The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states

TL;DR

This paper analyzes the mean-field equation for a leaky integrate-and-fire neural network, proving well-posedness, existence, and stability of steady states in different connectivity regimes using measure solutions and contraction methods.

Contribution

It establishes the global well-posedness and stability of the mean-field PDE for neural networks with moderate and weak connectivity, extending previous theoretical results.

Findings

Global well-posedness in measure space for moderate coupling

Existence of stationary solutions in the mean-field model

Uniqueness and exponential stability of steady states in weak connectivity

Abstract

Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical physics, the so-called mean-field theory. In this document, we investigate different issues related to the mean-field description of an excitatory network made up of leaky integrate-and-fire neurons. The description is written in the form a nonlinear partial differential equation which is known to blow up in finite time when the network is strongly connected. We prove that in a moderate coupling regime the equation is globally well-posed in the space of measures, and that there exist stationary solutions. In the case of weak connectivity we also demonstrate the uniqueness of the steady state and its global exponential stability. The method to show those mathematical results relies on a contraction argument of Doeblin's type in the linear case, which corresponds to a population of non-interacting units.