A new approach to mathematical modeling of chemical synapses

Abstract

The purpose of this work is to study a new mathematical model of a ring neural network with unidirectional chemical connections, which is a singularly perturbed system of differential-difference equations with delay. Methods. A combination of analytical and numerical methods is used to study the existence and stability of special periodic solutions in this system, the so-called traveling waves. Results. The proposed methods make it possible to show that the ring system under study allows the number of stable traveling waves to increase with the number of oscillators in the network. Conclusion. In this article, we rethink and refine the previously proposed method of mathematical modeling of chemical synapses. On the one hand, it was possible to fully take into account the requirement of the Volterra structure of the corresponding equations and, on the other hand, the hypothesis of saturating conductivity. This makes it possible to observe the principle of uniformity: the new mathematical model is based on the same principles as the previously proposed model of electrical synapses.

Keywords

• circular neural network
• chemical synapses
• relaxation cycles
• asymptotics
• stability