  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
non-unique stationary behaviors or non-stationary behaviors. To this end, using
Hopf-Cole transformation, we demonstrate that the solutions exponentially
concentrate around a singular Dirac measure as the connectivity parameter
diverges, centered at the zeros of a time-dependent continuous function. We
next characterize the points at which this measure concentrates, 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. 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. We illustrate these results with numerical simulations of
neural networks with a relatively modest number of neurons and finite coupling
strength, and show that away from the bifurcations of the limit system, the
asymptotic equation recovers the main properties of more realistic networks.
