A Deburghgraeve
email: a.deburghgraeve@brgm.fr

A Robinet
email: a.robinet@brgm.fr

S Lecacheux
email: s.lecacheux@brgm.fr

C Bouvier
email: c.bouvier@brgm.fr

E Simon
email: e.simon@catie.fr.

D Rosebery
email: david.rosebery@onf.fr.

MONITORING SHORELINE ROTATION WITH A COASTSNAP SYSTEM AT THE MESOTIDAL EMBAYED BEACH OF LAFITENIA, FRANCE

 is used to investigate the shoreline dynamics at the embayed beach of Lafitenia over a 9months period. After the validation of CoastSnap images rectification thanks to insitu surveyed data, a quantitative analyse of cross-shore position changes is undertaken using high tide waterline. The obtained results show apparent short time scale beach rotation for the first time at Lafitenia. This new shoreline mapping approach offers new perspectives to study short time coastal processes over long periods.

Introduction

Coastal areas are complex environments that act as a buffer, protecting biodiversity, human lives and settlements from strong waves and elevated water level. These land-sea interfaces are under an increasing pressure with both global mean sea level rise and a growing coastal population. Coastline position varies over a large range of temporal and spatial scales in response to a variety of forcing [START_REF] Stive | Variability of shore and shoreline evolution[END_REF]. The understanding of processes driving these changes is critical for effective and sustainable management of the coastal zone and require frequent and long term data acquisition (Nicolae [START_REF] Lerma | Decadal beach-dune profile monitoring along a 230-km high-energy sandy coast: Aquitaine, southwest France[END_REF].

Many measurement and observation methods are used by scientists and coastal managers for shoreline monitoring. Each one of them has its advantages and inconveniences. In situ methods such as DGPS surveying often provide high precision data but require significant logistic and available human operators (i.e. expensive and time consuming). For those reasons, it is difficult to maintain them over the long term on large spatial scale. Remote sensing techniques such as LiDAR also provide high precision data and can quickly cover wide areas [START_REF] Bossard | Geomorphologic Characteristic and Evolution of Managed Dunes on the South West Coast of France[END_REF]. However, its prohibitive cost and significant logistic remain limiting, especially for a frequent use. Video monitoring systems are frequently used to monitor shoreline dynamics at time scales from hours to years (Holman & Stanley, 2007). It has benefited from scientific advances in physic of the camera as an optical sensor to provide relevant shoreline monitoring data. Nevertheless, camera video stations require continuing energy supply and maintenance operations. Those requirements act as limiting factors in the deployment of cameras along coastlines. The use of satellite imaging to monitor costal dynamics has also strongly developed in recent years. Free access of both images and processing algorithms have made it a performant method suitable for shoreline monitoring at seasonal or decadal scale and at regional scale considering the wide areas covered by these images (e.g. [START_REF] Vos | CoastSat: A Google Earth Engine-enabled Python toolkit to extract shorelines from publicly available satellite imagery[END_REF]. However, large pixel footprint plus a regular cloud cover can make satellite images unusable for high-frequency monitoring of individual beaches. These above-mentioned weaknesses have encouraged scientists to keep exploring complementary alternatives. Following advances in video monitoring systems and worldwide smartphones growth, the CoastSnap system has been developed [START_REF] Harley | Shoreline change mapping using crowd-sourced smartphone images[END_REF]. Its aims at providing a quantitative approach of shoreline mapping using crowd-sourced images taken from a fixed station overlooking the beach. [START_REF] Harley | Shoreline change mapping using crowd-sourced smartphone images[END_REF] have successfully shown the capacity of CoastSnap to monitor the beach width of 2 Australian sandy beaches over a 1 km alongshore distance and a 7-months period of time. Besides providing high frequency data suitable to short time changes investigations, CoastSnap creates a link with local communities. Interested in these features, the Coastal Observatory of Nouvelle-Aquitaine region (France) decided to experiment the CoastSnap system at the embayed beach of Lafitenia (France) whose dynamics remain poorly known. In this study, CoastSnap images of Lafitenia were used to investigate the shoreline dynamics over a 9-months period. It has enabled to highlight short time beach dynamics that have not been identified yet with conventional measurement methods at Lafitenia. The study site is introduced in section 2, data and methods applied are presented in section 3, and results are presented and discussed in section 5.

Study site

The study was conducted at the open coast sandy beach of Lafitenia, located in Saint-Jean-de-Luz in southwest of France (Erreur ! Source du renvoi introuvable.). Bounded by two rocky headlands, this 350 m-long NW-facing embayed beach is exposed to high-energy waves generated in the North Atlantic Ocean. They propagate from W-NW sectors (average propagation is 307.3° TN) with an average significant wave height of 1.55 m and 10.9 s peak wave period measured at isobath 50 m. In this part of the Atlantic Ocean, atmospheric depressions and storms are more intense during the November to March period. The tide is meso-tidal with amplitude usually ranging from 2.80 m to 3.75 m for neap and spring tides, respectively. The sand lie on a rocky bedrock that is apparent on the surface at both extremities of the beach. The beach is surrounded by medium size cliffs made of relatively instable alterite.

The CoastSnap station has been installed on 25 th October 2021. It is located on top of the cliff, overlooking the beach of Lafitenia. The camera cradle is at an elevation of 37.8m FOD and provides a northeastward view of the beach extending to a distance of approximately 600 m alongshore. The important number of visitors and the proximity of the coastal trail ensure a minimum stream of contribution required for high frequency monitoring.

Data & Methods

Input Data

Crowd sourced images are regularly acquired and shared by citizens since station installation. All those images are centralized on a unique database. Date and time of acquisition are systematically associated to each image to allow retrieving the water level from astronomical tide prediction. The CoastSnap station enable to constraint both location and viewing angles of the image. These controlled parameters are combine with surveyed ground control points (GCP) to allow the rectification of each image in real world coordinates and quantitative analyses. Full topography of the beach was acquired using DGPS measurement on July 12 th 2022 to validate the spatial accuracy of waterlines extracted from rectified images. In addition, beach profiles measured annually at spring since 2009 by the Coastal Observatory of Nouvelle-Aquitaine (Nicolas [START_REF] Lerma | Decadal beach-dune profile monitoring along a 230-km high-energy sandy coast: Aquitaine, southwest France[END_REF] along three cross-shore transects within the embayment are used to compute the average beach slope (see section Methods). Astronomical tide levels are computed using tide harmonics derived from a time series of water levels simulated offshore Lafitenia by 10-m depth. Water levels are obtained from a regional-scale hindcast realized by the French Research Institute for Sea Resources (IFREMER) using the MARS2D hydrodynamic model [START_REF] Lazure | An external-internal mode coupling for a 3D hydrodynamical model for applications at regional scale (MARS)[END_REF]. Waves conditions are extracted from a wave hindcast realized by IFREMER using the WAVEWATCH III wave model (Roland and Ardhuin 2014).

Methods

To facilitate further processing, images are aligned on unique reference image. This alignment step is undertaken using Adobe Photoshop as suggested in [START_REF] Harley | Shoreline change mapping using crowd-sourced smartphone images[END_REF]. This set of images is then rectified using CoastSnap algorithms. Image rectification is a usual treatment that turns an oblique image with pixel coordinates, into a geo-referenced horizontal plan view with real world coordinates (e.g. [START_REF] Bouvier | Video monitoring of sandbarshoreline response to an offshore submerged structure at a microtidal beach[END_REF]. That transformation enable quantitative assessment of distances. However, it requires specifying the rectification altitude beforehand to be undertaken. Only points that are actually located on this rectification plane could be properly situated within the rectified image. In this study, water level at the time of image acquisition, provided by astronomical tide prediction, is used as rectification elevation. Geographical coordinates of each waterline are then obtained manually by digitizing on rectified images.

To assess the accuracy of the rectification, the digitized waterlines are compared to the tide corresponding contours computed from the measured beach topography (Figure 2). This comparison is undertaken using waterlines from 7 images acquired on July 11 th and July 12 th 2022. Wave energy offshore Lafitenia was very low on these 2 days (Hs < 0.5m). As it is the main factor driving shortterm beach changes, the beach configuration can be considered as spatially identical as when topography was acquired (see Input Data section). Cross-shore positions were compared along a series of profiles. The overall obtained RMSE and determination coefficient are respectively 3.86 and 0.79. Mean high waterline [START_REF] Boak | Shoreline Definition and Detection: A Review[END_REF] is used as a proxy of the shoreline as Lafitenia is a mesotidal beach. Indeed, the mean high waterline was shown to be a relevant shoreline proxy to study event-scale to inter-annual shoreline variability for such tidal regime (Castelle et al., 2014). Consequently, every low or mid tide images have been put aside in this study to keep only high tide images. Besides, when several images were acquired within a small period of time (e.g. few days) only the image exhibiting the lowest swash excursion is kept to reduce the difference between total water level and astronomical water level. This results in a set of 65 images between October 2021 and August 2022 from which high tide waterline can be extracted.

