Camille Boutin

Laurent Kodjabachian
email: laurent.kodjabachian@univ-amu.fr

BIOLOGY OF MULTICILIATED CELLS

Keywords: Multiciliated cells, centriole, cilia, deuterosome, polarity

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Introduction

Multiciliated cells are defined by the presence at their surface of few dozens to hundreds of cilia, which beat coordinately to generate robust polarized hydrodynamic forces (Fig. 1). Each cilium is templated by a basal body (BB), a modified centriole, which harbors two asymmetric appendages, a rootlet that plunges into the cytoplasm, and a basal foot that points in the beating direction (Fig. 1A). Through their interaction with cytoskeletal elements, these appendages link BBs together, thus coordinating cilia orientation [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF]. MCCs can be found as individual cells (e.g. cycad plant male gametes or Paramecium) or organized into specialized epithelia in metazoans [START_REF] Meunier | Multiciliated Cells in Animals[END_REF]. In small-sized aquatic invertebrate organisms, MCCs are mainly involved in locomotion, whereas in vertebrates, MCCs help propel biological fluids at the surface of specialized tissues. Three main models have helped advancing our knowledge of fundamental principles of MCC biology, the ciliated epidermis of the Xenopus embryo (Fig. 1B-E), the airway mucociliary and the brain ependymal (Fig. 1 B'-E') epithelia of the mouse.

Production of efficient directional fluid flows depends on several parameters integrated in time and space during development, as well as during regeneration. First, a correct number of MCCs must be produced, second a correct number of functional cilia must be assembled in each MCC, third cilia must coordinate their beating orientation within individual and between neighboring MCCs to produce robust and regular strokes. In humans, disruption of cilia production or beating in MCCs leads to hydrocephalus, respiratory and fertility pathologies [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF]. In this review, we shall focus on recent findings regarding the transcriptional control of MCC differentiation, the mechanisms of centriole synthesis, and the multiscale organization of polarized ciliary beating.

Establishment and maintenance of MCC identity

In all models, MCC fate adoption appears to be triggered by Notch pathway inhibition, which launches a specific gene regulatory network (GRN), whose key nodes are named Gemc1 (Gmnc), Mcidas (Multicilin), Myb, FoxJ1 and Rfx2/3 [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF] (Fig. 2). BMP pathway inhibition can also turn on the MCC GRN in Xenopus and in cultures of human airway epithelium [START_REF] Cibois | BMP signalling controls the construction of vertebrate mucociliary epithelia[END_REF]. Gemc1 and Mcidas belong to the Geminin family of nuclear factors, and together with E2F family transcription factors are necessary and sufficient to trigger multiciliogenesis [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF][START_REF] Kim | Multicilin and activated E2f4 induce multiciliated cell differentiation in primary fibroblasts[END_REF]. Both Gemc1 and Mcidas can activate Myb, which promotes centriole multiplication, as well as FoxJ1 and Rfx2/3, which coordinately control motile ciliogenesis [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF][START_REF] Quigley | Rfx2 Stabilizes Foxj1 Binding at Chromatin Loops to Enable Multiciliated Cell Gene Expression[END_REF]. A recent study revealed that in mouse, MCC fate is controlled by Gemc1, while Mcidas acts downstream to initiate centriole amplification, providing a rationale for the sequential involvement of two related genes that until now appeared to share the same activity [START_REF] Lu | Mcidas mutant mice reveal a two-step process for the specification and differentiation of multiciliated cells in mammals[END_REF]. The core MCC GRN has recently been enriched with additional regulators that branch at distinct points (Fig. 2). Among those, the chromatin factor Trrap appears to function downstream of Notch and upstream of Mcidas in human airway MCCs [START_REF] Wang | TRRAP is a central regulator of human multiciliated cell formation[END_REF]. Trrap was identified through a pool-based shRNA screen, demonstrating the value of unbiased phenotypic searches to identify additional MCC regulators. The p53 family member, p73, was shown to regulate about 100 ciliary genes, including FoxJ1 and Rfx2/3, and p73 mutant mouse display typical features linked to MCC dysgenesis such as hydrocephalus, sterility and chronic airway inflammation/infection [START_REF] Marshall | p73 Is Required for Multiciliogenesis and Regulates the Foxj1-Associated Gene Network[END_REF][START_REF] Nemajerova | TAp73 is a central transcriptional regulator of airway multiciliogenesis[END_REF]. Interestingly, it was suggested that p73 marks some basal cells in airway epithelium for MCC differentiation, lending to it a role as a pioneer or competence factor [START_REF] Marshall | p73 Is Required for Multiciliogenesis and Regulates the Foxj1-Associated Gene Network[END_REF]. Further supporting the importance of p73, it was also reported that it regulates ependymal MCC polarity through maintenance of actin and microtubule cytoskeletal networks [START_REF] Fuertes-Alvarez | p73 regulates ependymal planar cell polarity by modulating actin and microtubule cytoskeleton[END_REF]. It is important to mention that the MCC identity is inherently labile, as its maintenance requires constant FoxJ1 transcriptional activity, at least in murine ependymal MCCs [START_REF] Abdi | Uncovering inherent cellular plasticity of multiciliated ependyma leading to ventricular wall transformation and hydrocephalus[END_REF].

Centriole synthesis in MCCs

A key step of MCC differentiation is the large-scale production of centrioles that will subsequently be converted into BBs to template motile cilia. In vertebrate MCCs, it is achieved via two pathways: the parental centriole-dependent pathway, which is analogous to the duplication pathway active during cell cycle, but can produce 10-20 centrioles in MCCs; the deuterosome-dependent pathway, which produces 80-90% of total MCC centrioles [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF]12] (Fig. 3). Deuterosomes comprise non-centriolar electron-dense structures that can support the growth of single or multiple procentrioles according to their size, which may increase over differentiation time [13][14][15]. Despite its initial description in the late 70's, the deuterosome remained virtually unexplored for over 40 years. Recent molecular studies started to unveil the composition of Xenopus and mouse deuterosomes [13][14][15][16]. These studies have shown that the architecture of this organelle is based on a set of scaffolding proteins including Deup1 at its core, Pericentrin, g-tubulin and Cep152 at its periphery. The three latter proteins are wellknown components of the pericentriolar material (PCM) [17], which underlines an unexpected link between deuterosome and PCM. Refined mapping of Deup1, Pericentrin and g-tubulin revealed the existence of nested subdomains in deuterosomes that require further investigation [14]. For instance, it will be important to understand how these domains are assembled and evolve over deuterosome lifetime. In addition, the functional relevance of such organization for deuterosome function remains to be addressed.

Deciphering the sub-cellular origin of the deuterosome may provide interesting cues on how it is built. Questioning the long-held belief that centriole synthesis by deuterosomes occurs de novo (i.e. independently from centrosomal centrioles), live imaging of newly synthesized centrioles suggested that in cultured mouse ependymal cells, deuterosomes are seeded by the daughter centrosomal centriole [15]. However, using both chemical and genetic approaches, three independent studies now suggest that formation of deuterosomes and production of multiple centrioles can occur in absence of parental centrioles (Zhao et al., bioRxiv doi.org/10.1101/373662; Nanjundappa et al., bioRxiv doi.org/10.1101/478297; Mercey et al., bioRxiv doi.org/10.1101/503730). Thus, it is unclear whether deuterosomes are seeded by or transit through parental centrioles, which can only be resolved by live imaging of deuterosomes and centrioles in parallel. Seeking for the mechanisms of deuterosome construction, it is interesting to remember that this organelle is the second entity observed during mass centriole production, the first one being fibrous granules (FGs). Apart from early EM studies that described them as electron dense granules that condensate to form deuterosomes, FGs remain marginally characterized [12]. A notable exception is the identification of PCM1, a specific marker labelling FGs [18], but for which no role in multiciliogenesis could be detected [19]. However, a recent study showed that: i. Deup1 and PCM1 proteins co-localize during the initial steps of MCC differentiation; ii. after MCC transcriptional program is launched, E2F4, a co-activator of Multicilin, shuttles from the nucleus to the cytoplasm where it localizes in the PCM1 positive cloud of material and participates to deuterosome formation [20]. Therefore, further (re-)evaluating the role of FGs during MCC differentiation is likely to provide information about the construction and/or function of the deuterosome.

Imaging centriole and deuterosome markers in real-time and on fixed samples revealed three successive phases: templating of procentrioles, growth of procentrioles and disengagement of mature centrioles from deuterosomes and parental centrioles [13-15,21] (Fig. 3). Molecular

MCC polarized beating, a multi-step and multiscale problem

Multiciliated epithelia are specialized to ensure the vectorial transport of particles or cells along their surface. Their effective function relies on the coordination of ciliary beating within individual cells and at the tissue level. This is achieved through a multistep process involving events occurring both at the sub-micrometer scale and at the scale of the entire organ or organism.

Centrioles newly released in the cytoplasm are transported towards the apical surface.

Meanwhile, they are converted into BBs by the addition of rootlet and basal foot appendages [START_REF] Spassky | The development and functions of multiciliated epithelia[END_REF] (Fig. 3). The precise mechanisms of BB maturation and transport remain poorly understood but seem to implicate acto-myosin and the formation of vesicles at the distal appendages of BBs [24,25].

Once they arrive to their destination, BBs organize at the apical surface. Their distribution varies between different types of MCCs. In Xenopus epidermis, BBs are evenly dispersed at the surface of the MCC [26] (Fig. 1E). In contrast, BBs are organized in rows covering the entire apical surface in mouse tracheal MCCs [27]. In mouse ependymal MCCs, BBs are clustered in an off-centered patch, through a process called translational polarity (Fig. 1D'). Within this patch, BBs are organized in rows [28,29] (Fig. 1E'). Whether such variations in BB organization underlie functional differences remains to be determined. In all types of MCCs, BBs of a given cell are oriented in the same direction. This peculiar organization, referred to as rotational polarity, is essential to ensure the coordinated beating of cilia within individual MCCs.

Both distribution and orientation of BBs relies on their tight interaction with apical cytoskeletal elements that differentially contribute to the geometry of the array. Cortical actin in MCCs is organized in two structurally distinct apical and sub-apical networks [26,27,30,[START_REF] Tateishi | Three-dimensional Organization of Layered Apical Cytoskeletal Networks Associated with Mouse Airway Tissue Development[END_REF].

