# Damped oscillations of the probability of random events followed by absolute refractory period: exact analytical results

## TL;DR

This paper provides exact analytical formulas describing damped oscillations in the probability of random events following an absolute refractory period, using a stochastic neuron model as an example.

## Contribution

It introduces a novel, simplified analytical approach to describe transient oscillations in stochastic processes with refractory periods, avoiding renewal theory.

## Key findings

- Derived explicit formulas for oscillation amplitude damping.
- Demonstrated the approach with a stochastic neuron model.
- Provided insights into transient dynamics of refractory processes.

## Abstract

There are numerous examples of natural and artificial processes that represent stochastic sequences of events followed by an absolute refractory period during which the occurrence of a subsequent event is impossible. In the simplest case of a generalized Bernoulli scheme for uniform random events followed by the absolute refractory period, the event probability as a function of time can exhibit damped transient oscillations. Using stochastically-spiking point neuron as a model example, we present an exact and compact analytical description for the oscillations without invoking the standard renewal theory. The resulting formulas stand out for their relative simplicity, allowing one to analytically obtain the amplitude damping of the 2nd and 3rd peaks of the event probability.