As every selected image is acquired potentially at different stage of tide near the mean water level, waterlines have to be adjusted to this reference elevation to carry out a relevant comparison of their cross-shore position. A linear tidal correction is applied using water levels from tide data and measured beach face slope to translate each waterline to a reference elevation [START_REF] Vos | CoastSat: A Google Earth Engine-enabled Python toolkit to extract shorelines from publicly available satellite imagery[END_REF] and use it as an estimate of the shoreline. The beach face slope used in this study is obtained by cross-shore averaging the data surveyed over the last ten years between isobaths 1.5m and 3m as we focus on the high tide area. An average slope of 12.5%, considered as homogenous in longshore direction, is thus calculated and used to adjust every waterline position to the baseline Lafitenia beach exhibits distinct alongshore curvature that complicates the explicit definition of longshore and cross-shore directions and so the characterization of its dynamic. In these circumstances, a solution enable to remove the initial curvature of the beach and to facilitate further data processing [START_REF] Harley | A simple data transformation technique for pre-processing survey data at embayed beaches[END_REF]. It consists in using a log-spiral function fitted to the beach planform to transform Cartesian data into longshore-cross-shore coordinates system. In our study case, the beach planform is estimated by computing the average of high-tide waterlines along transects placed perpendicular to the local beach orientation. Thanks to this method, waterlines are projected in a new coordinates system and can easily be resampled with a regular step all along the longshore axis defined by the log-spiral baseline.

A linear fit of each shoreline (derived from the digitized waterlines as described above) in alongshore/cross-shore referential is also performed in order to compute the time series of angle between the overall shoreline direction and the alongshore axis (hereafter referred to the shoreline angle).

Results and discussion

The timestack of the shoreline cross-shore position (Figure 3a) exhibits complex patterns of change over a 9-month period, with alternating periods of beach accretion (yellow color) and erosion (blue color) in relation to the log-spiral baseline. Interestingly, the shoreline cross-shore fluctuations often show opposite directions of shoreline translation at both extremities, creating an apparent beach rotation. This is clearly visible between end of January and mid-February, with a 5 to 10-m shoreline advance at the south extremity of the beach (alongshore distance ≈ 250 m) synchronized with a 15 m retreat at the north extremity. Overall, the movements amplitude appears to be slightly more important during the first half of the time period (November to April) whereas a net decrease in shoreline fluctuations is observed starting from May.

Computing the time series of shoreline angle relative to the alongshore axis (Figure 3b) also highlights the occurrence of clockwise and anti-clockwise beach rotations at the scale of days to weeks with angle values ranging from -4° to +6°. In addition to these changes occurring at week scale, a seasonal pattern seems to emerge over those 9 months (Figure 2b). The winter period is made up of clockwise and anti-clockwise rotations of large amplitude while the summer period is characterized by reduced-amplitude and slower rotations with a beach remaining in anti-clockwise rotated state (negative shoreline angle).

Analysis of incident waves conditions (Figure 3c) usually helps understanding conditions under which these beach rotations develop (e.g. wave intensity or direction shift). The strong seasonally modulated wave climate of Lafitenia is likely to have a straight influence on the above mentioned seasonal trend. Larger shoreline orientation shifts could be related with the winter energetic wave climate, whereas the observed shoreline stability over the last 4 months could be seen as a result of the fair wave conditions. On shorter timescale (days), it is hard to say at what point the few large beach rotation observed in November or February (Figure 3a, b) are linked to peaks in wave energy and/or direction shifts. Longer time series of data would be necessary to establish consistent relations at both week and season scales and to increase the understanding of this beach rotation process.

CoastSnap rectified images enables to undertake quantitative insights about shoreline changes. Shoreline advances and retreats at Lafitenia are calculated using data such as waterline positions and the average beach slope whose accuracy could still be improved. Waterline positioning should be improved by carrying out in situ sensitivity study and integrating the effect of swash excursion while digitizing. The hypothesis of an homogeneous beach slope could also be reconsidered, especially in the alongshore direction.

Conclusion

The installation of a CoastSnap station at Lafitenia has enabled to characterize for the first time beach rotations occurring at this embayed beach. The high-frequency dataset, made up of 65 crowd-sourced images acquired over a 9months period, was used to detect and to quantify shoreline changes at short time scales (from day to season). Cross-shore dynamic of shoreline shows interesting patterns, with notable opposite movements of extremities. Despite some approximations, it is possible to obtain the shoreline position at a reasonable precision (see section Method). Future work will be carried out to better relate wave forcing to the shoreline dynamics. This will require continuous supply of citizen's contributions.

Figure 1 :

 1 Figure 1: Location of the CoastSnap station used in this study. (a) The city of Saint-Jean-de-Luz in the southwest of France. (b) The embayed beach of Lafitenia and the CoastSnap station location. (c) Example of smartphone image taken from the station

Figure 2 :

 2 Figure 2: (a) Waterline digitalization on rectified image. (b) Digitized waterline position (blue line) compared with the corresponding contour computed from beach topography (red line)

Figure 3 :

 3 Figure 3: (a) Shoreline timestack showing the cross-shore distance to the baseline as a function of time and alongshore distance. (b) Time series of beach angle with respect to the baseline angle; clockwise rotation at Lafitenia means that the beach gets more north oriented (c) Daily-averaged wave conditions offshore Saint-Jean-de-Luz (MARC, IFREMER)

Acknowledgements

This study was conducted within the Littoview project and funded by the European Regional Development Fund.


domain: phys spi

text:
Srisaran Venkatachalam

Xin Zhou
email: xin.zhou@cnrs.fr

Effects of stochastic forces on the nonlinear behaviour of a silicon nitride membrane nanoelectromechanical resonator

Keywords: stochastic switching, Duffing nonlinearity, silicon nitride membrane, nanoelectromechanical resonator I

In this work, we present the effects of stochastic force generated by white noise on the nonlinear dynamics of a circular silicon nitride membrane. By tuning the membrane to the Duffing nonlinear region, detected signals switching between low-and high-amplitudes have been observed. They are generated by noise-assisted random jumps between bistable states at room temperature and exhibit high sensitivity to the driving frequency. Through artificially heating different mechanical vibration modes by external input of white noise, the switching rate exhibits exponential dependence on the effective temperature and follows with Kramer's law. Furthermore, both the measured switching rate and activation energy exhibit sensitivity to the width of the hysteresis window in nonlinear response and the driving force, which is in qualitative agreement with the theoretical descriptions. Besides, white noise-induced hysteresis window squeezing and bifurcation point shifting have also been observed, which are attributed to the stochastic force modulation of the spring constant of the membrane. These studies are carried out in an all-electric operating scheme at room temperature, paving the way for the exploration of probability distribution-based functional elements that can be massively integrated on-chip.

INTRODUCTION

Stochastic switching between coexisting states has been observed in chemistry, physics, biology, and engineering systems. Its mechanism can be explained simply through descriptions of a fictive particle hopping randomly in bistable states of a double-well potential [START_REF] Gammaitoni | Stochastic resonance[END_REF]. Although these individual switching events are random, both distribution and probability of switching can be well predicted and controlled, exhibiting several potential applications. For instance, it brings an enhancement of signal-to-noise ratio [START_REF] Badzey | Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance[END_REF] and random number generators [START_REF] Vodenicarevic | Low-energy truly random number generation with superparamagnetic tunnel junctions for unconventional computing[END_REF], which could offer a means of enhancing signal processing . Besides, stochastic switching can directly perform stochastic computing, which are attractive for novel error-tolerant computing schemes in neuromorphic applications [START_REF] Gaba | Stochastic memristive devices for computing and neuromorphic applications[END_REF][START_REF] Zahari | Analogue pattern recognition with stochastic switching binary cmos-integrated memristive devices[END_REF]. Therefore, reserach interests are preserved in exploring stochastic switching in various systems, including Josephson junctions [START_REF] Wiesenfeld | Stochastic resonance and the benefits of noise: from ice ages to crayfish and squids[END_REF][START_REF] Muppalla | Bistability in a mesoscopic josephson junction array resonator[END_REF], protein folding [START_REF] Wales | Energy landscapes: Applications to clusters, biomolecules and glasses[END_REF], nanomagnets [START_REF] Spano | Experimental observation of stochastic resonance in a magnetoelastic ribbon[END_REF][START_REF] Wernsdorfer | Mea-surements of magnetization switching in individual nickel nanowires[END_REF].

Micro-or nano-mechanical resonator, which allows electrical and optical signals to couple with a mechanical degree of freedom, is one of interesting components for exploring stochastic switching [START_REF] Stambaugh | Noise-activated switching in a driven nonlinear micromechanical oscillator[END_REF][START_REF] Venstra | Stochastic switching of cantilever motion[END_REF]. Because of intrinsic nonlinearity coming from geomentry, nano-mechanical resonators can be driven as Duffing oscillators, providing the indispensable condition: bistable states. In addition, the idea of building multifunctional components based on nanoelectromechanical resonators has been driving researchers to explore fundamental aspects and potential applications beyond sensing [START_REF] Maillet | Nonlinear frequency transduction of nanomechanical brownian motion[END_REF][START_REF] Zhou | On-chip thermometry for microwave optomechanics implemented in a nuclear demagnetization cryostat[END_REF][START_REF] Dion | Reservoir computing with a single delay-coupled non-linear mechanical oscillator[END_REF][START_REF] Gazizulin | Surface-induced near-field scaling in the knudsen layer of a rarefied gas[END_REF][START_REF] Guerra | A noise-assisted reprogrammable nanomechanical logic gate[END_REF].