The apical pool is necessary for docking of BBs to the cell membrane [24,26]. The sub-apical network connects BBs and is necessary for their correct spacing [26,[START_REF] Park | Ciliogenesis defects in embryos lacking inturned or fuzzy function are associated with failure of planar cell polarity and Hedgehog signaling[END_REF]. Microtubules connecting BBs play an instrumental role in their alignment and orientation [26,27]. Finally, a network of intermediate filaments is also observed unsheathing BBs, although its precise functional contribution remains to be determined [27,[START_REF] Tateishi | Three-dimensional Organization of Layered Apical Cytoskeletal Networks Associated with Mouse Airway Tissue Development[END_REF].

Interplay between those cytoskeletal elements and their interacting proteins results in the BB organization described above [START_REF] Kunimoto | Coordinated ciliary beating requires Odf2-mediated polarization of basal bodies via basal feet[END_REF][START_REF] Antoniades | Making the connection: ciliary adhesion complexes anchor basal bodies to the actin cytoskeleton[END_REF][START_REF] Antoniades | Addressing the Functional Determinants of FAK during Ciliogenesis in Multiciliated Cells[END_REF][START_REF] Epting | The Rac1 regulator ELMO controls basal body migration and docking in multiciliated cells through interaction with Ezrin[END_REF][START_REF] Kulkarni | WDR5 Stabilizes Actin Architecture to Promote Multiciliated Cell Formation[END_REF]. Among those, some unexpected proteins were identified. For instance, FAK, Paxillin and Vinculin -well-known focal adhesion proteinsassociate with BBs and their striated rootlet to form ciliary adhesion complexes that connect BBs to the actin cytoskeleton [START_REF] Antoniades | Making the connection: ciliary adhesion complexes anchor basal bodies to the actin cytoskeleton[END_REF][START_REF] Antoniades | Addressing the Functional Determinants of FAK during Ciliogenesis in Multiciliated Cells[END_REF]. Likewise, the well-characterized chromatin modifier WDR5 acts as a scaffolding protein binding to BBs and modulating apical actin in MCCs [START_REF] Kulkarni | WDR5 Stabilizes Actin Architecture to Promote Multiciliated Cell Formation[END_REF].

Beyond its role in BB organization, the apical cytoskeleton of MCCs is thought to be important for propagation of the metachronal wave of ciliary beating [26]. It also confers a mechanical resistance to BBs to sustain the shear stress imposed by ciliary beating in mature MCCs [30,[START_REF] Bayless | Asymmetrically localized proteins stabilize basal bodies against ciliary beating forces[END_REF]. Ciliary activity itself provides an important feedback on BB organization. Although initially imperfectly coordinated, cilia beating instructs weak directional flow that improves coordination of BB polarity, which in turn reinforces the flow [START_REF] Guirao | Coupling between hydrodynamic forces and planar cell polarity orients mammalian motile cilia[END_REF][START_REF] Mitchell | A positive feedback mechanism governs the polarity and motion of motile cilia[END_REF]. However, the precise range of action of such hydrodynamic feedback remains to be evaluated. In addition, cilia motility is necessary for apical actin meshwork assembly, thereby reinforcing BB anchoring at the apical surface [30].

To generate a directional fluid flow and sustain the correct function of the organ, ciliary beating must be coordinated at the tissue-level. This is achieved through a planar cell polarity driven multistep process [29, [START_REF] Walentek | Planar cell polarity in ciliated epithelia[END_REF][START_REF] Mitchell | The PCP pathway instructs the planar orientation of ciliated cells in the Xenopus larval skin[END_REF][START_REF] Vladar | Microtubules enable the planar cell polarity of airway cilia[END_REF][START_REF] Chien | Mechanical strain determines the axis of planar polarity in ciliated epithelia[END_REF]. The initial cue that establishes organ-level polarity remains unknown in mouse ciliated epithelia. In contrast, it was shown that the mechanical strain transmitted from the mesoderm to the ectoderm during Xenopus gastrulation caused alignment of microtubule arrays along the rostro-caudal axis, which secondarily led to PCP protein asymmetric distribution along the same axis. Through this global polarization of the epithelium, individual MCCs acquire polarized membrane identity. This generates a compass in each MCC that is used to orient the BB array, thereby allowing cilia beating direction to align with global tissue polarity [29, [START_REF] Vladar | Microtubules enable the planar cell polarity of airway cilia[END_REF][START_REF] Kim | CLAMP/Spef1 regulates planar cell polarity signaling and asymmetric microtubule accumulation in the Xenopus ciliated epithelia[END_REF]. Unexpectedly, it was reported that maintenance of tissue-level polarity in airways requires the presence of a sufficient number of MCCs for cellcell transmission of PCP polarization [START_REF] Vladar | Airway epithelial homeostasis and planar cell polarity signaling depend on multiciliated cell differentiation[END_REF]. Consequently, diseases that cause loss of MCCs, such as cystic fibrosis, are characterized by poor tissue polarity, which may worsen mucociliary clearance [START_REF] Vladar | Airway epithelial homeostasis and planar cell polarity signaling depend on multiciliated cell differentiation[END_REF].

While tissue-and cell-level scales of organization start to be well described [START_REF] Walentek | Planar cell polarity in ciliated epithelia[END_REF], submicrometer scale events remains to be understood. For instance, structural and molecular chirality of BBs need to be resolved at very high resolution to understand their function (N'Guyen et al., biorXiV doi.org/10.1101/487330). Since BB structural organization varies between different types of MCCs, such studies will also help to understand functional differences between MCCs. At the opposite end of the spectrum, MCC activity needs to be integrated at the millimeter-scale to generate effective flows. Here, mechanical interactions between cilia and the fluid they propel become highly significant [START_REF] Khelloufi | Spatiotemporal organization of cilia drives multiscale mucus swirls in model human bronchial epithelium[END_REF]. To fully understand ciliated epithelium biology, future studies should aim at bridging sub-micrometer to millimeter scales, and at integrating molecular, structural and mechanical dimensions.

Conclusions

The field of MCC biology has advanced at fast pace in recent years. However, beyond the findings described above and the questions they open, many issues remain to be addressed.

A first problem regards the degree of universality of the principles identified so far. For instance, it is unclear whether MCCs in invertebrates use deuterosome-like structures to amplify their centrioles. In opposition to this idea, Mcidas and Deup1 genes appear to be absent in invertebrate genomes. This issue may receive light from planarians, which lack centrosomes, but make full-fledged epidermal MCCs required for locomotion [START_REF] Azimzadeh | Centrosome loss in the evolution of planarians[END_REF](Thi-Kim et al., bioRxiv doi.org/10.1101/324822). On a different note, the various steps driving centriole biogenesis were reported to be highly synchronous in individual cultured mouse ependymal MCCs [15,21]. It is unclear whether such synchrony applies to all types of MCCs, or reflects the peculiar organization of ependymal cilia into tufts of relatively constant size.

In future, it will be important to address the architecture and the biophysical nature of the deuterosome. Compartmentalization is used by cells to concentrate biochemical reactions. In most organelles, biochemical reactions are constrained by a membrane. Alternatively, organelles that lack delimiting membrane such as the PCM are formed through complex phase-separation mechanisms to create small volumes for specific biochemical reactions in the cytoplasm [START_REF] Wheeler | Controlling compartmentalization by non-membrane-bound organelles[END_REF]. Such liquid-like organelles have recently been reported in MCCs to serve as platforms of assembly of axonemal dynein arms to ensure ciliary beating [START_REF] Huizar | such particles are deprived of membranes and behave as phase-separated organelles. Some of the genes mutated in Primary Ciliary Dyskinesia patients encode components of these particles that are not transferred to cilia[END_REF]. It is tempting to speculate that based on its shared features with the PCM, the deuterosome may also comprise a phase-separated organelle, which would concentrate elementary parts and enzymes to build up multiple centrioles.

An overarching question in the field concerns the control of the final number of centrioles present in individual MCCs. Although this number can be highly variable in Xenopus epidermal MCCs or mouse tracheal MCCs, and less so in mouse ependymal MCCs, it always scales to the apical area of the cell (unpublished results; Nanjundappa et al., bioRxiv doi.org/10.1101/478297). Understanding such scaling represents a challenging but fascinating question, with fundamental as well as biomedical implications, for instance to better fight cancer cells, which are often characterized by excess number of centrioles.

It has become clear that centriole synthesis during the cell cycle and in MCCs share many molecular and regulatory principles. This leads to the idea that centriole duplication on the one hand and centriole mass production on the other hand are indeed extreme cases of one and the same biological process: centriole biogenesis. The question then becomes: what variations shape those pathways such that only one or hundreds of centrioles are produced, and can such variations account for centriole over-duplication in cancer ? This paper reports the striking finding that ciliary beating in ependymal MCCs is needed for actin fibers to form around BBs. In turn, actin fibers help BBs to sustain cilia beating force. When actin network is impaired, beating cilia and their associated BBs may detach from the cell surface.

  regulators and effectors operating in those phases have been identified. A set of major cell cycle regulators including CDK2, CDK1, PLK1 and APC/C control transitions between the different phases [21,22]. Highlighting more detailed mechanistic aspects of the disengagement phase, a recent study reported the existence of a proteolytic cascade involving the Separase protease and CDC20B to release centrioles from deuterosomes [14]. These complementary findings opened large avenues to understand deuterosome-mediated centriole production. For instance, the implication of cell cycle kinases calls for the identification of their targets in MCCs. Furthermore, it would be interesting to determine whether -similar to what happens during the cell cycle [23] -those kinases are counteracted by specific phosphatase activities.

Figure 1 :

 1 Figure 1: Multiciliated cells in Xenopus epidermis and mouse ependyma

Acknowledgements

We apologize to authors whose work could not be quoted due to space constraints. Work in LK's laboratory is supported by grants from Agence Nationale de la Recherche, Fondation pour la Recherche Médicale, Fondation ARC, and Ligue Nationale contre le cancer.


