Jonathan Touboul

Topological and dynamical complexity of random neural networks.

Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown and, similar to spin glasses, shall be fundamentally related to the behavior of the system. In this Letter, we investigate the explosion of complexity arising near that phase transition. We show that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system. Near criticality, we compute the exponential rate of divergence, called topological complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov exponent, a classical measure of dynamical complexity. This relationship unravels a microscopic mechanism leading to chaos which we further demonstrate on a simpler disordered system, suggesting a deep and underexplored link between topological and dynamical complexity.

Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions

The cortex is composed of large-scale cell assemblies sharing the same individual properties and receiving the same input, in charge of certain functions, and subject to noise. Such assemblies are characterized by specific space locations and space-dependent delayed interactions. The mean-field equations for such systems were rigorously derived in a recent paper for general models, under mild assumptions on the network, using probabilistic methods. We summarize and investigate general implications of this result. We then address the dynamics of these stochastic neural field equations in the case of firing-rate neurons. This is a unique case where the very complex stochastic mean-field equations exactly reduce to a set of delayed differential or integro-differential equations on the two first moments of the solutions, this reduction being possible due to the Gaussian nature of the solutions. The obtained equations differ from more customary approaches in that it incorporates intrinsic noise levels nonlinearly and make explicit the interaction between the mean activity and its correlations. We analyze the dynamics of these equations, with a particular focus on the influence of noise levels on shaping the collective response of neural assemblies and brain states. Cascades of Hopf bifurcations are observed as a function of noise amplitude, for noise levels small enough, and delays, in a finite-population system. The presence of spatially homogeneous solutions in law is discussed in different non-delayed neural fields and an instability, as noise amplitude is varied, of the homogeneous state, is found. In these regimes, very complex irregular and structured spatio-temporal patterns of activity are exhibited including in particular wave or bump splitting.

On a Kinetic Fitzhugh–Nagumo Model of Neuronal Network

We investigate existence and uniqueness of solutions of a McKean–Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh–Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we show exponential nonlinear stability in the small connectivity regime.

Heterogeneous Connections Induce Oscillations in Large-Scale Networks

Realistic large-scale networks display a heterogeneous distribution of connectivity weights that might also randomly vary in time. We show that, depending on the level of heterogeneity in the connectivity coefficients, different qualitative macroscopic and microscopic regimes emerge. We evidence, in particular, generic transitions from stationary to perfectly periodic phase-locked regimes as the disorder parameter is increased, both in a simple model treated analytically and in a biologically relevant network made of excitable cells.

Index distribution of the Ginibre ensemble

Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. In this Letter, we compute analytically the probability distribution of the number of eigenvalues

Local homeoprotein diffusion can stabilize boundaries generated by graded positional cues

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Large Deviations, Dynamics and Phase Transitions in Large Stochastic and Disordered Neural Networks

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Pinwheel-dipole configuration in cat early visual cortex.

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Spatially Extended Networks with Singular Multi-scale Connectivity Patterns

(the index) of a large

Macroscopic Equations Governing Noisy Spiking Neuronal Populations with Linear Synapses

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