So far, white noise assisted stochastic switching has been investigated by using cantilevers [START_REF] Venstra | Stochastic switching of cantilever motion[END_REF], double-clamped beams [START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF][START_REF] Aldridge | Noise-enabled precision measurements of a duffing nanomechanical resonator[END_REF], and membranes [START_REF] Dolleman | High-frequency stochastic switching of graphene resonators near room temperature[END_REF][START_REF] Chowdhury | Phase stochastic resonance in a forced nanoelectromechanical membrane[END_REF]. In these previous studies, nanomechanical resonators were operated using either magnetomotive detection at low temperature or optical readout at room temperature, which are not conducive to the further development of functional elements based on stochastic switching that can be integrated in large quantities on-chip. In very recent years, high-stress silicon nitride membrane electromechanical resonators emerge and offer high quality factors with resonance frequency in MHz ranges. They allow vibrating membranes to be strongly coupled to external electronic circuits, providing a highly sensitive actuation and detection scheme at room temperature [START_REF] Zhou | High-q silicon nitride drum resonators strongly coupled to gates[END_REF][START_REF] Pokharel | Capacitively coupled distinct mechanical resonators for room temperature phonon-cavity electromechanics[END_REF][START_REF] Yuan | Silicon nitride membrane resonators at millikelvin temperatures with quality factors exceeding 108[END_REF]. However, to date there are few studies of the effect of stochastic forces generated by white noise on the nonlinear behaviour of such membrane nanoelectromechanical resonators in a fully electric operating scheme at room temperature.

In this work, we present experimental studies of stochastic switching in a circular membrane nanoelectromechanical resonator, which consists of a silicon nitride drum, capacitively coupled to a suspended aluminum gate. At the room temperature experiment, the membrane resonator is frequency biased in Duffing nonlinear region, creating bi-stable states for mechanical displacements. The membrane is artificially heated by adding external white noise, and the switching rate of different mechanical modes as a function of the effective temperature has been measured. Its characteristics follow Kramer's law. Experimental parameters affect on this noise assisted jumping between bistable states have been analyzed by comparing our measurements with the theoretical descriptions. Besides, squeeze of hysteresis window in nonlinear responses has also been observed when the amplitudes of stochastic force increases.

II. METHODS

The device measured in this experiment is a silicon nitride membrane electromechanical resonator. It consists of a silicon nitride circular membrane with ∼ 33 µm in diameter and ∼ 80 nm in thickness, which is fabricated based on a substrate composed of a high tensile stress (∼ 1 GPa) silicon nitride thin film on top of the silicon wafer. The membrane is covered with an aluminum thin film, ∼ 25 nm in thickness, and is capacitively coupled to a suspended aluminium top-gate, as shown in Fig. 1 (a). The distance between the membrane and its gate is d ≈ 600 nm. Details of the fabrication process have been reported in the previous work [START_REF] Zhou | High-q silicon nitride drum resonators strongly coupled to gates[END_REF][START_REF] Pokharel | Capacitively coupled distinct mechanical resonators for room temperature phonon-cavity electromechanics[END_REF]. In this room temperature measurement, the membrane is set in a vacuum chamber (∼ 10 -6 mbar) and grounded through its top gate. In this capacitive coupling scheme, mechanical displacement x can be excited by an electrostatic force f d (t), which is generated through combining a dc signal V dc and a ac signal with a frequency Ω,

V ac • cos(Ω • t), which gives f d = 1 2 ∂(C g (x)(V dc +V ac •cos(Ω•t)) 2 )/∂x. The C g (x)
is the coupling capacitance between the membrane and its gate, and the x(t) is the mechanical displacement.

To study noise assisted random jumps between bistable states, white noise that is generated by an arbitrary waveform generator is injected to the membrane resonator, through a band-pass filter centred at the desirable frequency corresponding to the resonant mechanical mode. It yields the stochastic force f n (t) also driving on the membrane. Therefore, the mechanical displacement x(t) of this circular membrane can be described by a motion equation Eq.1,

m ef f (ẍ + γ m ẋ + Ω 2 m x + α m ef f x 3 ) = f n (t) + f d (t). (1)
Here, Ω m is the resonance frequency of mechanical resonator, the γ m is the linewidth, α is the Duffing coefficient, and m ef f is effective mass of the membrane. The α/m ef f term is the Duffing coefficient normalized with the effective mass, having the unit m -2 s -2 . For circular membrane resonators, the mechanical mode is designated by (i, j), where azimuthal mode number is indexed by the i (= 0, 1, ...) and the radial mode number is denoted by the j (= 1, 2, ...). In the following discussions, these parameters corresponding to the mechanical vibration modes will be specified.

A detection scheme is built on microwave interferometry, in which the tiny mechanical displacements excited by f d [Ω] are transduced by a microwave signal with frequency ω to be the signal having frequency at ω + Ω. The displacement x(t) is readout by a lock-in amplifier through a frequency down-conversion. The details of both driving and detection schemes are shown in 1 (b). The microwave interferometry allows to transduce the detected amplitude of electrical signals into mechanical displacement x by using the relation V out = GωZ 0 C g xV µw /(2d) [START_REF] Zhou | High-q silicon nitride drum resonators strongly coupled to gates[END_REF]. Here, the Z 0 = 50 Ohm is the impedance of the measurement line, C g ≈ 10 fF is the capacitance between the membrane and its coupled gate, V µw is the input amplitude of microwave signal for detection (in this measurement, V µw ≈ 300 mV p ), the V out is the detected amplitude of the microwave signal, and the G is the total gain in the detection chain in this experiment.

Figure 1 (c) shows linear responses of the mechanical fundamental mode (black line), whose resonance frequency is Ω m /2π ≈ 11.781 MHz with linewidth γ m /2π = 1015 Hz. When the mechanical resonator is driven by a periodic force with a large amplitude, the spring hardening (α > 0) makes the resonance frequency shift towards the higher values and frequency responses exhibit hysteric behavior between forward (red line) and backward (blue line) frequency sweeps. Bistable states exist in this region, which is defined between two saddle-node bifurcation points corresponding to the amplitude of mechanical response jump-down Ω bd in the forward frequency sweep and jump-up Ω bu in the backward sweep [START_REF] Venstra | Stochastic switching of cantilever motion[END_REF][START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF][START_REF] Aldridge | Noise-enabled precision measurements of a duffing nanomechanical resonator[END_REF]. It is a typical Duffing nonlinear phenomenon and has been observed in various mechanical resonators. Besides, the spectral density of the electromechanical resonator's amplitudes, driven by white noise in the form of stochastic force f n with different power, has also been measured, as shown in Fig. 1 (d). The spectral of stochastic force is given by

S f f (Ω)= < f n (t)f n (0) > e iΩt dt, exciting me- chanical displacement and creating spectrum S xx (Ω) = |χ m (Ω)| 2 S f f (Ω). The χ m (Ω) is the susceptibility of the membrane resonator, χ m (Ω) = 1 2m ef f Ωm(Ωm-Ω-iγm/2) .
It is well-known that the integration of the S xx allows to access the mean square amplitude < x(t) 2 >, < x(t) 2 >= 1 2π S xx (Ω)dΩ. According to equipartition theorem [START_REF] Albrecht | Frequency modulation detection using high-q cantilevers for enhanced force microscope sensitivity[END_REF][START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF][START_REF] Hauer | A general procedure for thermomechanical calibration of nano/micro-mechanical resonators[END_REF][START_REF] Zhou | Electric circuit model of microwave optomechanics[END_REF], the effective temperature T e of the mechanical resonator driven by stochastic forces, in the classical limit, is described by Eq.2,

T e = m ef f Ω 2 m < x(t) 2 > k B . (2)
The k B is the Boltzmann constant. In our measurement, the measured spectral (shown in Fig. 1 (d)) can be converted into displacement spectral density S xx according to the basic principle of microwave interferometry mentioned above [START_REF] Zhou | High-q silicon nitride drum resonators strongly coupled to gates[END_REF]. Then, by integration of the measured S xx , the T e is obtained based on the Eq.2. The corresponding effective temperatures are marked in Fig. 1 (d), which are obtained based on these expressions by taking parameters of m ef f ≈ 4 × 10 -14 kg and ω/(2π) = 7.01 GHz. We use this method to obtain the value of the T e when the membrane is artificially heated up by external white noise. In this work, the maximum vlues of the measured spectra density is ∼1 nm/ √ Hz.

III. EXPERIMENTAL RESULTS AND DISCUSSION

In order to investigate the switching dynamics of the system in this room temperature measurement, we prepare a small hysteresis window of nonlinear responses to prepare bistable states. In this experiment, it is a challenge to observe the detected signal switching between low-and high-amplitudes in the large hysteresis window, e.g. a width much larger than 100 Hz. It could be due to the mechanical resonance frequency drifting at room temperature. Thus, the width is controlled to be less than 100 Hz by regulating amplitude of the f d , as shown in Fig. 2 (a). The typical value of the detected mechanical displacement in this nonlinear region is ∼ 38 nm. Without injecting extra white noise, stochastic switching between bistable states has been observed by setting the frequency of the driving force in the hysteresis regime. Amplitudes of the detected signals switch between two states, corresponding to the low ("L") and the high ("H") amplitude, as shown in Fig. 2 (b)-(d). When the driving frequency is biasing at the center of the hysteresis window, the residence of each state exhibits almost equal probability, as shown in Fig. 2 (c). Whereas one of the states is more likely to be occupied when the frequency of the driving force is detuned from the center, even for only a few Hz, as presented in Fig. 2 (b) and (d). The prob-ability of the occupation states exhibits high sensitivity to the driving frequency. This is because the doublewell potential, in descriptions of bistable states, has been tilted by the frequency of the periodic driving force, f d .

