The dynamics of ensemble of neuron-like elements with excitatory couplings

Abstract

We study the phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model of coupling which is implemented by smooth function that approximate rectangular function. The proposed coupling depends on three parameters that define the beginning of activation of an element $\alpha$, the duration of the activation $\delta$ and the strength of the coupling $g$. We observed a rich diversity of types of neuron-like activity, including regular in-phase, anti-phase and sequential spiking activities. In the phase space of the system, these regular regimes correspond to specific asymptotically stable periodic motions (limit cycles). We also observed a chaotic anti-phase activity, which corresponds to a strange attractor that appears due to the cascade of period doubling bifurcations of limit cycles. We also provide the detailed study of bifurcations which lead to transitions between all these regimes and detect on the $(\alpha, \delta)$ parameter plane those regions that correspond to the above-mentioned regimes. We also show numerically the existence of bistability regions when various non-trivial regimes coexist. For example, in some regions, one can observe either anti-phase or in-phase oscillations depending on initial conditions. We also specify regions corresponding to coexisting various types of sequential activity.