Didier Daniel Cattaert

Didier Le Ray

D Le Ray
email: leray@lnf.cnrs-mrs.fr

NEURAL MECHANISMS OF REFLEX REVERSAL IN COXO-BASIPODITE DEPRESSOR MOTOR NEURONS OF THE CRAYFISH

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

In walking animals, sensory receptors are involved in the adaptation of posture and ongoing movements to external perturbations. In Vertebrates, the neural circuitry underlying stretch reflex and more complex spinal reflexes, has been extensively studied. Integration of sensory information is subject to considerable modification when involved in centrally programmed movements such as locomotion in both Vertebrates [START_REF] Forssberg | Phasic control of reflexes during locomotion in vertebrates[END_REF][START_REF] Grillner | Locomotion in vertebrates. Central mechanisms and reflex interaction[END_REF][START_REF] Rossignol | State-dependent response during locomotion[END_REF][START_REF] Rossignol | The role of sensory inputs in regulating patterns of rhythmical movements in higher vertebrates[END_REF], and invertebrates [START_REF] Bässler | Afferent control of walking in the stick insect Cuniculina impigra. II. Reflex reversal and the release of the swing phase in the restrained foreleg[END_REF]). However, compared to the complex organization of Vertebrate sensory-motor pathways, Arthropods are good models to study sensory-motor interactions, due to the reduced number of neurons involved. Therefore, a lot of studies on sensory-motor control have been carried out on insects [START_REF] Bässler | The femur-tibia control system of stick insect -a model system for the study of the neural basis of joint control[END_REF][START_REF] Burrows | Local circuits for the control of leg movements in an insect[END_REF]) and on crustaceans [START_REF] Bush | Proprioceptive reflexes in the legs of Carcinus maenas[END_REF][START_REF] Cannone | Reflexes mediated by non-impulsive afferent neurons of the thoracic-coxal muscle receptor organ in the crab Carcinus maenas. I. Receptor potentials and promotor motorneuron responses[END_REF][START_REF] Cattaert | Direct evidence for presynaptic inhibitory mechanisms in crayfish sensory afferents[END_REF]El Manira et al. 1991a,b;[START_REF] Wiens | Reflex pathway of the crayfish claw[END_REF]. In Arthropods, chordotonal organs are a principal source of proprioceptive information from limb joints [START_REF] Mill | Structure and function of Proprioceptors in the Invertebrates[END_REF]. They mediate intrajoint resistance reflexes where stretch or release of the receptor excites the motor neurons (MNs) innervating muscles that resist the movement of the joint [START_REF] Bush | Leg reflexes from chordotonal organ in the crab Carcinus maenas[END_REF]. The resistance reflex is a negative feedback reflex, as is the vertebrate stretch reflex, mediated by direct connections between primary sensory afferents and MNs (El Manira et al. 1991a;[START_REF] Skorupski | Reflex pathways responsive to depression of the locust coxotrochanteral joint[END_REF]. Stretch receptors are also involved in the control of rhythmic motor patterns [START_REF] Clarac | How do sensory and motor signals interact during locomotion?[END_REF]) by providing phasic inputs to the central motor network.

The in vitro preparation of the thoracic ganglia of the crayfish provides a good model to study sensory-motor interactions, involved in motor control of leg movements. Generally, this preparation produces a spontaneous tonic motor activity, in which the "classical" resistance reflexes are elicited in response to stretch and release of the coxo-basipodite chordotonal organ (CBCO). The CBCO is an elastic strand in which about 40 sensory cells are inserted. The CBCO neurons can be divided into two groups: 20 stretch-sensitive and 20 release-sensitive cells, the axons of which compose the CBCO nerve and project to the ipsilateral thoracic hemiganglion. The CB joint allows upward and downward movements of the leg and is controlled by levator (Lev) and depressor (Dep) MNs (El Manira et al. 1991a).

Within the hemiganglion, sensory afferent terminals from the CBCO connect with Lev and Dep MNs (20 and 12 neurons respectively: [START_REF] Bévengut | Synaptic connections of the common inhibitory motoneurone within the fifth thoracic ganglion of crayfish[END_REF]. These connections are responsible for reflex responses: Stretching the CBCO activates Lev MNs while releasing the CBCO activates Dep MNs. However, as is the case for other receptors, these negative feedback reflexes vary in a phase-dependent way during centrally programmed rhythmic activities (El Manira et al. 1991b;[START_REF] Skorupski | Heterogeneity and central modulation of feedback reflexes in crayfish motor pool[END_REF]). The reflex may even reverse in sign, so that the reflex assists the ongoing movement. This "assistance reflex" has been described in many species [START_REF] Bässler | Reversal of a reflex to a single motorneuron in the stick insect Carausius morosus[END_REF][START_REF] Dicaprio | Reversal of a walking leg reflex elicited by a muscle receptor[END_REF][START_REF] Forssberg | Phase dependent reflex reversal during walking in chronic spinal cats[END_REF][START_REF] Rossignol | Phasic modulation of reflexes during rhythmic activity[END_REF][START_REF] Skorupski | Phase-dependent reversal of reflexes mediated by the thoracocoxal muscle receptor organ in the crayfish Pacifastacus leniusculus[END_REF]. The neural mechanisms underlying the phasedependent modulation and reversal of reflexes during voluntary movements remain unclear in any species.

Nevertheless, this requires that the central network is able to select the suitable sensory inputs and direct them towards the appropriate MNs. At least three levels in the reflex pathway have to be considered in the analysis of this phenomenon. The first level concerns the sensory input that can be blocked by presynaptic inhibition [START_REF] Baev | Polarization of primary afferent terminals of lumbosacral cord elicited by the activity of spinal locomotor generator[END_REF][START_REF] Boyan | Presynaptic inhibition of identified wind-sensitive afferents in the cercal system of the locust[END_REF][START_REF] Cattaert | Direct evidence for presynaptic inhibitory mechanisms in crayfish sensory afferents[END_REF][START_REF] Eccles | Presynaptic inhibition of the spinal monosynaptic reflex pathway[END_REF][START_REF] Kennedy | Presynaptic inhibition: primary afferent depolarization in crayfish neurones[END_REF]. This blockade of sensory inflow is essential to block the monosynaptic pathway from primary afferents to MNs. However, at least some sensory inputs may be not blocked by presynaptic inhibition in order to allow the polysynaptic assistance reflex. The blockade may be limited to some particular primary afferents or some branches of each. The inputs not subjected to presynaptic inhibition would use alternative interneuronal pathways, which represent the second level. This second level has been analyzed in different preparations [START_REF] Jankowska | The effect of DOPA on the spinal cord. 6. Half centre organization of interneurons transmitting effects from the flexor reflex afferents[END_REF][START_REF] Reichert | Integration of nonphaselocked exteroceptive information in the control of rhythmic flight in the locust[END_REF][START_REF] Sillar | A neuronal mechanism for sensory gating during locomotion in a vertebrate[END_REF]. A third level concerns the MN pool itself where gating could take place [START_REF] Lund | Interaction between the jaw-opening reflex and mastication[END_REF].

In this study, we focused on the Dep MN pool, because of its accessibility and its small number of well localized neurons, to investigate the last two levels. We have established a The in vitro preparation consisted of the last three thoracic ganglia, the two first abdominal ganglia and the motor nerves innervating the two proximal joints of the 5 th leg (the two sets of # J-515-6 6 antagonistic motor nerves Promotor/Remotor and Levator/Depressor, Fig. 1A). The CBCO was dissected out together with its sensory nerve, which encodes the vertical movements of the leg. The preparation was pinned down dorsal side up in a Sylgard-covered Petri dish and superfused with oxygenated crayfish saline (NaCl: 195 mM, KCl: 5 mM, CaCl 2 : 13 mM, MgCl 2 : 2 mM). In some experiments, divalent cation concentration was raised (CaCl 2 : 34 mM, MgCl 2 : 6.4 mM) with the sodium concentration reduced accordingly. Saline solutions were buffered with 3 mM Hepes and pH adjusted at 7.7 at 15°C. In some experiments (n=5), oxotremorine 10 -5 M (oxo) was added in normal saline in order to induce fictive walking activity. However, since this study aimed to describe the wiring of the sensory-motor pathway, and due to the great variability of neuronal discharge induced by oxo, all intracellular responses presented have been recorded in the absence of oxo.

Monopolar extracellular recordings and nerve stimulations were performed using platinum pin electrodes contacting the nerves, isolated from the bath with Vaseline, and directed to a 4-channel differential AC amplifier (A-M System). Single and paired intracellular recordings from MNs and INs were made with thin-walled glass micro-electrodes filled with a potassium chloride solution (3M) and having a 25-30 M resistance. The signals were amplified by an Axoclamp 2B (Axon Instruments). Intracellular current pulses, delivered through the recording micro-electrodes, and nerve stimulations were controlled by an eight channel digital stimulator (A.M.P.I). A home-made electro-mechanical puller was coupled to the distal end of the CBCO strand attached to provide mechanical stimuli. These were characterized by cyclic stretch-release of the CBCO strand, according to a sinusoidal or a ramp protocol (i.e. succession of ramps and steps which permitted separation of movementand position-related responses). Sinusoidal and ramp stimulations were performed from the most released position of the CBCO strand. Total movement amplitude was one third of the most relaxed CBCO strand length (i.e. 1 -1.8 mm). All signals (physiological recordings and movement control voltage traces) were monitored on an 8-channel oscilloscope, a 4-channel digital oscilloscope (Yokogawa DL 1200) and stored on DAT tapes (BioLogics digital tape recorder) and digitized on PC-based computer through an A/D interface (Cambridge Electronic Device, CED 1401PLUS). Intracellular and extracellular recordings were digitized at 5-10 kHz and written to disk. Signals were analyzed using the CED programs SPIKE2 for spike sorting and SIGAVG for spike triggered averaging.

Dep MNs were identified after penetration with a micro-electrode by the following procedure: i) The antidromic spike evoked by electrical stimulation of its axon in the identified nerve could be recorded by the micro-electrode. ii) There was a one to one correlation between MN intracellular spikes and motor nerve extracellular spikes during spontaneous activity. iii) Intracellular injection of depolarizing current into the MN evoked orthodromic spikes correlated one to one with extracellular spikes recorded in the corresponding motor nerve.

ARINs were identified by the following criteria: i) Electrical stimulation of any motor or sensory nerve of the leg never evoked any antidromic spikes in the intracellular recording.

ii) Injection of depolarizing current into the ARIN evoked an excitatory response of Dep MNs, recorded extracellularly from the Dep motor nerve and/or intracellularly from Dep MN. iii) Stretching movement applied to the CBCO strand evoked a dual depolarizing and hyperpolarizing response in the intracellularly recorded ARIN. Intracellular recordings from ARINs were performed in 6 experiments. However, paired recordings from ARIN and Dep MN were difficult to perform and only two successful experiments were used in this study.

RESULTS

# J-515-6 8 CBCO-induced reflex activities.

In quiescent preparations superfused with normal saline, we observed that the motor nerves exhibited low frequency tonic discharge. During sinusoidal stimulation of the CBCO strand (Fig. 1B), Dep and Lev nerves produced a resistance reflex response: stretching the CBCO strand (that is analogous to the depression of the leg) elicited the activation of one to three distinct units in the Lev nerve in a given burst whereas releasing the CBCO strand (that mimics upward leg movements) induced the firing of two to four units in the Dep nerve.

