Axel Hutt

Daniel Trotter

Aref Pariz

Taufik A Valiante

Jérémie Lefebvre

Taufik

Diversity-induced trivialization and resilience of neural dynamics

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

I. INTRODUCTION

Complex biological systems, such as the neural circuits of the brain, are surprisingly resilient over time [START_REF] Kitano | Biological robustness[END_REF] . The dynamics of these networks resist change, persisting in the face of perturbations, such as neuromodulation, stimuli, pathological insults, and plasticity occurring during development and learning. Failure to compensate for such perturbations would otherwise undermine the reliability of brain function and predispose these circuits to pathological states, such as those seen in epilepsy [START_REF] Rich | Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony[END_REF] . What makes those systems so stable in spite of their complexity? Identifying the mechanisms and/or principles responsible for such robustness is a formidable experimental and math-ematical problem with important implications for brain function and disease. Enduring stability is a desirable, yet challenging trait to implement in engineering. Optimizing organizational and societal designs for efficiency often jeopardizes resilience, leading to fragile systems prone to collapse. [START_REF] Scheffer | Critical transitions in nature and society[END_REF][START_REF] Taleb | Antifragile: Things that gain from disorder[END_REF][START_REF] Scheffer | Anticipating critical transitions[END_REF][START_REF] Manheim | The fragile world hypothesis: Complexity, fragility, and systemic existential risk[END_REF] .

Despite important advances in our understanding of complex systems stability [START_REF] Scheffer | Critical transitions in nature and society[END_REF][START_REF] Scheffer | Anticipating critical transitions[END_REF][START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] May | Stability in multispecies community models[END_REF][START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Gao | Universal resilience patterns in complex networks[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Tang | Correlation between interaction strengths drives stability in large ecological networks[END_REF][START_REF] Fraccascia | Resilience of complex systems: State of the art and directions for future research[END_REF][START_REF] Elmqvist | Response diversity, ecosystem change, and resilience[END_REF] , the manifest resilience of neural circuits remains difficult to reconcile with their widespread and constantly evolving diversity. Efforts in identifying the features imparting resilience to neural networks have historically been oriented towards their topology (i.e. the properties of their connectivity), yet results remain perplexing [START_REF] May | Stability in multispecies community models[END_REF][START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Fraccascia | Resilience of complex systems: State of the art and directions for future research[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF] . Indeed, the hallmark of complex systems is their high susceptibility to instability: traits that accompany complexity, such as increased dimensionality [START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Stone | The feasibility and stability of large complex biological networks: a random matrix approach[END_REF][START_REF] Ipsen | Consequences of dale's law on the stabilitycomplexity relationship of random neural networks[END_REF] , strength of connections [START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF] , motifs richness [START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF] , connectance [START_REF] May | Stability in multispecies community models[END_REF][START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF] and degree heterogeneity [START_REF] Yan | Degree heterogeneity and stability of ecological networks[END_REF] generally make complex systems fragile [START_REF] Taleb | Antifragile: Things that gain from disorder[END_REF] and prone to instability through a so-called May-Wigner state transition [START_REF] May | Stability in multispecies community models[END_REF][START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Tang | Correlation between interaction strengths drives stability in large ecological networks[END_REF][START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Fyodorov | Topology trivialization transition in random nongradient autonomous odes on a sphere[END_REF][START_REF] Fyodorov | topology trivialization and large deviations for the minimum in the simplest random optimization[END_REF] . Indeed, increases in such topological characteristics most often hinder resilience instead of promoting it [START_REF] Scheffer | Anticipating critical transitions[END_REF][START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] Elmqvist | Response diversity, ecosystem change, and resilience[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Ives | Stability and diversity of ecosystems[END_REF][START_REF] Landi | Complexity and stability of adaptive ecological networks: a survey of the theory in community ecology[END_REF] . In light of the constant reorganization and complexification during development and learning, neural circuits require compensatory strategies to preserve the robustness of brain dynamics and function over time.

One way to quantify the resilience of a given dynamical system is to measure the number of its equilibria [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF][START_REF] Lacroix-A-Chez-Toine | Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates[END_REF][START_REF] Zeng | Counting equilibria of large complex systems by instability index[END_REF] .

The number of fixed points is indeed directly linked to a system's susceptibility to multistability and potential for bifurcations, reflecting a predisposition to transitions between states in the wake of disturbances. One may indeed interpret the dynamics of complex systems as evolving on a (a priorily unknown) high-dimensional landscape, occupied by various optima, corresponding to either stable or unstable fixed points. The topology of this landscape -that is the number of equilibria and the size of their respective basins of attraction -determines the response of such systems to perturbations. Resilience decreases with increasing number of fixed points: the more equilibria are present, the more complex the landscape is, predisposing this system to transitions between qualitatively different dynamics. Seminal studies [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF] have shown that, in general, neural networks are susceptible to May-Wigner transitions in which the number of equilibria populating such landscapes increases exponentially as they transition towards instability, notably as a result of increase in connectivity [START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Elmqvist | Response diversity, ecosystem change, and resilience[END_REF][START_REF] Landi | Complexity and stability of adaptive ecological networks: a survey of the theory in community ecology[END_REF][START_REF] May | Stability and complexity in model ecosystems[END_REF] . The existence and ubiquity of such transitions, commonly accompanying the emergence of chaotic dynamics [START_REF] Sompolinsky | Chaos in random neural networks[END_REF] , are a priori difficult to reconcile with the manifest resilience and robustness of neural circuits.

For many of these studies, however, one central assumption is that individual neurons, or nodes of these neural networks, are either indistinguishable from each other or alternatively break down into discreet classes (e.g. excitatory pyramidal cells, inhibitory interneurons) sharing similar excitability profiles and responses to stimuli. However, biophysical heterogeneity amongst neurons is one of the most salient features of neural tissue [START_REF] Turrigiano | Homeostatic plasticity in the developing nervous system[END_REF][START_REF] Zhang | The other side of the engram: Experience-driven changes in neuronal intrinsic excitability[END_REF][START_REF] Beck | Plasticity of intrinsic neuronal properties in CNS disorders[END_REF][START_REF] Tripathy | Intermediate intrinsic diversity enhances neural population coding[END_REF][START_REF] Padmanabhan | Intrinsic biophysical diversity decorrelates neuronal firing while increasing information content[END_REF] , even within the same cell type [START_REF] Cembrowski | Continuous variation within cell types of the nervous system[END_REF][START_REF] Cembrowski | Heterogeneity within classical cell types is the rule: lessons from hippocampal pyramidal neurons[END_REF][START_REF] Scala | Phenotypic variation of transcriptomic cell types in mouse motor cortex[END_REF][START_REF] Zeng | Neuronal cell-type classification: challenges, opportunities and the path forward[END_REF][START_REF] Zeng | What is a cell type and how to define it?[END_REF][START_REF] Tasic | Shared and distinct transcriptomic cell types across neocortical areas[END_REF] , whose phenotypes are far from identical and instead distributed along a continuum. Pioneering experimental 2,44-52 and theoretical 2,53-61 studies have explored such heterogeneity, especially intrinsic variability in excitability, and demonstrated that it imparts neural circuits with optimized coding capabilities [START_REF] Mejias | Optimal heterogeneity for coding in spiking neural networks[END_REF][START_REF] Mejias | Differential effects of excitatory and inhibitory heterogeneity on the gain and asynchronous state of sparse cortical networks[END_REF][START_REF] Gast | Effects of neural heterogeneity on spiking neural network dynamics[END_REF][START_REF] Gast | Macroscopic dynamics of neural networks with heterogeneous spiking thresholds[END_REF][START_REF] Volo | Optimal responsiveness and information flow in networks of heterogeneous neurons[END_REF] and robustness [START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF][START_REF] Tang | Robustness of a rhythmic circuit to short-and long-term temperature changes[END_REF][START_REF] Goaillard | Ion channel degeneracy, variability, and covariation in neuron and circuit resilience[END_REF][START_REF] Marder | New insights from small rhythmic circuits[END_REF][START_REF] Marder | Neuromodulation of neuronal circuits: back to the future[END_REF] , whose dynamics remain qualitatively invariant over a wide range neuromodulatory perturbations and/or disturbances in connectivity [START_REF] Hutt | Intrinsic neural diversity quenches the dynamic volatility of neural networks[END_REF] . Notably, excitability heterogeneity -a feature originating from various combinations of single cell biophysical attributeshas been shown to stabilize neural dynamics away from pathological brain states 2 .

These results echo the conclusions of numerous studies in the physical [START_REF] Skardal | Optimal synchronization of complex networks[END_REF][START_REF] Skardal | Disorder induces explosive synchronization[END_REF][START_REF] Hermann | Heterogeneous connections induce oscillations in large-scale networks[END_REF][START_REF] Scafuti | Heterogeneity induces emergent functional networks for synchronization[END_REF] and biological [START_REF] Touboul | Noiseinduced synchronization and antiresonance in interacting excitable systems: Applications to deep brain stimulation in parkinson's disease[END_REF][START_REF] Gu | Heterogeneity induces rhythms of weakly coupled circadian neurons[END_REF] sciences demonstrating that noise and/or disorder may help quench complexity to support reliability [START_REF] Hutt | Intrinsic neural diversity quenches the dynamic volatility of neural networks[END_REF][START_REF] Villegas | Intrinsic noise and deviations from criticality in boolean gene-regulatory networks[END_REF] . Such studies make an eloquent argument as to the counter intuitive role of randomness and variability in promoting the predictability and reliability of complex systems dynamics. Interestingly, statistical mechanics approaches have also convincingly demonstrated that under specific conditions, nonlinear systems may simplify through so called topological trivialization [START_REF] Fyodorov | Topology trivialization transition in random nongradient autonomous odes on a sphere[END_REF][START_REF] Fyodorov | topology trivialization and large deviations for the minimum in the simplest random optimization[END_REF] hereafter called trivialization. This term, introduced by Fyodorov and Le Doussal, refers to the process by which the total number of equilibria -and hence net complexity of the aforementioned landscapeis quenched. An eloquent example for this phenomenon is the special case of gradient systems, whose dynamics evolve in highly non-linear and frequently random potentials. Through trivialization, the geometry of such potentials is restructured and its quadratic component enhanced, constraining the dynamics around a limited set of equilibria [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Fyodorov | Topology trivialization transition in random nongradient autonomous odes on a sphere[END_REF] . As a corollary, the net stability and resilience of such system are hence both reinforced. Interestingly, trivialization has been shown to result from intrinsic heterogeneity in spin glass systems [START_REF] Lacroix-A-Chez-Toine | Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates[END_REF] , suggesting that diversity in complex biological system might serve a similar functional role, limiting the dynamical repertoire of otherwise highly unstable systems to promote their resilience.

