Non-equilibrium dynamics of stochastic point processes with refractoriness

TL;DR

This paper extends renewal theory to model non-stationary stochastic point processes with refractoriness, revealing complex dynamic behaviors like oscillations and resonances in response to time-varying inputs.

Contribution

It introduces a novel framework using occupation numbers and delay differential equations to analyze non-stationary renewal processes with refractoriness, applicable to various physical and biological systems.

Findings

Demonstrates oscillations and phase jumps in response to periodic inputs

Reveals resonances and frequency doubling phenomena

Provides exact solutions for non-stationary renewal processes

Abstract

Stochastic point processes with refractoriness appear frequently in the quantitative analysis of physical and biological systems, such as the generation of action potentials by nerve cells, the release and reuptake of vesicles at a synapse, and the counting of particles by detector devices. Here we present an extension of renewal theory to describe ensembles of point processes with time varying input. This is made possible by a representation in terms of occupation numbers of two states: Active and refractory. The dynamics of these occupation numbers follows a distributed delay differential equation. In particular, our theory enables us to uncover the effect of refractoriness on the time-dependent rate of an ensemble of encoding point processes in response to modulation of the input. We present exact solutions that demonstrate generic features, such as stochastic transients and oscillations in the step response as well as resonances, phase jumps and frequency doubling in the transfer of periodic signals. We show that a large class of renewal processes can indeed be regarded as special cases of the model we analyze. Hence our approach represents a widely applicable framework to define and analyze non-stationary renewal processes.