Persistence in a large network of locally interacting neurons

TL;DR

This paper introduces a stochastic neural network model with local interactions, demonstrating a phase transition between activity extinction and persistence as the network size grows infinitely large.

Contribution

It presents a novel meanfield limit for a neural network with local interactions and proves the existence of a phase transition in neural activity.

Findings

The network converges to a jump-type stochastic differential equation in the infinite limit.

A phase transition exists where activity either dies out or persists forever.

The model explains how persistent activity can emerge from weak inputs in large networks.

Abstract

This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of local interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear meanfield process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks.