Glyzin S.D., Preobrazhenskaia M.M. (2021) Ring of Unidirectionally Synaptically Coupled Neurons with a Relay Nonlinearity. In: Kryzhanovsky B., Dunin-Barkowski W., Redko V., Tiumentsev Y. (eds) Advances in Neural Computation, Machine Learning, and Cognitive Research IV. NEUROINFORMATICS 2020. Studies in Computational Intelligence, vol 925. Springer, Cham.
The dynamic of ring of m unidirectionally coupled neurons with electrical coupling are considered. The system of relay differential-difference equations was chosen as the mathematical model of this ring. The unidirectional synaptic coupling was modeled based on the idea of fast threshold modulation (FTM). We show that the system has at least m stable cycles, which are discrete traveling waves. Moreover, the solution corresponding to each of the oscillators is a periodic function with a predetermined number of bursts per period. Also in case of even m it is proved existence and stability of impulse-refractive periodic mode. This is a such solution that oscillators with odd indexes have high periodic bursts and with even indexes have a refractoriness behaviour.