Inspired by these results, we here explore this hypothesis from the perspective of heterogeneity, and how it might enhance the resilience of large-scale neural networks to perturbations through trivialization. That is, intrinsic neural diversity may promote stability by reducing the number of fixed points of a generic class of neural networks, thereby counter-intuitively quenching their dynamical repertoire. We first summarize the elegant theoretical framework developed [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Fyodorov | topology trivialization and large deviations for the minimum in the simplest random optimization[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF][START_REF] Zeng | Counting equilibria of large complex systems by instability index[END_REF] to quantify the number of equilibria of a system susceptible to May-Wigner stability transitions. Using this framework we demonstrate how the dispersion in excitability profile delays such transitions while suppressing the number of equilibria through trivialization. We discuss these results and explore their potential functional implications, notably in stabilizing neural dynamics during development and learning.

II. TRIVIALIZATION: THEORY

A. Neural Network Model

Determining the number of fixed points for highdimensional non-linear systems is a notoriously challenging problem [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Fyodorov | Topology trivialization transition in random nongradient autonomous odes on a sphere[END_REF][START_REF] Fyodorov | topology trivialization and large deviations for the minimum in the simplest random optimization[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF][START_REF] Zeng | Counting equilibria of large complex systems by instability index[END_REF] , that possesses a large range of applications in optimization, physics and biology. Fortunately, such calculations remains tractable given some simplifying assumptions. Our analysis is motivated by a widely used type of neural network [START_REF] Sompolinsky | Chaos in random neural networks[END_REF][START_REF] Rajan | Eigenvalue spectra of random matrices for neural networks[END_REF] in which the activity u j of a neuron obeys the following set of coupled evolution equations

d dt u i (t) = L[u i ] + N ∑ j=1 w i j f j [u j ] (1)
where L[u] = -u is a linear relaxation operator and

f j [u] = 1 2 (1 + erf[β (u + h j )]
) is a nonlinear sigmoidal activation function with gain β , where h j sets the individual excitability of neurons. The synaptic weights w i j are random, normally distributed with mean µ W and variance σ 2 w > 0 and connection probability ρ. One may further consider balanced connectivity [START_REF] Rajan | Eigenvalue spectra of random matrices for neural networks[END_REF] , in which strength of these synaptic connections are individually Gaussiandistributed with mean µ e and µ i , variance σ 2 e and σ 2 i and with probability density functions p e (x) and p i (x), respectively, with x ∈ R. The relative density of excitatory versus inhibitory connections is parameterized by g, 0 ≤ g ≤ 1, so that the probability density function of synaptic weights w i j may be written as

p(x) = ρgp e (x) -ρ(1 -g)p i (x) .
Based on these criteria, the mean and variance of the synaptic connectivity reads 73

µ w ≡ E[w i j ] = ρ(gµ e -(1 -g)µ i ) σ 2 w ≡ Var[w i j ] = ρ(gσ 2 e + (1 -g)σ 2 i + gµ 2 w /(1 -g)) .
This connectivity motif corresponds to a weighted Erdös-Rényi random graph; we emphasize, however, that the following results may be easily extended to other topologies, e.g. see the work of Yann et al. [START_REF] Yan | Degree heterogeneity and stability of ecological networks[END_REF] . The neural network, whose dynamics are described by Eq. ( 1), has been extensively studied for vanishing average connectivity (µ w = 0) in the homogeneous case, i.e. neurons with identical excitability profiles (h j = 0). Such systems are well known to exhibit May-Wigner-type transitions [START_REF] Sompolinsky | Chaos in random neural networks[END_REF][START_REF] Rajan | Eigenvalue spectra of random matrices for neural networks[END_REF][START_REF] Van Vreeswijk | Chaos in neuronal networks with balanced excitatory and inhibitory activity[END_REF][START_REF] Jirsa | Will a large complex system with time delays be stable?[END_REF] and an associated exponential increase in the number of equilibria [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF] whenever the spectral radius Γ becomes larger than unity, i.e. Γ > 1. Note that a detailed calculation for Γ will be provided below. This phenomenon echoes numerous seminal studies linking complexity and stability in high-dimensional biological networks [START_REF] Scheffer | Critical transitions in nature and society[END_REF][START_REF] Scheffer | Anticipating critical transitions[END_REF][START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] May | Stability in multispecies community models[END_REF][START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Tang | Correlation between interaction strengths drives stability in large ecological networks[END_REF][START_REF] Fraccascia | Resilience of complex systems: State of the art and directions for future research[END_REF][START_REF] Elmqvist | Response diversity, ecosystem change, and resilience[END_REF][START_REF] Stone | The feasibility and stability of large complex biological networks: a random matrix approach[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF][START_REF] Yan | Degree heterogeneity and stability of ecological networks[END_REF][START_REF] Ives | Stability and diversity of ecosystems[END_REF][START_REF] Landi | Complexity and stability of adaptive ecological networks: a survey of the theory in community ecology[END_REF][START_REF] May | Stability and complexity in model ecosystems[END_REF] whose results hold on the spectral properties of random Gaussian matrices [START_REF] Girko | Circular law[END_REF][START_REF] Wigner | Characteristic vectors bordered matrices with infinite dimensions[END_REF][START_REF] Ginibre | Statistical ensembles of complex, quaternion, and real matrices[END_REF] .

Figure 1. Example of trivialization in a gradient system. A

The dynamics of the system in Eq. 3 are constrained within a potential V whose shape changes with excitability heterogeneity. In absence of heterogeneity (σ 2 h = 0; black line) the potential is bimodal, reflecting the multistability of Eq. 3 and its susceptibility to state transitions. Increase in heterogeneity (σ 2 h > 0; blue lines) causes the potential V to become purely quadratic. B Reduction of the number of fixed points of Eq. 3 as excitability heterogeneity is increased. Such fixed points correspond to the (vanishing) maxima of the potential V seen in panel A. Parameters are given β = 15, x o = 0.6, µ h = 0.25 and σ 2 w = 0.

B. Gradient systems

Diversity-induced trivialization can be easily observed in simple gradient networks. Let us consider the ensemble average of the system state ū ≡ ⟨u i ⟩ N = ∑ N i=1 u i /N. Then Eq. ( 1) yields

1 N N ∑ i=1 du i dt = 1 N N ∑ i=1 L[u i ] + 1 N N ∑ i, j=1 w i j f j [u j ] d ū dt = L[ ū] + N ∑ j=1 W j f j [u j ]
with the mean connectivity at each node W j = ∑ N i=1 w i j /N. Assuming that firing rates (i.e. f j [u j ]) are independent of synaptic weights (i.e. w i j ), which holds only whenever N is large [START_REF] Hutt | Intrinsic neural diversity quenches the dynamic volatility of neural networks[END_REF][START_REF] Hutt | Phase coherence induced by additive Gaussian and non-Gaussian noise in excitable networks with application to burst suppression-like brain signals[END_REF][START_REF] Hutt | Coherence resonance in random Erdos-Renyi neural networks : mean-field theory[END_REF] , and by virtue of the network connectivity properties defined above, W j = µ w + X j with random perturbations X j ∼ N (0, σ 2 w /N) yields

d ū dt = L[ ū] + N ∑ j=1 (µ w + X j ) f j [u j ] = L[ ū] + Nµ w ⟨ f j ⟩ N + N⟨X j f j [u j ]⟩ N . (2)
Since, in general,

⟨YW ⟩ N = ⟨Y ⟩ N ⟨W ⟩ N + S and S ∼ N (0, Var[YW ]/N) with Var[YW ] = Var[Y ]Var[W ] + Var[Y ]⟨W ⟩ 2 N + Var[W ]⟨Y ⟩ 2
N for two statistically independent random variables Y, W , Eq. ( 2) reads

d ū dt = L[ ū] + Nµ w ⟨ f j ⟩ N + σ w Var[ f j ] + ⟨ f j ⟩ 2 N Z
with the random variable Z ∼ N (0, 1).

Assuming that the excitability h j of individual neurons in Eq. ( 1) are normally distributed with mean µ h and variance σ 2 h , one may consider setting w i j = x i j /N yielding µ w = x o /N and σ 2 w = σ 2 x /N 2 . Then in the limit N → ∞ 14,79-81 Eq. ( 1) admits a well-defined mean field

d dt ū(t) = L[ ū] + x o F[ ū] . (3)
where F[ ū] corresponds to the statistical average

⟨ f j ⟩ N ≈ F[ ū] = ∞ -∞ 1 2 (1 + erf(β ( ū + h)) p h (h)dh = 1 2 1 + erf β ( ū + µ h ) γ ( 4
)
with p h (h) as the Gaussian distribution with mean µ h and variance σ 2 h and γ = 1 + 2β 2 σ 2 h . Equation (4) holds true whenever N is sufficiently large [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF][START_REF] Lefebvre | Induced synchronisation by endogenous noise modulation in finite-size random neural networks: a stochastic mean-field study[END_REF] . Now Eq. ( 3) may be written in terms of the gradient of a scalar potential V ( ū), i.e

d dt ū(t) = - dV ( ū) d ū defined by V ( ū) = ū2 2 - x o 2 √ πβ γe -β 2 (µ h -ū) 2 γ + √ πβ   (µ h -ū)erf   β (µ h -ū) 1 + 2β 2 σ 2 h   + ū    (5)
Figure 1 depicts trivialization that occurs in Eq. ( 3) once excitability heterogeneity is increased. As can be seen in Fig. 1A, for σ 2 h = 0 the potential V ( ū) displays multiple optima, corresponding to the fixed points of Eq. ( 3). Further increases of σ 2 h alters the shape of the potential V ( ū), converging towards a purely quadratic form in which there is only one equilibrium. Hence, large heterogeneity yields trivialization.

