Hopf bifurcation in a Mean-Field model of spiking neurons

TL;DR

This paper analyzes a mean-field model of spiking neurons using non-linear stochastic differential equations, identifying conditions for periodic solutions via Hopf bifurcation and spectral analysis.

Contribution

It introduces a novel spectral condition framework for detecting Hopf bifurcations in neuron models driven by Poisson measures.

Findings

Spectral conditions for periodic solutions are explicitly characterized.

A Markov Chain model links spike phases to periodic solution shapes.

Analytical checks are demonstrated on a toy model.

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.