In preparations superfused with oxo, the global activity of the preparation is increased, MNs and INs being more excitable [START_REF] Cattaert | Cholinergic control of the walking network in the crayfish Procambarus clarkii[END_REF]. The antagonistic Lev and Dep nerves fire bursts of action potential in rhythmical alternation. In these conditions, a similar stimulation of the CBCO (Fig. 1C) elicited an assistance reflex, characterized by the Dep MNs being excited during stretching of the CBCO strand, and Lev MNs being activated during releasing of the CBCO strand. On the Lev neurogram, two units (one with a large extracellular spike and one with a smaller spike) were added to the three units observed in the tonic preparation, and fired with a higher frequency during the releasing phase of the CBCO mechanical stimulation. In the Dep nerve, eight different units fired at high frequency during stretching of the CBCO strand (three large amplitude units and one medium amplitude unit were added to the four units that were observed in the tonic preparation).The total number of MN units activated in assistance reflex is however variable (generally 3-5 Lev and 4-9 Dep MNs), depending upon the state of the preparation. Thus, oxo not only induced rhythmic activity, but also caused the reversal of the reflex response. In the Dep MN pool, this consisted of (i) a reversal of the reflex response of individual units that, in quiescent preparation, were activated by movements in a resistance manner and (ii) the appearance of an assistance reflex response in units previously silent.

# J-515-6 9

Evidence for a monosynaptic assistance reflex.

Each hemi-ganglion has been previously reported to contain 12 distinct Dep MNs that can be identified according to the size and shape of their extracellular action potential, and their conduction velocity [START_REF] Bévengut | Synaptic connections of the common inhibitory motoneurone within the fifth thoracic ganglion of crayfish[END_REF]. Fig. 2 shows intracellular recordings of these 12 MNs that were impaled successively in the same experiment. Mechanical stimulation of the CBCO elicited two types of responses from the Dep MNs in a quiescent preparation: either a resistance reflex (n=8) or an assistance reflex (n=1); 3 Dep MNs didn't respond to the mechanical stimulation applied to the CBCO. The resistance reflex observed in 8 MNs was characterized by membrane potential depolarizations of 0.5 to 4 mV, resulting of the summation of excitatory post-synaptic potentials (EPSPs) during the release of the CBCO strand. Generally, the MNs that received the larger EPSPs were responsible for resistance responses recorded extracellularly. However, in some quiet preparations, most of the MNs were hyperpolarized and their membrane potential kept under threshold for spiking (Fig. 2A) Even though each of these 8 MNs displayed a different response to CBCO strand release, there was nevertheless a clear correlation between movement phases and membrane potential depolarizations (see vertical doted lines in Fig. 2A). Traces in Fig. 4B were obtained by averaging the Dep MN responses to all stretching or releasing ramps from 4 cycles of CBCO mechanical stimulation. In each case, starting membrane potentials were offset. The 8 resistance MNs exhibited depolarizations (mean values from 0.4 to 1.4 mV) during the release of the CBCO strand (right column in Fig. 2B). Each compound EPSP could be related to release sensitive CBCO afferent (according to the protocol used in Fig. 5A). Moreover, it is noticeable that most of the resistance Dep MNs received weak phasic depolarizing inputs # J-515-6 10 (0.05 to 0.3 mV) during the stretching movements applied to the CBCO strand (left column in Fig. 2B). However, these weak assistance responses disappeared in the presence of high divalent cation concentration; therefore, these depolarizations had a polysynaptic origin [START_REF] Berry | Criteria for distinguishing between monosynaptic and polysynaptic transmission[END_REF].

Finally, one of the Dep MN was characterized by an assistance reflex response to the stimulation of the chordotonal organ: depolarizations (from 1.2 to 3 mV) were observed during stretching of the CBCO strand (bottom trace in Fig. 2A). The averaged trace in Fig. 2B shows a mean value of 1.5 mV. When perfusing the preparation with high Ca ++ and high Mg ++ , the CBCO stimulation was still capable of producing the assistance reflex response in this Dep MN (upper trace in Fig. 3A), whereas in the other Dep MNs, only the resistance response was maintained (middle trace on Fig. 3A). Therefore, the EPSPs observed in the « assistance » Dep MN (aDep MN) were elicited by monosynaptic connections from a mean of 4 stretch-sensitive CBCO afferents (number calculated according to the protocol used in Fig. 5A). This aDep MN displayed up to 10 mV depolarizations in response to CBCO stretching movements whereas no significant variation of its membrane potential was observed during the other phases of imposed movement.

The response of the a Dep MN was speed-dependent, as shown on Fig. 3B. When continuous ramp stimuli (without any steps) were applied, the aDep MN exhibited membrane potential depolarizations (about 4.5 mV) due to summations of EPSPs during stretching movements. A five fold increase of the stretch velocity resulted in a two fold increase of the stretch-induced depolarization (V2 = 12 mV against V1 = 4.7 mV, V1 and V2 being measured for an equal amplitude of movements). It appeared then that the amplitude of the depolarizing response was closely related to the speed of the movement.

Interneurone mediating polysynaptic assistance response.

We have investigated a possible mechanism by which Dep MNs may switch from a resistance reflex to an assistance reflex in response to CBCO stimulation during fictive locomotion, namely an interneuronal stage fulfilling two criteria:

(i) it receives excitation from stretch-sensitive CBCO afferents;

(ii) it is excitatory to Dep MNs.

Such an assistance reflex interneuron (ARIN) has been found in 6 preparations in the neuropilar region of Dep MNs, in a slightly more ventral position. Its main features and its relations with one post-synaptic Dep MN are illustrated in Figs. 4 to 8. To date, our results suggest that there may be only one such ARIN per MN pool in the hemi-ganglion.

In a quiescent preparation perfused with normal saline, the ARIN responded in an opposite way to resistance Dep MNs. ARIN exhibited subthreshold depolarizing events during stretch movements applied to the CBCO (Fig. 4A). Insert 4B presents average traces (n = 36 ramps) of both neuron reflex responses. ARIN received weak hyperpolarizing influences during the release of the CBCO strand while the Dep MN responded by a large depolarization (upper traces). In contrast, during the stretch of the CBCO strand (lower traces), the Dep MN exhibited a very weak depolarization (< 0.1 mV) while ARIN exhibited a large depolarizing response (the mean amplitude of which decreased along the ramp). By increasing the velocity of the mechanical stimulation applied to the CBCO (Fig. 4C), the responses of both neurons were emphasized. ARIN especially exhibited a compound hyperpolarization (summation of inhibitory post-synaptic potentials) upon termination of each of the small successive high velocity stretching movements. This observation suggests that the compound IPSP recorded in ARIN originates from velocity sensitive CBCO afferents, responding during and immediately following stretch movements. Fig. 4 insert D shows averaged traces obtained from 36 high # J-515-6 12 velocity ramps. It appears that the hyperpolarization of ARIN during release movements increased with the velocity of the release ramp as did the depolarizing resistance response of Dep MN (upper traces, Fig. 4D cf. B). The increase in stimulus velocity unmasked the hyperpolarizing influence that ARIN received during the stretching phase (lower traces, Fig. 4C,D) which may explain the declining depolarization recorded during slow stretch movements (Fig. 4 insert B, lower traces). The Dep MN also exhibited with this stretch velocity a greater depolarizing response (1 or 2 mV), which appeared with a greater latency than its release-related response, suggesting a polysynaptic origin. The presence of both responses suggests that the MN reflex response to CBCO stimulation is the result of the balance between monosynaptic and polysynaptic inputs.

In order to characterized the sensory units connected to each intracellularly recorded neuron, we analyzed the extracellular action potentials recorded from the CBCO nerve during slow stepwise movements (presented in Fig. 4A) imposed on the CBCO strand (Fig. 5A and 5B from the same preparation). From three to six CBCO afferents (mean: 4; n=71), active during CBCO release, evoked a time locked EPSP (delays from 4 to 6 ms, amplitudes from 7 to 12 mV) in Dep MNs (Fig. 5A). The ARIN was found to receive excitatory inputs from about eight stretch sensitive CBCO units that evoked EPSPs having a 6 to 14 mV amplitude with a 2.5 to 7 ms delay (Fig. 5B). The delays are consistent with conduction delays observed by El Manira et al. (1991a). The stability (amplitude and delay) of each EPSP evoked in ARIN and Dep MN makes it likely that they were monosynaptic. However, large differences exist between CBCO units that elicit an EPSP in the Dep MN or in the ARIN: some display large amplitude extracellular spikes, while others display very small extracellular spikes. Moreover, the large range of delays between sensory extracellular spikes and EPSPs recorded from the ARIN and Dep MN, which are not correlated to the amplitude of extracellular spikes, would This corroborates the results illustrated in Fig. 4A and suggests that the larger EPSPs are locked more closely to movement sensitive than to position sensitive afferents.

Characterization of ARIN response to injection of depolarizing currents.

Although nonspiking in the conventional sense, depolarization of ARIN nevertheless revealed active membrane properties: As shown on Fig. 7A, injection of increasing depolarizing current pulses (+1 to +22 nA by steps of 1 nA, duration 35 ms) caused a biphasic variation of ARIN membrane potential. An early transient phase of depolarization developed for current intensities above +3 nA and peaked 15.1 ms (+4 nA stimulation) to 3.5 ms (stimulation above +15 nA) after the pulse onset. The time to peak of this transient phase decreased with increasing current (see graph 1 on Fig. 7B). The second depolarizing phase, that developed from the lowest currents, is the commonly observed passive response due to capacitive charge of the membrane (no reliable value could be measured from +16 nA). The membrane potential of this second phase varied in nearly the same parabolic manner for positive currents as that of the transient phase (see graph 2 of Fig. 7B).

# J-515-6 14 Although the second depolarizing phase is commonly observed, the first, transient phase, is unusual and is likely to be different from a simple Na + spike since (i) it is a graded depolarizing response, and (ii) its time to peak decreases with increasing currents.

Synaptic relation between ARIN and Dep MNs.

Since ARINs are not spiking neurons (positive current injection never triggered any spike in the ARIN, Fig. 7A), the synaptic relation between ARINs and Dep MNs was studied with the following protocol (Fig. 8A

DISCUSSION

The monosynaptic assistance reflex.