C. Non-gradient systems

Conversely, general systems subscribing to Eq. ( 1) do neither possess a well-defined mean field nor are they gradient, being instead governed by random potentials [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF] . We here summarize the theoretical framework proposed in [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF] , highlighting the relationship between the May-Wigner transition threshold and the number of equilibria of large-scale nonlinear systems such as those in Eq. ( 1) in absence of a well-defined potential. For generality, we focus on the well-studied case µ w = 0; we note however that this condition may be relaxed and yield identical results under some conditions [START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Ipsen | Consequences of dale's law on the stabilitycomplexity relationship of random neural networks[END_REF] . After the review of major results on May-Wigner transitions in the following paragraphs, sub-section II C 2, (Diversity Induced Trivialization), provides new results on trivialization in random networks with excitability heterogeneity.

Theory of May-Wigner transitions

One may indeed estimate the mean number of equilibria of Eq. ( 1), denoted by ⟨N ⟩, through the celebrated Kac-Rice relation [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Fyodorov | Topology trivialization transition in random nongradient autonomous odes on a sphere[END_REF][START_REF] Fyodorov | topology trivialization and large deviations for the minimum in the simplest random optimization[END_REF][START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF]

⟨N ⟩ = R N du ⟨|det (-I + J(u)) | × δ (L[u] +W f [u])⟩ (6)
where δ stands for the Dirac delta function, J the Jacobian associated with Eq. (1), W is the weight matrix with (W ) i j = w i j , (f ) = f i , I is the N-dimensional identity matrix and ⟨•⟩ corresponds to the average over realizations of the matrix J . In the case where the Jacobian J is independent of u, Eq. ( 6) simplifies further to [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF] ⟨N ⟩ = ⟨|det(-I + J )|⟩ (7) Following the rationale detailed in [START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF] , we know that for spectral radii Γ < 1, Eq. ( 1) possesses a unique equilibrium denoted by u o . One further notices that

log |det(-I + J)| = ∑ λ ∈Λ log |λ -1| (8)
where λ ∈ C are discrete eigenvalues and Λ corresponds to the spectrum associated with the Jacobian J . Dividing both sides of Eq. ( 8) by N and taking the limit N → ∞, one obtains

1 N ∑ λ ∈Λ log |λ -1| = C log |s -1|ρ(s)ds (9)
where ρ(s) is the probability density of eigenvalues in the complex plane. For the system defined in Eq. ( 1), according to Girko's circular law of random matrices [START_REF] Girko | Circular law[END_REF][START_REF] Wigner | Characteristic vectors bordered matrices with infinite dimensions[END_REF][START_REF] Ginibre | Statistical ensembles of complex, quaternion, and real matrices[END_REF] , the spectrum Λ is uniformly distributed in the complex plane over a disk of radius Γ. For s = re iθ , ρ(s) = ρ(r) = (πΓ 2 ) -1 for r ≤ Γ and zero otherwise. Hence, one may write Eq. ( 9) as And hence the right hand side of Eq. ( 10) simplifies to

C log |s -1|ρ(s)ds = 1 πΓ 2 2π 0 Γ 0 r log |re iθ -1|drdθ (10)
2Γ -2 Γ 1 r log r dr = log Γ + 1 2 1 Γ 2 -1
The expected number of equilibria ⟨N ⟩ is hence related, to the first order, to the spectral radius Γ by the following well known [START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF] relationship

⟨N ⟩ = 1 0 < Γ < 1 exp[N log Γ + 1 2 1 Γ 2 -1 ]
Γ ≥ 1 (11) which confirms that whenever the spectral radius Γ > 1, an exponential increase in the number of equilibria accompanies the May-Wigner transition.

Diversity-Induced Trivialization

The quenching effect of excitability heterogeneity on the number of equilibria for arbitrary non-gradient systems can be quantified through the same approach described above. As per Eq. ( 11), ⟨N ⟩ depends, to the first order, exclusively on the spectral radius Γ, which can be computed explicitly in the presence of heterogeneity [START_REF] Hutt | Intrinsic neural diversity quenches the dynamic volatility of neural networks[END_REF] . This allows us to revisit the above analysis and quantify how ⟨N ⟩ scales with the dispersion of intrinsic neuronal properties.

One may note that below the May-Wigner transition (Γ < 1), as per the Banach fixed point theorem, a unique stable fixed point u o = [u o 1 , u o 2 , ..., u o N ] exists, satisfying the set of implicit equations

u o i = N ∑ j=1 w i j f j [u o j ] (12)
Now let us consider neuron-specific excitability thresholds h j ̸ = 0 in Eq. ( 1) where h j obey the Gaussian probability density function ρ(h) with vanishing mean and non-vanishing variance σ 2 h . Then the resulting unique fixed point u o is not trivial and distributed with a probability density function ρ(u o ). This function's first two moments can be easily computed, given a few simplifying assumptions. Indeed, assuming statistical independence between firing rates, synaptic weights and heterogeneity, and whenever N is sufficiently large, the mean over the nodes in Eq. ( 12) may now be approximated by the following statistical average

µ u o ≡ ⟨u o i ⟩ N = N ∑ j=1 w i j f j [u o j ] N = μw ∞ -∞ f [u o ]ρ(u o )du o = 0 (13)
with μw = Nµ w and by independence of connectivity weights and fixed point coordinates. Equation ( 13) is an implicit equation for µ u o and holds true since f [u o ] and ρ(u o ) are anti-symmetric and symmetric to the origin, respectively. The variance σ 2 u o can be computed similarly 65

σ 2 u o ≡ (u o i ) 2 N = N ∑ j,k=1 w i j w ik f j [u o j ] f j [u o k ] N = Nσ 2 W 4   1 - 2 4 + β 2 π 2 σ 2 h   . ( 14
)
We proceed by first observing that elements of the Jacobian J associated with Eq. ( 12) can be written as

J i j = -δ i j + w i j ∂ f j ∂ u j | u o j = -δ i j + w i j β √ π e -β 2 (u o j +h j ) 2 (15)
Substituting two normally distributed and independent random variables s and q whose statistics are identical to those of u o i and h j i.e. ρ(s) → ρ(u o ) and ρ(q) → ρ(h), one obtains a surrogate yet statistically identical Jacobian J i.e. Ji j = J i j + δ i j = wi j (16) where we have used the change of variable, effectively re-scaling the synaptic weights,

wi j = w i j β √ π e -β 2 (s+q) 2
Through this substitution, the calculation of the spectral radius follows 9,10,14,15

Γ = (N -1)ρσ 2 w , (17)
i.e. the fixed point u o is stable if Γ < 1.

In addition, whenever N is sufficiently large, the variance of the re-scaled synaptic weights becomes,

σ 2 W = σ 2 W Var β √ π e -β 2 (s+q) 2 = σ 2 W β 2 π ∞ -∞ ∞ -∞ e -β 2 (s+q) 2 ρ(s)ρ(q)dsdq = σ 2 W β 2 π √ γ (18)
with

γ = 1 + 4β 2 (σ 2 u o + σ 2 h ) .
Hence the spectral radius reads

Γ = σ 2 W (N -1)ρβ 2 π √ γ . ( 19
)
utilizing Eq. ( 14). We emphasize that Eq. ( 19) applies for Erdös-Renyi connectivity matrices, however, our results can easily be extended to other topologies, such as multi-modal, scalefree or cascade models, through a proper re-scaling of the spectral radius [START_REF] Yan | Degree heterogeneity and stability of ecological networks[END_REF][START_REF] Landi | Complexity and stability of adaptive ecological networks: a survey of the theory in community ecology[END_REF] .

By virtue of the statistical equivalence of the Jacobian J and J and their independence with respect to u o , the expected number of fixed points may hence be approximated through Eq. ( 7)

⟨N ⟩ (σ 2 h ) ≈ |det(-I + J)|
and Eq. ( 11) follows directly. We may hence substitute Eq. ( 19) in Eq. ( 11) to obtain an explicit expression for the expected number of equilibria ⟨N ⟩ as a function of the variance in excitability thresholds. Specifically, we may expand Eq. ( 11) at the stability threshold Γ ∼ 1 and obtain

⟨N ⟩ (σ 2 h ) ≈ 1 + N σ 2 W (N -1)ρβ 2 π √ γ -1 2 . (20)
Obviously, the mean number of fixed points ⟨N ⟩ is a decaying function with respect to the excitability heterogeneity variance σ 2 h . This confirms that trivialization occurs whenever intrinsic excitability increases its variance. This translates into an effective shift in the May-Wigner transition threshold occurring at Γ = 1 for a critical heterogeneity level 54,56,62 σ 2 h,c . For σ 2 h << 1, one may expand Eq. ( 14) to first order in σ 2 h to obtain an explicit expression for the critical heterogeneity for trivialization, i.e.

σ 2 h,c ≈ 8 π 2 β 2 N 2 β 4 σ 4 W -π 2 Nσ 2 W π 2 β 2 + 32 (21)
It has been shown [START_REF] Hutt | Intrinsic neural diversity quenches the dynamic volatility of neural networks[END_REF] that dispersion of excitability heterogeneity (i.e. σ 2 h > 0) effectively contracts the spectral radius, stabilizing the dynamics and decreasing the probability of state transitions. This phenomenon translates here directly to trivialization: as can be seen in Fig. 2, both numerical and analytical calculations jointly predict a decrease in the number of expected equilibria ⟨N ⟩ as the dispersion of excitability thresholds increases. Such diversity seems to limit the exponential explosion of fixed points and appears to delay the May-Wigner transition generically observed whenever Γ ≥ 1.

III. APPLICATIONS

A. Resilience

We have demonstrated in the previous section, that excitability heterogeneity induces trivialization. Since trivialization is strongly related to the resilience of a system to perturbations, the current section aims to demonstrate enhanced resilience by stronger excitability heterogeneity. We assume that a system is in its stationary state and receives a short strong perturbation impulse I(t), that repels the system from its stationary state. After the perturbation impulse, the system evolves according to its unperturbed dynamics and relaxes to a new stationary state. We state that the system is resilient if the new stationary state is the same as the original state, where the system was located before the perturbation. We quantify the degree of resilience, and compute the distance D of the newly approached state to the original state. If D ≈ 0, then the system is resilient, whereas D > 0 indicates lower resilience.

