A mean-field model of Integrate-and-Fire neurons: non-linear stability of the stationary solutions

We consider a stochastic network of Integrate-and-Fire spiking neurons, in its mean-field asymptotic. Given an invariant probability measure of the McKean-Vlasov equation, we give a sufficient condition to ensure the local stability of this invariant measure. Our criteria involves the location of the zeros of an explicit holomorphic function associated to the considered invariant probability measure. We prove that when all the complex zeros have negative real part, local stability holds.