In many different preparations, negative feedback responses (resistance reflex) have been demonstrated to be mediated largely by monosynaptic sensory-motor pathways. In Vertebrates, the direct connection that Ia afferents have with homonymous MNs produces the "stretch reflex" which opposes to the imposed movement that provoked it [START_REF] Henneman | Functional organization of motoneuron pool and its input[END_REF]. In invertebrates resistance reflexes are also mediated by monosynaptic connections between sensory afferents and MNs, as shown in the crayfish walking system (El Manira et al. 1991a) as well as in the locust flight network [START_REF] Burrows | Monosynaptic connections between wing stretch receptors and flight motorneurons in the locust[END_REF]). Here we provide evidence for a monosynaptic sensory-motor assisting pathway to the Dep MN pool (Fig. 3), in the walking network of crayfish. Our experiments showed the existence of different MN reflex responses to the same CBCO stimulation. In the fifth thoracic ganglion of the crayfish, the characterization of all the monosynaptic reflex responses of a specific pool of MNs to a complex movement could be performed due to the small number of neurons (only 12 Dep MNs -Fig. 2). Generally, the response observed was the "classical" resistance reflex but it was always possible to identify one Dep MN that presented a non-typical monosynaptic assisting response to the CBCO stimulation. Previous studies on the fourth thoracic ganglion have described such a positive feedback reflex involving TCMRO and remotor MNs [START_REF] Skorupski | Synaptic connections between nonspiking afferent neurons and motor neurons underlying phase-dependent reflexes in crayfish[END_REF]). The T-fiber of the TCMRO, an muscle receptor organ that controls the antero-posterior movements of the leg, connects monosynaptically with promotor MNs to elicit a resistance reflex response during the stretch of the organ. It has been shown that this stretch sensitive fiber was also able to excite directly a specific pool of remotor MNs, evoking then a # J-515-6 16 monosynaptic assistance response. The functional significance of this connection remains uncertain. Nevertheless, at least two non-exclusive hypotheses can be proposed:

(1) MN pools are heterogeneous and could be considered as sets of more or less independently driven MNs. Within a pool, each set could be activated or inactivated by the central network, depending upon the required behavioral situation. Such differential aminergic control of MN sets has recently been demonstrated to exist in the promotor and remotor MN pools of the crayfish [START_REF] Skorupski | Octopamine induces steady state reflex reversal in crayfish thoracic ganglia[END_REF]. In this view, assistance and resistance MNs would not be simultaneously active.

(2) In rhythmic walking activity, the recruitment of the different MNs in a given MN pool is generally progressive. The units of small spike amplitude being activated before the larger ones, the transition from resistance to assistance, or assistance to resistance, could also be considered as being a part of this pattern of activity of the MN pool. For example during stance, mainly resistance reflexes occur whereas during swing, assistance reflexes are involved. This phase-dependent reflex reversal has been described in many vertebrate and invertebrate preparations [START_REF] Rossignol | State-dependent response during locomotion[END_REF][START_REF] Rossignol | The role of sensory inputs in regulating patterns of rhythmical movements in higher vertebrates[END_REF][START_REF] Skorupski | Phase-dependent reversal of reflexes mediated by the thoracocoxal muscle receptor organ in the crayfish Pacifastacus leniusculus[END_REF].

The active transient depolarization in ARIN.

Injection of a depolarizing current in ARIN elicited an active transient depolarization which was graded with current intensity. Such an active transient response has already been described in the T-fiber of the TCMRO where, as in our model, this "spikelike transient" was sometimes sufficient to produce an EPSP in the promotor MNs [START_REF] Skorupski | Synaptic connections between nonspiking afferent neurons and motor neurons underlying phase-dependent reflexes in crayfish[END_REF]). In the same way, active depolarizing transients have been shown in the lobster stomatogastric ganglion [START_REF] Graubard | Synaptic transmission without action potential: input-output properties of a non-spiking presynaptic neuron[END_REF], in the crab T-fiber [START_REF] Blight | The non-impulsive stretch-receptor complex of a crab: a study of depolarization-release coupling at a tonic sensorimotor synapse[END_REF] and in the locust nonspiking INs [START_REF] Laurent | Voltage-dependent nonlinearities in the membrane of locust nonspiking local interneurons, and their significance for synaptic integration[END_REF]). The nature of the conductances underlying the graded spikes described was clarified neither in these preparations nor in ours. According to [START_REF] Blight | The non-impulsive stretch-receptor complex of a crab: a study of depolarization-release coupling at a tonic sensorimotor synapse[END_REF], it may be due to a calcium current, although [START_REF] Bush | Non-impulsive stretch receptors in crustaceans[END_REF], in the same preparation, suggested a TTX-sensitive sodium current was involved. The physiological role of this graded active depolarization is unclear. We can assume it may operate in a "pre-integration" of the sensory signals by the IN, functioning as an amplifier of the amplitude and/or the duration of the sensory EPSP.

Involvement of ARIN in the polysynaptic sensory-motor pathway.

Our experiments aimed at a better characterization of the MN reflex responses to the CBCO mechanical stimulation. We found 3 to 6 (average 4) sensory afferents connecting with one Dep MN (n>70 Dep MNs) with some heterogeneity in the nature of those CBCO afferents (movement/position selectivity and conduction velocity). Concerning the INs, it appears that only movement sensitive afferents connect to ARINs (Fig. 6B). It also appears that a greater number of afferents control ARIN: 8 monosynaptic excitatory CBCO afferents (Fig. 5B) and at least 1 inhibitory (probably disynaptic) influence from release-coding CBCO afferents responsible for compound IPSPs (Fig. 4C,D). This finding is in agreement with El Manira et al. (1991a): these authors showed that electrical stimulation of the CBCO nerve elicited a short latency EPSP for weak intensities (1.3 V), followed by a long latency EPSP (or IPSP) for higher intensities (4 V). The increase of intensity recruited more CBCO afferents and we can suppose that this number (for 4 V) was sufficient to activate the polysynaptic sensory pathways. Then, there may be a convergence of the sensory inputs to a small number of # J-515-6 18

ARINs (perhaps only one per group of MNs), and subsequently, a divergence to the MNs involved in the expected assistance reflex activity.

The real connectivity, especially the number of ARINs and related postsynaptic MNs, remains to be determined. According to [START_REF] Bässler | The femur-tibia control system of stick insect -a model system for the study of the neural basis of joint control[END_REF], a single nonspiking IN is able to excite one pool of MNs (for example the Dep pool) and inhibit another antagonistic pool (the Lev one). Another possibility would be the existence of two INs, one excitatory, the other inhibitory on the same pool of MNs [START_REF] Burrows | Proprioceptive inputs to nonspiking local interneurons contribute to local reflexes of a locust hindleg[END_REF]. The systematic investigation of the CBCO units connecting both kinds of neurons (Fig. 5) allowed us to measure the latency of appearance of the sensory EPSPs in Dep MNs and ARINs. It appeared that latencies in both neurons were quite similar, and compatible with a monosynaptic connection in both cases.

Moreover, the synaptic delay (about 2 ms) calculated between ARIN and the postsynaptic motor neuron (Fig. 8) is consistent with the timing of the polysynaptic part of the response observed by El Manira et al. (1991a). Therefore, we may assume that ARIN can be the only link between the CBCO afferents and the motor neurons in the polysynaptic reflex pathway; it may therefore be a disynaptic pathway.

CBCO inhibitory input to ARIN.

We have demonstrated that ARINs were activated by movement coding CBCO neurons (Figs. 4,6). Moreover, the amplitude of the compound EPSP is velocity sensitive: the faster the movement, the larger the EPSP. However, the response observed during fast movement is also more variable and may be partly masked by IPSPs. These characteristics explain why the average traces (Fig. 4D cf. B) do not reflect the velocity-dependence observed in raw data (Fig. 4C cf. A). It is striking that the IPSP only became evident with the faster movements studied. This inhibitory connection limits the amount of depolarization induced by # J-515-6 19 the summation of unitary EPSPs. It is noticeable that for slow movements imposed to the CBCO strand, EPSPs do not summate much with each other, so that the amplitude of the response does not exceed 10 mV. By contrast, during faster movements, summation would actually occur and depolarization would reach more than 20 mV. In reality, due to the IPSP, this response is limited to 10-15 mV (see inserts B and D in Fig. 4). This velocity sensitive inhibition could play at least two roles:

(1) IPSPs could represent a gain control mechanism that limits the efficacy of the assistance (positive feedback) reflex that otherwise could result in an "explosive-like" reaction.

(2) IPSPs could be mediated by assistance reflex controlling INs (ARCINs) controlled by both the central locomotor network and velocity coding CBCO afferents. Such central control could therefore work in parallel with presynaptic control of primary afferents, in such a way that monosynaptic resistance reflexes or polysynaptic assistance reflexes are selected.

Phase-dependent modulation of reflex transmission.

From all these data, and those from previous work on presynaptic inhibition of primary afferents in this preparation [START_REF] Cattaert | Direct evidence for presynaptic inhibitory mechanisms in crayfish sensory afferents[END_REF], we can propose a more complete organization for the sensory-motor reflex network controlling the coxo-basipodite joint of the leg. Fig. 9 presents a schema of this organization concerning exclusively the Dep MN reflex activities. Both cases, quiescent and active preparation, are presented. In the first case (Fig. 9A), the central pattern generator (CPG) is not rhythmic and some of its outputs tonically excite () the ARCIN responsible for the inhibition (-) of ARIN (Fig. 4A andC). In that case, upward movement of the leg would activate release sensitive CBCO fibers that stimulate monosynaptically most of the Dep MNs (Fig. 9A1); the Dep reflex expressed in the preparation would be a resistance response. However, downward movement of the leg would activate stretch sensitive afferents that stimulate the aDep MN, resulting in a weak assisting influence in the Dep activity (Fig. 9A2); at the same time, the concomitant activation of ARCIN by stretch sensitive CBCO fibers and tonic outputs from the CPG would result in a strong inhibition (---) of the ARIN, and subsequently of the polysynaptic transmission of the stretch signal (Fig. 9A2). In contrast, when the CPG is rhythmically active (Fig. 9B), the central inhibition of ARIN through ARCIN is supposed to be removed. During the swing phase (Fig. 9B1), the CBCO strand is released by upward movement of the leg but Dep MNs stay silent, due to presynaptic inhibition [START_REF] Cattaert | Direct evidence for presynaptic inhibitory mechanisms in crayfish sensory afferents[END_REF]. During the stance phase, Dep MNs are centrally activated. The CBCO strand is stretched during downward movement of the leg and ARIN transmits efficiently the stretch inputs to the Dep MNs, which therefore respond in the same way as the aDep MN. The Dep response expressed in the preparation would be a strong assistance reflex.

