Abstract
We investigate the dynamics of a limit of interacting FitzHugh-Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a nonlinear evolution partial differential equation representing the probability distribution of one given neuron in a large network. The case of weak connectivity previously studied displays a unique, exponentially stable, stationary solution. Here, we consider the case of strong connectivities, and exhibit the presence of possibly nonunique stationary behaviors or nonstationary behaviors. To this end, using Hopf-Cole transformation, we demonstrate that the solutions exponentially concentrate, as the connectivity parameter diverges, around singular Dirac measures centered at the zeros of a time-dependent continuous function satisfying a complex partial differential equation. We next characterize the points at which this measure concentrates. We show there are infinitely many possible solutions and exhibit a particular solution corresponding to a Dirac measure concentrated on a time-dependent point satisfying an ordinary differential equation identical to the original FitzHugh-Nagumo system. We conjecture that the system selects only this particular solution and converges to it, through informed heuristic arguments and numerical simulations. This solution may thus feature multiple stable fixed points or periodic orbits, respectively corresponding to a clumping of the whole system at rest, or a synchronization of cells on a periodic solution. Numerical simulations of neural networks with a relatively modest number of neurons and finite coupling strength agree with these predictions away from the bifurcations of the limit system, showing that the asymptotic equation recovers the main properties of more realistic networks.