters of the marine diatom Pseudo-nitzschia delicatissima[END_REF]. Accumulation of triacylglycerol (TAG) could also be an effective way to prevent photooxidative damages from altered P. delicatissima photosynthetic chain. TAG can be a sink for electrons of photosynthetic reactions [START_REF] Solovchenko | Probing the effects of high-light stress on pigment and lipid metabolism in nitrogen-starving microalgae by measuring chlorophyll fluorescence transients: Studies with a Δ5 desaturase mutant of Parietochloris incisa (Chlorophyta, Trebouxiophyceae)[END_REF]. Once favorable conditions return, the energy stored under the form of TAG can also easily be mobilized for cells to divide. The drastic increase in lipid content might explain the higher C/N ratio under Cu starvation, although it could also be partially attributed to a decrease of the protein content per cell: a strategy used by the diatom Thalassiosira oceanica to decrease cellular Cu demand [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF].

Beside the overall increase in neutral lipids, neutral FA composition (exclusively composed of FFA and triglycerides) was also modified underlying physiological adaptations. The percentage of FA with 16 and 18 carbons were the most affected by Cu starvation. Within the neutral FA, the proportion of the saturated FA 16:0 and 18:0 were the most affected with a decrease in absence of Cu. Neutral 16:0 were likely converted in 16:1n-7 (the main FA form for triglycerides; [START_REF] Lu | Regulation of the cholesterol biosynthetic pathway and its integration with fatty acid biosynthesis in the oleaginous microalga Nannochloropsis oceanica[END_REF]) and in small proportions of 16:2n-4, while 18:0 were likely mobilized for the formation of 18:1n-7 and 18:1n-5 as their proportion significantly increased in polar and neutral lipids. Eventually, the increase in 18:1n-7 might originate from the elongation 16:1n-7 as suggested by [START_REF] Cañavate | Common and species-specific effects of Phosphate on marine microalgae fatty acids shape their function in phytoplankton trophic ecology[END_REF]. In the green microalgae Dunaliella, Cu limitation induced a similar desaturation of FA with 16 and 18 carbons [START_REF] Ayala | Changes in carotenoids and fatty acids in photosystem II of Cudeficient pea plants[END_REF]. Modification of neutral FA metabolism mainly reflects a storage of energy under the form of the mono-unsaturated FA 16:1n-7. Other modifications may also originate from the de-novo synthesis of FA to meet the demand of new membrane lipids, especially for photosynthetic membranes to adapt to Cu starvation, or to replace FA peroxidated by ROS coming from photooxidation [START_REF] Kong | Transcriptomes of an oceanic diatom reveal the initial and final stages of acclimation to copper deficiency[END_REF].

Cu starvation induced a shift in phytosterols. Cu starvation modified proportions of phytosterols, the ratio cholesterol/24-methylenecholesterol was reversed under Cu starvation with an increase of 24-methylenecholesterol. Cholesterol and 24-methylenecholesterol only differ from a doubled bond within the radical chain of the molecule. While the role of phytosterols under Cu starvation is unknown, sterols have many roles such as for example regulation of plasmic membrane fluidity and permeability, signal transduction, modulation of enzyme activity, or precursors of secondary metabolites [START_REF] Dufourc | Sterols and membrane dynamics[END_REF][START_REF] Hartmann | Plant sterols and the membrane environment[END_REF]. Among the roles that could be related to Cu starvation, sterols are essential for the functioning of cellular membranes, photosynthesis [START_REF] Piepho | Simultaneous effects of light intensity and Phosphorus supply on the sterol content of phytoplankton[END_REF][START_REF] Lu | Regulation of the cholesterol biosynthetic pathway and its integration with fatty acid biosynthesis in the oleaginous microalga Nannochloropsis oceanica[END_REF] and are likely involve in FA biosynthesis as highlighted in

Nannochloropsis oceanica [START_REF] Lu | Regulation of the cholesterol biosynthetic pathway and its integration with fatty acid biosynthesis in the oleaginous microalga Nannochloropsis oceanica[END_REF]. However, their precise role during Cu starvation remains to be elucidated in further analyses.

Conclusion

In the present study we investigated the physiological responses of a strain of P. delicatissima to Cu starvation with a special interest for biochemical changes (lipids and pigments) in relation with the overall physiology. Despite Cu starvation, the diatom was able to grow and maintain some photosynthetic activity. The lipidomic approach revealed two physiological adaptations to counteract Cu starvation a) membrane rearrangements and b) energetic adaptation. Rearrangement of membrane lipids was highlighted by modifications in phytosterol composition and increases in lipids and pigments associated with photoprotective responses. These responses must have helped maintaining the photosynthetic efficiency and mitigate damages on photosystem II. In parallel, the energy captured by photosynthesis was partially diverted from microalgal growth to lipid production. Cells were bigger and stored energy as triglycerides (with 16:1n-7 being the main FA stored). The stored energy could be mobilized for cells to resume growth when conditions become favorable (i.e., when Cu starvation is relieved). This modification in C allocation may have ecological relevance. The drastic increase in lipids (TAG content was 40 times higher in Cu-starved cells), other things being equal, may change the overall density of the cells [START_REF] Norici | Impact of irradiance on the C allocation in the coastal marine diatom Skeletonema marinoi Sarno and Zingone*[END_REF] and, therefore, their position in the water column [START_REF] Smetacek | Role of sinking in diatom life-history cycles: ecological, evolutionary and geological significance[END_REF][START_REF] Raven | The evolution of silicification in diatoms: inescapable sinking and sinking as escape?[END_REF]. This effect on buoyancy could be reinforced or counteracted by the accumulation of other cell components. In diatoms, an appreciable contribution to cell density is given by the silica frustule [START_REF] Raven | The evolution of silicification in diatoms: inescapable sinking and sinking as escape?[END_REF]. However, for this specific strain of P. delicatissima (Pd08RB), Cu starvation did not affect the biogenic silica content of the cells in a previous study [START_REF] Boutorh | Effect of trace metal-limited growth on the postmortem dissolution of the marine diatom Pseudonitzschia delicatissima[END_REF] and would not counteract the buoyancy effect of lipids. Although this study revealed an adaptation strategy to Cu starvation, one has to be cautious and should not generalize these results to different strains. Different strains from the same species can have contrasting physiological responses to Cu stress [START_REF] Hippmann | Contrasting effects of copper limitation on the photosynthetic apparatus in two strains of the open ocean diatom Thalassiosira oceanica[END_REF] and further study should investigate the strain variability of P. delicatissima responses.

Figure 1 :

 1 Figure 1: Effect of Cu depletion on photosystem II (PSII) of P. delicatissima. A) Maximum PSII quantum yield in the control (green) and in the Cu-starved conditions (red). Statistical differences between the two treatments are indicated as followed "NS" not significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01. B) Relative electron transport through PSII (rETRPSII) in the control (green) and in the Cu-starved conditions (red). Error bars represent the standard deviation. Other measured photosynthetic parameters can be found in Supplementary files.

Figure 2 :

 2 Figure 2: Composition of A-B) polar (membranes) fatty acids (FA), C-D) neutral (storage) FA of P. delicatissima in the control (green) and in the Cu-starved conditions(red). Zooms of the significant differences of FA in small proportions are given in B) and D). Results are expressed as mean ± standard deviation. Statistical differences between the two conditions are indicated as followed "NS" not significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01, "***" p-value < 0.001.

Figure 3 :

 3 Figure3: Composition of phytosterols of P. delicatissima in the control and in the Cu starved conditions. Results are expressed as mean ± standard deviation. Statistical differences between the two conditions are indicated as followed "NS" not significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01, "***" p-value < 0.001.

T

  . oceanica under Cu limitation[START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF]. While the maximum PSII quantum yield only highlights one step of the photosynthetic chain, the relative electron flow at PSII (rETRPSII) provides a better picture of photosynthesis. The inhibition of rETRPSII of P. delicatissima observed in the present study highlights a blockage of the electron flow in the photosynthetic chain upstream or downstream of PSII, at the Calvin-Benson-Bassham cycle for instance. Between the PSII b6-f, the photosynthetic chain of diatoms can contain plastocyanin (PCn), a Cu-containing protein carrying electrons. The decrease in rETRPSII suggests that P. delicatissima may contain some PCn, which functioning might have been limited by Cu. A lack of PCn electron carrier might have resulted in a bottleneck limiting the electron flow.

