Relaxation in Time Elapsed Neuron Network Models in the Weak Connectivity Regime

Abstract

In order to describe the firing activity of a homogenous assembly of neurons, we consider time elapsed models, which give mathematical descriptions of the probability density of neurons structured by the distribution of times elapsed since the last discharge. Under general assumption on the firing rate and the delay distribution, we prove the uniqueness of the steady state and its nonlinear exponential stability in the weak connectivity regime. In other words, total asynchronous firing of neurons appears asymptotically in large time. The result generalizes some similar results obtained in Pakdaman et al. (Nonlinearity 23(1):55–75, 2010) and Pakdaman et al. (SIAM J. Appl. Math. 73(3):1260–1279, 2013) in the case without delay. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators.