Exponential stability of the stationary distribution of a mean field of spiking neural network

Audric Drogoul 1 Romain Veltz 1
1 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in \cite{de_masi_hydrodynamic_2015,fournier_toy_2016}. We complement the convergence result proved in \cite{fournier_toy_2016} using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation.
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Audric Drogoul, Romain Veltz. Exponential stability of the stationary distribution of a mean field of spiking neural network. [Research Report] RR-8899, INRIA Sophia Antipolis - Méditerranée. 2018. ⟨hal-01290264v2⟩

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