On a toy model of interacting neurons

TL;DR

This paper extends the analysis of a stochastic interacting neuron model by removing initial condition restrictions, establishing convergence rates, and exploring the long-term behavior of the limiting nonlinear stochastic differential equation.

Contribution

It improves previous results by removing initial support constraints, providing a convergence rate of 1/√N, and analyzing the limit equation's invariant distributions and equilibrium behavior.

Findings

Propagation of chaos established as N→∞

Convergence rate of 1/√N proved

Existence and uniqueness of a non-trivial invariant distribution

Abstract

We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-L\"ocherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of De Masi-Galves-L\"ocherbach-Presutti (2014), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in $1/\sqrt N$. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique non-trivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.