Long time behavior of an age and leaky memory-structured neuronal population equation

TL;DR

This paper analyzes the long-term stability of a two-dimensional mean-field neuronal model that includes a leaky memory variable, showing it behaves like simpler models in weak connectivity regimes and establishing stability using ergodic theory.

Contribution

It extends the analysis of population equations by incorporating a leaky memory variable and proves asymptotic stability in the weak connectivity regime.

Findings

Two-dimensional models relax to a unique stationary state in weak connectivity.

The stability proof uses Harris' ergodic theorem and perturbation methods.

Emergent behaviors like population bursts are not observed in this regime.

Abstract

We study the asymptotic stability of a two-dimensional mean-field equation, which takes the form of a nonlocal transport equation and generalizes the time-elapsed neuron network model by the inclusion of a leaky memory variable. This additional variable can represent a slow fatigue mechanism, like spike frequency adaptation or short-term synaptic depression. Even though two-dimensional models are known to have emergent behaviors, like population bursts, which are not observed in standard one-dimensional models, we show that in the weak connectivity regime, two-dimensional models behave like one-dimensional models, i.e. they relax to a unique stationary state. The proof is based on an application of Harris' ergodic theorem and a perturbation argument, adapted to the case of a multidimensional equation with delays.