Theory of non-Markovian Stochastic Resonance

TL;DR

This paper develops a comprehensive theoretical framework for non-Markovian stochastic resonance using renewal process theory, deriving new integral equations and response functions that extend previous models to non-stationary, driven systems.

Contribution

It introduces a stochastic path integral approach to non-Markovian SR, generalizes renewal equations for non-stationary conditions, and formulates a response theory beyond linear approximation.

Findings

Derived general integral equations for non-Markovian SR

Established a response theory beyond linear response

Applied the theory to ion channels with fractal kinetics

Abstract

We consider a two-state model of non-Markovian stochastic resonance (SR) within the framework of the theory of renewal processes. Residence time intervals are assumed to be mutually independent and characterized by some arbitrary non-exponential residence time distributions which are modulated in time by an externally applied signal. Making use of a stochastic path integral approach we obtain general integral equations governing the evolution of conditional probabilities in the presence of an input signal. These novel equations generalize earlier integral renewal equations by Cox and others to the case of driving-induced non-stationarity. On the basis of these new equations a response theory of two state renewal processes is formulated beyond the linear response approximation. Moreover, a general expression for the linear response function is derived. The connection of the developed approach with the phenomenological theory of linear response for manifest non-Markovian SR put forward in [ I. Goychuk and P. Hanggi, Phys. Rev. Lett. 91, 070601 (2003)] is clarified and its range of validity is scrutinized. The novel theory is then applied to SR in symmetric non-Markovian systems and to the class of single ion channels possessing a fractal kinetics.