From the previous reports [START_REF] Dykman | Theory of fluctuational transitions between stable states of a nonlinear oscillator[END_REF], we know that the switching rate Γ in the stochastic switching between the bistable states follows Kramer's law,

Γ = Γ 0 exp( -E a k B T e ). (3)
The Γ 0 is the maximum switching rate, E a is an activation energy corresponding to the barrier height between bistable states [START_REF] Aldridge | Noise-enabled precision measurements of a duffing nanomechanical resonator[END_REF]. For Duffing oscillators, the width of hysteresis window |Ω bd -Ω bu | and the driving frequency Ω d are supposed to play important role in manipulating the E a and the Γ 0 . In the switching process, the distribution of residence time (τ ) corresponding to the "L" or "H" state follows a Poisson law,

N (τ ) = I • Γ exp(-τ Γ). (4)
Here, the I is a parameter for normalizing the residence time distribution. The probability of residence in each state is therefore sensitive to the driving frequency. Thus, we first measure the distributions of the residence time for both "L" and "H" occupation states corresponding to each value of the T e . To do so, we exploit two forces to drive the membrane. The periodic driving force f d with frequency Ω d is used to control the width of hysteresis window and bistable states are prepared by driving membrane at the frequency Ω d ∼ (Ω bd -Ω bu )/2. The stochastic force f n generated by white noise is employed to control the effective temperature T e of the mechanical mode, by artificially "heating up" the membrane. By

G 0 » 23 Hz f 1 (H) f 1 (L) f 2 (H) f 2 (L) Exponential fit Switching rate G (Hz) 1/T eff (K -1 )
G 0 » 30 Hz measuring the time trace of the detected signal amplitudes, we can make statistics of residence time. Figure 3 (a) shows distributions of residence time for both "L" and "H" states exhibit equal counts, which are measured at T e ≈ 3 × 10 4 K. The inset figure shows amplitudes of the signal in a two-quadrature space, which presents bistable states of the membrane. These results indicates that potential well is controlled to be symmetric. Then, by using Poisson's law to fit the distribution of residence time, we can obtain the transition rate Γ corresponding to different T e .

Figure 3 (b) shows exponential plots of Γ as functions of 1/T e for the fundamental mode (0,1), corresponding to different widths of the hysteresis window controlled by the driving force f d . The maximum switching rate Γ 0 can be obtained by exploiting Kramer's law as described by Eq.3. The value of Γ 0 ≈ 23 Hz is obtained when the width of the hysteresis window (Ω bd -Ω bu )/2/π is controlled to be 18 Hz by the driving force f 1 . The relatively higher value of Γ 0 ≈ 30 Hz is obtained by using the higher force f 2 , yielding the relatively wider hysteresis window with width (Ω bd -Ω bu )/2/π = 67 Hz. Besides, we also measured the switching rate of the higher mechanical modes, (1,1) and (0,2), as shown in Fig. 3 (c). In the measurement of the mode (1,1), a similar width of hysteresis window, around 62 Hz, has been set and we obtain Γ 0 ≈ 40 Hz. While, for the mode (0,2), the hysteresis window is controlled to be quite small, less than 10 Hz, and the measured values of Γ 0 are ≈ 3.1 Hz and 3.8 Hz, well below the damping rate of mechanical resonator, γ m /(2π). These measurement results indicate that the maximum value of switching rate between bistable states relies on the hysteresis window (Ω bd -Ω bu ), in accordance with theoretical descriptions [START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF][START_REF] Dykman | Theory of fluctuational transitions between stable states of a nonlinear oscillator[END_REF]. Besides, we also found that the slope of these plots is sensitive to the amplitude of the driving force f d , as shown in the Fig. 3(b)-(c). From Kramer's law, we know that these slopes are proportional to the activation energy E a . Therefore, these measurement results demonstrate that the activation energy E a is also sensitive to the driving force, which is in accordance with Dykman's theoretical descriptions [START_REF] Dykman | Theory of fluctuational transitions between stable states of a nonlinear oscillator[END_REF].

In theoretical analysis of stochastic switching in Duffing oscillators, it has been predicted that both of E a and Γ 0 rely on several experimentally accessible parameters, such as driving force f d , and frequency detuning between the driving frequency Ω d and the bifurcation point Ω bd [START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF][START_REF] Dykman | Theory of fluctuational transitions between stable states of a nonlinear oscillator[END_REF]. Therefore, we make a quantitative comparison between our experimental results and the theoretical expressions, the barrier height

E a ∝ f 2 d (Ω bd -Ω d ) 3 2 /(Ω bd -Ω m ) 5
2 /γ m and the switch- [START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF]. From slopes of two curves shown in the Fig. 3 (b), we have obtained E a (f 2 )/E a (f 1 ) ≈ 19.4, in accordance with calculation results of 18.2. However, for the switching rate Γ 0 , the calculation result of Γ 0 (f 1 ) ≈ 148 Hz and Γ 0 (f 2 ) ≈ 71 Hz are nearly ≈ 5 times and ≈ 2.3 times higher than our measurement results, respectively. Consequently, it gives Γ 0 (f 2 )/Γ 0 (f 1 ) ≈ 2, higher than the value of ≈ 1.3 obtained in measurement. Besides of Duffing nonlinear term in 1, other nonlinearities could cause this discrepancy, such as nonlinear damping or resorting forces [START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF]. Although it is difficult to have a quantitative agreement in comparisons, measurement results exhibit qualitative accordance with theoretical descriptions. Besides of the noise assisted random jumps in bistable states, white noise induced stochastic force affecting on mechanical properties in nonlinear region are also deserved to be investigated. Therefore, we track the variations of hysteresis window in a fixed sweeping range for both the frequency and the V ac , when increase the amplitude of stochastic force in the form of the effective temperature T e . Figure 4 (a)-(e) show the 2D plots of hysterics regime, corresponding to different T e . Each hysteric region is obtained by subtracting the data of the backward frequency sweep from the forward one. With increasing stochastic force, it is clear that the hysteresis window shifts towards to the higher frequency re-gion. Here, we choose the hysteresis region, which consisting of forward and backward frequency sweeps and are measured at the V ac = 40 mV, for further analysis. By fitting each nonlinear curve, the resonance frequency Ω f it is obtained and we find that it shifts towards the higher frequencies as T e increases. Moreover, variations of bifurcation points as a function of T e have also been observed through definitions of δΩ = Ω bd -Ω f it and δΩ = Ω bu -Ω f it . These analyses are shown in Fig. 4(f).

ing rate Γ 0 = |Ω bd -Ω d | 1 2 (Ω bd -Ω m ) 1 2 /(2π)
Compared with Ω bu , the Ω bd seems to be more sensitive to stochastic forces. For a Duffing oscillator, driven by a periodic force f d (t), it is well-known that the bifurcation point Ω bd (for the case of spring hardening effect) can be generally described by Ω bd = Ω m (1 + 3α 8k x 2 max ) [START_REF] Lifshitz | Nonlinear dynamics of nanomechanical and micromechanical resonators[END_REF]. It depends on the Duffing parameter α, the maximum mechanical displacement x max , and the spring constant k = Ω 2 m m ef f . From Fig. 4 (b)-(e), we did not observe clearly variations in amplitudes at Ω bd . Besides, for circular membrane, Duffing (hardening) nonlinear coefficient α, is mainly dominated by geometry [START_REF] Cattiaux | Geometrical nonlinearity of circular plates and membranes: An alternative method[END_REF]. Therefore, the displacements of the hysteresis window can be attributed to the modulations of spring constant by stochastic forces. Besides observations of hysteresis windows shifting, we also find that the width of hysteresis window starts to be reduced as the amplitude of stochastic force increases. Similar phenomenon has been reported in previous studies of doubly-clamped beams with [START_REF] Aldridge | Noise-enabled precision measurements of a duffing nanomechanical resonator[END_REF] and micro-cantilever [START_REF] Venstra | Stochastic switching of cantilever motion[END_REF] increasing amplitude of stochastic force, which also has been attributed to stochastic forces affecting the spring constant of the mechanical resonator. However, compared with these previous reports, we did not observe the hysteresis quenched even though the membrane is artificially heated at same orders of the T e . We suppose that it requires the much higher value of the f n to quench hysteresis regime. Because this high stressed silicon nitride membrane, with a fully clamped scheme, has the higher spring constant k ∼ 90 N/m [START_REF] Zhou | High-q silicon nitride drum resonators strongly coupled to gates[END_REF].

