On born of in-phase limit cycle in ensemble of excitatory coupled FitzHugh-Nagumo elements

We proposed and studied numerically efficient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling. This coupling is defined by function that depends on phase of active element and that is smooth approximation of rectangular impulse function. Above-mentioned coupling depends on three parameters that define the beginning of element activation, the duration of the activation and the coupling strength. We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements. It is proved that this limit cycle is a result of supercritical Andronov-Hopf bifurcation. The chart of activity regimes is depicted on the plane of parameters that define beginning and duration of activation. The boundaries of bifurcations that lead to birth of this cycle are found.