Table 1 :

 1 Physiologic and morphometric parameters of P. delicatissima in the control and under Cu starved conditions.

Table 2 :

 2 Cellular C and N content of P.delicatissima in the control and in the Cu-starved conditions. Results are expressed as mean ± standard deviation. Statistical differences between the two conditions are indicated as followed "NS" not

	significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01.
		Control	Cu-starved	p-value
	C (pmol cell -1 )	0.73 ± 0.03	1.0 ± 0.1	**
	C (mol dm -3 of cells)	9.6 ± 0.5	11.0 ± 0.9	NS
	N (pmol cell -1 )	0.12 ± 0.01 0.15 ± 0.01	*
	N (mol dm -3 of cells)	1.6 ± 0.1	1.6 ± 0.1	NS
	C/N	5.9 ± 0.1	7.0 ± 0.3	**

Table 4 :

 4 Total (membrane and storage) lipid classes composition and productivity of P. delicatissima in the control and inthe Cu-starved conditions. Results are expressed as mean ± standard deviation. Statistical differences between 20Cu and 0Cu

	are indicated as followed "NS" not significant, "*" 0.01 < p-value < 0.05, "**" 0.001 < p-value < 0.01, "***" p-value <
	0.001.
			Control	Cu-starved	p-value
		% SQDG	19.8 ± 0.7	17.7 ± 0.7	*
		% DGDG	5.9 ± 0.2	10.2 ± 0.4	***
		% MGDG	44.1 ± 2.2	45.4 ± 1.3	NS
		% PG	18.1 ± 0.1	11.9 ± 0.9	***
	Membranes	% PE	0 ± 1	1.9 ± 0.8	NS
		% phytosterols	12.4 ± 0.8	13.0 ± 1.5	NS
		Content (pg cell -1 )	2438 ± 210 3691 ± 244	***
		Productivity (pg cell -1 day -1 )	2461 ± 189 2450 ± 150	NS
		Content/volume of cells (pg µm -3 )	32 ± 3	71 ± 9	*
		% FFA	27.5 ± 3.0	6.1 ± 6.7	**
		% TAG	72.5 ± 3.0	93.9 ± 6.7	**
	Storage	Content (pg cell -1 )	55 ± 12	1648 ± 302	**
		Productivity (pg cell -1 day -1 )	55 ± 12	1093 ± 192	*
		Content/volume of cells (pg µm -3 )	0. 7 ± 0.2	17 ± 3	*
		Content (pg cell -1 )	2493 ± 215 5339 ± 540	***
	Total
		Productivity (pg cell -1 day -1 )	2516 ± 194 3543 ± 337	**

SQDG: sulfoquinovosyl-diacylglycerol, DGDG: digalactosyl-diacylglycerol, MGDG: monogalactosyl-diacylglycerol, PG: phosphatidylglycerine, PE: phosphatidylethanolamine, FFA: free fatty acids, TAG: triglycerides

Acknowledgements

This work was supported by the ANR program (ICOP, ANR-10-JCJC-606), the French ministry of research (MENRT grant), and Region Bretagne grant, SAD Volet 1, PSEUDOPE to HH and CNRS EC2CO Microbien to PS.

The dataset is available at publicly available at: https://www.seanoe.org/data/00833/94472/


Chouaib Farhati

Souhaila Fki

Abdeldjalil Aïssa-El-Bey

Fatma Abdelkefi

Automatic modulation classification for multi-criteria generic channel equalization

Keywords: Blind equalization, AMC, pdf, neural network

In this paper, we study blind equalization techniques to mitigate inter-symbol interference (ISI), and mainly, we are focused on generic blind equalizer (GBE). A GBE has no prior information about the transmission channel or the used constellation. To solve this challenge, a joint generic blind equalizer, based on a new multi-criteria cost function and automatic modulation classification (AMC) is proposed. The new multi-criteria cost function is based on the probability density fitting (PDF) and the k-nearest neighbor (KNN) algorithm is used for the AMC stage. Thus, using a neural architecture, the new criterion is demonstrated in its linear and nonlinear context. Simulation results support our claims with Quadrature Amplitude Modulation (QAM) transmitted signals in single input single output (SISO) communication system and they show a better performance in terms of mean square error (MSE) and symbol error rate (SER) compared to other GBE from the literature.

I. Introduction

Blind equalization is a very interesting research topic aiming to develop effective and low-complexity algorithms that reduce inter-symbol interference (ISI) and avoid bandwidth waste resulting from training data. Generic blind equalization is needed when constructing intelligent receivers that do not know the channel and the transmitted modulation type. To estimate the transmitted signal, the receiver has to blindly equalize the channel, then, it has to reliably classify the transmitted modulation by considering an adequate AMC technique. Although massive access has a great interest in the Internet of Things (IoT) communication system, the SISO model is well used and we have shown our contribution with a SISO model first. In the literature, among the works addressing joint blind equalization and AMC, we can refer to [START_REF] Barbarossa | Classification of digital constellations under unknown multipath propagation conditions[END_REF]. In this paper, the authors suggested using different equalization branches, each adapted to a specific constellation. This yields a complex architecture system where the filter providing the smallest cost function indicates the right constellation. In [START_REF] Shi | Blind equalization and characteristic function based robust modulation recognition[END_REF], the constant modulus algorithm (CMA) was used to produce a generic equalizer with a normalized radius. They used the CMA cost function relative to the quadrature phase-shift keying (QPSK) constellation to equalize any transmitted constellation belonging to the phase-shift keying modulation (PSK) or the QAM modulation. As a result, the equalizer output is symbols belonging to the transmitted constellation but compressed in the unit radius. After the classification by the amplitude of the equalized signal characteristic function (CF), the equalized signal is multiplied by a scaling factor to restore the emitted constellation amplitudes before deciding and exploiting the transmitted data. Moreover in [START_REF] Fki | Blind equalization and automatic modulation classification based on PDF fitting[END_REF], authors follow the same idea as [START_REF] Shi | Blind equalization and characteristic function based robust modulation recognition[END_REF] but instead of using the CMA criterion, they suggested considering the PDF with the multi-modulus stochastic quadratic distance (MSQDℓp) criterion [START_REF] Fki | Blind equalization based on pdf fitting and convergence analysis[END_REF]. In this case, during the convergence phase, the equalizer fits the PDF of the received symbols to match with the QPSK constellation PDF regardless of the emitted modulation. Concerning the AMC, three methods were proposed that detect the modulation which has the PDF that best fits, in some sense, that of the equalizer output. In all these mentioned works, AMC is applied after equalization of the received signal. However, this paper proposes a new generic equalizer architecture that combines jointly blind equalization and AMC. In [START_REF] Fki | Blind equalization and automatic modulation classification based on PDF fitting[END_REF], generic equalizer MSQD-ℓ2 𝑔𝑒𝑛 based on multi-modulus stochastic quadratic distance criterion outperforms generic equalizer CMA 𝑔𝑒𝑛 based on constant modulus algorithm criterion in [START_REF] Shi | Blind equalization and characteristic function based robust modulation recognition[END_REF]. For this reason, we have considered MSQD-ℓ2 𝑔𝑒𝑛 as a benchmark in its linear and neural version. For the equalization step, we propose a new multi-criteria generic cost function based on the PDF with the MSQD-ℓp criterion since it outperforms those based on high order statistical properties criterion [START_REF] Farhati | Blind channel equalization based on Complex-valued neural network and probability density fitting[END_REF]. In addition, we implemented the proposed criterion in linear and nonlinear case banks complex-valued neural networks (CVNN). For the AMC step, in the literature, various techniques for modulation classification are considered such us Maximum-likelihood [START_REF] Sills | Maximum-likelihood modulation classification for psk/qam[END_REF] [7] and order cyclic cumulants [8] [9]. In this work, we apply the KNN algorithm with the fourth-order cumulants [START_REF] Ali | An M-QAM signal modulation recognition algorithm in AWGN channel[END_REF] [START_REF] Hashim | Recognition of QAM signals with low SNR using a combined threshold algorithm[END_REF]. The main contributions of this paper are :

The rest of the paper is organized as follows. In Section II, we introduce the system model. In Section III, we detail our multicriteria generic cost function based on the MSQD-ℓ𝑝 algorithm in its linear and neural versions. Simulation results are expounded in Section IV. Finally, Section V concludes our work.

