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Stochastic point processes with refractoriness appear frequently in thequantitative analysis of physical and biological systems, such as thegeneration of action potentials by nerve cells, the release and reuptake ofvesicles at a synapse, and the counting of particles by detector devices. Herewe present an extension of renewal theory to describe ensembles of pointprocesses with time varying input. This is made possible by a representation interms of occupation numbers of two states: Active and refractory. The dynamicsof these occupation numbers follows a distributed delay differential equation.In particular, our theory enables us to uncover the effect of refractoriness onthe time-dependent rate of an ensemble of encoding point processes in responseto modulation of the input. We present exact solutions that demonstrate genericfeatures, such as stochastic transients and oscillations in the step responseas well as resonances, phase jumps and frequency doubling in the transfer ofperiodic signals. We show that a large class of renewal processes can indeed beregarded as special cases of the model we analyze. Hence our approachrepresents a widely applicable framework to define and analyze non-stationaryrenewal processes.

Non-equilibrium dynamics of stochastic point processes with refractoriness