On a toy model of interacting neurons

We continue the study of a stochastic system of interacting neurons introduced in De Masi, Galves, Locherbach and Presutti (J. Stat. Phys. 158 (2015) 866-902). 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 average potential. We prove propagation of chaos of the system, as N -> infinity, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of (J. Stat. Phys. 158 (2015) 866-902), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in 1/root N. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique nontrivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.