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Hopf bifurcation in a Mean-Field model of spiking neurons

Quentin Cormier 1, 2 Etienne Tanré 2 Romain Veltz 3 
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
3 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
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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Submitted on : Wednesday, November 25, 2020 - 4:07:41 PM
Last modification on : Friday, July 8, 2022 - 10:10:02 AM


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



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