Relaxation Cycles in a Model of Synaptically Interacting Oscillators

Abstract

We study the mathematical model of a circular neural network with synaptic interaction between the elements. The model is a system of scalar nonlinear differential-difference equations, the right parts of which depend on large parameters. The unknown functions included in the system characterize the membrane potentials of the neurons. The search for relaxation cycles within the system of equations is of interest. Thus, we postulate the problem of finding its solution in the form of discrete travelling waves. This allows us to study a scalar nonlinear differential-difference equation with two delays instead of the original system. We define a limit object which represents a relay equation with two delays by passing the large parameter to infinity. Using this construction and the step-by-step method, we show that there are six cases for restrictions on the parameters. In each case there exists a unique periodic solution to the relay equation with the initial function from a suitable function class. Using the Poincaré operator and the Schauder principle, we prove the existence of relaxation periodic solutions of a singularly perturbed equation with two delays. We find the asymptotics of this solution and prove that the solution is close to the solution of the relay equation. The uniqueness and stability of the solutions of the differential-difference equation with two delays follow from the exponential bound on the Fréchet derivative of the Poincaré operator.