A multiple time renewal equation for neural assemblies with elapsed time model. (29th August 2022)

• Record Type: Journal Article
• Authors: Torres, Nicolás; Perthame, Benoît; Salort, Delphine
• Is Part Of: Nonlinearity. Volume 35:Number 10(2022)
• Page Start: 5051
• Page End: 5075
• Publication Date: 2022-08-29
• Subjects: structured equations -- renewal equation -- mathematical neuroscience -- neural networks -- Doeblin theory; 35B40 -- 35F20 -- 35R09 -- 92B20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac8714
• Languages: English
• ISSNs: 0951-7715

Abstract

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 last discharge. In this extension, we incorporate the elapsed time since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system, we prove convergence with exponential rate to stationary state by means of Doeblin's theory in the case of weak non-linearities using 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 the classical and extended models.