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

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

“…For a reference on this method, see Perthame et al [30,36] and for some applications in the nonlinear elapsed time model see for instance [26,33]. Following these ideas, we can prove this property for the linear system (18). Proposition 1 (Generalized relative entropy).…”

“…Assume (2) and (3). Let n(t, s, a) be a solution of (18), then there exist t 0 > 0, α ∈ (0, 1) and a probability density ν ∈ L 1 such that n(t 0 , s, a) ≥ αν(s, a).…”

“…Proposition 1 (Generalized relative entropy). Assume there exists a steady state solution of the linear system (18) with n X , N X > 0. Then for all convex functions H : [0, ∞) → [0, ∞) with H(0) = 0, the solution n of the linear system (18)…”

“…From a mathematical and analytical point of view, the elapsed-time model has been studied by several authors using different techniques such as Cañizo et al in [6], Kang et al [26], Mischler et al in [31,32] and the pioneering works of Pakdaman et al in [33][34][35]. Moreover, different extensions of the elapsed time model have been considered by incorporating new elements such as spatial dependence with connectivity kernel in Salort et al [45], a leaky memory variable in Fonte et al [18] and a stochastic extension that accounts for finite population size in Schmutz et al [42] and Schwalger et al [44].…”

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

“…The relation between integrate-and-fire and the elapsed time model was studied in Dumont et al [9,10]. Different extensions of the elapsed time model have been considered by incorporating new variables such as spatial dependence and a connectivity kernel in Salort et al [29] or a leaky memory variable in Fonte et al [11]. The aim of the present work is to extend the classical elapsed time model by taking into account the elapsed time since the penultimate discharge in addition to the last one.…”