A multiple time renewal equation for neural assemblies with elapsed time model

TL;DR

This paper extends the classical elapsed time neuron model to include the time since the penultimate discharge, resulting in a more complex system of equations, with convergence proofs and numerical simulations demonstrating diverse firing behaviors.

Contribution

It introduces a novel extended model incorporating the second last discharge time and proves convergence to stationarity under certain conditions.

Findings

Convergence to stationary state proven for weak non-linearities.

Numerical simulations show diverse firing rate behaviors.

Comparison with classical model highlights new dynamics.

Abstract

We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since the last discharge, i.e., the refractory period. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin's theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical simulations to observe how different firing rates can give different types of behaviors and to contrast them with theoretical results of both classical and extended models.