Asymptotic behaviour of neuron population models structured by elapsed-time

TL;DR

This paper analyzes two neuron population models based on elapsed time or fatigue, proving existence, steady states, and exponential convergence to equilibrium, with a novel use of probability theory for spectral gap analysis.

Contribution

It introduces a new approach using Doeblin's theorem to establish spectral gap properties in neuron models with structured variables.

Findings

Existence of solutions and steady states in measure spaces.

Exponential convergence to equilibrium in weak connectivity regimes.

Application of probability theory to spectral analysis in neural models.

Abstract

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In the first model, the structuring variable $s$ represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic "state". We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e., weak connectivity). The main innovation is the use of Doeblin's theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.