Emergence of Oscillatory Behaviors for Excitable Systems with Noise and Mean-Field Interaction: A Slow-Fast Dynamics Approach

We consider the long-time dynamics of a general class of nonlinear Fokker–Planck equations, describing the large population behavior of mean-field interacting units. The main motivation of this work concerns the case where the individual dynamics is excitable, i.e. when each isolated dynamics rests in a stable state, whereas a sufficiently strong perturbation induces a large excursion in the phase space. We address the question of the emergence of oscillatory behaviors induced by noise and interaction in such systems. We tackle this problem by considering this model as a slow-fast system (the mean value of the process giving the slow dynamics) in the regime of small individual dynamics and by proving the existence of a positively stable invariant manifold, whose slow dynamics is at first order the dynamics of a single individual averaged with a Gaussian kernel. We consider applications of this result to Stuart–Landau and FitzHugh–Nagumo oscillators, and to the Cucker–Smale alignment model.