On the dynamics of random neuronal networks

TL;DR

This paper analyzes the behavior of large, randomly connected neuronal networks using a mean-field approach, revealing how network activity transitions between quiescent and self-sustained states depending on connectivity.

Contribution

It introduces a stochastic model incorporating intrinsic firing randomness and derives the mean-field limit and stability properties of the network's stationary distributions.

Findings

Network activity exhibits phase transitions based on connectivity.

Existence of stable trivial and self-sustained activity states.

The model captures the impact of intrinsic neuronal randomness.

Abstract

We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated to finite networks are investigated. We derive the limit in distribution of the sample path of the state of a neuron of the network when its size gets large. The invariant distributions of this limiting stochastic process are analyzed as well as their stability properties. 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 in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.