Excitable systems with noise and delay with applications to control: renewal theory approach

TL;DR

This paper develops an analytical approach using renewal theory to study noise-induced dynamics in excitable systems with time delay, incorporating control mechanisms based on delayed feedback.

Contribution

It introduces a novel analytical method combining renewal theory with delay-based control in excitable systems, providing insights into their stochastic behavior.

Findings

Analytical expressions for system dynamics under noise and delay.

Insights into control effects on excitable systems with delay.

Framework applicable to various noise-driven delayed systems.

Abstract

We present an approach for the analytical treatment of excitable systems with noise-induced dynamics in the presence of time delay. An excitable system is modeled as a bistable system with a time delay, while another delay enters as a control term taken after [Pyragas 1992] as a difference between the current system state and its state "tau" time units before. This approach combines the elements of renewal theory to estimate the essential features of the resulting stochastic process as functions of the parameters of the controlling term.