Discrete traveling waves in a relay system of differential-difference equations modeling a fully connected network of synaptically connected neurons

Abstract

Purpose. Consider a system of differential equations with delay, which models a fully connected chain of m + 1 neurons with delayed synoptic communication. For this fully connected system, construct periodic solutions in the form of discrete traveling waves. This means that all components are represented by the same periodic function u(t) with a shift that is a multiple of some parameter Δ (to be found). Methods. To search for the described solutions, in this work we move from the original system to an equation for an unknown function u(t), containing m ordered delays, differing with step Δ. It performs an exponential substitution (typical of equations of the Volterra type) in order to obtain a relay equation of a special form. Results. For the resulting equation, a parameter range is found in which it is possible to construct a periodic solution with period T depending on the parameter Δ. For the found period formula T = T(Δ), it is possible to prove the solvability of the period equation, that is, to prove the existence of non-zero parameters — integer p and real Δ — satisfying the equation (m + 1)Δ = pT(∆). The constructed function u(t) has a bursting effect. This means that u(t) has a period of n high spikes, followed by a period of low values. Conclusion. The existence of a suitable parameter Δ ensures the existence of a periodic solution in the form of a discrete traveling wave for the original system. Due to the choice of permutation, the coexistence of (m + 1)! periodic solutions is ensured. 

Keywords

• differential-difference equations
• fully coupled system
• discrete traveling waves
• bursting effect
• periodic solutions
• neuron modeling