II. System model

In this section, we introduce the baseband model of a transmission system with a generic equalizer in its linear and nonlinear forms and AMC as it is Fig. 1, where 𝑠(𝑛) 𝑛∈𝑍 is an M-QAM transmitted symbol, at time 𝑛, through a multi-path channel with a finite impulse response (FIR) with length 𝐿 ℎ , denoted h = [ℎ 0 , ℎ 1 , ..., ℎ 𝐿 ℎ-1 ].

Fig. 1: Transmission system baseband model with a generic equalizer and AMC. 𝑏(𝑛) is an independent and identically distributed (i.i.d) additive white Gaussian noise, 𝑥(𝑛) is the input of the equalizer, 𝐺 is the equalizer function such as :

• Linear context : 𝐺 = [𝑤 0 , 𝑤 1 , ..., 𝑤 𝐿 𝑤-1 ] 𝑇 is the equalizer impulse response, with length 𝐿 𝑤 and 𝑦(𝑛) is the equalized signal at time 𝑛, 𝑦(𝑛) = ∑ 𝐿 𝑤-1 𝑖=0 𝑤 𝑖 𝑥(𝑛 -𝑖) = 𝑤 𝑇 𝑥(𝑛) and x(n) = [𝑥(𝑛), 𝑥(𝑛 -1), ..., 𝑥(𝑛 -𝐿 𝑤+1 )] 𝑇 where 𝑥(𝑛) = ∑ 𝐿 ℎ-1 𝑖=0 ℎ 𝑖 𝑠(𝑛 -𝑖) + 𝑏(𝑛).

• Nonlinear context: 𝐺 is a neural network that will be detailed in section III.B.

III. Equalizer model

In this section, two contributions are proposed. The first one is a generic linear multi-criteria equalizer, based on a simultaneous mixture of the AMC and blind equalization. The second contribution is a generic nonlinear multi-criteria equalizer based also on a simultaneous mixture of blind equalization and AMC and using a neural network. The new multi-criteria cost function for the equalization part and the algorithm used for the AMC for each contribution will be detailed.

A. Multi-criteria MSQD-ℓ 𝑝 𝑔𝑒𝑛 (MC-MSQD-ℓ 𝑝 𝑔𝑒𝑛 ) in linear case

The equalizers whose criterion exploits the PDF outperform the equalizers whose criterion exploits the high-order statistical properties [START_REF] Lázaro | Stochastic blind equalization based on pdf fitting using parzen estimator[END_REF]. Moreover, decomposing the equalization criterion into an in-phase component and a quadrature component is more efficient than processing two components together, with respect to a phase shift introduced by the transmission channel [START_REF] Fki | Blind equalization based on pdf fitting and convergence analysis[END_REF]. These two reasons lead us to propose a new multi-criteria generic equalization based on the MSQD-ℓ 𝑝 algorithm [START_REF] Fki | Blind equalization based on pdf fitting and convergence analysis[END_REF]. Specially for 𝑝 = 2 the MSQD-ℓ 2 has the following cost function :

𝐽 𝑀𝑆𝑄𝐷-ℓ 2 (𝑤) = - 1 𝑁 𝑠 𝑁 𝑠 ∑ 𝑘=1 𝐾 𝜎 (|𝑦 𝑟 | 2 -|𝑠 𝑟 (𝑘)| 2 ) - 1 𝑁 𝑠 𝑁 𝑠 ∑ 𝑘=1 𝐾 𝜎 (|𝑦 𝑖 | 2 -|𝑠 𝑖 (𝑘)| 2 ), (1)
where 𝑁 𝑠 is the number of complex symbols in the considered constellation and 𝐾 𝜎 0 is a Gaussian Kernel with zero mean and variance 𝜎 0 which is referred to Kernel width. 𝑦 𝑟 and 𝑦 𝑖 and 𝑠 𝑟 and 𝑠 𝑖 are the real and imaginary parts of the equalized output and the transmitted constellation, respectively. The equalizers based on PDF fitting aim to minimize a specific distance between the data distribution at the output of the equalizer and a target distribution that corresponds to the transmitted constellation one. Since, in our case, we assume that we do not know the transmitted constellation, we defined a new multicriteria generic cost function as a sum of several variations of equation ( 1), one for each constellation order multiplied by an updated penalty factor. So, the criterion that we propose is :

𝐽 𝑀𝐶-𝑀𝑆𝑄𝐷-ℓ 2 𝑔𝑒𝑛 (𝑤) = 𝐶 ∑ 𝑚=1 𝛼 𝑚 𝐽 𝑀𝑆𝑄𝐷-ℓ 2 𝑚 (𝑤), (2)
where C is the number of the considered constellations orders.

The penalty factor 𝛼 𝑚 in ( 2) is updated in each iteration according to the following expression such that we reach, over the iterations, the transmitted constellation cost function :

𝛼 𝑚 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑐𝑙𝑎𝑠𝑠 𝑚 ) (3)
where 𝑐𝑙𝑎𝑠𝑠 𝑚 is the number of times where the 𝑚 𝑡ℎ constellation is successful in the classification result after the classification step.

The equalizer coefficients are updated using the stochastic gradient descendent (SGD) algorithm as:

𝑤(𝑛 + 1) = 𝑤(𝑛) -𝜇▽ 𝑤 𝐽 𝑀𝑆𝑄𝐷-ℓ 2 𝑔𝑒𝑛 (𝑤) = 𝑤(𝑛) -𝜇 𝐶 ∑ 𝑚=1 𝛼 𝑚 ▽ 𝑤 𝐽 𝑀𝑆𝑄𝐷-ℓ 2 𝑚 (𝑤), (4)
where 𝜇 is the step size.

In the sequel, we will focus on the explicit expression of (2). For this purpose, we begin by calculating the gradient of (1) which leads to that of (2). On the other hand, 𝑦(𝑛) = 𝑤(𝑛) (5) In order to update the cost function, each equalized symbol is classified using the KNN algorithm with fourth-order cumulant [START_REF] Marchand | Classification of linear modulations by a combination of different orders cycliC cumulants[END_REF] [9] as features. So that we do jointly the equalization and the AMC. The expressions of the 𝑝 𝑡ℎ order cumulant and moment are respectively as follows :

𝐶 𝑝𝑞 = 𝑐𝑢𝑚(𝑥 𝑝-𝑞 (𝑥 * ) 𝑞 ) 𝑎𝑛𝑑 𝑀 𝑝𝑞 = 𝐸[𝑥 𝑝-𝑞 (𝑥 * ) 𝑞 ],
where 𝐸[.] is the expectation operator. In particular, consider :

𝐶 40 = 𝑀 40 -3𝑀 2 20 𝐶 42 = 𝑀 42 -|𝑀 20 | 2 -2𝑀 2 21
as they are more suitable for M-QAM modulations [START_REF] Marchand | Multiple hypothesis modulation classification based on cycliC cumulants of different orders[END_REF]. A baseline containing 𝐶 40 and 𝐶 42 values for noisy symbols according to various values of signal-to-noise ratio (SNR) and belonging to {16, 32, 64, 128, 256}-QAM modulations is prepared. the classifier follows the following steps :

• Calculate 𝑆 1 = 𝐶 40 +𝐶 42 the sum of the last 1000 equalized symbols.

• Calculate the Euclidean distances between 𝑆 1 and each sum in the reference base. • A set of the 𝑘 nearest neighbors from the baseline is created.

• Since each neighbor matches a specific constellation order, then we select the order is the most repeated order in the previous set.

Finally, according to the AMC, we update the penalty factors following (3). Then (2) will be updated.

B. Multi-criteria neural network NNMSQD-ℓ 𝑝 𝑔𝑒𝑛 (MC-NNMSQD-ℓ 𝑝 𝑔𝑒𝑛 )

The neural network used for our equalizer is the complexvalued feed-forward which is trained using the complex backpropagation (CBP) algorithm [13] [14] [15]. This neural network is combined with a KNN classifier as it is illustrated in Fig. 2 Fig. 2: A complex-valued feed-forward multi-criteria neural network equalizer combined with a KNN classifier.