IV. CONCLUSION

In conclusion, we have investigated the effect of stochastic forces on the Duffing nonlinear behaviors of silicon nitride membrane nanoelectromechanical resonators, by using microwave interferometry detection scheme at room temperature. In the hysteresis region, the probability of occupying a state exhibits high sensitivity to the driving frequency, demonstrating the potential to explore nonlinear-based sensing. The stochastic switching of the membrane, corresponding to different mechanical modes, has been found to follow Kramer's law. Our measurements are in qualitative agreement with the theoretical descriptions of the maximum switching rate Γ 0 and activation energy E a influenced by the experimental setting of the driving force and the width of the hysteresis window. Besides, squeeze of Duffing hysteresis window and shifts of bifurcation points have also been observed when increasing the effective temperature T e and been attributed to stochastic forces influencing the spring constant of the membrane. Although the typical values of the Γ 0 obtained in this experiment is in the orders of tens Hz, these studies pave way for explorations of stochastic switching based novel functions beyond sensing, such as random number generators, stochastic computing, and logic gates [START_REF] Vodenicarevic | Low-energy truly random number generation with superparamagnetic tunnel junctions for unconventional computing[END_REF][START_REF] Gaba | Stochastic memristive devices for computing and neuromorphic applications[END_REF][START_REF] Zahari | Analogue pattern recognition with stochastic switching binary cmos-integrated memristive devices[END_REF][START_REF] Guerra | A noise-assisted reprogrammable nanomechanical logic gate[END_REF].

  FIG. 1. (a) Left, an optical image of a silicon nitride membrane, covered with an Al thin film ∼25 nm in thickness. The silicon nitride (SiN) membrane is released from Si substrate by reactive ion etching un-wanted silicon nitride parts, circular holes (∼ 300 nm in diameter), followed with XeF2 selective etching of the Si layer below the silicon nitride layer through those holes. Right, schematic diagram of the device in lateral view. Note that SiN is silicon nitride. (b) The measurement setup. A silicon nitride membrane is driven by means of an electrostatic force through combing dc and ac signals, and stochastic forces generated by a Gaussian white noise which is filtered to have a band-pass around resonance frequency of the desired mechanical mode. All measurements are performed at room temperature, in vacuum (pressure < 10 -6 mbar). (c) Black line, linear response of silicon nitride membrane, measured with V dc = 2 V, Vac = 1 mVp for the fundamental mode Ω01. Both blue and red lines show the Duffing responses, which are measured with the higher driving amplitude, V dc = 2 V, Vac = 75 mVp. The hysteresis regime is formed between forward (red line) and the backward (blue line) frequency sweeps. (d) Spectral density of the membrane's displacements, at different effective temperature Te.

FIG. 2 .

 2 FIG. 2. (a)Left, without adding extra white noise, nonlinear responses of the forward and backward frequency sweeps, which are measured with a high driving force generated by V dc = 2 V and Vac = 9 mVp, . Inset, zoom of the hysteresis frequency window, around 15 Hz in width. Right, schematic diagram of double wells corresponding to three different driving frequencies in bistable region, which is marked on the inset figure as "l", "m", and "r" respectively. (b)-(d) The left side of figures show amplitudes as a function of the measurement time in this stochastic system, driving respectively at "l", "m", and "r", as marked in the inset of the (a). Their histograms of the distributions in amplitudes are shown on the right side.

FIG. 3 .

 3 FIG. 3. (a) Residence time distributions for both "L" and "H" states at Te ≈ 3 × 10 4 K. The red curves are fit by Poisson distribution function, giving a value of switching rate Γ ≈ 2.2 Hz. The inset figure is amplitude distributions of detected signals shown in the two quadrature space and each point is one sample in the measurement of amplitude as a function of time. (b) Switching rate Γ of both states as a function of T -1 e , corresponding to different driving forces. In both (a) and (b), the driving force f1, f2 are generated through combing V dc = 2 V with Vac = 14 mVp, and 18 mVp, respectively. (c) Left and Right show Γ versus 1/Te for mechanical mode (1,1) and (0,2). These driving forces are normalized with f1.

FIG. 4 .

 4 FIG. 4. The measurement of the nonlinear responses for the first mode is performed at a fixed dc voltage V dc = 2V . (a)-(e) Hysteresis of 2D (two dimensional) plots corresponding to different noise drive power, in the form of the effective mode temperature. Each hysteresis curve is obtained from both forward and backward frequency sweeps at a fixed Vac voltage. All 2D plots share the same color bar of the (e). Here, the sweeping frequency is normalized by the starting point. (f) δΩ as a function of mechanical mode temperature, where δΩ = Ω bd -Ω f it are plotted as blue squares for the forward and = Ω bu -Ω f it are marked as red squares for backward sweeps respectively, measured at Vac = 40 mV. The black dots correspond to δΩ = Ω f it -Ωm. Here, we choose Ωm as the reference frequency when there is no extra white noise.

ACKNOWLEDGMENTS

We would like to acknowledge financial support from STaRS-MOC Project No. 181386 the Region Hauts-de-France, from ISITE-MOST Project No. 201050, and the French National Research Agency, ANR-MORETOME, No. ANR-22-CE24-0020-01. This work was partly supported by the French Renatech network.


Sébastien Orange

Nathalie Verdière

Loïs Naudin

Target observability of nonlinear complex networks with linear and nonlinear couplings

Keywords: Observability, complex networks, nonlinear couplings, algorithm, neurons

Many real-life systems can be modeled as a complex network made up of nodes whose dynamics are governed by nonlinear differential equations. Observability on such networks is one fundamental problem consisting in the capability of inferring the states of all/some target nodes from the knowledge of some node states. In this paper, we present a new approach for studying the full and the target observability in networks of nonlinear systems with linear and/or nonlinear couplings. The proposed method is based on a mathematical result ensuring the existence of specific local relations obtained from equations of each node.

Two consequences in terms of observability are deduced from this theoretical result and are used to elaborate an algorithm. This algorithm, named TargetObservability, determines sets of nodes ensuring the observability of a given target set. We exemplify our approach on a biological neural network of C. elegans, made up of Hodgkin-Huxley type models coupled through linear and nonlinear terms. This provides a testable hypothesis that is likely to accelerate the discovery and analysis of the biological circuitry in C. elegans.

Introduction

Many real-life systems can be modeled as a complex network, made up of nodes connected through edges [START_REF] Newman | Networks[END_REF]. In such a configuration, each node has its own dynamics governed by a set of differential equations, and the edges describe the interactions between the nodes. A neuronal network is an example of complex network in which the nodes represent the neurons, and the edges represent the synaptic connections. The best way to monitor the state of a complex network would be to measure the state of all its nodes separately. However, in practice, we are often restricted to a limited set of measurements due to technical and experimental impediments. Therefore, we can ask which nodes of the system can contain sufficient information about the other nodes to retrieve the network complete state, making the system full observable.

However, in many applications, the full observability is not the right concept since retrieving the state of all the nodes in the network is not expected. Indeed, we might want to infer the state of only a subset of nodes, called target nodes, since only their states would describe a desired property of the network. As an example, in neuroscience, muscle states form the basis of the locomotor behavior, and are modulated by the activity of neurons in the nervous system. In this case, we might be interested in identifying an optimal set of neurons from which we can infer the state of the muscles, without having to retrieve the state of all neurons in the entire nervous system.

There is a long tradition and a substantial literature that deal with the observability and its dual concept, the controllability, of linear complex systems [START_REF] Meng | Control energy scaling for target control of complex networks[END_REF][START_REF] Aguirre | Controllability and synchronizability: Are they related?[END_REF][START_REF] Wang | Controllability and observability analysis for vertex domination centrality in directed networks[END_REF][START_REF] Gao | Target control of complex networks[END_REF][START_REF] Leitold | Controllability and observability in complex networks-the effect of connection types[END_REF][START_REF] Czeizler | Structural target controllability of linear networks[END_REF][START_REF] Montanari | Functional observability and target state estimation in large-scale networks[END_REF]. In such a case, the nodal dynamics and couplings are considered linear, despite the fact that they are fundamentally nonlinear in nature. If these simplifications facilitate the mathematical studies of systems, they are not without consequences [START_REF] Aguirre | Structural, dynamical and symbolic observability: From dynamical systems to networks[END_REF]. Indeed, some recent works demonstrate the irrelevance of the linear observability and controllability to nonlinear dynamical networks. First, the linearized system could be not controllable, while the original nonlinear system is actually controllable [START_REF] Liu | Control principles of complex systems[END_REF]. Also, linear controllability theory tends to generate information that is not useful for nonlinear control of many complex systems.

For instance, linear controllability may give great importance to some nodes for the controllability of the system while these nodes are theoretically unimportant [START_REF] Jiang | Irrelevance of linear controllability to nonlinear dynamical networks[END_REF]. Last, linear observability can highly underestimate the number of sensors necessary to retrieve the state of the variables of the original nonlinear system [START_REF] Letellier | A symbolic network-based nonlinear theory for dynamical systems observability[END_REF]. Taken together, these evidences urged the development of mathematical tools capable of dealing with the nonlinear observability.

