Some recent results on mean fields of networks of spiking neurons

Romain Veltz (Inria, Cronos)

Lundi 18 mars, 11h00, salle Coriolis (Galois).

Abstract. In this talk, I will present recent results regarding the dynamics of networks of stochastic spiking neurons. This network is an elaboration of the one introduced in [De Masi et al. 2014] by generalizing the dynamics of the individual neurons.

After a brief overview of neuronal biology, I will explain why it is important to study the dynamics of populations of interacting stochastic firing neurons and to make the link between the microscopic characteristics of the neurons and the macroscopic behavior of the population. To carry out this study, we study the limiting case of an infinite number of neurons under the hypothesis of a McKean-Vlasov type mean field approximation. This limiting model can be seen as a nonlocal transport equation.

The model presents some challenges. It is a nonlinear Piecewise Deterministic Markov process with unbounded total rate function. I will discuss some theoretical results regarding the linearized equation (well posedness, ergodicity) and highlight the difficulties associated to the nonlinear one.

A second part of the talk will be dedicated to the numerical study of these mean fields based on a Finite Volume method. I will provide a study of the (nonlinear) invariant distributions, their stability and the existence of periodic solutions.