We assume 𝑁 𝑘 neurons in the 𝑘 𝑡ℎ layer and we denote by 𝜙 𝑘 𝑗 and 𝑥 𝑘+1 𝑗 the input and the output of the 𝑘 𝑡ℎ neuron, such that :

𝜙 𝑘 𝑗 = 𝑁 𝑘-1 ∑ 𝑖=1
𝑤 𝑘 𝑖𝑗 𝑥 𝑘 𝑖 + 𝜃 𝑘 𝑗 and ( 6)

𝑥 𝑘+1 𝑗 = 𝑓 𝑘 (Re(𝜙 𝑘 𝑗 )) + 𝑗𝑓 𝑘 (Im(𝜙 𝑘 𝑗 )), (7)
where 𝑥 𝑘 𝑖 is the 𝑘 𝑡ℎ layer output, 𝑤 𝑘 𝑖𝑗 is the weight between the 𝑖 𝑡ℎ neuron in the 𝑘 𝑡ℎ layer and the 𝑗 𝑡ℎ neuron in the (𝑘 + 1) 𝑡ℎ layer, 𝜃 𝑘 𝑗 and 𝑓 𝑘 (⋅) are the 𝑘 𝑡ℎ layer bias and activation function. In this paper, we consider a complex-valued feed-forward equalizer including two main steps for each iteration. The first step is the forward pass from the input layer to the output layer. This processing consists in computing the network output by applying an activation function on the weighted sum in each layer. The second step is the CBP to update the synaptic weights according to our equalizer criterion. The network includes three layers: input, hidden, and output layers. We assume in this paper that the activation function is as in [START_REF] Wang | Generalized derivation of neural network constant modulus algorithm for blind equalization[END_REF] :

𝑓(𝑥) = 𝑥 + 𝛼 * sin(𝜋 * 𝑥),
where 𝜋 is a mathematical constant approximately equal to 3.14159, 𝛼 a positive number between 0 and 1 and sin is the trigonometric function. The cost function for the generic multicriteria neural network equalizer is the same as that of (2) :

𝐽 𝑀𝐶-𝑁𝑁𝑀𝑆𝑄𝐷-ℓ 2 𝑔𝑒𝑛 (𝑤) = 𝐶 ∑ 𝑚=1 𝛼 𝑚 𝐽 𝑀𝑆𝑄𝐷-ℓ 2 𝑚 . ( 8
)
The penalty factor 𝛼 𝑚 is calculated and updated as in the MC-MSQD-ℓ 2 𝑔𝑒𝑛 algorithm and for each iteration, we follow the same steps. First, we equalize the received symbol, then we classify it, afterward, we update the neural network weights and finally, we update the penalty factors respectively. The weights of the neural network are updated as:

𝑤 𝑘 𝑖𝑗 (𝑛 + 1) = 𝑤 𝑘 𝑖𝑗 (𝑛) -𝜇 𝐶 ∑ 𝑚=1 𝛼 𝑚 𝜕𝐽 𝑀𝑆𝑄𝐷-ℓ 2 𝑚 𝜕𝑤 𝑘 𝑖𝑗 (𝑛) . ( 9
)
In the sequel, we focus on the derivation of ( 8). We begin by calculating the gradient of (1) which leads to that of [START_REF] Marchand | Classification of linear modulations by a combination of different orders cycliC cumulants[END_REF]. To update the output layer weights, we use the same procedure as in [START_REF] Wang | Generalized derivation of neural network constant modulus algorithm for blind equalization[END_REF] for simplification purposes, we introduce 𝑄 𝑟 , 𝑄 𝑖 and 𝛿 𝑜 𝑝 which are expressed as the following :

𝑄 𝑟 = 1 𝑁 𝑠 √ 2Π𝜎 𝑁 𝑠 ∑ 𝑘=1 𝑒 - (|𝑦 𝑟 (𝑛)| 2 -|𝑠 𝑟 (𝑘)| 2 ) 2 2𝜎 2 (|𝑦 𝑟 (𝑛)| 2 -|𝑠 𝑟 (𝑘)| 2 ) 𝜎 2 , 𝑄 𝑖 = 1 𝑁 𝑠 √ 2Π𝜎 𝑁 𝑠 ∑ 𝑘=1 𝑒 - (|𝑦 𝑖 (𝑛)| 2 -|𝑠 𝑖 (𝑘)| 2 ) 2 2𝜎 2 (|𝑦 𝑖 (𝑛)| 2 -|𝑠 𝑖 (𝑘)| 2 ) 𝜎 2 , 𝛿 𝑜 𝑝 = 𝑄 𝑟 𝑦 𝑟 (𝑛)𝑓 𝑜 ′ (𝜑 𝑜 𝑟 ) + 𝑗𝑄 𝑖 𝑦 𝑖 (𝑛)𝑓 𝑜 ′ (𝜑 𝑜 𝑖 ). (10)
The weights of the output layer are updated as :

𝑤 𝑜 𝑗 (𝑛 + 1) = 𝑤 𝑜 𝑗 (𝑛) -𝜇 𝐶 ∑ 𝑚=1 𝛼 𝑚 𝛿 𝑜 𝑝 𝑚 𝐼 * 𝑝𝑗 , (11)
where 𝜑 𝑜 𝑟 , 𝜑 𝑜 𝑖 , are the real and imaginary inputs of the output layer, and 𝐼 𝑝𝑗 is the output of the hidden layer. 𝑓 𝑜 ′ (𝜑 𝑜 𝑟 ) and 𝑓 𝑜 ′ (𝜑 𝑜 𝑖 ) are the derivatives of the activation function in the output layer applied in the real and imaginary parts of the output layer input.

To update the hidden layer weights, we consider the same reasoning as in [START_REF] Wang | Generalized derivation of neural network constant modulus algorithm for blind equalization[END_REF]. We simplify the final expression by introducing 𝛿 ℎ 𝑝𝑗 ,

𝛿 ℎ 𝑝𝑗 = 𝑓 ℎ ′ 𝑗 (𝜑 ℎ 𝑟 )ℜ(𝛿 𝑜 𝑝 𝑤 𝑜 * 𝑗 ) + 𝑗𝑓 ℎ ′ 𝑗 (𝜑 ℎ 𝑖 )ℑ(𝛿 𝑜 𝑝 𝑤 𝑜 * 𝑗 ). (12)
Similarly, the weights of the hidden layer are updated as :

𝑤 ℎ 𝑖𝑗 (𝑛 + 1) = 𝑤 ℎ 𝑖𝑗 (𝑛) -𝜇 𝐶 ∑ 𝑚=1 𝛼 𝑚 𝛿 ℎ 𝑝𝑗 𝑚 𝑥 * 𝑝𝑖 , (13)
where 𝜑 ℎ 𝑟 , 𝜑 ℎ 𝑖 , are the real and imaginary inputs of the hidden layer, and 𝑥 𝑝𝑖 is the input vector of the equalizer. 𝑓 ℎ ′ 𝑗 (𝜑 ℎ 𝑟 ) and 𝑓 ℎ ′ 𝑗 (𝜑 ℎ 𝑖 ) are the derivatives of the activation function of the hidden layer applied in the real and imaginary parts of the hidden layer input at the 𝑗 𝑡ℎ neuron. AMC process is applied using the same principle as for MC-MSQD-ℓ 2 𝑔𝑒𝑛 .