Our model for the control of reflex activities in the crayfish walking system is comparable to those described in the locust [START_REF] Burrows | Proprioceptive inputs to nonspiking local interneurons contribute to local reflexes of a locust hindleg[END_REF] or in the stick insect [START_REF] Bässler | The femur-tibia control system of stick insect -a model system for the study of the neural basis of joint control[END_REF]). Nevertheless, it is interesting to remark that in the crayfish, the interneuronal level seems to be specialized in the reversal of the reflex activities: the monosynaptic sensorymotor connections elicit resistance reflexes while the interneuronal pathway, when permitted by the CPG, participates in the expression of assisting reflex responses in the motor nerves. It appears therefore that the reflexes expressed by the preparation are the result of a CPGcontrolled balance between resistance and assistance patterns (as is suggested by Fig. 4).

These results indicate that the CPG not only recruits sets of MNs, but also controls the sensory-motor pathways in such a way as to adapt them to the prevailing motor behavior. B: Velocity-dependence of the Dep MN assistance response. Release of the CBCO produced a depolarization whose amplitude increased with increasing velocity of the movement: about 12 mV (V2) for the higher speed (0.25 mm.s -1 ) against about 5 mV (V1) for the lower speed (0.05 mm.s -1 ); the values were measured for the same amplitude of movement.

  wiring diagram based upon systematic intracellular recordings from all twelve Dep MNs. The Dep MN pool appears to be constituted of three types of MNs with respect to their direct connections with CBCO afferents: resistance unit, assistance unit and not connected to CBCO. Assistance reflex interneurons (ARINs) are also described which are involved in the disynaptic assistance reflex pathway. These INs are recruited by velocity sensitive CBCO afferents. Our results indicate that the recruitment of the disynaptic reflex pathway is controlled by inhibitory sensory inputs that prevent ARIN from activating assistance reflexes in quiescent central network. Finally, both pathways coexist and MN reflex responses observed during locomotion may be the result of a centrally modulated balance between assisting and resisting influences. METHODS Results are based on 83 intracellular recordings from MNs and INs (24 experiments) that were performed on adult male and female crayfish, Procambarus clarkii and Pacifastacus leniusculus. Animals were maintained in aquarium at 17-18°C and fed once a week.

  heterogeneity in diameters, and hence conduction velocities of CBCO fibers that connect with the ARIN or Dep MNs.The temporal occurrence of each of the EPSPs produced in both post-synaptic neurons was analyzed by triggering on all the CBCO nerve extracellular spikes eliciting a response in the impaled neurons, irrespective of spike size and EPSP latency. This procedure allowed us to measure the amplitude of individual EPSPs. As shown on Figs.6A and B, the EPSPs induced by the CBCO units were larger during the movement phases of the CBCO stimulation (respectively releasing movements for the Dep MN and stretching movements for the ARIN).

  ): positive current pulses which were injected intracellularly into the ARIN elicited EPSPs in the post-synaptic Dep MN. Fig. 8B1 presents an overdraw of the averaged EPSP recorded for each current pulse intensity. The post-synaptic EPSP developed gradually for intensities of current pulses above +4 nA and reached its maximum value (about 5 mV) with current intensities above +15 nA. The amplitude of the MN EPSP as a function of the peak of the ARIN transient shows a sigmoidal relation (see graph 2 on Fig. 8B). The EPSP appeared for transient amplitudes above 2.8 mV and was maximum for transient amplitudes larger than 26 mV. As the current pulse intensity increased, the latency of the MN EPSP decreased from 9 to 5.4 ms as shown on Fig. 8B1. The synaptic delay between the two cells is expressed in graph 3 on Fig. 8B: the relation between the time to peak of the transient in ARIN and the latency of the EPSP development is linear (y= ax + b, with a=1.080.05, and b=1.670.03 being the synaptic delay) indicating a constant delay. The fitting of linear regression is excellent (r=0.99) in the whole range of current intensities and gives a 1.67 ms synaptic delay.

LEGENDSFig. 1 :

 1 Fig. 1: Intrajoint reflex patterns in quiescent and rhythmic preparations.A: Schematic drawing of the in vitro preparation, consisting in the 3 rd , 4 th and 5 th thoracic

Fig. 2 :

 2 Fig. 2: Responses of each of the 12 Dep MNs to ramp stimulation of the CBCO in quiescent

Fig. 3 :

 3 Fig. 3: The monosynaptic assistance reflex.

Fig. 4 :

 4 Fig. 4: "Assistance reflex interneuron" (ARIN).

Fig. 5 :

 5 Fig. 5: Monosynaptic CBCO inputs to Dep MN and ARIN.

Fig. 6 :

 6 Fig. 6: Dep MN and ARIN sensory EPSPs occurred during movements of the CBCO.

Fig. 8 :

 8 Fig. 8: Evidence for connection of ARIN with postsynaptic Dep MN.

Fig. 9 :

 9 Fig. 9: Functional schema of the sensory-motor network that controls the Dep MN reflexes.

Acknowledgments: This study was supported by the EEC contract number CT930190 and CNRS. We thank Dr. J. Y. Barthe , Dr. P. Skorupski and Dr. L. Vinay for helpful discussions and improving the manuscript. We are grateful to Dr. F. Clarac for constructive comments and laboratory facilities.


Benjamin Monnet

Sergio Ciliberto

Ludovic Bellon
email: ludovic.bellon@ens-lyon.fr

Extended Nyquist formula for a resistance subject to a heat flow

The Nyquist formula quantifies the thermal noise driven fluctuations of voltage across a resistance in equilibrium. We deal here with the case of a resistance driven out of equilibrium by putting it in contact with two thermostats at different temperatures. We reach a non-equilibrium steady state where a heat flux is flowing through the resistance. Our measurements demonstrate anyway that a simple extension of the Nyquist formula to the non uniform temperature field describes with an excellent precision the thermal noise. For a metallic ohmic material, the fluctuations are actually equivalent to those of a resistance in equilibrium with a single thermostat at the mean temperature between the hot and cold sources.

I. INTRODUCTION

With the Fluctuation-Dissipation Theorem (FDT), statistical physics offers a powerful tool to describe the fluctuations of an observable of a system in equilibrium [START_REF] Callen | Irreversibility and generalized noise[END_REF]. For example, applications of the FDT to the Brownian motion of micrometer sized systems include micro-rheology measurements with optical tweezers [START_REF] Mizuno | Active and passive microrheology in equilibrium and nonequilibrium systems[END_REF][START_REF] Pesce | Microrheology of complex fluids using optical tweezers: a comparison with macrorheological measurements[END_REF], or the calibration of the stiffness of atomic force microscopy probes [START_REF] Butt | Calculation of thermal noise in atomic force microscopy[END_REF]. For electrical systems, the FDT is expressed by the celebrated Nyquist formula [START_REF] Nyquist | Thermal agitation of electric charge in conductors[END_REF], describing the Johnson voltage noise [START_REF] Johnson | Thermal agitation of electricity in conductors[END_REF] across a resistance R at temperature T :

S V = V 2 ∆f = 4k B T R (1)
with S V the Power Spectrum Density (PSD) of the voltage V across the resistance, ∆f the frequency bandwidth, k B the Boltzmann constant and . stands for the statistical average. In non equilibrium situations however, such a relation between fluctuations and dissipation is not granted, and excess noise is usually expected to be observed with respect to an equilibrium state [START_REF] Conti | Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators subject to heat flux[END_REF][START_REF] Conti | Thermal noise of mechanical oscillators in steady states with a heat flux[END_REF][START_REF] Li | Small-angle rayleigh scattering from nonequilibrium fluctuations in liquids and liquid mixtures[END_REF][START_REF] Cugliandolo | Energy flow, partial equilibration, and effective temperatures in systems with slow dynamics[END_REF][START_REF] Li | Concentration fluctuations in a polymer solution under a temperature gradient[END_REF][START_REF] Tomás | Observation of fluctuation-dissipation-theorem violations in a structural glass[END_REF][START_REF] Bellon | Violation of the fluctuation-dissipation relation during the formation of a colloidal glass[END_REF][START_REF] Berthier | Nonequilibrium dynamics and fluctuation-dissipation relation in a sheared fluid[END_REF][START_REF] Hérisson | Fluctuation-dissipation ratio of a spin glass in the aging regime[END_REF][START_REF] Grenard | The fluctuationdissipation relation on a melde string in a turbulent flow; considerations on a 'dynamical temperature[END_REF][START_REF] Monchaux | Fluctuationdissipation relations and statistical temperatures in a turbulent von kármán flow[END_REF][START_REF] Marini | Fluctuationdissipation: Response theory in statistical physics[END_REF][START_REF] Loi | Non-conservative forces and effective temperatures in active polymers[END_REF][START_REF] Santamaría-Holek | Mean-field "temperature" in far from equilibrium systems[END_REF][START_REF] Cugliandolo | The effective temperature[END_REF][START_REF] Kasas | Detecting nanoscale vibrations as signature of life[END_REF][START_REF] Dieterich | Single-molecule measurement of the effective temperature in non-equilibrium steady states[END_REF]. We will focus here on the case where the system is in a Non-Equilibrium Steady State (NESS), because it is submitted to a constant heat flux. This is the case for example of a conductor submitted to a large temperature gradient, or of a resonator whose extremity are receiving an external radiation. In this latter case, contradictory observations have been made: in a experiment by L. Conti and collaborators [START_REF] Conti | Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators subject to heat flux[END_REF][START_REF] Conti | Thermal noise of mechanical oscillators in steady states with a heat flux[END_REF], an excess of thermal noise has been measured, while in a similar experiment in our group [START_REF] Geitner | Low thermal fluctuations in a system heated out of equilibrium[END_REF], lower fluctuations than those expected from the system average temperature have been observed.

To give further insights into these puzzling physical phenomena, we explore in this article the thermal fluctuation in such a NESS, focusing on the simple example of voltage noise across an electrical resistance. In a first part, we present the simple extension of the Nyquist formula that could be applied to this system. In the following parts, we describe the experimental setup and the results of the thermal noise measurements. The last section concludes this work by a discussion on the results.