In more detail, we assume model (1) with L[u] = -u and µ w = 1.0, σ 2 w = 0.05, µ h = 1.0 and σ 2 h > 0. The distance reflecting the degree of resilience is defined as

D = 1 N ∑ i=1 |u original i -u perturbed i | . ( 22
)
We simulated Eq. ( 1) with initial condition u i (t = 0) = ξ i , i=1. . . ,N, where ξ i ∈ [0; 1] has been uniform distributed. Temporal integration with time step ∆ = 0.05 yielded a stationary state if

∑ N i=1 |∆u i |/N < 10 -3 with ∆u i = -u i + ∑ N j=1 w i j f j [u j ].
Then a random uniformdistributed instantaneous perturbation Y i ∈ [-500; 500] is applied at each network node i, after which the system approaches the new stationary state u perturbed i . This new state is reached if ∑ N i=1 |∆u i |/N < 10 -4 . For each excitability variance σ 2 h , we have run T = 400 simulation trials, where each trial draws new random connections w i j and new excitability values h i from their corresponding distributions. This yields a value of D for each σ 2 h and we have computed this value T times. Figure 3 shows all obtained values of D for difference values of the excitability heterogeneity variance σ 2 h . We observe that systems with weak heterogeneity (low values of σ 2 h ) exhibit different non-zero values of D and thus a low degree of resilience, whereas increasing heterogeneity yields better and better resilience.

B. Neural development and learning

The current work is strongly motivated by the role of intrinsic diversity during development, through which synaptic connectivity motifs become increasingly dense and complex. What maintains neural circuit stability? Indeed, during development and learning, the number of synapses and their effective weights change, notably through long-term synaptic potentiation (LTP) and depression (LTD) [START_REF] Lynch | Long-term potentiation and memory[END_REF] . Such increase in complexity, parameterized by gradual increase in connection probability and/or density of synaptic connections and/or changes to the number of neurons and synapses, was repeatedly found to foster instability by increasing the risk of May-Wigner type transitions [START_REF] May | Stability in multispecies community models[END_REF][START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Stone | The feasibility and stability of large complex biological networks: a random matrix approach[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF][START_REF] Yan | Degree heterogeneity and stability of ecological networks[END_REF][START_REF] Aljadeff | Transition to chaos in random networks with cell-type-specific connectivity[END_REF] . Similarly, synaptic plasticity models in recurrent neural networks commonly face unstable regimes which need to be compensated by/stabilized through various feedback processes [START_REF] Turrigiano | Homeostatic plasticity in the developing nervous system[END_REF][START_REF] Zenke | Synaptic plasticity in neural networks needs homeostasis with a fast rate detector[END_REF][START_REF] Zenke | Hebbian plasticity requires compensatory processes on multiple timescales[END_REF][START_REF] Abraham | Metaplasticity: tuning synapses and networks for plasticity[END_REF] . While many neurophysiological processes, such as synaptic inhibition, adaptation, modularity, and circuit motifs are certainly involved in stabilizing neural circuits' function, we believe that intrinsic excitability heterogeneity is also involved through trivialization. To emulate such a developmental/learning process for networks whose dynamics obey Eq. ( 1), we considered a population of N initially isolated neurons, whose mutual connectivity would increase gradually in time (cf. Fig. 4). To implement this, we simulated the solutions of Eq. (1) while the connection probability ρ linearly increases from 0 to 5% over an extended time scale (Fig. 4A). Note that while synaptic weights would also be expected to increase during development/learning, the influence of this parameter on stability would be identical (cf. Eq. ( 19)). Hence we focused on the connection probability for simplicity. Results presented in Fig. 4B confirm the destabilizing effect of increase in connectivity and the resurgence of a May-Wigner instability: the network dynamics reflects a Response of a homogeneous network, in which all neurons have the same excitability profile (i.e. h j = 0 ∀ j). Chaotic dynamics can be observed, indicative that the system resides beyond the May-Wigner transition threshold B Associated eigenvalues in the complex plane (black dots) alongside the theoretically derived spectral radius (19) plotted around λ = -1 (blue circle). Some eigenvalues occupy the right hand side of the imaginary axis, confirming instability. C-D. Dynamics and eigenspectrum of the same network whenever heterogeneity is increased to σ 2 h = 0.0001 and; E-F σ 2 h = 0.001. Heterogeneity stabilizes the system by contracting the spectral radius in the complex plane, yielding stable dynamics. Blue dots in C,E illustrate different values of h j . G Spectral radius Γ as a function of excitability heterogeneity. As σ 2 h increases and passes the critical point σ 2 h,c (vertical dashed line), the May-Wigner transition threshold (i.e. Γ = 1; horizontal line) is crossed, stabilizing the dynamics (cf Panels A-F). Numerical results averaged over 50 independent realizations of the network connectivity (black dotted line; grey shaded area represents ± SD) are plotted alongside theoretical predictions (solid blue line). H Expected number of equilibria ⟨N⟩ as a function of σ 2 h , revealing trivialization and a shift in the May-Wigner instability threshold induced by excitability heterogeneity. Numerical results (black dotted line) and theoretical predictions (solid blue line) coincide at and beyond the critical point, but diverge otherwise due to error propagation through an exponential (see Eq. ( 11)). I Variance of the steady state probability density function as a function of excitability heterogeneity, confirming that σ 2 h does increase the dispersion of the steady state, while showing good alignment between numerical (black dotted line; grey shaded area represents ± SD) and theoretical (solid blue line) calculations. J Variance of the dynamics for the individual neurons (i.e. u j ; grey shaded lines) across various levels of excitability heterogeneity, shown alongside their average over independent trials (black dotted line). As the system transits from chaotic to stable dynamics by crossing the critical threshold σ 2 h,c (cf. panels A-F), the system activity variance decreases. In other words, increasing the excitability heterogeneity reduces the system fluctuations and increases system order. Parameters are N = 100, ρ = 0.05,g = 0.8, µ e = 0.005, β = 25,

µ i = g g-1 µ e , σ 2
W,e = σ 2 W,i = 0.0015. Whenever applicable, averages were computed over 50 independent trials of duration T = 200 [a.u.].

highly unstable, multi-modal oscillatory state. Introducing excitability heterogeneity in Fig. 4C was in contrast found to stabilize the dynamics by dispersing the neurons' activity. Spectral radii calculations affirmed these numerical observations ( Fig. 4D) and further revealed that heterogeneity delays the May-Wigner transition and its associated exponential increase in equilibria ( Fig. 4E).

Trivialization was further found to scale with the level of excitability heterogeneity (i.e. σ 2 h ): as can be seen in Fig. 4F-G, increased level of heterogeneity decreases 22) reflects the degree of the system's resilience: the smaller D, the closer are the original and perturbed state and the more resilient is the system towards perturbations. We observe that strong heterogeneity renders the system more resilient to perturbations. For each value of σ 2 h , 400 values of D are plotted to correspond with the same number of simulation trials.

the spectral radius and further limits the number of fixed points. Taken together, these findings suggest that heterogeneity quenches the dynamical repertoire of non-linear networks as their complexity increases to maintain their resilience, suggesting that neural diversity plays a role in the persistence of brain function.

IV. DISCUSSION

Excitability heterogeneity affects stability of random networks and hence can induce trivialization and enhance resilience. This is shown theoretically in Eq. ( 5) in gradient systems and in the corresponding application shown in Fig. 3. In non-gradient systems, Equation (11) confirms that an exponential increase in fixed points occurs whenever the May-Wigner instability threshold is met i.e. Γ ≥ 1 [START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Fyodorov | Topology trivialization transition in random nongradient autonomous odes on a sphere[END_REF][START_REF] Fyodorov | topology trivialization and large deviations for the minimum in the simplest random optimization[END_REF][START_REF] Wainrib | Topological and dynamical complexity if random neural networks[END_REF] . An important corollary of this equation is that the number of equilibria scales with the spectral radius, whose magnitude has been shown to increase with a variety of topological traits, such as increased dimensionality [START_REF] May | Will a large complex system be stable ?[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Stone | The feasibility and stability of large complex biological networks: a random matrix approach[END_REF] , strength of connections [START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF] , motifs richness [START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Levin | Some mathematical questions in biology[END_REF][START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Allesina | Stability criteria for complex ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF] , connectance 8,9,14,18,21 and degree heterogeneity [START_REF] Yan | Degree heterogeneity and stability of ecological networks[END_REF] . Interestingly, our results as well as others [START_REF] Hutt | Intrinsic neural diversity quenches the dynamic volatility of neural networks[END_REF] indicate that increased intrinsic diversity in cellular excitability represents an exception, contracting the spectral radius and hence promoting stability through a reduction of the number of equilibria. This suggests that variability amongst cells of a network -as opposed to heterogeneity in their connectivity -promotes resilience, by reducing the probability of multistability and bifurcations.

It is interesting to conjecture on the potential role of trivialization in neural circuits' dynamics and function. Intuitively, increasing the number of control parameters by enabling cells with different excitability profiles should enrich a system's phase space -however, our results sug-gest the opposite. Numerous experimental [START_REF] Rich | Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony[END_REF][START_REF] Markram | Interneurons of the neocortical inhibitory system[END_REF][START_REF] Mody | Diversity of inhibitory neurotransmission through gabaa receptors[END_REF][START_REF] Soltesz | Diversity in the neuronal machine: order and variability in interneuronal microcircuits[END_REF][START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF][START_REF] Tomassy | Distinct profiles of myelin distribution along single axons of pyramidal neurons in the neocortex[END_REF][START_REF] Matias | Astrocyte heterogeneity: impact to brain aging and disease[END_REF][START_REF] Moradi Chameh | Diversity amongst human cortical pyramidal neurons revealed via their sag currents and frequency preferences[END_REF][START_REF] Goaillard | Ion channel degeneracy, variability, and covariation in neuron and circuit resilience[END_REF] and theoretical studies 2,53-59 have examined the role of cell-to-cell heterogeneity in brain dynamics and function. For instance, several neural population models considered heterogeneous single cell properties [START_REF] Goldobin | Reduction methodology for fluctuation driven population dynamics[END_REF][START_REF] Stefanescu | A low dimensional description of globally coupled heterogeneous neural networks of excitatory and inhibitory neurons[END_REF][START_REF] Zerlaut | Heterogeneous firing rate response of mouse layer v pyramidal neurons in the fluctuation-driven regime[END_REF] . Collectively, these indicate that intrinsic diversity may both enhance resilience [START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF][START_REF] Tang | Robustness of a rhythmic circuit to short-and long-term temperature changes[END_REF][START_REF] Goaillard | Ion channel degeneracy, variability, and covariation in neuron and circuit resilience[END_REF][START_REF] Marder | New insights from small rhythmic circuits[END_REF][START_REF] Marder | Neuromodulation of neuronal circuits: back to the future[END_REF] and optimize coding performance [START_REF] Mejias | Optimal heterogeneity for coding in spiking neural networks[END_REF][START_REF] Mejias | Differential effects of excitatory and inhibitory heterogeneity on the gain and asynchronous state of sparse cortical networks[END_REF][START_REF] Gast | Effects of neural heterogeneity on spiking neural network dynamics[END_REF][START_REF] Gast | Macroscopic dynamics of neural networks with heterogeneous spiking thresholds[END_REF][START_REF] Volo | Optimal responsiveness and information flow in networks of heterogeneous neurons[END_REF] . Our complementary results show that variations in cellular excitability represents a stabilizing mechanism, quenching the dynamical repertoire of balanced neural networks. This surprising feature arises notably through the effective linearization of input-output response functions [START_REF] Rich | Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony[END_REF][START_REF] Fyodorov | Non-linear analogue of the maywigner instability transition[END_REF][START_REF] Rich | Neurostimulation stabilizes spiking neural networks by disrupting seizurelike oscillatory transitions[END_REF] , promoting graded response to perturbations and subsequent increasingly linear relaxation towards a limited number of stable states.

