The impulse-refractive mode in the neural network with ring synaptic interaction

In the paper, a mathematical model of a neural network with an even number of ring synaptic interaction elements is considered. 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 of special impulse-refraction cycles within the system of equations is of interest. The functions with odd numbers of the impulse-refraction cycle have an asymptotically high pulses and the functions with even numbers are asymptotically small. Two changes allow to study a two-dimension nonlinear differential-difference system with two delays instead of the system. Further, a limit object that represents a relay system with two delays is defined by a large parameter tending to infinity. There exists the only periodic solution of the relay system with the initial function from a suitable function class. This is structurally proved, by using the step method. Next, the existence of relaxation periodic solutions of the two-dimension singularly perturbed system is proved by using the Poincare operator and the Schauder principle. The asymptotics of this solution is constructed, and it is proved that the solution is close to the decision of the relay system. Because of the exponential estimate of the Frechet derivative of the Poincare operator it implies the uniqueness and stability of solutions of the two-dimension differential-difference equation with two delays. Furthermore, with the help of reverse replacement the proved result is transferred to the original system.