IV. Simulation results

We have simulated our approach using Matlab 2018. We have considered M=16, 32, 64, 128, and 256 M-QAM constellations. So, the penalty factors are equally initialized to 0.2. An important factor for the convergence of the equalizers is the learning rate 𝜇. It is fixed to 10 -3 /𝑃 𝑥 , where 𝑃 𝑥 is the received signal power. The total number of symbols used during the training step is 70000 and the equalizers were trained for epoch numbers equal to 1. We evaluate the performance of the proposed equalizers in terms of mean square error (MSE), symbol error rate (SER), and accuracy rate when using a one typical digital radio channel ℎ 𝑐𝑜𝑚𝑝 that was used in [START_REF] Fki | Blind equalization based on pdf fitting and convergence analysis[END_REF]. ℎ 𝑐𝑜𝑚𝑝 (𝑧) = (0.0410 + 𝑗0.0109) + (0.0495 + 𝑗0.0123)𝑧 -1 + (0.0672 + 𝑗0.0170)𝑧 -2 + (0.0919 + 𝑗0.0235)𝑧 -3 + (0.7920 + 𝑗0.1281)𝑧 -4 + (0.3960 + 𝑗0.0871)𝑧 -5 + (0.2715 + 𝑗0.0498)𝑧 -6 + (0.2291 + 𝑗0.0414)𝑧 -7 + (0.1287 + 𝑗0.0154)𝑧 -8 + (0.1032 + 𝑗0.0119)𝑧 -9 . MC-MSQD-ℓ2 𝑔𝑒𝑛 and MSQD-ℓ2 𝑔𝑒𝑛 equalizers size is 𝐿 𝑤 = 31 and their weights are initialized with null values, except the middle element which is set to 1. Then, we have simulated MC-NNMSQD-ℓ2 𝑔𝑒𝑛 and NNMSQD-ℓ2 𝑔𝑒𝑛 equalizers with a complex-valued feed-forward neural network using three layers. The number of neurons in the input, the hidden, and the output layers are [𝐿 𝑖 = 15 (input), 𝐿 ℎ = 9 (hidden), 𝐿 𝑜 = 1 (output)]. The activation function is 𝑓(𝑥) = 𝑥 + 𝛼𝑠𝑖𝑛(Π𝑥) where 𝛼 is fixed to 0.1 and is independent of the modulation size, signal-to-noise ratio and layers order. We have initialized the two matrices of synaptic weights 𝑤 𝑜 [𝐿 𝑜 , 𝐿 [ ℎ] and 𝑤 ℎ [𝐿 ℎ , 𝐿 𝑖 ] with small values. Except for 𝑤 𝑜 [(𝐿 𝑜 + 1)/2, (𝐿 ℎ + 1)/2] and 𝑤 ℎ [(𝐿 ℎ + 1)/2, (𝐿 𝑖 + 1)/2] that were set to 1.5. Equalizers have the same convergence speed and we compared performances in terms of MSE, SER, and accuracy rate. Computation burdens for MC-MSQD-ℓ2 𝑔𝑒𝑛 and MC-NNMSQD-ℓ2 𝑔𝑒𝑛 are defined in Tab I. It can be seen that for the considered transmitted constellation, our linear approach is more performant than the linear approach in [START_REF] Fki | Blind equalization based on pdf fitting and convergence analysis[END_REF] and its developed neural version. Also, the proposed neural approach MC-NNMSQD-ℓ2 𝑔𝑒𝑛 outperforms the MC-MSQD-ℓ2 𝑔𝑒𝑛 . Figures (4. Figure 5 shows that the classification rate reaches an acceptable value when the SNR exceeds 15dB. The classification rate is better with our new multi-criteria in its neural and linear versions.

Equalizer

V. Conclusion

In this paper, we have proposed a new blind generic equalization approach. This technique is based on simultaneous joint blind equalization and AMC. For the equalization stage, a new generic multi-criteria cost function is derived. Differently from the work in the literature, where the equalization criterion is static, this new criterion is dynamic and updated during the process. For the AMC stage, the KNN algorithm is used with the fourth-order cumulant as a feature. The obtained simulation results prove that our proposed techniques lead to outperforming the MSQD-ℓ2 𝑔𝑒𝑛 in terms of MSE, SER, and accuracy rate in linear and nonlinear form. As a perspective, we aim at generalizing our approach to other constellations and extending this approach for multiple inputs multiple outputs (MIMO) architecture.

  a) and(3.b) show the performance of the different equalizers in terms of MSE according to the SNR, when we consider the value of MSE after the convergence of the algorithm.

  a) and (4.b) prove that our generic equalizers guarantee interesting performance in terms of SER and it has the best performance for the different constellations.

Fig. 3 :

 3 Fig. 3: MSE versus SNR:(a) 16-QAM, (b) 64-QAM.

Fig. 4 :

 4 Fig. 4: SER versus SNR:(a)16-QAM, (b)64-QAM.

Fig. 5 :

 5 Fig. 5: Accuracy rate for 64-QAM and ℎ 𝑐𝑜𝑚𝑝 .

  𝜕𝑦(𝑛)/𝜕𝑤 𝑟 = 𝑥(𝑛) and 𝜕𝑦(𝑛)/𝜕𝑤 𝑖 = 𝑗𝑥(𝑛). Consequently, the derivative of 𝐽 𝑀𝑆𝑄𝐷-ℓ2 (𝑤) with respect to the equalizer weights is :

	𝑇 𝑥(𝑛) can be expressed as 𝑖 𝑥 𝑖 (𝑛)] + 𝑗[𝑤 𝑇 𝑟 𝑥 𝑟 (𝑛) -𝑤 𝑇 𝑦(𝑛) = [𝑤 𝑇 𝑟 𝑥 𝑖 (𝑛) + 𝑤 𝑇 𝑖 𝑥 𝑟 (𝑛)]. It results that : ▽ 𝑤 𝐽 𝑀𝑆𝑄𝐷-ℓ2 (𝑤) = 𝜕𝐽 𝑀𝑆𝑄𝐷-ℓ2 (𝑤) 𝜕𝑤 𝑟 + 𝑗 𝜕𝐽 𝑀𝑆𝑄𝐷-ℓ2 (𝑤) 𝜕𝑤 𝑖 = 1 √ 2𝜋𝑁 𝑠 𝜎 3 𝑁 𝑠 ∑ 𝑘=1 (𝑠𝑖𝑔𝑛(𝑦 𝑟 (𝑛))|𝑦 𝑟 (𝑛)| (|𝑦 𝑟 (𝑛)| 2 -|𝑠 𝑟 (𝑘)| 2 ) 𝑒 -(|𝑦 𝑟 (𝑛)| 2 -|𝑠 𝑟 (𝑘)| 2 ) 2 /2𝜎 2 + 𝑗𝑠𝑖𝑔𝑛(𝑦 𝑖 (𝑛))|𝑦 𝑖 (𝑛)|(|𝑦 𝑖 (𝑛)| 2 -|𝑠 𝑖 (𝑘)| 2 ) 𝑒 -(|𝑦 𝑖 (𝑛)| 2 -|𝑠 𝑖 (𝑘)| 2 ) 2 /2𝜎 2 )𝑥

* (𝑛).

TABLE I :

 I Computational complexityFigures(3.

• Proceeding blind equalization and AMC simultaneously.• Derivation of a new multi-criteria cost function for a generic equalization.


Alexandre Fouré

Augustin C Ogier

Arnaud Le Troter

Thorsten Feiweier

Christophe Vilmen

Maxime Guye

Julien Gondin

Pierre Besson

Bendahan David

Le Troter

assessed with Ultra High-Field Diffusion Tensor Imaging and Tractography: Sensitivity to Sex-Difference and Intramuscular Variability

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

3D muscle architecture assessed from diffusion tensor imaging with ultra-high-field MRI and tractography are sensitive to sex but only slight differences were detected in lower leg muscles fiber length and pennation angle of young healthy populations. The combination of diffusion tensor imaging-derived parametric maps 3D coregistration and statistical parametric mapping analysis shed light on additional differences along lower leg muscles in both sexes. Assessment at once and on a large spatial coverage of all the lower leg muscle volume, diffusion properties and 3D architecture with 7T-MRI provides reproducible parameters that might help in diagnostic of muscle tissue alterations leading to muscle function impairments.

Conclusion Materials & Methods Results

[1] Lieber RL, Friden J. Clinical significance of skeletal muscle architecture. Clin Orthop Relat Res. 2001(383):140-51. Muscle segmentation. Regions of interest were manually drawn on T 1 -weighted images using FSLview software (Fig. 1). Muscle areas were segmented every 5 slices, and missing slices were automatically interpolated [5]. Muscle volume was calculated by summing the volume of each muscle in all the slices.

3D muscle architecture.

From muscle fiber tractography, muscle fiber length (L F ) and pennation angle (θ) were characterized. θ was determined as the angle between the muscle's principal axis and the global direction of each fiber tracked within the muscle. L F and θ were averaged over all fiber tracts within each voxel.

Statistical analysis.

