On a Kinetic Fitzhugh–Nagumo Model of Neuronal Network

International audienceWe investigate existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh-Nagumo neurons. This equation is hypoelliptic , nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover , we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually , using a semi-group factorisation method , we show exponential nonlinear stability in the small connectivity regime