In this context, some approaches dealing with the nonlinear observability have emerged from different perspectives. [START_REF] Liu | Observability of complex systems[END_REF] proposed a graph-theoretic approach, based on an inference diagram, that only yields a strict lower bound for the size of the minimum sensor set, which still tend to severely underestimate their number to get a good observability (Letellier et al., 2018a). Therefore, Letellier et al. (2018a) slightly modified the inference diagram from [START_REF] Liu | Observability of complex systems[END_REF] by removing all the nonlinear couplings, thus taking into account only the linear ones. If this approach proved to be efficient in some particular cases, it turns out to be inapplicable for complex systems with only or mostly nonlinear connections. Another approach is based on the symbolic formalism [START_REF] Bianco-Martinez | Symbolic computations of nonlinear observability[END_REF][START_REF] Letellier | Symbolic observability coefficients for univariate and multivariate analysis[END_REF][START_REF] Letellier | A symbolic network-based nonlinear theory for dynamical systems observability[END_REF]) that suffers from a high computational cost: the number of variable combinations to investigate increases exponentially with the system dimension.

In this paper, we propose a work on the target observability in network of nonlinear systems with non linear couplings avoiding such a high computationnal cost. Further, this paper has a twofold contribution. The first is a theoretical result: from the differential equations corresponding to one node, this result ensures the existence of relations involving only one state variable of the node, the coupling terms and eventual controls. Two observability properties propagating in the network are deduced from this result. These two properties constitute the basis of the second contribution which consists in an algorithm, called TargetObversability, that determines sets of nodes which should be observed to deduce the state of a given target set of nodes. Many models fall within the scope of the present work in neuroscience [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF][START_REF] Fitzhugh | Mathematical models of excitation and propagation in nerve[END_REF][START_REF] Hindmarsh | A model of neuronal bursting using three coupled first order differential equations[END_REF]Naudin et al., 2022b,a).

To show the suitability of our approach, we apply it for the target observability of C. elegans muscles involved in a chemotaxis behavior [START_REF] Costalago-Meruelo | Emulation of chemical stimulus triggered head movement in the c. elegans nematode[END_REF]. The C. elegans worm is a well-known model organism in neuroscience due to its simple nervous system and its fully mapped connectome (White et al., 1986;[START_REF] Varshney | Structural properties of the caenorhabditis elegans neuronal network[END_REF]. In particular, we determine a set of neurons from which the state of C. elegans muscles can be inferred.

The remainder of the paper is organized as follows. In Section 2, we define the problem of the target observability in mathematical terms. Section 3 presents a theoretical result centered on local differential algebraic relations from which two corollaries giving observability conditions are deduced in Section 4. These corollaries form the basis of the TargetObservability algorithm developed in Section 5. Finally, this algorithm is applied in Section 6 on a C. elegans neuronal network, and the perspectives are drawn in Section 7.

Problem statement

In this paper, we consider directed complex networks formed of N nonlinear nodes coupled through linear and nonlinear terms. Some nodes can be controlled using inputs u i and the other nodes are coupled between them and to the controlled ones (Figure 4.A). Here the coupling acts on a single variable of the node, the one characterizing the node dynamics. Moreover, the dynamics of each node are governed by a nonlinear dynamical system described by a set of differential equations. These differential systems are not necessarily identical. The state variable X i = (x i,1 , . . . , x i,n ) T of the ith node (i = 1, . . . , N ) takes the form:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ẋi,1 = f i,1 (X i , Θ i ) + ∑ j∈N - i c j (x i,1 , x j,1 ) + u i ẋi,2 = f i,2 (X i , Θ i ), ⋮ ẋi,n = f i,n (X i , Θ i ),
(1) where • f i, j (X i , Θ i ) are linear combinations of the state variables x i,2 , . . . , x i,n whose coefficients are analytical functions in Θ i , x i,1 and its derivatives A ;

• the initial states X i (t 0 ), i = 1, . . . , N , of the network are supposed to be known at time t 0 ;

• ∑ j∈N - i c j (x i,1 , x j,1
) is a coupling term with N - i the in-neighbors set of the node i (that is, the nodes whose edge comes into the node i), and

c j ∶ R 2 → R is an infinitely differentiable such that, for all x i,1 ∈ R, x j,1 → c j (x i,1 , x j,1
) is a one to one function. It is worth noting that System (1) describes a large class of models in real-life problems. In this paper, we emphasize on the neuroscience field in which most of neuron models describe the dynamics of the neuron voltages following a system of the form (1). Some examples are the large class of conductance-based models that use the Hodgkin-Huxley formalism [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF], the Hindmarsh-Rose model [START_REF] Hindmarsh | A model of neuronal bursting using three coupled first order differential equations[END_REF], the Fitzhugh-Nagumo model [START_REF] Fitzhugh | Mathematical models of excitation and propagation in nerve[END_REF], or the phenomenological non-spiking models (Naudin et al., 2022b,c).

Now, we aim to study the observability of a set of nodes of the network, called target afterwards.

To do this, from the solutions of some nodes, we set the following target observability definition.

Definition 2.1. Let N and T be two sets of nodes of the network, the second one being the target.

The set of nodes T is N -observable if there exists a surjective function on the solution set of (X i ) i∈N in the solution set of (X i ) i∈T .

A An example of such a system is given by System (5). A system in which function f

i,2 (X i , θ i ) = ẋi,1 x i,2 + x i,1 x 2 i,3
does not belong to the framework of the present work since it is not linear in x i,3 . with the set of dynamical systems associated with each node, and the intra-node (in black) and inter-node (in orange) couplings. This representation highlights the inter-node couplings through the first variable x i1 , i = 1, . . . , N , and the heterogeneity of dynamical systems governing the behavior of nodes of the network.

From the known solutions of the set of nodes N , our goal is to determine whether it is possible to infer the solutions of the set of nodes T . Our approach is based on specific relations linking the first state variable of a node and the first variables of its in-neighbors. More precisely, we deduce an ordinary differential equation ensuring that the first variable of System (1) can be inferred from its in-neighbor nodes and vice versa. This is the subject of the following section.

Analytical redundancy relations of nodes

The result presented in this section gives the general form of the differential equation linking the node i to the nodes j ∈ N - i . In the literature, these relations linking the parameters, the outputs, and the inputs (and their derivatives) of the system are classically called Analytical Redundancy Relations (ARR) [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF][START_REF] Travé-Massuyes | Diagnosability analysis based on componentsupported analytical redundancy relations[END_REF][START_REF] Verdière | Functional diagnosability and detectability of nonlinear models based on analytical redundancy relations[END_REF].

Proposition 3.1. Let i ∈ [1, N ] and X -i be the vector (x j,1 ) j∈N - i . There exists an ARR of the form

Pi (x i,1 , X -i , Θ i ) = P i (x i,1 , Θ i ) + α ∑ k=0 h i,k (x i,1 , Θ i )u (k) i + ∑ j∈N - i α ∑ k=0 h i,k (x i,1 , Θ i )c j (x i,1 , x j,1 ) (k) = 0 (2)
where

• h i,k are analytical functions in Θ i , x i,1 and its derivatives;

• α < n with n the dimension of System (1).

Proof. To simplify the proof, let us set

w i = ∑ j∈N - i c j (x i,1 , x j,1 ) + u i .
Under the hypotheses of Section 2, System (1) can be written

(3) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ẋi,1 = g i,1,1 (x i,1 , Θ i ) + g i,1,2 (x i,1 , Θ i )x i,2 + ⋯ + g i,1,n (x i,1 , Θ i )x i,n -w i ẋi,j = g i,j,1 (x i,1 , Θ i ) + g i,j,2 (x i,1 , Θ i )x i,2 + ⋯ + g i,j,n (x i,1 , Θ i )x i,n , j = 2, ⋯, n, (3a) (3b)
with g i,j,k analytical functions in Θ i , x i,1 and its derivatives. Using Equation (3b), the successive derivatives of

w i = -ẋi,1 + g i,1,1 (x i,1 , Θ i ) + g i,1,2 (x i,1 , Θ i )x i,2 + ⋯ + g i,1,n (x i,1 , Θ i )x i,n (see Equation 3a)
can be expressed as linear combinations of x i,2 , . . . , x i,n :

w (l) i = m 1,l (x i,1 , Θ i ) + m 2,l (x i,1 , Θ i )x i,2 + ⋯ + m n,l (x i,1 , Θ i )x i,n , l ∈ N (4)
We have (w i , ẇi , . . . , w

(n-1) i ) T = M (1, x i,2 , . . . , x i,n
) T where M = (m j,l ) 1≤j≤n, 0≤l≤n-1 . If det(M ) = 0 then a linear combination of the rows of M is null. The last equality shows that the same linear combination of the (w i , ẇi , . . . , w

(n-1) i ) T is null. If det(M ) ≠ 0 then M (w i , ẇi , . . . , w (n-1) i ) T = det(M ) (1, x i,2 , . . . , x i,n
) T where M is the transposed comatrix of M . This last equality shows, in particular, that a linear combination of w i , ẇi , . . . , w

(n-1) i is equal to det(M ).
In both cases, we obtain an ARR of the form

P i (x i,1 , X -i , Θ i ) = P i (x i,1 , Θ i ) + α ∑ k=0 h i,k (x i,1 , Θ i )w (k) i = 0 with α < n.
Equation ( 2) is then a direct consequence of this last equality and of the definition of w i .