II. EXTENDED NYQUIST FORMULA

To introduce our approach, let us first discuss the case of two resistances R 1 and R 2 connected in series, each being in equilibrium with a different thermostat at temperature T 1 and T 2 respectively. The PSD of the voltage noise across each resistance is described by the Nyquist formula (eq. 1). The voltage across the total resistance

R = R 1 + R 2 is V = V 1 + V 2 ,
where we neglect the resistance of the conductor linking the two dipoles. Voltage fluctuations V 1 and V 2 are statistically uncorrelated, thus the measured PSD should write:

S V = (V 1 + V 2 ) 2 ∆f = S V1 + S V2 = 4k B (T 1 R 1 + T 2 R 2 )
(2) The voltage noise is thus proportional to the sum of the products between temperature and resistance. The fact that there are two thermostats is decoupled here from the thermal noise analysis, since the heat flux between them only takes place in a conductor with negligible resistance. If we consider a similar case where N resistances R n in equilibrium at temperature T n are connected in series, we will simply derive the following formula:

S V = 4k B N n=1 T n R n (3)
Let us now consider the continuum limit of this approach, where each segment dx of the resistor is at a local temperature T (x). Under this hypothesis of local equilibrium, a possible extension of eq. 3 is:

S V = 4k B L x=0 T (x)ρ T (x) dx (4)
where we suppose that the resistance is distributed in one dimension (along x, from x = 0 to x = L), with a linear resistivity ρ(T ):

R = L x=0 ρ T (x) dx (5)
This description is however not granted, since now each segment is submitted to a heat flux, thus equilibrium laws may not apply to this NESS. The goal of our experiments is to test this extended Nyquist formula, using a resistor in contact with two different thermostats at each of its extremities.

In the following, instead of comparing directly the measured PSD S V to eq. 4, we will characterise the amplitude of thermal noise by T fluc , defined with:

T fluc := S V 4k B R (6)
T fluc is the effective temperature one would infer from measuring both the amplitude of the thermal noise and the value of the resistance, regardless of its equilibrium state. At equilibrium, this should be the thermostat temperature. Out of equilibrium, if the extended Nyquist formula is valid, we expect that

T fluc = 1 R L x=0 T (x)ρ T (x) dx (7)
In other words, we expect T fluc to be the temperature field averaged with a weight proportional to the local resistivity. Although the systems are very different (purely dissipative versus resonant system), this formula extends to electrical observables our work on the mechanical thermal noise of a micro-cantilever in a NESS [START_REF] Geitner | Low thermal fluctuations in a system heated out of equilibrium[END_REF]. K. Komori and collaborators came independently to a similar expression for the thermal noise of a generic mechanical system in a NESS [START_REF] Komori | Direct approach for the fluctuationdissipation theorem under nonequilibrium steady-state conditions[END_REF].

III. MEASURING THE THERMAL NOISE OF A RESISTOR IN A THERMAL GRADIENT

In the experiments, we measure the voltage fluctuations using a home made ultra low noise amplifier [START_REF] Cannatà | An ultralow noise preamplifier for low frequency noise measurements[END_REF][START_REF] Ciliberto | Statistical properties of the energy exchanged between two heat baths coupled by thermal fluctuations[END_REF], featuring a voltage noise of only 10 -18 V 2 /Hz, flat down to 10 Hz (see Fig. 5). This is equivalent to the thermal noise of a 60 Ω resistance at room temperature. Its current noise is negligible in our experiments (order of magnitude: 10 -30 A 2 /Hz). Though we will subtract the amplifier noise spectrum from our measured PSD, having a reasonable signal to noise ratio implies using resistances greater than a few 100 Ω.

Ideally, we would like to mesure the voltage fluctuations across a resistance as sketched in Fig. 1. However, to avoid having local thermostats along the resistance (which would bring us close to the equilibrium case of eq. 3), we would like to avoid any material not part of the resistance to be in contact with it. We may not use a standard film resistor for instance, where most of the dipole is made of an insulating ceramic. We therefore choose to work with a resistive wire, where electrical and thermal conductivities are mainly due to the electronic transport. The material, length L and diameter D of this resistive wire need to accommodate the following criteria:

1. The total resistance R = 4ρ v L/πD 2 is greater than a few 100 Ω (with ρ v the volume resistivity).

2. The heat flux by conduction within the resistance is much greater than that due to radiation.

3. The heat flux by conduction within the resistance is much greater than that with the surrounding atmosphere.

A. Resistance criterium

To meet the first criterium, we need a long and thin wire of high electrical resistivity. If we choose for example Ni-Fe alloys (among the worst metallic conductors), their bulk resistivity is around ρ v = 5 × 10 -7 Ωm. Thus a wire of diameter D = 25 µm will need to be at least L = 30 cm long to reach R = 300 Ω.

B. Negligible radiation criterium

Let us estimate the balance between conduction and radiation in a cylindrical wire bridging a cold thermostat at temperature T c with a hot one at temperature T h = T c + ∆T . The equation governing the temperature field is derived by an analysis along the wire of the heat fluxes J λ by conduction and J σ emitted by radiation. The first follows the Fourier law:

J λ = -λ π 4 D 2 ∂ x T (8)
with λ the thermal conductivity. We neglect any variation of λ with T for this estimation of the effect of radiation. As for J σ , we suppose that the resistor is exchanging thermal photons with a half space at temperature T c , and a second half space at temperature T h . For a element of length dx, we thus write

J σ = eσ(T 4 -T 4 σ )πDdx (9)
with σ = 5.67 × 10 -8 Wm -2 K -4 the Stephan-Boltzmann constant, e the emissivity of the material, and

T σ = (T 4 c /2 + T 4 h /2) 1/4
. The temperature profile is a balance between the fluxes for the element dx:

λ π 4 D 2 ∂ 2 x T = eσ(T 4 -T 4 σ )πD (10) 1 T σ ∂ 2 x T = 4eσT 3 σ λD ( T 4 T 4 σ -1) (11)
We define the characteristic length L σ by

L σ = λD 4eσT 3 σ (12)
Normalising temperatures by T σ (θ = T /T σ ) and positions by L (X = x/L), we end up with the following adimensional equation:

∂ 2 X θ = 2 (θ 4 -1) (13)
with = L/L σ . Heat conduction will thus dominate if 1, and radiation will dictate the temperature field otherwise.

The second criterium thus favours long and thin wires, of low thermal conductivity. Given the Wiedemann-Franz law for metallic material stating that thermal and electrical conductivities are proportional [START_REF] Franz | Über die wärmeleitungsfähigkeit der metalle[END_REF][START_REF] Jones | Theoretical Solid State Physics[END_REF], the two first criteria are exactly antinomic ! For Ni-Fe alloys for example, the thermal conductivity is around λ = 20 Wm -1 K -1 , using e = 1 (worst case corresponding to black body emission), T σ = 350 K and D = 25 µm, we compute L σ = 7 mm, which is much smaller than L = 30 cm imposed by the first criterium. A naive geometry is thus not compatible with the phenomenon we want to probe, and we must look at some alternative design.

High thermal conductivity and strong electrical resistance are not achievable with a linear geometry. Our design compensates this incompatibility by using small resistances in parallel for heat fluxes but in series for the electrical resistance. This is achieved by wrapping the resistive wire around two cylindrical thermostats (M2 brass screws, 2r ≈ 1.6 mm in diameter), only l = 3 mm appart (axis distance). With N ≈ 30 loops, we end up with the required 30 cm length. To avoid electrical shortcuts, the FIG. 2. Sketch of the experimental setup (a), equivalent electrical circuit (b) and temperature profile in the resistance (c). The resistance is a resistive wire (in blue) wrapped about 30 times around 2 brass screws (in yellow), which are part of the 2 thermostats (in gray at T0 and red at T0 + ∆T ). Pt1000 thermistances are placed inside the thermostats to measure their temperature. A foil heater (in brown) is used to tune ∆T . The voltage V is measured across the whole resistance, but their thermal fluxes are in parallel in each free standing part of the wire.

resistive wire is coated with a thin insulating layer. Thermal conducting grease on the brass screws ensure good thermal contact between the resistance and the thermostat. The resulting electrical system, sketched in Fig. 2, is the series of resistances

R = N (R c + R f + R h + R f ),
where the three subscripts stand for the part of the resistive wire in contact with the cold thermostat (c), the hot one (h), and the free standing part bridging the two (f ). R c and R h share the same length l c = l h = πr ≈ 2.5 mm, comparable to that of R f :

l f = l = 3 mm.
As illustrated in Fig. 3, numerically solving eq. 13 for R f in this specific case (l = 3 mm, D = 25 µm, λ = 20 Wm -1 K -1 , e = 1) shows that deviations to the purely conductive temperature profile are below 1 %. To estimate if the thermal photon bath can be considered as a local thermostat, we can also compare the conduction heat flux per unit surface j λ = -λ∂ x T to that of emitted radiations j σ = eσT 4 . In figure Fig. 3(c), we observe that this ratio j σ /j λ is at most a few percent, and even much smaller for the largest ∆T .

As for thermal noise measurements, each of the N loops around the thermostats has an equivalent contribution both for the total resistance (R = N R loop ) and the total voltage PSD (S V = N S V loop ), hence T fluc defined by eq. 6 is the same for one loop or the full resistance. Finally, the extended Nyquist Formula in this geometry is equivalent to:

T fluc = 1 R loop L loop x=0 T (x)ρ T (x) dx (14) = R c T c + R h T h + 2R f T fluc f R c + R h + 2R f (15)
with

T fluc f = 1 R f l x=0 T (x)ρ T (x) dx (16)
To the first order in ∆T , we can neglect the variation of ρ and λ with temperature, hence the temperature profile is linear (T (x) = T c + ∆T x/L) and eqs. 15 and 16 simply becomes:

T fluc = T fluc f = T c + ∆T 2 (17)

C. Negligible convection criterium

We now estimate the balance between conduction inside the resistive wire and heat flowing to or from the surrounding atmosphere. This air layer is confined in our setup between two horizontal plates (the thermostats), the hotter one being on top. This configuration is stable with respect to convection, the temperature will thus be linear in space in the medium around the resistance. As detailed in the previous paragraph, to the first order in ∆T the temperature profile due to conduction is also linear in space inside the resistance. The temperatures being equal at the same position, no net heat exchange with the environment is thus happening.

