Probabilistic models for systems of interacting spiking neurons and some words about their mean field limits

E. Löcherbach

I will give an overview of recent results about mean field limits for systems of interacting point processes modeling spiking (biological) neurons. I will start with a short introduction to the functioning of neurons and the modeling of their spiking activity by stochastic integrate and fire models and then focus on a particular class of models which are stochastic intensity based processes. Then we will discuss mean field limits and propagation of chaos results for large homogeneous systems of neurons and see how the limit is described by a McKean-Vlasov type equation driven by Poisson random measure. A second part of the talk is devoted to the study of systems with random synaptic weights in a diffusive scaling. We will see how this setting leads to conditional propagation of chaos and how the convergence can be obtained by means of a new martingale problem. Finally, if time permits, I will also discuss the longtime behavior both of the finite and the limit system of neurons. 

The second part of the talk is based on joint work with Xavier Erny and Dasha Loukianova. 

Strong conditional propagation of chaos for systems of interacting particles with nearly stable jumps

E. Marini