Proof of Proposition 3.1 gives a way to obtain such ARRs and the expression form (2) will permit to define criteria for the target observability. However, the proposed method can be difficult to put in place since it requires the inversion of a formal matrix leading to complex calculus even in simple examples. When the functions f i,j of System (1) are differential polynomials in the components of X i , ARRs can be found in using the Rosenfeld-Groebner elimination algorithm implemented in the package Differential Algebra of Maple [START_REF] Boulier | Study and implementation of some algorithms in differential algebra[END_REF][START_REF] Verdière | Functional diagnosability and detectability of nonlinear models based on analytical redundancy relations[END_REF][START_REF] Verdière | Diagnosability in the case of multi-faults in nonlinear models[END_REF]. Let us see with the FitzHugh-Nagumo model how to obtain the ARR of Proposition 3.1 by this kind of procedure.

Example 1. Consider a complex network made up of three neurons whose dynamics are governed by the FitzHugh-Nagumo (FHN) model [START_REF] Fitzhugh | Mathematical models of excitation and propagation in nerve[END_REF]. In this network, the first two nodes are controlled by a constant input current I i that models a sensory information (see Figure 2). The entire system of the complex network, noted Γ, is then given by

Γ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Γ 1 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ1,1 = x 1,1 - x 3 1,1 3 -x 1,2 + I 1 , ẋ1,2 = 1 τ (x 1,1 + a -bx 1,2 ). Γ 2 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ2,1 = x 2,1 - x 3 2,1 3 -x 2,2 + I 2 , ẋ2,2 = 1 τ (x 2,1 + a -bx 2,2 ). Γ 3 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ3,1 = x 3,1 - x 3 3,1 3 -x 3,2 + e 31 (x 1,1 -x 3,1 ) + e 32 (x 2,1 -x 3,1 ), ẋ3,2 = 1 τ (x 3,1 + a -bx 3,2 ).
(

) 5
where x 1,1 , x 2,1 and x 3,1 represent the membrane potential of the neuron 1, 2 and 3 respectively.

Moreover, assume that x 3,1 is the output of the system Γ, and denote Θ ∶= (a, b, τ ).

For System Γ 3 , x 1,1 and x 2,1 can be considered as inputs. The Gröbner-Rosenfeld algorithm applied to System Γ 3 with the elimination order

[Θ] ≺ [x 3,1 , x 1,1 , x 2,1 ] ≺ [x 3,2 ] (that consists in
eliminating first the state variable x 3,2 then x 2,1 and x 1,1 ) returns:

P3 (x 3,1 ) = P 3 (x 3,1 ) + e 31 (3b(x 1,1 -x 3,1 ) + 3τ ( ẋ1,1 -ẋ3,1 )) + e 32 (3b(x 2,1 -x 3,1 ) + 3τ ( ẋ2,1 -ẋ3,1 )) = 0 (6) with P 3 (x 3,1 ) = -3 τ x 2 3,1 ẋ3,1 + 3 τ ẋ3,1 -3 τ ẍ3,1 -3 x 3,1 -b x 3 3,1 + 3 b x 3,1 -3 b ẋ3,1 -3 a.
Identifying ( 2) and ( 6) leads to set h 3,0 = 3b, h 3,1 = 3 τ , c 1 (x 3,1 , x 1,1 ) = e 31 (x 1,1 -x 3,1 ) and c 2 (x 3,1 , x 2,1 ) = e 32 (x 2,1 -x 3,1 ),

Consequences on target observability

In this section, the ARR ( 2) is used to present consequences on the target observability from a given set of nodes N . The ARR, Pi is a differential equation linking x i,1 and x j,1 for j ∈ N - i and is supposed to be determined for each node of the network. Recall that the initial condition X i (t 0 ), i = 1, . . . , N is assumed to be known in System (1) and also the derivatives of X i at t 0 at any order.

The following corollaries are deduced from these ARRs and the initial condition. They give conditions ensuring the N -observability of a given node from a set of nodes N -observable.

These corollaries are illustrated by Figure 3.A and 3.B.

Corollary 4.1. Let N a given set of nodes. If, for all j ∈ N - i , the j th node is N -observable, then the i th node is N -observable.

Proof. By hypothesis, there exists a surjection from the set of solutions of (X j ) j∈N on the set of solutions of (X j ) j∈N - i . The Picard-Lindelöf theorem, applied to Equation (2), ensures the existence of a surjection from (X j ) j∈N - i to X i . Consequently, there exists a surjection from the set of solutions of (X j ) j∈N to X i .

Example 2. Let take again Example 1. The initial value problem formed of polynomial P3 (see Equation ( 6)) and initial conditions x 3,1 (t 0 ), ẋ3,1 (t 0 ) admits a unique solution x 3,1 according to the Picard-Lindelöf theorem.

Corollary 4.2. Let N be a given set of nodes and j 0 ∈ N - i . If the i th node and all the j th nodes for j ∈ N - i ∖ {j 0 } are N -observable then the j th 0 node is N -observable.

Proof. By isolating terms depending on j 0 , the ARR (2) can be rewritten

Q(x i,1 , X -i , Θ i ) + α ∑ k=0 h i,k (x i,1 , Θ i )z (k) = 0 (7)
where z = c j 0 (x i,1 , x j 0 ,1 ). Equation ( 7) can be solved with respect to z and z is N -observable. Since

x j 0 ,1 → c j 0 (x i,1 , x j 0 ,1 ) is a one to one function, X j 0 is also N -observable.

Consequently, there exists a surjection from the set of solutions of (X j ) j∈{i}∪N - i ∖{j 0 } to X j 0 . As in the proof of Corollary 4.1, we can construct a surjection from the set of solution (X j ) j∈N to X j 0 .

Remark 4.1. Note that ARR (2) may not depend explicitly on x j 0 for j 0 ∈ N - i . In this case, there exists a differential equation linking x j 0 and x i,1 . Consequently, it stays possible to infer the set of solutions of X j 0 from the solutions of X i and the initial condition.

Example 3. Consider again Example 1. Let i = 3 and j 0 = 2 and suppose that nodes 1 and 3 are N -observable. The initial value problem composed of the differential equation P3 and the known initial conditions x 2,1 (t 0 ) and ẋ2,1 (t 0 ) admits a unique solution x 2,1 .

The set of the instances

Node

A B

The 5 Target observability algorithm

In this section, in order to simplify the notation, the nodes are identified to their indexes.

The algorithm TargetObservability, presented afterwards, is based on Corollaries 4.1 and 4.2.

This algorithm determines sets of nodes from which the state of a set of target nodes can be inferred.

Let N ′ be a given set of nodes and T the set of target nodes. The algorithm TargetObservability returns the set E of all the subsets N ⊂ {1, . . . , N } ∖ T containing N ′ , minimal for inclusion, such that T is N -observable. The sets of E are minimal in the sense that, for any subset N ′′ of {1, . . . , N } ∖ T such that T is N ′′ -observable, there exists at least an element N of E contained in N ′′ . In order to compute the set E, the algorithm performs a tree transversal to construct recursively an increasing sequence of sets

N ′ = N 0 ⊂ N 1 ⊂ ⋯ ⊂ N m = N until T is N -observable.
Further, the algorithm TargetObservability proceeds as follows. Initially, E is assigned to the empty set {} and the algorithm is called with a set N ′ of supposed observed nodes. Now, let us describe any recursive call performed by the algorithm. From a subset Ñ of {1, . . . , N } ∖ T , this algorithm tests whether a set of E is included in Ñ . If it is the case, T is Ñ -observable but Ñ is not minimal for inclusion and no other computation is performed. Otherwise, the algorithm calls the function ObservableNodes that determines the set O ⊂ {1, . . . , N } of Ñ -observable nodes.

Note that the computation of O is based on Corollaries 4.1 and 4.2. Two cases appear:

• T ⊂ O. In this case, T is Ñ -observable; Ñ is inserted to E and sets of E that are not minimal for inclusion are removed from E.

• T / ⊂ O. In this case, for each of the nodes i ∈ {1, . . . , N } ∖ (O ∪ T ) B , a recursive call of this algorithm is performed with Ñ ∪ {i} as input.

The above description is summarized in the following pseudo-code.

TargetObservability ( Ñ ) If Ñ is not included in one of the set of E Then O := ObservableNodes( Ñ ); If T ⊂ O Then E ∶= E ∪ { Ñ } ;
Remove sets of E which are not minimal for inclusion ;

Else

for i ∈ {1, . . . , N } ∖ (O ∪ T ) do (*) E = TargetObservability ( Ñ ∪ {i}); end do ; end if ; end if ;
Return E ;

In the next section, we apply the algorithm TargetObservability described in this section on a C. elegans chemotaxis neuronal network.

B If {1, . . . , N } ∖ (O ∪ T ) = ∅ then Ñ can not be completed into a set N ′ ⊂ {1, . . . , N } ∖ T so that T is N ′observable. In such a case, the loop (*) is not executed.

