Comparison principles and asymptotic behavior of delayed age-structured neuron models

TL;DR

This paper studies the long-term behavior of age-structured neuron models with delays, using comparison principles for Volterra equations to show convergence to equilibrium.

Contribution

It introduces a simpler method based on comparison principles to analyze the asymptotic behavior of delayed age-structured neuron models.

Findings

Proves convergence to equilibrium for models with discrete delays

Establishes convergence for models with distributed delays

Provides a new, straightforward analytical approach

Abstract

In the context of neuroscience the elapsed-time model is an age-structured equation that describes the behavior of interconnected spiking neurons through the time since the last discharge, with many interesting dynamics depending on the type of interactions between neurons. We investigate the asymptotic behavior of this equation in the case of both discrete and distributed delays that account for the time needed to transmit a nerve impulse from one neuron to the rest the ensemble. To prove the convergence to the equilibrium, we follow an approach based on comparison principles for Volterra equations involving the total activity, which provides a simpler and more straightforward alternative technique than those in the existing literature on the elapsed-time model.