On the dynamics of random neuronal networks

Abstract

We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal networks. In contrast with the classical integrate-and-fire neuron, we take into account explicitly the intrinsic randomness of firing times. We analyze the behavior of this system for finite networks, and show that this jump process is well-posed and that no explosion occurs, contrasting with the integrate-and-fire model. This well-posedness persists in the thermodynamic limit, and we derive the McKean-Vlasov jump process governing the dynamics of the limit. Stationary distributions are investigated: we show that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.