Long Time Behavior of an Age- and Leaky Memory-Structured Neuronal Population Equation

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, such as spike-frequency adaptation or short-term synaptic depression. Even though two-dimensional models are known to have emergent behaviors, such as 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's ergodic theorem and a perturbation argument, both adapted to the case of a multidimensional equation with delays.

Convergence towards equilibrium for a model with partial diffusion

“…For this reason, we propose here an entirely different approach based on an adaptation of the so-called Doeblin-Harris method. The latter has proved very performant in a number of contexts recently, as a growing number of papers have focused on the use of this method for many equations from neuroscience or physics [4,3,10,16,17,27,19]. Following these ideas, we eventually prove exponential convergence to the stationary state for Equation (1.1).…”

“…The study of partial differential equations (PDE) associated to neurosciences has seen a growing interest in recent years. These models often involve a mixture of effects leading to singular terms, heterogeneities, nonlocal terms, jumps, partial diffusion mechanisms and long term memory effects, which overall make their qualitative study somewhat challenging (see for instance [25,17,2,24,16,5,13,18,22]). In this paper, we propose to study the following toy model, which is directly derived from a model in [21].…”