Multiple mechanisms have been proposed to regularize the dynamics of complex networks beyond heterogeneity. Noise and disorder -ubiquitous in neural systems [START_REF] Faisal | Noise in the nervous system[END_REF] -represent important drivers of regularization, where randomlike perturbations may paradoxically yield highly predictable dynamics over a wide range of physical [START_REF] Skardal | Disorder induces explosive synchronization[END_REF][START_REF] Hermann | Heterogeneous connections induce oscillations in large-scale networks[END_REF][START_REF] Scafuti | Heterogeneity induces emergent functional networks for synchronization[END_REF] and biological [START_REF] Touboul | Noiseinduced synchronization and antiresonance in interacting excitable systems: Applications to deep brain stimulation in parkinson's disease[END_REF][START_REF] Gu | Heterogeneity induces rhythms of weakly coupled circadian neurons[END_REF] systems. This form of complexity collapse [START_REF] Tavakoli | Multi-delay complexity collapse[END_REF] has important implications for neural circuit function. These suggest a trade-off between topological complexity -which has been extensively studiedand intrinsic diversity, balancing coding efficiency and resilience. As such, trivialization resulting from either noise, disorder or heterogeneity, should be considered as a control mechanism, tuning stability towards an optimal dynamical state [START_REF] Cocchi | Criticality in the brain: A synthesis of neurobiology, models and cognition[END_REF][START_REF] Ma | Cortical circuit dynamics are homeostatically tuned to criticality in vivo[END_REF][START_REF] Wang | Sustained activity in hierarchical modular neural networks: self-organized criticality and oscillations[END_REF][START_REF] Levina | Dynamical synapses causing self-organized criticality in neural networks[END_REF][100] . For instance, destabilizing changes in connectivity during development and learning may be compensated for through adaptive increase in the level of fluctuations, biological diversity or both, to optimize neural circuit performance.

Intrinsic plasticity [START_REF] Turrigiano | Homeostatic plasticity in the developing nervous system[END_REF][START_REF] Zhang | The other side of the engram: Experience-driven changes in neuronal intrinsic excitability[END_REF][START_REF] Beck | Plasticity of intrinsic neuronal properties in CNS disorders[END_REF][START_REF] Tripathy | Intermediate intrinsic diversity enhances neural population coding[END_REF][START_REF] Padmanabhan | Intrinsic biophysical diversity decorrelates neuronal firing while increasing information content[END_REF] refers to a non-synaptic form of plasticity through which individual cells' biophysical characteristics and/or excitability change in experience and stimuli-dependent ways, notably through adjustments in ionic conductances [START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF][START_REF] Goaillard | Ion channel degeneracy, variability, and covariation in neuron and circuit resilience[END_REF][START_REF] Marder | New insights from small rhythmic circuits[END_REF]101 . While speculative, our results nonetheless suggest that intrinsic plasticity could play an important role in stabilizing neural dynamics and function, compensating for the notoriously destabilizing effect of synaptic plasticity [START_REF] Turrigiano | Homeostatic plasticity in the developing nervous system[END_REF][START_REF] Zenke | Synaptic plasticity in neural networks needs homeostasis with a fast rate detector[END_REF][START_REF] Zenke | Hebbian plasticity requires compensatory processes on multiple timescales[END_REF][START_REF] Abraham | Metaplasticity: tuning synapses and networks for plasticity[END_REF] .

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request. 19). The spectral disks' radius remains below the May-Wigner transition threshold (Γ = 1; red circle) only in the heterogeneous case. E Expected number of equilibria ⟨N ⟩ for the homogeneous (σ 2 h = 0; black line) and heterogeneous (σ 2 h = 0.001; blue line) network as the connection probability increases. In the presence of heterogeneity, the number of equilibria remains small through trivialization, and increases exponentially otherwise. F Spectral radius as a function of the variance of the synaptic weight matrix W , reflecting destabilizing influence of increase in connectivity and the compensatory effect of heterogeneity (blue lines). One notices that the May-Wigner transition threshold (Γ = 1; horizontal grey line ) is delayed. G Associated reduction in the number of equilibria ⟨N ⟩ through trivialization at increasing degrees of excitability heterogeneity σ 2 h (blue lines). Aside from the connection probability ρ which is increasing, all other parameters are identical to those used in Fig.

  Noting that by residue calculus, one has 2π 0 log |re iθ -1| dθ = 0 0 < r < 1 2π log r r ≥ 1

Figure 2 .

 2 Figure 2. Example of diversity-induced trivialization in a random neural network A.Response of a homogeneous network, in which all neurons have the same excitability profile (i.e. h j = 0 ∀ j). Chaotic dynamics can be observed, indicative that the system resides beyond the May-Wigner transition threshold B Associated eigenvalues in the complex plane (black dots) alongside the theoretically derived spectral radius(19) plotted around λ = -1 (blue circle). Some eigenvalues occupy the right hand side of the imaginary axis, confirming instability. C-D. Dynamics and eigenspectrum of the same network whenever heterogeneity is increased to σ 2 h = 0.0001 and; E-F σ 2 h = 0.001. Heterogeneity stabilizes the system by contracting the spectral radius in the complex plane, yielding stable dynamics. Blue dots in C,E illustrate different values of h j . G Spectral radius Γ as a function of excitability heterogeneity. As σ 2 h increases and passes the critical point σ 2 h,c (vertical dashed line), the May-Wigner transition threshold (i.e. Γ = 1; horizontal line) is crossed, stabilizing the dynamics (cf Panels A-F). Numerical results averaged over 50 independent realizations of the network connectivity (black dotted line; grey shaded area represents ± SD) are plotted alongside theoretical predictions (solid blue line). H Expected number of equilibria ⟨N⟩ as a function of σ 2 h , revealing trivialization and a shift in the May-Wigner instability threshold induced by excitability heterogeneity. Numerical results (black dotted line) and theoretical predictions (solid blue line) coincide at and beyond the critical point, but diverge otherwise due to error propagation through an exponential (see Eq. (11)). I Variance of the steady state probability density function as a function of excitability heterogeneity, confirming that σ 2 h does increase the dispersion of the steady state, while showing good alignment between numerical (black dotted line; grey shaded area represents ± SD) and theoretical (solid blue line) calculations. J Variance of the dynamics for the individual neurons (i.e. u j ; grey shaded lines) across various levels of excitability heterogeneity, shown alongside their average over independent trials (black dotted line). As the system transits from chaotic to stable dynamics by crossing the critical threshold σ 2

Figure 3 .

 3 Figure 3. Resilience by excitability heterogeneity The distance D taken from Eq. (22) reflects the degree of the system's resilience: the smaller D, the closer are the original and perturbed state and the more resilient is the system towards perturbations. We observe that strong heterogeneity renders the system more resilient to perturbations. For each value of σ 2 h , 400 values of D are plotted to correspond with the same number of simulation trials.

Figure 4 .

 4 Figure 4. Trivialization may stabilize neural dynamics through development and learning. A To model circuit modifications occurring during development and learning, a network of N neurons was examined as its connection probability ρ increased linearly from 0 to 0.05. B. In the absence of excitability heterogeneity (σ 2 h = 0), the network gradually converges towards instability. As the connection probability increases passed the critical point (vertical dashed line), the network dynamics become unstable. C This does not occur if heterogeneity is present in the network (σ 2 h = 0.001) and the dynamics remain stable throughout. D Spectral disks for the homogeneous (σ 2 h = 0; black circle) and heterogeneous excitability (σ 2 h = 0.001; blue circle) calculated at intervals of 500 a.u. through Eq. (19). The spectral disks' radius remains below the May-Wigner transition threshold (Γ = 1; red circle) only in the heterogeneous case. E Expected number of equilibria ⟨N ⟩ for the homogeneous (σ 2 h = 0; black line) and heterogeneous (σ 2 h = 0.001; blue line) network as the connection probability increases. In the presence of heterogeneity, the number of equilibria remains small through trivialization, and increases exponentially otherwise. F Spectral radius as a function of the variance of the synaptic weight matrix W , reflecting destabilizing influence of increase in connectivity and the compensatory effect of heterogeneity (blue lines). One notices that the May-Wigner transition threshold (Γ = 1; horizontal grey line ) is delayed. G Associated reduction in the number of equilibria ⟨N ⟩ through trivialization at increasing degrees of excitability heterogeneity σ 2 h (blue lines). Aside from the connection probability ρ which is increasing, all other parameters are identical to those used in Fig.2.

  Figure 4. Trivialization may stabilize neural dynamics through development and learning. A To model circuit modifications occurring during development and learning, a network of N neurons was examined as its connection probability ρ increased linearly from 0 to 0.05. B. In the absence of excitability heterogeneity (σ 2 h = 0), the network gradually converges towards instability. As the connection probability increases passed the critical point (vertical dashed line), the network dynamics become unstable. C This does not occur if heterogeneity is present in the network (σ 2 h = 0.001) and the dynamics remain stable throughout. D Spectral disks for the homogeneous (σ 2 h = 0; black circle) and heterogeneous excitability (σ 2 h = 0.001; blue circle) calculated at intervals of 500 a.u. through Eq. (19). The spectral disks' radius remains below the May-Wigner transition threshold (Γ = 1; red circle) only in the heterogeneous case. E Expected number of equilibria ⟨N ⟩ for the homogeneous (σ 2 h = 0; black line) and heterogeneous (σ 2 h = 0.001; blue line) network as the connection probability increases. In the presence of heterogeneity, the number of equilibria remains small through trivialization, and increases exponentially otherwise. F Spectral radius as a function of the variance of the synaptic weight matrix W , reflecting destabilizing influence of increase in connectivity and the compensatory effect of heterogeneity (blue lines). One notices that the May-Wigner transition threshold (Γ = 1; horizontal grey line ) is delayed. G Associated reduction in the number of equilibria ⟨N ⟩ through trivialization at increasing degrees of excitability heterogeneity σ 2 h (blue lines). Aside from the connection probability ρ which is increasing, all other parameters are identical to those used in Fig.2.

