Concentration phenomena in Fitzhugh-Nagumo's equations: A mesoscopic approach

Abstract

We consider a spatially extended mesoscopic FitzHugh-Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of Dirac distribution concentrated to an averaged membrane potential.Our approach is based on techniques widely developed in kinetic theory (Wasserstein distance, relative entropy method), where macroscopic quantities of the mesoscopic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between microscopic and reaction-diffusion models.

Global derivation of a Boussinesq-Navier-Stokes type system from fluid-kinetic equations

“…In [17,18], the authors consider a regime of strong local interaction between neurons, which asymptotically leads to a somewhat monokinetic distribution in some of the variables. At the limit, one obtains a macroscopic model which is a reaction-diffusion system (see also [7]). This hydrodynamic limit shares some similarities with the one described for the kinetic Cucker-Smale model.…”