Asymptotic behaviour of neuron population models structured by elapsed-time

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman et al (2010 Nonlinearity 23 55–75) and Pakdaman et al (2014 J. Math. Neurosci. 4 1–26). In the first model, the structuring variable s represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic 'state'. We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e. weak connectivity). The main innovation is the use of Doeblin's theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.MTM2014-52056-P, MTM2017-85067-P, "la Caixa" Foundatio

Impact of demography on extinction/fixation events

International audienceIn this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright–Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov’s tranform