A 3D spatial normalization and statistical parametric mapping (SPM) were performed on the basis of an original methodology (Fig. 2) previously described [6]. Muscle volume was then divided in three equal parts (i.e., distal, central and proximal) to discriminate potential change in architecture parameters along muscle and across sex.

alexandre.foure@hotmail.fr alexandre.foure@univ-amu.fr Significant sex differences were detected in muscle volume (+ 21.7 % in men for the entire lower leg, P = 0.008).

Corrections of DWI.

Each diffusion-weighted dataset was denoised using BM4D algorithm. Geometric and signal distortions were estimated on b = 0 s/mm² images with the topup function from FSL software and modifications were applied to correct images from these distortions and Eddy currents using FSL eddy function.

Intensity of non-distorted diffusion-and T 1 -weighted images were corrected from B 1 homogeneities using N4ITK algorithm from ants library. Finally, each T1-weighted image was registered with the image obtained for b = 0 s/mm² using 3D rigid-body transformation, subsequently applied to muscle segmentation masks. For instance, L F was significantly higher in the central part for GL, FHL and POP but lower in the distal part for GM, SO sup , FHL, LCLL and ACLL. θ was lower in proximal part of SO sup , TP and ACLL Differences were found independently of sex in architecture along several muscles of the lower leg (Fig. 3).

Significant sex differences were found in fiber length of all DPCLL muscles (i.e., L F men > L F women for FHL, FDL, TP and POP) and pennation angle of FDL but SPM analysis revealed only few significant local differences regarding to sex.

[ 2 ]Fig. 3 :

 23 Fig.3: Fiber length (L F ) and pennation angle (θ) in the distal (DIST), central (MID) and proximal (PROX) parts of the lower leg muscles in men (blue) and women (red). L F and θ was altered along muscles (gastrocnemius medialis [GM], gastrocnemius lateralis [GL], soleus [SO], flexor hallucis longus [FHL], flexor digitorum longus [FDL], tibialis posterior [TP], popliteus [POP] and muscles included in the lateral [LCLL] and anterior [ACLL] compartment of the lower leg). *: significant difference between parts and sex (post-hoc comparison from length × sex interaction effect). †: significant difference between parts (post-hoc comparison from length main effect).

Fig. 1 :

 1 Fig.1 : Segmentation of the lower leg muscles

Fig. 2 :

 2 Fig.2 :Pipeline of data processing for the 3D spatial normalization


Le Troter

AREAS OF MUSCLE TISSUE ALTERATION CAN DIFFER FROM ACTIVATED REGIONS DURING ELECTRICALLY-INDUCED ISOMETRIC CONTRACTIONS

On the basis of immediate and long lasting T 2 changes recorded after an isometric NMES session, the present results clearly demonstrated that muscle areas activated during the contractions can be spatially different than the damaged muscle regions.

The discrepancies between activation areas and alteration regions in the vastus lateralis implies that tissues other than active skeletal muscle fibers were altered during unaccustomed NMES-induced isometric contractions.

Conclusion Materials & Methods Results

Bibliography

Neuromuscular electrical stimulation (NMES) has been shown to induce severe muscle damage under isometric conditions. Structural alterations [1] have been assessed in the muscles located beneath the stimulation electrodes. These findings have been supported by muscle transverse relaxation time (T 2 ) mapping using magnetic resonance imaging (MRI)[2]. Although muscle activation is considered to be mainly superficial using NMES [3], MRI investigations have demonstrated a relative spatial heterogeneity regarding T 2 changes [4]. Therefore, two main contributing causes could be linked to isometric NMES-induced muscle damage. A direct damage of weaker sarcomeres as suggested by the popping sarcomere hypothesis [5] in the activated muscle areas during the NMES exercise and/or an indirect damage of less activated muscle areas associated to intramuscular shear strain during the NMES exercise.

This study aimed at comparing the localization of activated and damaged muscle areas after a NMES session in order to distinguish these potential causes related to isometric NMES-induced muscle damage.

Introduction

Population. Thirteen subjects (26 ± 3 y, 173 ± 8 cm, 70 ± 9 kg, 4 women) volunteered to participate in this study. They were fully informed about the nature and the aim of the study and gave their informed written consent to participate in this study conducted in conformity with the Declaration of Helsinki Alexandre.foure@hotmail.fr David.bendahan@univ-amu.fr Significant decrease in MVC was found at POST (-26 ± 9%) and in delayed time at D7 (-20 ± 6%) and D14 (-12 ± 5%).

The authors thank the Assistance Publique des Hôpitaux de Marseille (APHM), the Centre National de la Recherche Scientifique (CNRS UMR 7339) and all the volonteers involved in this study.

Exercise. During NMES exercise inducing muscle damage, subjects were positioned on vastus lateralis (VL) and vastus medialis (VM) as shown in Figure 2. NMES exercise consisted of 40 isometric contractions (5s on and 35s off) evoked with pulses delivered at 100Hz (pulse duration of 400µs). Intensity was increased until maximally tolerable level. T 2 quantification in deep and superficial parts of the VM and VL muscles. Two depth levels (i.e., superficial and deep) were considered for VM and VL the most damaged muscles located beneath the stimulation electrodes. As previously described [START_REF] Fouré | Localization and quantification of intramuscular damage using statistical parametric mapping and skeletal muscle parcellation[END_REF], a polar coordinate system (with the bone center used as the reference) was used to determine the thickness of damaged muscles at different angles and the frontiers between superficial and deep parts was then characterized in each slice [START_REF] Fouré | Localization and quantification of intramuscular damage using statistical parametric mapping and skeletal muscle parcellation[END_REF]. Regarding muscle length, three levels were considered (i.e., distal from slice #1 to #7, central from slice 8 to #13 and proximal from slice #14 to #20). T 2 was then quantified for each region and at each time point

Statistics. Normality of the data distribution was initially investigated using Shapiro-Wilk test. One-way ANOVA (time) were used (Statistica, Statsoft, Tulsa, USA) to assess immediate (i.e., PRE vs. POST) and remaining (i.e., PRE vs. D7 to D21) changes in volume and T 2 of each muscle. In addition, two-way ANOVAs (muscle depth * time) were used to assess T 2 differences between superficial and deep parts over time for both VL and VM and so for each muscle length level (i.e., proximal, mid and distal). A Tukey's HSD post hoc analysis was performed when appropriate.

Although activation of muscle was statistically detected through T 2 increase at POST in superficial parts of the two muscles located beneath the stimulation electrodes (i.e., vastus lateralis and vastus medialis), alterations quantified in a delayed time from increased T 2 were mainly located in the deep muscle region of the vastus lateralis (+57 ± 24%) and superficial area of the vastus medialis (+24 ± 16%) at D7 and were still observed in whole muscle at D21 (Table 1).

Figure 2 :

 2 Figure 2: Subject position during the neuromuscular electrostimulation session

Table 1 :Figure 1 :

 11 Figure 1: Schematic representation of the experimental design


A Yacoub
email: aznam.yacoub@lsis.org

M Hamri

C Frydman

Formal Methods and Discrete-Event Simulations Méthodes Formelles et Simulation à Evénements Discrets

Keywords: Méthodes Formelles, Model-Checking, Vérification formelle, Vérification et Validation, Modélisation et Simulation, SPIN, PROMELA, Systèmes à événements discrets Formal Methods, Model-Checking, Formal Verification, Verification and Validation, Modeling and Simulation, SPIN, PROMELA, Discrete-event Systems

Les méthodes formelles (FM) représentent un formidable outil pour la Vérification et la Validation (V&V) de logiciels et de systèmes électroniques, voire de systèmes au sens large, grâce à l'emploi d'un raisonnement logique rigoureux. En particulier, les techniques dites de model-checking explorent de manière exhaustive l'espace d'états de ces systèmes et démontrent leur validité par rapport à certaines spécifications. Toutefois, ces techniques sont inefficaces, et même inapplicables sur des systèmes complexes, tels que les systèmes temporelles ou à événements discrets. A contrario, le domaine de la Modélisation et Simulation (M&S) fournit des outils et des techniques matures pour la V&V de ces systèmes. La mise au point d'un cadre de travail opérationnel combinant FM et M&S semble alors être une approche prometteuse permettant d'améliorer qualitativement la V&V de modèles, et ainsi augmenter la confiance placée aux systèmes étudiés ou conc ¸us. En particulier, nos travaux portent sur la combinaison entre SPIN, un outil reconnu de model-checking, et la simulation à événements discrets pour la V&V de programmes.

