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Journal Articles Electronic Journal of Probability Year : 2021

Hopf bifurcation in a Mean-Field model of spiking neurons

Abstract

We study a family of non-linear McKean-Vlasov SDEs driven by a Poisson measure, modelling the mean-field asymptotic of a network of generalized Integrate-and-Fire neurons. We give sufficient conditions to have periodic solutions through a Hopf bifurcation. Our spectral conditions involve the location of the roots of an explicit holomorphic function. The proof relies on two main ingredients. First, we introduce a discrete time Markov Chain modeling the phases of the successive spikes of a neuron. The invariant measure of this Markov Chain is related to the shape of the periodic solutions. Secondly, we use the Lyapunov-Schmidt method to obtain self-consistent oscillations. We illustrate the result with a toy model for which all the spectral conditions can be analytically checked.
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Dates and versions

hal-02925025 , version 1 (28-08-2020)
hal-02925025 , version 2 (25-11-2020)

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Quentin Cormier, Etienne Tanré, Romain Veltz. Hopf bifurcation in a Mean-Field model of spiking neurons. Electronic Journal of Probability, 2021, 26, ⟨10.1214/21-EJP688⟩. ⟨hal-02925025v2⟩
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