Application on a C. elegans chemotaxis network

Here we apply our proposed algorithm TargetObservability for the target observability of muscles in the C. elegans worm. C. elegans is a well-known model organism in neuroscience due to its simple nervous system, made up of 302 neurons and about 7000 synaptic connections, and its fully mapped connectome (White et al., 1986;[START_REF] Varshney | Structural properties of the caenorhabditis elegans neuronal network[END_REF]. Despite its simplicity, the nematode surprisingly shares many of the general essential human biological features using similar neurotransmitters, channels, and developmental genes (Altun et al., 2020). Moreover, principles that underlie behaviors in C. elegans may also be similar in more complex animals such as humans [START_REF] Chalasani | Dissecting a circuit for olfactory behaviour in caenorhabditis elegans[END_REF]. For these reasons, C. elegans has become a model of reference to investigate how behavior emerges from its underlying physiological processes [START_REF] Sarma | Openworm: overview and recent advances in integrative biological simulation of caenorhabditis elegans[END_REF]. Thus, recent efforts have been made to build conductance-based models [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF]) that reproduce the experimental dynamics of individual neurons in the worm [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF](Naudin et al., , 2022a)). These models take the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C V = -g Ca m Ca h Ca (V -E Ca ) -g Kir h Kir∞ (V )(V -E K ) -g K m K (V -E K ) -g L (V -E L ) + I ṁCa = m Ca∞ (V ) -m Ca τ m Ca , m Ca∞ (V ) = (1 + exp ( V m Ca 1/2 -V k m Ca )) -1 ḣCa = h Ca∞ (V ) -h Ca τ h Ca , h Ca∞ (V ) = (1 + exp ( V h Ca 1/2 -V k h Ca )) -1 ṁK = m K∞ (V ) -m K τ m K , m K∞ (V ) = (1 + exp ( V m K 1/2 -V k m K )) -1 h Kir∞ (V ) = (1 + exp ( V h Kir 1/2 -V k h Kir
))

-1

(8) where:

• V , m Ca , h Ca and m K are the four state variables of the system. In particular, V is the membrane potential of the neuron, which is the variable that characterizes the behavior of the neuron, and m Ca , h Ca and m K are the activation and inactivation variables of the ion channels;

• I is the control that models a sensory information produced by the environment, or an artificial injection current applied by the experimenter;

• g ion and E ion , ion ∈ {Ca, K, Kir, L}, are parameters, as well as τ x , V x 1/2 , k x with x ∈ {m Ca , m K , h K , h Kir }. The values of these parameters depend on the considered neuron [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF](Naudin et al., , 2022a)).

Chemotaxis network and coupling. Some neuronal networks underlying specific behaviors in C. elegans have been cracked using data from the anatomical connectome (White et al., 1986;[START_REF] Varshney | Structural properties of the caenorhabditis elegans neuronal network[END_REF], together with powerful experimental techniques and computational tools.

Here, we use a neuronal network associated with a chemotaxis behavior [START_REF] Costalago-Meruelo | Emulation of chemical stimulus triggered head movement in the c. elegans nematode[END_REF] (Figure 4). Each neuron is arbitrarily named with three capital letters for convention, and a fourth letter L (left) or R (right) (White et al., 1986). The dynamics of each neuron are governed by a model of the form (8). The electrical and chemical synapses are modeled by coupling terms corresponding to the variable I of the first equation of System (8). The electrical synapses are linear, modeled as ohhmic resistances, while chemical synapses are nonlinear with channels gated in the postsynaptic membrane. Their respective expressions are then given by

I gap ij (V i , V j ) = g gap (V i -V j ) I syn ij (V i , V j ) = g syn s ∞ (V j )(V i -E ij )
where i and j denote a post-and pre-synaptic neuron, respectively; g gap , g syn and E ij are parameters; and s ∞ is a sigmoid function of the form

s ∞ (V j ) = 1 1 + exp ( V th -V j V slope )
with V th and V slope two other parameters.

Therefore, using the notations of System (1), the variable I of System ( 8) is substituted by

the coupling term ∑ j∈N - i c j (V i , V j ) + I i with c j (V i , V j ) = I gap ij (V i , V j ) + I syn ij (V i , V j ),
The union of N ′ = {ASEL, ASER} and of any of these sets turns T into an observable node C . C Note that, by construction of Algorithm TargetObservability, any set N ′′ containing nodes ASEL and ASER such that T is an N ′′ -observable contains at least one of the sets of E.

The CPU time needed to obtain this output is approximately 1.07 second with a Maple implementation of algorithm TargetObservability on a Intel Quad Core 2.50GHz Processor with 8 Go of RAM.

An example of the propagation of the observability property through the network.

In this paragraph, we focus on the propagation of the observability property in the network.

Let us consider the node T = {M uscle} which is N = {ASEL, ASER, AIAL, AIAR}-observable (See ( 9)). By tracking the computations performed by the function ObservableNodes, it is possible to obtain each of the steps leading to the observability of T from the set N . In this particular case, a propagation of the observability property can be summed up in Table 1.

Conclusions and perspectives

Summary. This paper focuses on the target observability, that consists in determining which nodes are needed to infer the state of a target subset. To that end, we present theoretical results based on specific local analytical redundancy relations. These results lead to two observability properties used to develop the algorithm TargetObservability that automatically identify sets of nodes that can infer the state of target ones. Finally, we apply our algorithm for the target observability of a C. elegans muscle involved in a chemotaxis behavior.

Perspectives. The observability property test does not tell us how to numerically reconstruct the states of target nodes and it is well-known that different sets of nodes do not provide the same reconstruction quality [START_REF] Letellier | On the non-equivalence of observables in phase-space reconstructions from recorded time series[END_REF][START_REF] Montanari | Observability of network systems: A critical review of recent results[END_REF][START_REF] Sysoeva | Reconstruction of coupling structure in network of neuron-like oscillators based on a phase-locked loop[END_REF]. Nevertheless, our approach currently returns different sets of nodes from which the target set can be reconstructed. A first extension to this work will be to develop methods quantifying the quality of these sets of nodes by developing metrics to choose the best option to reconstruct the target nodes states. Since this paper provides some building blocks for a practical reconstruction of the nodes states, a second extension of the present work will be the development of a state reconstructor, which is still an open challenge for nonlinear systems [START_REF] Liu | Control principles of complex systems[END_REF].

Figure 4 .

 4 Figure 4.B illustrates the full complex network presented in Figure 4.A with the set of differential equations associated with each node, and the intra-and inter-node couplings.

Figure 1 :

 1 Figure1: (A) Example of a complex network composed of 4 nodes (N = 4) coupled through directed edges, and controlled through the first node. (B) Full complex network presented in (A) with the set of dynamical systems associated with each node, and the intra-node (in black) and inter-node (in orange) couplings. This representation highlights the inter-node couplings through the first variable x i1 , i = 1, . . . , N , and the heterogeneity of dynamical systems governing the behavior of nodes of the network.

Figure 2 :

 2 Figure 2: Coupling of three neurons governed by the FitzHugh-Nagumo (FHN) model, where the first two nodes are controlled by constant input currents I 1 and I 2 .

Figure 3 :

 3 Figure 3: (A) Illustration of Corollary 4.1. The N -observability of node i is deduced from the one of its in-neighbors (N - i ). (B) Illustration of Corollary 4.2.The N -observability of node j 0 is deduced from the N -observability of the node i and of its in-neighbors, N - i (except the j 0 th).

Figure 4 :

 4 Figure 4: A neural network underlying a chemotaxis behavior in C. elegans (Costalago-Meruelo et al., 2018).

  E = {{RIAR}, {RM D}, {AIAL, AIAR}, {AIAL, AIBL}, {AIAL, AIY L}, {AIAL, AIY R, }, {AIAL, AIZL}, {AIAL, AIZR}, {AIAR, AIBL}, {AIAR, AIY L}, {AIAR, AIY R}, {AIAR, AIZL}, {AIAR, AIZR}, {AIBR, AIY L, AIZR}, {AIBR, AIY R, AIZR}, {AIBR, AIZL, AIZR}, {AIY L, AIZR, AV AR}, {AIY L, AIZR, RIBR}, {AIY R, AIZR, AV AR}, {AIY R, AIZR, RIBR}, {AIZL, AIZR, AV AR}, {AIZL, AIZR, RIBR}, {AIBL, AIBR, AIZR, AV AR}, {AIBL, AIBR, AIZR, RIBR}, {AIBL, AIZR, AV AR, RIBR}}.

Table 1 :

 1 Propagation of the {ASEL, ASER, AIAL, AIAR}-observability property in the network to the target T = {M uscle} performed by Algorithm TargetObservability.

	Node considered	In-neighbors of the node	Corollary used Deduced observable node
	ASER	{AIBL,AIAL,AIAR}	Cor. 4.2	AIBL
	AIAL	{ASEL,ASER,AIAR,AIZL}	Cor. 4.2	AIZL
	AIAR	{ASER,AIAL,AIZR}	Cor. 4.2	AIZR
	AIBR	{ASEL,ASER,AIAR,AIZL,AIZR}	Cor. 4.1	AIBR
	ASEL	{AIYR.AIZL}	Cor. 4.2	AIYR
	AIYR	{ASEL,ASER,AIYL}	Cor. 4.2	AIYL
	AIBL	{ASEL,ASER,AIAL,AIZL,AIZR,RIBR}	Cor. 4.2	RIBR
	RIBR	{AIBR,AIAL}	Cor. 4.2	AVAR
	RIAR	{ASEL,AIYR,AIZR,RIBR,AVAR}	Cor. 4.1	RIAR
	RMD	{RIAR}	Cor. 4.1	RMD
	Muscle	{RMD}	Cor. 4.1	Muscle

Acknowledgments

This work was supported by the ERDF (XTerm Project), Normandie Region (France) and by The LMAH, FR-CNRS-3335.

Declaration of competing interests

The authors declare no competing interests.