Let us anyway model this heat exchange J a if a difference were to appear between the resistance temperature T (x) and the air temperature T a , by introducing the heat exchange coefficient h [START_REF] Bergman | Fundamentals of Heat and Mass Transfer[END_REF]:

J a = h(T -T a )πDdx ( 18
)
where we considered a small element of length dx, as for radiation. For free convection at the vertical surface of a solid in contact with air around room temperature, the order of magnitude is h ∼ 10 Wm -2 K -1 [START_REF] Bergman | Fundamentals of Heat and Mass Transfer[END_REF]. For a stationary solution, the balance between the fluxes for the element dx writes:

λ π 4 D 2 ∂ 2 x T = h(T -T a )πD ( 19
)
∂ 2 x T = 4h λD (T -T a ) (20)
The characteristic length L a for this equation is

L a = λD 4h ∼ 4 mm (21)
If significative temperature differences between the air and the resistive wire were to be present, the characteristic length L a over which the profile T (x) would be impacted is thus larger than the wire actual free standing length l = 3 mm. Again, thanks to the sample design, heat exchanges with air can thus be neglected, and the resistive wire environment cannot be considered as a local thermostat.

IV. MEASUREMENT RESULTS

A. Experimental setup and measurement of temperature coefficient

Guided by the design considerations above, our sample is made of a wire of Ni70Fe30 [START_REF]GoodFellow SARL under reference[END_REF], of diameter D = 25 µm and length around L = 30 cm. The wire is electrically insulated by a thin Polyimide layer (thickness 3.5 µm). Its total resistance at room temperature is R = 295 Ω. It is wrapped approximately 30 times around the two brass screws that act as thermostats. Each screw is half buried (along its length) in a thick aluminium plate whose temperature is controlled: the low temperature thermostat is in contact with a one square meter large metallic plate at room temperature, and the high temperature thermostat is heated by a foil electrical resistance (sketch on Fig. 2). Pt1000 temperature sensors are placed inside the aluminium plates, as close as possible to the brass screws, to measure T c and T h = T c + ∆T . We checked that the temperature was uniform within 0.2 K in each thermostat. The mid point between these two readings gives an estimation of the average temperature of the resistance: In a first set of experiments, we characterise the variation of resistance (hence resistivity) with temperature. To this aim, we thermally shortcut the two thermostats and insulate them from the environment to reach a uniform temperature for all the system. All the readings are uniform within ∆T < 1.7 K, as displayed with the horizontal error bars in Fig. 4. We fit the measurement with a linear law:

T m := T c + T h 2 = T c + 1 2 ∆T (22)
R = R 0 [1 + α(T -T 0 )] (23)
with α = 4.1 × 10 -3 K -1 and R 0 = 300.4 Ω at T 0 = 300 K. This law applies as well to resistivity:

ρ = ρ 0 [1 + α(T -T 0 )].
Using this linear dependency, we can infer an average value of the temperature of the resistance from its value when ∆T = 0: let us define T avg by

T avg := T 0 + 1 α ( R R 0 -1) (24)
Straightforward calculations show indeed that:

T avg = 1 L L x=0 T (x)dx (25)
To the first order in ∆T , T (x) being linear in space, we have T avg = T m (with T m defined by eq. 22). In the measurements, we will thus be able to compare the amplitude of thermal noise, measured with T fluc , with two independent measurements of the mean temperature of the system (T m and T avg ). The statistical properties on the voltage fluctuations are illustrated in Figs. 6 and5, where we plot the Probability Distribution Function (PDF) of the signal, and its Power Spectrum Density (PSD).

In Fig. 5, we plot the power spectrum density of the voltage signals, acquired during a minute long acquisition at 6 kHz for various ∆T . The amplifier voltage noise has been subtracted from the displayed PSD. Except for a peak corresponding to line frequency, the spectra are flat in the 5 Hz to 2 kHz bandwidth displayed here. At higher frequencies, the amplifier input capacitance starts short-cutting the resistance and filtering the signal. At lower frequencies, the amplifier 1/f noise can be an issue. We therefore restrict ourselves to the displayed frequency range, and exclude the 50 Hz peak to compute the mean value of the plateau. The statistical uncertainty on its value is then around 0. The statistical properties on the voltage fluctuations are also illustrated in Fig. 6, where we plot the Probability Distribution Function (PDF) of the signal, for various ∆T . To avoid artefacts from slow drifts in the temperature of the sample or to the amplifier noise (low frequency, line frequency), the PDF are computed from the signal digitally filtered with a high pass filter at 0.3 Hz, and a notch filter with a 2 Hz stop band around 50 Hz. For all observables and all probed ∆T , the noise is perfectly white: its PDF is gaussian (skewness lower than 6 × 10 -3 , excess kurtosis lower than 10 -2 ), and its PSD is flat. Its variance, or its power spectrum density plateau, hence T fluc , is thus enough to describe the voltage fluctuations.

In figure 7, we finally report the measured values for T fluc as a function of ∆T . We compare to the two estimation of the average temperature of the resistance T m and T avg . Experimental uncertainties are around 1 K: they correspond mainly to the statistical uncertainty for T fluc , to the uncertainty in α for T avg , and to temperature stability for T m . Within those error bars, all three temperatures stand very close, and give credit to the simple extension of the Nyquist Formula of eq. 7.

V. DISCUSSION AND CONCLUSIONS

Taking into account the experimental results, to the first order in ∆T , our approach seems very reasonable. Let us evaluate in the following lines if higher order effects could be present in our system.

The Wiedemann-Franz law for metals states that thermal and electrical conductivities are proportional [START_REF] Franz | Über die wärmeleitungsfähigkeit der metalle[END_REF][START_REF] Jones | Theoretical Solid State Physics[END_REF], which may be written in our case as:

ρ(T )λ(T ) = ρ 0 λ 0 (26)
As illustrated by the measurements, the resistance has a noticeable variation on the temperature range we probe. Thus thermal conductivity present a similar variation, meaning that the temperature gradient is not uniform in space, even with negligible radiation losses. Evaluating T fluc analytically from eq. 7 seems thus non trivial in the general case. However, neglecting radiation, let us write that the heat flux is constant:

J λ = π 4 D 2 λ(T )∂ x T (27)
Combining eqs. 26 and 27, we have ρ(T ) = A∂ x T , where A is a constant. Integrating this equation from x = 0 to L, we immediately get A = R/∆T , hence:

ρ(T ) = R ∆T ∂ x T (28)
Let us now report this expression of the resistivity in the extended Nyquist prediction of eq. 16:

T fluc f = 1 R f l x=0 T (x)ρ T (x) dx (29) = 1 ∆T l x=0 T (x)∂ x T (x)dx (30) = T c + 1 2 ∆T = T m (31)
Under quite broad hypothesis (constant heat flux, Wiedemann-Franz law), the result is surprisingly simple: the amplitude of thermal noise corresponds to a temperature exactly at the mid-point between the two thermostats. Note that this result is valid even if the temperature field is non linear, and the resistivity non uniform.

In our experiment, this formula applies only to the free standings parts of the resistive wire, thus to T fluc f . For the actual measurement, using eq. 15 and injecting the temperature dependence of R h = R c (1 + α∆T ), we get to the second order in ∆T :

T fluc = T c + 1 2 ∆T + R c 4(R c + R f ) α∆T 2 (32)
In our configuration, the ratio R c /4(R c + R f ) can be evaluated by l c /4(l c + l f ) ≈ 0.1. The quadratic term is then only a 5 K correction at the end of the explored ∆T range. Other effects of similar amplitude but opposite direction (such as the contribution of the few centimeters of the resistive wire between the wrapped part and the connector to the amplifier, mainly at room temperature) make this contribution inobservable in our system.

To summarise our work, we have presented in this article an extension of the Nyquist formula to a resistance bridging two thermostats at different temperatures. On the basis of reasonable hypotheses we have analytically shown that the thermal noise in this NESS is equivalent to that of a resistance in equilibrium at the mean temperature between the two thermostats. Within an experimental accuracy of a few percent, we then demonstrated that this extended Nyquist Formula is valid in our setup. Beyond the results presented in this article, we probed a commercial resistor of 640 kΩ (film resistor with ceramic substrate) and reached exactly the same conclusion. From a more general perspective, the thermal fluctuations are adequately described by the average of the temperature field weighted by the local dissipation. Beyond electrical observables, we expect this approach to hold in other expressions of the fluctuation-dissipation theorem.

FIG. 1 .

 1 FIG.1. Principle of the experiment: we measure the thermal noise driven voltage fluctuation across a resistance submitted to a temperature difference. The signal is amplified with a low noise voltage amplifier (gain G = 10 3 ) and acquired at 6 kHz with a 24 bits data acquisition card (NI PXI 4461).

1 FIG. 3 .

 13 FIG.3. Simulated temperature profile (a), deviation to the purely conductive case (b), and ratio between radiation and conduction heat fluxes (c), for various ∆T , with eq. 13 using the following parameters: l = 3 mm, D = 25 µm, λ = 20 Wm -1 K -1 , e = 1, and Tc = 295 K. In this specific geometry, radiation is negligible versus conduction.

FIG. 4 .

 4 FIG. 4. Resistance of the sample as a function of its uniform temperature. The dependency is linear: R = R0[1+α(T -T0)] with α = 4.1 × 10 -3 K -1 and R0 = 300.4 Ω at T0 = 300 K.

8 FIG. 5 .

 85 FIG.5. Power Spectrum Density (PSD) of voltage fluctuations across the resistor for various ∆T . The amplifier voltage noise (lower curve) has been subtracted from the displayed spectra. The noise is white from a few Hz to 2 kHz, and increases with ∆T . The thin lines correspond to the measured spectra, while the thick dashed lines are the mean value of the spectra on this frequency interval.

6 FIG. 6 .

 66 FIG.6. Probability Distribution Function (PDF) of voltage fluctuations across the resistor for various ∆T . Each PDF correspond to a minute long acquisition at 6 kHz, filtered (AC + notch filter at 50 Hz). The dashed lines are centered normal distribution fits of the experimental PDF: no deviation from gaussian noise can be detected. The variance of the signal increases with ∆T .

FIG. 7 .

 7 FIG. 7. Temperatures T m (mean of the two thermostats), T avg (spacial average) and T fluc (from noise amplitude) as a function of the imposed temperature difference ∆T . The proximity of the three measurements gives credit to the simple extension of the Nyquist formula in this NESS.

ACKNOWLEDGMENTS

We thank F. Vittoz and M. Tanaze for technical support, B. Huard and Q. Ficheux for stimulating discussions. This work has been performed using the low noise chamber of the Optolyse project, funded by the French région Auvergne-Rhône-Alpes (Optolyse, CPER2016).