Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation

Abstract

Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge.In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two “extreme” cases.Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20.

A Microscopic Spiking Neuronal Network for the Age-Structured Model

“…If the initial datum has a fast decay (in the sense described in Theorem 4), and if hypotheses (10) and (13) hold, then the previous convergence results remain true.…”

“…The aim of the present work is to give a new microscopic point of view of the age structured equation considered in Pakdaman-Perthame-Salort [11][12][13]. Therein, the model is proposed as a reinterpretation of the well known renewal equation and the microscopic derivation is omitted.…”

Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

“…The exponential decay of the L 1 norm was obtained by analytical methods (functional inequalities) in [38,27,35] and probabilistic methods (coupling arguments) in [5]. However the convergence is controlled by a distance between the initial distribution and the asymptotic profile which is stronger than the L 1 norm.…”

Asymptotic behaviour of neuron population models structured by elapsed-time

“…The second model has many similarities with a class of partial differential equations called growth-fragmentation equations. The nonlinear version we study here was introduced in Pakdaman et al (2014), but on this general type of equations we also mention the works in Doumic-Jauffret and Gabriel (2010); Michel (2006); Perthame and Ryzhik (2005); Calvez et al (2010); Engler et al (2006); Farkas and Hagen (2007); Gabriel (2012); Laurençot and Walker (2007); Simonett and Walker (2006).…”