Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing

TL;DR

This paper analyzes a theta neuron with delayed self-feedback, deriving explicit conditions for self-sustained pulsations and multistability, serving as a fundamental model for pulsating dynamics in excitable systems.

Contribution

The study provides an analytical framework for understanding self-sustained oscillations in a theta neuron with delay, highlighting its role as a prototypical model for pulsating behavior.

Findings

Explicit expressions for existence and stability of periodic solutions.

Identification of multistability with increasing delay.

Complete parameter space characterization for self-sustained oscillations.

Abstract

We consider a single theta neuron with delayed self-feedback in the form of a Dirac delta function in time. Because the dynamics of a theta neuron on its own can be solved explicitly -- it is either excitable or shows self-pulsations -- we are able to derive algebraic expressions for existence and stability of the periodic solutions that arise in the presence of feedback. These periodic solutions are characterized by one or more equally spaced pulses per delay interval, and there is an increasing amount of multistability with increasing delay time. We present a complete description of where these self-sustained oscillations can be found in parameter space; in particular, we derive explicit expressions for the loci of their saddle-node bifurcations. We conclude that the theta neuron with delayed self-feedback emerges as a prototypical model: it provides an analytical basis for understanding pulsating dynamics observed in other excitable systems subject to delayed self-coupling.