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.
Highlights
We consider a mean-field model of spiking neurons
We assume that b(0) ≥ 0 and that X0 ≥ 0, such that the dynamics lives on R+
We study the existence of periodic solutions t → L(Xt) where (Xt) is the solution of (1.1), near a non-stable invariant measure να∞0
Summary
We consider a mean-field model of spiking neurons. Let f : R+ → R+, b : R+ → R and let N(du, dz) be a Poisson measure on R2+ with intensity the Lebesgue measure dudz. We assume that b(0) ≥ 0 and that X0 ≥ 0, such that the dynamics lives on R+ This SDE is non-linear in the sense of McKean-Vlasov, because of the interaction term E f (Xt) which depends on the law of Xt. Let ν(t, dx) := L(Xt) be the law of Xt. Let ν(t, dx) := L(Xt) be the law of Xt It solves the following non-linear Fokker-Planck equation, in the sense of measures:. For the limit equation the long time behavior is richer: for fixed values of the parameters there can be multiple invariant measures (see [5] and [6] for some explicit examples) and, as shown here, there can exist periodic solutions (see Figure 1)
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