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  stationary solution. Here, we consider the case of strong connectivities, and exhibit the presence of complex, non-necessarily 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 the measure concentrates, and exhibit a particular solution corresponding to a Dirac measure at a time satisfy an ordinary differential equation identical to the original FitzHugh-Nagumo system. Classically, this solution may 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.
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.
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