Relaxation cycles in a model of synaptically interacting oscillators

In this paper the mathematical model of a neural network with a ring synaptic interaction elements is considered. The model is a system of scalar nonlinear differential-difference equations, the right parts of which depend on a large parameter. The unknown functions included in the system characterize the membrane potentials of the neurons. The search of relaxation cycles within the system of equations is interested. To this end solutions of the task are finded in the form of discrete traveling waves. It allows to research a scalar nonlinear differential-difference equations with two delays instead of system. Further, a limit a object that represents a relay equation with two delays is defined by large parameter tends to infinity. There are six cases of restrictions on the parameters. In every case exist alone periodic solution of relay equation started from initial function from suitable function class. It is structurally proved by using the step method. Next, the existence of a relaxation periodic solutions of a singularly perturbed equation with two delays is proved by using Poincare operator and Schauder principle. The asymptotics of this solution is constructed, and then it is proved that the solution is close to decision of the relay equation. Because of the exponential estimate Frechet derivative of the Poincare operator implies the uniqueness and stability of solutions of differential-difference equation with two delays.