Relaxation Oscillations in a System of Two Pulsed Synaptically Coupled Neurons

Abstract

In this paper, we consider a mathematical model of synaptic interaction between two pulse neuron elements. Each of the neurons is modeled by a singularly perturbed difference-differential equation with delay. Coupling is assumed to be at the threshold with the time delay being taken into account. The problems of existence and stability of relaxation periodic movements for the systems derived are considered. It turns out that the critical parameter is the ratio between the delay caused by internal factors in the single-neuron model and the delay in the coupling link between the oscillators. The existence and stability of a uniform cycle for the problem are proved in the case where the delay in the link is less than the period of a single oscillator, which depends on the internal delay. As the delay grows, the in-phase regime becomes more complex; specifically, it is shown that, by choosing an adequate delay, we can obtain more complex relaxation oscillations and, during a period, the system can exhibit more than one high-amplitude splash. This means that the bursting effect can appear in a system of two synaptically coupled neuron-type oscillators due to the delay in the coupling link.