Introduction

Making reliable software or systems 1 is nowadays becoming more difficult. Systems involve complex behaviours between several components with various specifications. For instance, one of the differences between components typically concerns time representation. When one designs a system or tries to understand it, making a representation of this system is thus needed: this is the modeling process. The question is thus how increasing the confidence put into these models. These last decades, many techniques of design were proposed in the literature in order to answer this question. Verification and Validation (V&V) procedures have been well-defined, and rely on two domains which can appear as opposing: on the one hand, the Formal Verification (FV) methodology groups rigourous mathematical methods which show the correctness of a model by using formal proofs; on the other hand, the Theory of Modeling and Simulation (M&S) provides methodologies and tools allowing accurate representation, verification and validation of systems by focusing on their behaviour. Then, while FV appears as a set of rigorous and exhaustive methods, M&S seems to be similar to an empirical experiment, although these domains share many terms and have common objectives. However, FV and M&S both suffer of limits. The proof techniques are limited by the complexity of the systems under study (which can also impact the quality of the design) and by the computational power, while the simulations strongly depend on the played scenarii and on the experimental cases. That is why techniques based on the combined use of FV and M&S have begun to be studied in the literature since few years. This is in this context that we propose in this work a general approach for combining FV and M&S in a same framework, especially by combining model checking tools and discrete-event simulation (DEVS), in order to improve the procedures of V&V. This leads us to define a new formalism, called Discrete-Event Protocol Meta Language (DEv-PROMELA), to represent discrete-event systems (DES). This new formalism allows combining formal verification and simulation-based verification, and is also usable for validation purposes.

The first section of this paper is about the state of the art, explaining why combined methods are interesting in the context of V&V. Next, we introduce the DEv-PROMELA formalism and talk about our contributions, before concluding and presenting the future work.

State of the Art

Like said in introduction, V&V is an important process in the development of a system. It increases the reliability that one can put in this system. V&V techniques are various, and based on FV and M&S domains. And if we analyze in depth the literature about FV, M&S and V&V, we can remark these domains are really interlaced. This is thus important to precise definitions before speaking about existing techniques and methodologies.

Verification and Validation

Verification and Validation are two independent procedures.

They are used together to check that a system meets initial requirements and specifications, and does what why it was designed for. The PMBOK guide [Project-Management-Institute2011] gives two standard definitions:

Validation is the assurance that a product, service, or system meets the needs of the customer and other identified stakeholders. It often involves acceptance and suitability with external customers. Contrast with verification.

Verification is the evaluation of whether or not a product, service, or system complies with a regulation, requirement, specification, or imposed condition. It is often an internal process. Contrast with validation.

Verification thus answers the question "Is this system being built right?", whereas validation answers the question "Is this the right system which is building?" That meens Validation gives an important information about the correctness of the specifications used to build the system. Among V&V techniques, Formal Verification, Test Cases and Simulation are widely used.

The M&S community provides two others, but close, definitions of V&V [Missile-Defense-Agency2008].

M&S Verification is the process of determining that a computer model, simulation, or federation of models and simulations implementations and their associated data accurately represent the developer's conceptual description and specifications.

M&S Validation is the process of determining the degree to which a model, simulation, or federation of models and simulations, and their associated data are accurate representations of the real world from the perspective of the intended use(s).

In that sense, verification of simulation models concerns the correctness of the computerized simulation model against a conceptual model (using FV techniques), whereas validation of simulation models gives informations about the accuracy of the representation of the real system. Sargent goes further by precising the relationship between real world and simulation world [Sargent2004]. The conceptual model is thereby obtained by modeling theories and assumptions about the system under study. This conceptual model specifies the simulation model specifications, which are then used to implement the computerized simulation model. M&S Verification is thus done at two levels: between the conceptual model and the simulation model specifications on the one hand; and between the simulation model and the specifications on the other hand.

It is important to keep in mind these definitions. Indeed, our objective is not to improve V&V of simulation models, which is well studied in the the literature, but V&V at large.

Introduction to Formal Verification

Formal verification is the act of proving or disproving the correctness of a system against properties and specifications, using formal methods. Formal methods (FM) are a set of formal notation and tools that allow a strict and rigourous description of the system under study, with formal semantics and an automatic proof mechanism [Bowen and Hinchey1995].

FM are divided into two families:

• Automated theorem proving methods show that a set of statements of a system can be deducted from another set of statements. Formally, we consider Γ, a set of logical properties describing the system (we called them axioms and hypothesis), and φ a set of specifications (that we called conjectures). Theorem proving methods try to find a proof that Γ φ, in other words, that we can syntactically deduce specifications from properties of the system.

• Model

Checking methods [START_REF] Huth | Logic in computer science: modelling and reasoning about systems[END_REF]Ryan2005] [Baier andKatoen2008] show that a system satisfies a set of properties. Formally, we consider M , a model (in the mathematical sense) of the system, and φ, a set of logical properties. Model Checking methods check whether M |= φ: all models M syntactically and semantically satisfy φ. In fact, because the system is generally modelled by a finite automaton, model checking tools systematically explore the entire state space of the system model, inducting to the well-known state space explosion problem, which is extensively treated in the literature [Clarke2008].

In practice, software formal verification with formal methods can be done in several manners. First of them is modeling software using a formal specification language, then translating this model into code. Another fashion consists of extracting the model from the code and doing the verification on the resulting model [Holzmann and Smith1999].

According to these definitions, one can easily understand why FM are considered as powerful methods. But, FM are facing heaviness and are not really applicable on large and complex systems [Heitmeyer1998]. FM indeed require strong assumptions and strong abstractions in order to be efficient. That is why formal methods impose strong restrictions (that involve finiteness for instance) to their modeling language. Their expressive capability is thus reduced. As a result, many systems, like timed or event-based systems, can't be verified against temporal properties.

Introduction to M&S and Discrete-Event Simulation

M&S domain has been explored since the early 1960s, but was really theorised by [Zeigler1976]. This theory tried to make uniform these two notions used extensively in many disciplines like medicine, physics, etc; it also defines a global and universal framework and methodology that is not dependent on the domain of application. As the name suggests, the two key concepts behind M&S are "Model" and "Simulation". A model is a semantic interpretation of a structure, while a simulation is "executing a model to generate its behaviour" [Zeigler et al.2000], by acting on inputs and parameters of the model. Zeigler defined also an Experimental Frame (EF) as a set of conditions under which the real system is observed. This notion is also important because it implies a certain level of abstraction.

However, Zeigler also introduced an unique and universal formalism to describe discreteevent system in a generic manner. Discrete-Event system Specifications (DEVS) formalism is a symbolic representation of system semantics, unlike syntactic formalisms used in FV approaches. This involves that DEVS models are focusing on the behaviour of the systems which they represent, without any other constraints. Models can also be interpreted in only one way, which is not necessarily the case in FV approaches.

The main problem with the M&S framework is that simulation strongly depends on the EF. Because the model is a result of the view of the real system from the point of view of the EF, the correctness of the simulation essentially comes from the accuracy of the assumptions made under the EF. As a result, simulation-based verification methods cannot guarantee values outside of the domain of the EF.

Complementarity between Simulation and Formal Methods

Complementarity between Simulation and Formal Methods has been already shown in the literature. The FV community agrees that "In order to improve the quality of the model, a simulation prior to the model checking can take place. Simulation can be used effectively to get rid of the simpler category of modeling errors." [Baier and Katoen2008]. Like viewed in the previous section, M&S Verification is also explored in order to increase the credibility of simulation models [Sargent1998] [Sargent2001] [Kuhn et al.2003]. Many methods to transform certain DEVS subclasses into Timed Automata for purpose of static verification were developed [START_REF] Dacharry | A formal verification approach for devs[END_REF]Giambiasi2007] [Saadawi andWainer2009]. Other approaches tend to integrate Z into DEVS models [Trojet et al.2009] [Trojet2010] by transformation.

But other approaches [Abdulhameed et al.2014] [ Li et al.2005] try to combine formal verification and simulation by deriving specifications into two different specification languages, for simulation and formal verification purposes. However, these approaches don't take into account the heterogeneous aspects of systems.

