Abstract

We consider the solution to a non-linear mean-field equation modeling a FitzHug-Nagumo neural network. The non-linearity in this equation arises from the interaction between neurons. We suppose that these interactions depend on the spatial location of neurons and we focus on the behavior of the solution in the regime where short-range interactions are dominant. The solution then converges to a Dirac mass. The aim of this talk is to characterize the blow-up profile: we will prove that it is Gaussian. More precisely, we will compare several approaches:  we will first present a weak convergence result, based on a analytic coupling method for Wasserstein distances, then we will strengthen this result by obtaining strong convergence estimates, using relative entropy methods and we will conclude by presenting a different approach, inspired from the analysis of Hamilton Jacobi equations, which enables to obtain L infinity convergence estimates.