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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