A cycle with a "short" period in the phenomenological model of a single neuron

TL;DR

This paper analyzes a phenomenological neuron model described by a differential-difference equation, constructing solutions with specific initial conditions and proving the existence of a short-period unstable cycle.

Contribution

It extends previous work by constructing solutions with initial functions having at most two zeros and identifying a short-period unstable solution.

Findings

Existence of a short-period unstable solution.

Construction of solutions with initial functions containing no more than two zeros.

Extension of stability analysis to new initial conditions.

Abstract

We consider the relay version of the generalized Hutchinson's equation as a phenomenological model of an isolated neuron. After an exponential substitution, the equation takes the form of a differential-difference equation with a piecewise-constant right-hand side. In the work of A. Yu. Kolesov and others, this equation was studied with a negative continuous initial function, and the existence and orbital stability of a periodic solution with a period longer than the delay were proven. In the present work, all possible solutions with continuous on the interval of the delay length initial functions, containing no more than two zeros, are constructed. It is proven that the equation has a periodically unstable solution, the period of which is shorter than the delay.