ACKNOWLEDGMENTS

We thank the National Sciences and Engineering Research Council of Canada (NSERC Grants RGPIN-2017-


Aurélie Antunes

Denis Servent

Rémy Béroud

Michel De Waard

Evelyne Benoit

Evelyne Benoit Venomous Ani- Mal

Venomous animal peptides: an innovative approach to treat pain

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

Venomous animal peptides: An innovative approach to treat pain (Ingénieurire des Protéines et Cibles Thérapeutiques)

Aurélie Antunes 1,2 , Denis Servent 1 , Rémy Beroud 2 , Michel De Waard 2,3 , Evelyne Benoit 1 Pain management is a world health priority, and current pain medications, while effective in many circumstances, are frequently associated with adverse side effects. Targeting dorsal root ganglion (DRG) neurons in the peripheral nervous system is a promising strategy for managing pain and limiting central side effects. These neurons are well-known to express various painrelated proteins, including the 1.7 subtype of voltage-gated sodium channels (NaV1.7), a genetically proven target for pain management. In this context, several small molecules have been described in the literature as NaV1.7 blockers but their limited selectivity and/or potency often prevented their therapeutic development. Concurrently, small reticulated peptides produced by venomous animals for predation or defense have been recognized as an original source of antinociceptive peptides. My PhD project is to identify and then evaluate a large number of new antinociceptive venom peptides with high selectivity and efficacy for NaV1.7 using multi-scale experiments (from cell to whole organism). An electrophysiological screening on cells overexpressing human NaV1.7 was used to uncover novel hits from Smartox Biotechnology's unique and vast venom library. The current studies on some selected peptides aim (1) to chemically synthesize these peptides, (2) to characterize their selectivity profile and safety issue, particularly on cell lines overexpressing other NaV subtypes (from 1.1 to 1.8) and on native systems such as mouse cardiomyocytes and neuromuscular system, (3) to specify their efficiency and mechanism of action on mouse DRG neurons, and (4) to evaluate in vivo their analgesic properties in vigil mice. (5) to define their pharmacophore and optimize their functional characteristics, and (6) to provide evidence for the clinical application of the hit candidates.


Romain Baudat

Evelyne Benoit

Mathilde Keck

Pascal Kessler

Denis Servent

Denis Servent Venom

Venom peptides: emerging biological tools for lung cancer imaging

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

Venom peptides: emerging biological tools for lung cancer imaging Certain ionic channels are overexpressed in lung cancer cell lines. These channels can be targetted with high affinity and selectivity by some venom toxins that are easily tagged with fluorescent or radioactive group. This study will participate to the development of new lung cancer biomarkers

Romain BAUDAT (1) , Evelyne BENOIT (1) , Mathilde KECK (1) , Pascal KESSLER (1) , Charles TRUILLET (2) , Denis SERVENT (1) Overexpression of nAChR α7 and Na V 1.7 in cancerous cells and nAChR α9 only in A549 cells ChTx-TAMRA interacts with K Ca 3.1 channels. Noted that the fluorescence intensity is significantly higher in A549 cells and lower in H460 cells, compared to BEAS-2B cells.

RTq-PCR

Cell lines

HwTx-IV inhibited 44% of the peak Na V 1.7 current.

Exemplified by ChTx-TAMRA effect on cancerous cell lines

Exemplified by HwTx-IV effect on HEK-293 cell line stably overexpressing hNa V 1.7 Many cancers, and the non-small cell lung cancer (NSCLC) in particular, are associated with significant overexpression of various ion channels such as certain subtypes of sodium, calcium and potassium channels, pH-sensitive channels (ASICs for acid sensing ion channels) or nicotinic acetylcholine receptors. Activation of these ion channels plays an important role in cancer cell aggressiveness and tumor invasion, their overexpression seeming to be correlated with the appearance of metastases. Although imaging approaches (radiography, CT-scan for computed tomography scanning, PET-scan for positron emission tomography) are commonly used in the anatomical characterization of lung cancer, some of their limitations require the development of new imaging probes more specific of overexpressed molecular targets in lung adenocarcinomas. Peptide toxins, which have unique pharmacological properties, are well known to act selectively and with high affinity on certain overexpressed ion channels in cancer cells. Therefore, a promising solution towards the development of new lung cancer biomarkers appears to be the chemical synthesis of these toxins allowing them to be labelled by "click" chemistry using various fluorescent or radioactive probes.

44% of inhibition

A S I C N a V 1 . 7 K C a

  Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, Département Médicaments et Technologies pour la Santé (DMTS), Service d'Ingénierie Moléculaire pour la Santé (SIMoS), EMR CNRS/CEA 9004, Gif-sur-Yvette, France -(2) Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, Service Hospitalier Frédéric Joliot (SHFJ), Laboratoire d'Imagerie biomédicale multimodale Paris Saclay (BioMaps), Orsay, France

Biomarkers 1. Peptide synthesis 2. Functional characterization 3. Fluo/TEP imaging Solid-Phase Peptide Synthesis (SPPS)

Tagged α-IM1 conotoxin


Romain Baudat

Evelyne Benoit

Mathilde Keck

Pascal Kessler

Denis Servent

Denis Servent Venom

Venom peptides: emergent biological tools for lung cancer imaging

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

10:45 Oral communication session 2: "ion channels and cancer"

Organized by Claire Legendre (Angers, France) VENOM PEPTIDES: EMERGENT BIOLOGICAL TOOLS FOR LUNG CANCER IMAGING. Romain BAUDAT; Evelyne BENOIT; Mathilde KECK; Pascal KESSLER; Denis SERVENT. Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, Département Médicaments et Technologies pour la Santé (DMTS), Service d'Ingénierie Moléculaire pour la Santé (SIMoS) Many cancers, and the non-small cell lung cancer in particular, are associated with a disturbance of the mechanisms controlling ion homeostasis due, especially, to dysfunction and/or overexpression of many ion channels such as certain subtypes of sodium (NaV 1.7 ), potassium (KCa 3.1 ) channels, pH-sensitive channels (ASICs for acid sensing ion channels) or nicotinic acetylcholine receptors (nAChr α7 , nAChr α9/α10 ). In particular, the activation of these ion channels plays an important role in cancer cell aggressiveness and tumor invasion, their overexpression seeming to be correlated with the appearance of metastases. Although imaging approaches (radiography, CT-scan for computed tomography scanning, PET-scan or positron emission tomography) are commonly used in the anatomical characterization of lung cancer, some of their limitations require the development of new imaging probes more specific of overexpressed molecular targets in lung adenocarcinomas. Peptide toxins, which have unique pharmacological properties, are well known to act selectively and with high affinity on certain overexpressed ion channels in cancer cells. Therefore, a promising solution towards the development of new lung cancer biomarkers appears to be the chemical synthesis of these toxins allowing them to be marked by "click" chemistry using various fluorescent or radioactive probes.


Romain Baudat

Evelyne Benoit

Mathilde Keck

Pascal Kessler

Charles Truillet

Denis Servent

Venom peptides: promising tools for lung cancer imaging

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

Beas

Venom peptides: promising tools for lung cancer imaging

Certain ionic channels are overexpressed in lung cancer cell lines. These channels can be targetted with high affinity and selectivity by some venom toxins that are easily tagged with fluorescent or radioactive group. The results show that the Na V 1.7 and HwTx-IV-Cyanine 5 couple is promising to detect tumors, especially invasive ones. To complete these results, TEP imaging will be done on xenografted mice.

This study will participate to the development of new lung cancer biomarkers

Romain BAUDAT (1) , Evelyne BENOIT (1) , Mathilde KECK (1) , Pascal KESSLER (1) , Charles TRUILLET (2) , Denis SERVENT (1) Overexpression of nAChR α7 and Na V 1. Many cancers, and the non-small cell lung cancer (NSCLC) in particular, are associated with significant overexpression of various ion channels such as certain subtypes of sodium, calcium and potassium channels, pH-sensitive channels (ASICs for acid sensing ion channels) or nicotinic acetylcholine receptors. Activation of these ion channels plays an important role in the aggressiveness of cancer cell and tumor invasion, their overexpression seeming to be correlated with the appearance of metastases. Although imaging approaches (radiography, CT-scan for computed tomography scanning, PET-scan for positron emission tomography) are commonly used in the anatomical characterization of lung cancer, some limitations of these approaches require the development of new imaging probes more specific of overexpressed molecular targets in lung adenocarcinomas. Peptide toxins, which possess unique pharmacological properties, are well known to act selectively and with high affinity on certain overexpressed ion channels in cancer cells. Therefore, a promising solution towards the development of new imaging agents oflung cancer biomarkers appears to be the chemical synthesis of these toxins allowing them to be labelled by "click" chemistry using various fluorescent or radioactive probes.

44% of inhibition

A S I C N a V 1 . 7 K C a 3 . 1 n A C h R α 9 n A C h
- B.

E. C.

nM HwTx -Cyanine 5 Concanavaline A (Membrane)

Hoechst (Nucleus)

Beas-2B A549 H460

A.

A. Maximum gray intensity at the membrane of ten cells for each cell line, representing the intensity of HwTx-IV-Cyanine 5 fixed on Na V 1.7

No detectable sodium current in Beas-2B and A549 cells  Weak sodium current in H460 (unexploitable for HwTx-IV studies) → Use of HEK-293 cell line stably overexpressing hNa V 1.7 → Experiments with HwTx-IV-Cyanine 5 to be done HwTx-IV-Cyanine 5 interacts with Na V 1.7 channels. Noted that the fluorescence intensity is significantly higher in H460 cells as compared to A549 and Beas-2B cells

D.