Contributions

General Approach for V&V using Combined Formal Methods and Simulation

The main contribution of our work is to provide a new generic framework for V&V of systems, and which combines the use of formal methods and simulation. This framework is based on the M&S and DEVS formalism defined by Zeigler. It consists into introducing a clear operationnal semantics in formalisms used by formal methods. That then allows combining FV and simulation by transforming specifications expressed in the new formalism into specifications expressed in a verifiable formalism on the one hand, and into specifications expressed in a simulable formalism in the other hand. Four steps can be summarized in:

1. Firstly, it is necessary to make a rigorous description of the formalisms in order to understand concepts and notions involved.

We call source formalism f s , the formalism used by a formal method (typically a model checker). Target formalism f t is a chosen formalism of simulation. For a given couple (f s , f t ), we determine a metamodel of each formalism. This allows establishing a relation between the concepts of each formalism and identifying missing notions, which can be related to the semantics employed. It is also in this step that we can easily define a transformation language between the two formalisms;

2. If a notion is missing in the metamodel of f s , we construct an extended source formalism f s with these concepts. In this way, we can easily add operational semantics to a formalism which has none.

We then define the transformation language from f s to f t . Note that defining a new formalism can lead to redefining the grammar of the language of f s ; defining the transformation language is also defining morphisms between models at each level of system specification as given by [Zeigler et al.2000];

3. We redefine the system model M s to transform f s into f s if needed. Let us call M s the system model in f s .

4. We apply transformation to the model M s and get the model M t . Simulation-based verification and validation is then performed on M t . If transformation is correct, M t must respect requirements checked by M s with the formal method.

Using the modularity properties of the M&S theory, this framework allows us modular specifications and enables a kind of interoperability. By this, we mean that a system can be splitted into small components which can be verified and validated alone; each component is then composed with the others in order to model the entire system; this one is then validated by simulation. Modularity also allows replacement of each subcomponent without breaking the global behaviour of the system. Components can consequently modeled at different level of abstraction.

First contributions of this framework were validated through the transformation of specifications from PROMELA to FDDEVS and PROMELA to TSM [Yacoub et al.2014a] [Yacoub et al.2014b]. Recall that the given framework was successfully applied to verify and validate, by formal verification and simulation, the specifications of a commercial soccer video game.

DEv-PROMELA, a Formalism for Discrete-Event Modeling and Verification

As a part of a demonstration of our framework, we are currently developing a new formalism based on the well-known Protocol Meta Language (PROMELA) and its model checker SPIN [Holzmann2003], initally designed for the verification of concurrent protocols. Discrete-Event PROMELA (DEv-PROMELA) allows modeling of discrete-event algorithms by introducing discrete-event concepts into PROMELA. The resulting specifications are more accurate for the representation of discrete-event algorithms and combine the advantages of both DEVS and PROMELA formalisms. DEv-PROMELA indeed provides a clear operationnal semantics and a clear syntax that allows simulation and formal verification. A DEv-PROMELA model is then translated into a PROMELA specification, which preserve structural properties, for formal verification purposes using the SPIN model checker. And, the DEv-PROMELA specifications are also translated into a DEVS conceptual model, that preserves behavioural properties, and which can be simulated for verification and validation of behaviour the initial model.

Conclusion and Future Work

V&V approach combining Formal Methods and Simulation is a promising approach increasing confidence put into models and systems. This approach fills the weakness of formal verification on the one hand, and simulation one the other hand, by relying on the strengths of both domains. Also, modularity proposed by the M&S Framework allows incremental development of models.

Future work is about the adaptation of formal verification algorithms to DEv-PROMELA and making an integrated environment for modeling using the DEv-PROMELA formalism.

We use in this paper the term system at large. System is thus used to denote software, hardware, algorithms, or any complex systems.

Acknowledgements

The authors want to gratefully thank the Process Control System Team in Rousset. This research is part of the RD project "MAGE", from French Investing for the Future national program.


Hassan Haghighi
email: hassan.haghighi@lis-lab.fr

Maamar El

Amine Hamri
email: amine.hamri@lis-lab.fr

Macroscopic Calibration of Microscopic Queue Model Interpretation for Traffic Simulation of Electrical Vehicles *

Keywords: traffic model, discrete simulation, calibration, optimization

Introduction

Modeling & Simulation (M&S) is considered as a useful method in non-reference situations, as opposed to methods based on experimental data. For reliable simulation of the microscopic patterns based on the macroscopic results, our main equations are the macroscopic flow dynamics equation, Lighthill-Whitham-Richards (LWR) [START_REF] James | On kinematic waves II. A Theory of Traffic Flow on Long Crowded Roads[END_REF][START_REF] Paul | Shock Waves on the Highway[END_REF], the microscopic Gibbs model [START_REF] Peter | A behavioral Car-Following Model for Computer Simulation[END_REF], and the continuity equation for energy [START_REF] Goujon | Hamri Impact of Eco-Driving on Energy Consumption of an Electric Vehicle[END_REF] where queuing systems rely on discrete event simulation systems. Since these systems operate in an open loop structure and are time-consuming in terms of run time, it is possible to reach a model for microscopic simulation by using the results of the macroscopic system in an inner high-frequency closed loop structure. As a result, using a bidirectional optimization structure, the key parameters of the model can be calibrated.

Mathematics model

The objective is to increase the accuracy of the microscopic model by using the macroscopic effects while extracting microscopic patterns from the macroscopic solution. According to the velocity v as a function of traffic density ρ ∈ [0, ρ max ], the flow dynamics is Q = min{V ρ, w(ρ max -ρ)} where the congestion wave speed w is a function of critical density ρ cri and traffic free velocity V as w = ρcrivmax ρmax-ρcri . Therefore, a simplified continuity LWR model for ρ and charging energy e as the state of the charge (SOC) at time t and location x, as well as a queue model of velocity and location as the Gipps model for n th vehicle {n ∈ N|n ≥ 2}, can then be described as follows:

dρ dt + dQ dx = 0 (1) de dt = F (v, Ω, θ) (2) v n (t + τ ) = min v n (t) + 2.5a n τ 1 - v n (t) V n 0.025 + v n (t) V d n , b n τ + b 2 n τ 2 -b n 2 (δx(t) -s n-1 ) -v n (t)τ - v n-1 (t) 2 b ( 3
)
where F as the battery discharge function depends on velocity v, engine shaft rotation Ω, and road ramp angle θ. The Gibbs model is dependent on several parameters, such as reflection time τ , desired maximum acceleration a, safety distance s, maximum desired velocity in traffic free conditions V , which are different for each vehicle based on its conditions and driver's traffic behavior. Let to consider the mentioned parameters as random variables normally distributed with mean µ and standard deviation σ, as (1.58, 0.45) Error order

τ n ∼ N (µ τ , σ 2 τ ), a n ∼ N (µ a , σ 2 a ), s n ∼ N (µ s , σ 2 s ),
O(10 -7 ) (µ * s , σ * s ), m (6.51, 1.11) (µ * V , σ * V ), m s (20.1, 3.21) (α * , β * ), m

J = min

{µτ ,µa,µs,µ V ,στ ,σa,σs,σ V ,α,β}

(Q mac -Q mic ) 2 s.t. v mac ≤ v, ρ ≤ ρ max , (µ τ , µ a , µ s , µ V , σ τ , σ a , σ s , σ V ) > 0 -2.0 ≤ α < 0, -3.0 ≤ β < 0 (4)

Conclusions and simulation results

Microscopic simulation for two types of dynamic flow is shown in Fig. 2 for 20 vehicles. Fig. 3 and Tab. 1 present the optimal values of Calibration. The model and optimization performance show high efficiency in the behavior of output error.

FIG. 1 :FIG. 2 :FIG. 3 :

 123 FIG. 1: Optimization diagram

s 2 (

 2 -0.374, -1.79) TAB. 1 -Optimal output and V n ∼ N (µ V , σ 2 V ). Also we can estimate the maximum breaking deceleration b n = αa n , and frontal vehicle maximum deceleration b = min{β, bn-β 2 }. Calibrating these parameters with macroscopic results takes place in the closed loop as shown in Fig. 1, considering the objective to find the optimal values of µ * τ , σ * τ , µ * a , σ * a , µ * s , σ * s , µ * V , σ * V , α * , and β * as following cost function for macroscopic flow Q mac and microscopic flow Q mic .

* This work was supported by the French agency ANR under the chair in artificial intelligence MASSALIA.