Supported by a « FOCUS Biomarqueurs » grant

  -IV-Cyanine 5 effect on lung cell lines Exemplified by HwTx-IV effect on HEK-293 cell line stably overexpressing hNa V 1

  Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, Département Médicaments et Technologies pour la Santé (DMTS), Service d'Ingénierie Moléculaire pour la Santé (SIMoS), EMR CNRS/CEA 9004, Gif-sur-Yvette, France -(2) Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, Service Hospitalier Frédéric Joliot (SHFJ), Laboratoire d'Imagerie biomédicale multimodale Paris Saclay (BioMaps), Orsay, France Beas-2B Non cancerous cell line A549 Cancerous cell line H460 Cancerous and strongly metastatic cell line

  Global potassium and sodium voltage-dependant currents recorded in lung cell lines. B. Representative current traces of sodium current recorded from HEK overexpressing hNa V 1.7 without (black) and with (red) HwTx-IV (9 nM) C. Representative current traces of sodium current recorded from HEK overexpressing hNa V 1.7 as a function of membrane potential. D. Kinetics of HwTx-IV effects on the peak sodium current. E. Sodium current-voltage curve without (black) and with (red) HwTx-IV (9 nM) A.


domain: sdv sdv

text:
Sophie Comtet-Marre

Pierre Peyret
email: pierre.peyret@uca.fr

Keywords: Preterm birth, Vaginal microbiome, Predictive diagnosis, Deep neural network, Artificial intelligence, Machine learning, Pregnancy, Microbial signature, Clinical data, Phenotype prediction, Model explainability

In recent decades, preterm birth (PTB) has become a significant research focus in the healthcare field, as it is a leading cause of neonatal mortality worldwide. Using five independent study cohorts including 1290 vaginal samples from 561 pregnant women who delivered at term (n = 1029) or prematurely (n = 261), we analysed vaginal metagenomics data for precise microbiome structure characterization. Then, a deep neural network (DNN) was trained to predict term birth (TB) and PTB with an accuracy of 84.10% and an area under the receiver operating characteristic curve (AUROC) of 0.875 ± 0.11. During a benchmarking process, we demonstrated that our DL model outperformed seven currently used machine learning algorithms. Finally, our results indicate that overall diversity of the vaginal microbiota should be taken in account to predict PTB and not specific species. This artificial-intelligence based strategy should be highly helpful for clinicians in predicting preterm birth risk, allowing personalized assistance to address various health issues.

DeepMPTB: a vaginal microbiome-based deep neural network as artificial intelligence strategy for efficient preterm birth prediction Oshma Chakoory 1 , Vincent Barra 2 , Emmanuelle Rochette 3 , Loïc Blanchon 4 , Vincent Sapin 4,5 , Etienne Merlin 3 , Maguelonne Pons 3 , Denis Gallot 4,6 , Sophie Comtet-Marre 1* and Pierre Peyret 1*

To the editor

Preterm birth (PTB) is a leading cause of neonatal mortality worldwide and the second most common cause of child deaths under the age of five years [START_REF] Liu | Global, regional, and national causes of child mortality in 2000-13, with projections to inform post-2015 priorities: an updated systematic analysis[END_REF]. Additionally, premature neonates are at risk of numerous health complications, including neurological damage in early childhood but also respiratory and gastrointestinal disorders. Existing diagnostic methods involve the collection of maternal obstetric history and cervical measurements via transvaginal ultrasound imaging conducted in the first and second trimesters of pregnancy. However, diagnoses are often inaccurate, as physician experience varies and the processes can be time-consuming.

Existing literature suggests that vaginal microbial communities could be involved in the pathophysiology of PTB delivery [START_REF] Menon | Research to achieve a reduction in the global rate of preterm birth needs attention: Preface to the special issue by the preterm Birth International Collaborative (PREBIC)[END_REF]. This microbiome is extremely important to the host tissue, as it maintains an acidic environment, inhibits the growth of pathogenic bacteria, and modulates inflammation by cross-kingdom signalling. Despite the efforts of longitudinal studies and metaanalyses, no clear distinct microbial signatures have been characterized to identify the risk of PTB [START_REF] Feehily | Shotgun sequencing of the vaginal microbiome reveals both a species and functional potential signature of preterm birth[END_REF]. We propose DeepMPTB, a vaginal microbiota-based deep neural network (DNN) for efficient PTB prediction (Fig. 1; Supplementary Material).

A total of 234 786 trainable parameters were optimized and the optimal hyperparameter combination for the final model (Fig. 2A, Fig. S2), included 416 units (neurons) in the 1st hidden layer and a total of 3 hidden layers, with the number of units in each layer set to half that in the preceding layer (Fig. S3). To deal with class imbalance (1029 TB and 261 PTB) in our datasets, we evaluated model performance using multiple metrics (Supplementary Material). The 20 most important features contributing to these results were also determined by the SHAP explainer (Fig. 2B). Interestingly, low-abundance species were also observed to contribute to PTB classification. Moreover, these contributing features included clinical and demographic data.

The performance of DeepMPTB was compared with that of seven state-of-the-art classification algorithms, namely, the decision tree (DT), K-nearest neighbour (KNN), random forest (RF), naïve Bayes (NB), extreme gradient boosting (XGBoost), logistic regression (LR), S1) [START_REF] Feehily | Shotgun sequencing of the vaginal microbiome reveals both a species and functional potential signature of preterm birth[END_REF][START_REF] Fettweis | The vaginal microbiome and preterm birth[END_REF][START_REF] Goltsman | Metagenomic analysis with strain-level resolution reveals fine-scale variation in the human pregnancy microbiome[END_REF][START_REF] Pace | Complex species and strain ecology of the vaginal microbiome from pregnancy to postpartum and association with preterm birth[END_REF][START_REF] Tortelli | The structure and diversity of strain-level variation in vaginal bacteria[END_REF]. The RiboTaxa pipeline [START_REF] Chakoory | RiboTaxa: combined approaches for rRNA genes taxonomic resolution down to the species level from metagenomics data revealing novelties[END_REF] was used to obtain taxonomic profiles from the metagenomics datasets using the SILVA SSU 138.1 NR99 database. Vaginal microbiota profiles differed greatly (Welch's t-test, p < 0.05) within individual cohorts, illustrating the heterogeneity of the vaginal population. No significant difference in the α-diversity measure was found between the TB or PTB groups. All the output taxonomy tables were grouped into a single table containing all the bacterial and eukaryotic specieslevel profiles of 1290 samples. In addition, the clinical data of each sample were considered. The normalized species abundances (Fig. S1) and vectorized clinical data were used to train and optimize the neural network. Features contributing to explaining the model were extracted and visualized using SHAP. To predict the phenotype based on new unknown vaginal microbiota samples (step 2), a list of features with important biomarkers contributing to the prediction was output and support vector machine (SVM) models, which were trained and optimized based on the same input data as the DNN (Supplementary Material). Deep-MPTB outperformed all other prediction models, with an AUROC score of 0.877 ± 0.11 (p < 0.05 for ANOVA test) and an accuracy of 84.10% (Table S2).

The model trained based on the third trimester data displayed the highest accuracy of 88%, suggesting that samples collected during the third trimester may lead to better prediction rates, although the models trained based on first and second trimester data obtained also very good accuracies of 71% and 83%, respectively (Fig. S4). We also argue that input data quality has a significant impact on model performance (Fig. S5). We compared performance of three DNN trained using species coupled to their relative abundances determined with RiboTaxa [START_REF] Chakoory | RiboTaxa: combined approaches for rRNA genes taxonomic resolution down to the species level from metagenomics data revealing novelties[END_REF] or two other popular metagenomics classifiers (p < 0.05 for ANOVA test) using only the biggest cohort (Supplementary Material). The DNN trained based on input data from RiboTaxa [START_REF] Chakoory | RiboTaxa: combined approaches for rRNA genes taxonomic resolution down to the species level from metagenomics data revealing novelties[END_REF] showed the best performance, with an AUROC score of 0.898 ± 0.09. The DNN trained based on DeepMicrobes [START_REF] Liang | DeepMicrobes: taxonomic classification for metagenomics with deep learning[END_REF] and MetaPhlAn3 [START_REF] Beghini | Integrating taxonomic, functional, and strain-level profiling of diverse microbial communities with bioBakery 3[END_REF] data showed an AUROC score of 0.838 ± 0.14 and 0.795 ± 0.08, respectively. When only microbiome data obtained with RiboTaxa (without metadata, keeping phenotype) were used for model training, the AUROC value decreased to 0.831 ± 0.12 (p > 0.05 for Mann-Whitney U test).

To show the generality of this model, we used a completely new set of 694 vaginal metagenomic data (430 TB and 264 PTB cases) from Baud et al. [START_REF] Baud | Microbial diversity in the vaginal microbiota and its link to pregnancy outcomes[END_REF]. Overall, the optimized DNN successfully identified 80% of TB samples and 66% of the PTB samples. Importantly, phenotype prediction, especially in the case of PTB, is not determined by the presence of the same species or group of species (Fig. S6).

In conclusion, the present study presents a cuttingedge deep learning model to efficiently predict TB and PTB using vaginal microbiome data of pregnant women combined to clinical data. This new model based on data from 5 cohorts outperforms previously published machine learning-based model for PTB prediction [START_REF] Baud | Microbial diversity in the vaginal microbiota and its link to pregnancy outcomes[END_REF][START_REF] Park | Predicting preterm birth through vaginal microbiota, cervical length, and WBC using a machine learning model[END_REF]. Continued accumulation of high-quality microbiome data and complete phenotypic data in perfectly controlled cohorts will certainly improve the individual phenotype prediction performance of deep learning models. Furthermore, including virome information, known to drive microbiota dynamics, would help to reach better performances. Finally, DNN enables to distinguish complex interindividual microbial interactions related to term and preterm deliveries, to highlight in-depth microorganisms potentially associated to phenotype. Interestingly we observed that different microbial profiles led to the same phenotype. This efficient TB and PTB predictive diagnosis should be highly helpful for clinicians in a personalized medicine context.

Fig. 1

 1 Fig. 1 Overview of model training and phenotype prediction. For model training (step 1), the shotgun metagenomics sequences of 1290 vaginal samples from 561 pregnant women were retrieved from public databases in the form of fastq files (TableS1)[START_REF] Feehily | Shotgun sequencing of the vaginal microbiome reveals both a species and functional potential signature of preterm birth[END_REF][START_REF] Fettweis | The vaginal microbiome and preterm birth[END_REF][START_REF] Goltsman | Metagenomic analysis with strain-level resolution reveals fine-scale variation in the human pregnancy microbiome[END_REF][START_REF] Pace | Complex species and strain ecology of the vaginal microbiome from pregnancy to postpartum and association with preterm birth[END_REF][START_REF] Tortelli | The structure and diversity of strain-level variation in vaginal bacteria[END_REF]. The RiboTaxa pipeline[START_REF] Chakoory | RiboTaxa: combined approaches for rRNA genes taxonomic resolution down to the species level from metagenomics data revealing novelties[END_REF] was used to obtain taxonomic profiles from the metagenomics datasets using the SILVA SSU 138.1 NR99 database. Vaginal microbiota profiles differed greatly (Welch's t-test, p < 0.05) within individual cohorts, illustrating the heterogeneity of the vaginal population. No significant difference in the α-diversity measure was found between the TB or PTB groups. All the output taxonomy tables were grouped into a single table containing all the bacterial and eukaryotic specieslevel profiles of 1290 samples. In addition, the clinical data of each sample were considered. The normalized species abundances (Fig.S1) and vectorized clinical data were used to train and optimize the neural network. Features contributing to explaining the model were extracted and visualized using SHAP. To predict the phenotype based on new unknown vaginal microbiota samples (step 2), a list of features with important biomarkers contributing to the prediction was output

Fig. 2

 2 Fig. 2 Performance of DeepMPTB based on the 20% test set (239 samples). A During model evaluation, the three metadata features (age, ethnicity and trimester of sample collection) were associated with each sample. For each sample, the true and predictive phenotypes were compared to evaluate the performance of DeepMPTB. B A summary plot for the SHAP values was generated to understand the contributions of the first 20 features in this performance analysis. Features related to clinical/demographic metadata are indicated by an asterisk. Each dot represents a sample. Negative and positive SHAP values are associated to TB and PTB prediction, respectively. Low and high SHAP values are shown in blue and red, respectively. TB: Term birth; PTB: Preterm birth

Acknowledgements

We are grateful to the Mésocentre Clermont Auvergne University and AuBi platform for providing help and computing and storage resources. Computations were performed on the supercomputer facilities of the Mésocentre Clermont Auvergne University.

Abbreviations

Supplementary Information

The online version contains supplementary material available at https://doi. org/

Data availability

All raw sequencing data and metadata analysed during this study were obtained from these published articles: Feehily et al. [START_REF] Feehily | Shotgun sequencing of the vaginal microbiome reveals both a species and functional potential signature of preterm birth[END_REF]. under the BioProject PRJEB34536 (61.49 Gb), Tortelli et al. [START_REF] Tortelli | The structure and diversity of strain-level variation in vaginal bacteria[END_REF] under the BioProject PRJNA639592 (8.52 Gb), and Goltsman et al. [START_REF] Goltsman | Metagenomic analysis with strain-level resolution reveals fine-scale variation in the human pregnancy microbiome[END_REF]. under the BioProject PRJNA288562 (115.53 Gb). Raw data (2.77 Tb) and metadata for Fettweis et al. [START_REF] Fettweis | The vaginal microbiome and preterm birth[END_REF]. cohort were received from the authors of the study following data access approval from National Institute of Health (NIH). Raw sequencing for the Pace et al. [START_REF] Pace | Complex species and strain ecology of the vaginal microbiome from pregnancy to postpartum and association with preterm birth[END_REF] cohort were available under the BioProject PRJNA451212 (15.92 Gb) and the metadata were received from the authors of the study. Supplementary material describes methods and complementary results.

Declarations Competing interests

The authors declare that they have no conflicts of interests.

Consent for publication

Not applicable.

Ethics approval and consent to participate

Not applicable.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


est destinée au dépôt

Cyclic imine toxins (CiTXs) are a family of worldwide-distributed marine macrocyclic phycotoxins comprising different groups of lipophilic compounds: gymnodimines (GYMs), spirolides, pinnatoxins (PnTXs), pteriatoxins (PteTXs), portimines, prorocentrolides and spiroprorocentrimine. CiTXs and the acyl derivative products of shellfish metabolism contaminate and bioaccumulate in filter-feeding bivalves, edible mollusks and other marine invertebrates and thus, represent a risk for shellfish consumers and a menace for wildlife conservation. All CiTXs studied interact and block, with different affinities, embryonic (α1 2 βγδ) and mature (α1 2 βεδ) nicotinic acetylcholine receptors (nAChRs) expressed in skeletal muscles. 3,4-Diaminopyridine (3,4-DAP) is a well-known K + -channel blocker that prolongs the presynaptic action potential and indirectly enhances the entry of Ca 2+ into nerve terminals and, thereby, increases quantal ACh release at the neuromuscular junction. 3,4-DAP is used effectively to treat muscular weakness in various human myasthenic syndromes. The aim of this work was to determine whether 3,4-DAP, which increases quantal ACh release, is able to reverse the action of CiTXs on skeletal muscle expressing α1 2 βεδ nAChRs. The use of isolated mouse phrenic-nerve-hemi-diaphragm muscle preparations enabled monitoring nerve-evoked isometric muscle contraction in the presence of GYM-A, PnTX-A, PnTX-G and PteTX-A and, afterwards, the action of 3,4-DAP (10-100 µM) in the continuous presence of the respective CiTXs. The results show that 3,4-DAP improves neuromuscular function in a time-and concentration-dependent manner and, at the highest concentration used (100 µM), completely reverses the neuromuscular block caused by the CiTX studied. These results support the view that GYM-A, PnTX-A, PnTX-G and PteTX-A are competitive and reversible inhibitors of muscle-type α1 2 βεδ nAChRs. In fact, CiTXs can be typified as competing with the binding of ACh (the natural ligand) to the nAChR. The ability of 3,4-DAP to antagonize the action of CiTXs is related to the increased release of ACh that is able to displace the respective CiTX from their nAChR binding sites.

Keywords: cyclic imine toxin, 3,4-diaminopyridine, K + -channel block, muscle contraction, neuromuscular junction, nicotinic acetylcholine receptor, nerve terminal.

3, 4 -

 4 Diaminopyridine antagonizes the block of muscle nicotinic acetylcholine receptor caused by cyclic imine toxins at isolated mouse diaphragm preparations Nicolas Lamassiaude, Évelyne Benoit, Rómulo Aráoz, Armen Zakarian, Denis Servent, Jordi Molgó To cite this version: Nicolas Lamassiaude, Évelyne Benoit, Rómulo Aráoz, Armen Zakarian, Denis Servent, et al.. 3,4-Diaminopyridine antagonizes the block of muscle nicotinic acetylcholine receptor caused by cyclic imine toxins at isolated mouse diaphragm preparations. RT29 -29th Meeting of the French Society of Toxinology -Toxins: From the Wild to the Lab, Nov 2023, Paris, France. hal-04474247

-Diaminopyridine antagonizes the block of muscle nicotinic acetylcholine receptor caused by cyclic imine toxins at isolated mouse diaphragm preparations

	Toxines: De la nature au labo -Toxins: From the wild to the lab 29 èmes Rencontres en Toxinologie -29 th Meeting on Toxinology -RT29	58
	Institut Pasteur, Paris -30 novembre -01 décembre 2023 -November 30 -December 01, 2023
	Session "Toxin functional exploration"
		Poster
	3,4Nicolas LAMASSIAUDE 1 , Évelyne BENOIT 1 , Rómulo ARAOZ 1 , Armen ZAKARIAN 2 ,
	Denis SERVENT 1 , Jordi MOLGÓ 1,*
	1 Université Paris-Saclay, CEA, Département Médicaments et Technologies pour la Santé (DMTS), Service
	d'Ingénierie Moléculaire pour la Santé (SIMoS), Equipe Mixte de Recherche CNRS/CEA (EMR 9004), 91191 Gif-sur-
	Yvette, France. 2 University of California Santa Barbara, Department of Chemistry and Biochemistry, Santa Barbara,
	CA 93106, USA.

*Corresponding author, e-mail: jordi.molgo@cea.fr


Romain Baudat

Evelyne Benoit

Pascal Kessler

Denis Servent

Venom peptides: emerging biological tools for lung cancer diagnosis

Beas

Venom peptides: promising tools for lung cancer diagnosis

Certain ionic channels are overexpressed in lung cancer cell lines. These channels can be targetted with high affinity and selectivity by some venom toxins that are easily tagged with fluorescent or radioactive group. The results show that the Na V 1.7 and HwTx-IV-Cyanine 5 couple is promising to detect tumors, especially invasive ones. To complete these results, TEP imaging will be done on xenografted mice.

This study will participate to the development of new lung cancer biomarkers

Romain BAUDAT (1) , Evelyne BENOIT (1) , Pascal KESSLER (1) , Denis SERVENT (1) Overexpression of nAChR α7 and Na V 1. Many cancers, and the non-small cell lung cancer (NSCLC) in particular, are associated with significant overexpression of various ion channels such as certain subtypes of sodium, calcium and potassium channels, pH-sensitive channels (ASICs for acid sensing ion channels) or nicotinic acetylcholine receptors. Activation of these ion channels plays an important role in the aggressiveness of cancer cell and tumor invasion, their overexpression seeming to be correlated with the appearance of metastases. Although imaging approaches (radiography, CT-scan for computed tomography scanning, PET-scan for positron emission tomography) are commonly used in the anatomical characterization of lung cancer, some limitations of these approaches require the development of new imaging probes more specific of overexpressed molecular targets in lung adenocarcinomas. Peptide toxins, which possess unique pharmacological properties, are well known to act selectively and with high affinity on certain overexpressed ion channels in cancer cells. Therefore, a promising solution towards the development of new imaging agents of lung cancer biomarkers appears to be the chemical synthesis of these toxins allowing them to be labelled by "click" chemistry using various fluorescent or radioactive probes.

Fluorescence imaging

  Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, Département Médicaments et Technologies pour la Santé (DMTS), Service d'Ingénierie Moléculaire pour la Santé (SIMoS), EMR CNRS/CEA 9004, Gif-sur-Yvette, France Beas-2B Non cancerous cell line A549 Cancerous cell line H460 Cancerous and strongly metastatic cell line

  current in Beas-2B and A549 cells  Weak sodium current in H460 (unexploitable for HwTx-IV studies) → Use of HEK-293 cell line stably overexpressing hNa V 1.7 → Experiments with HwTx-IV-Cyanine 5 HwTx-IV-Cy5 interacts with Na V 1.7 channels. Noted that the fluorescence intensity is significantly higher in cancerous than Beas-2B cells. Excess of unlabbeled toxins markedly decrease the signal indicating the specificity of interactions.