# Evolution of moments and correlations in non-renewal escape-time processes

## TL;DR

This paper introduces a numerical method based on solving the Fokker-Planck equation to analyze non-renewal stochastic systems, providing insights into moments and correlations beyond traditional approaches.

## Contribution

The authors develop a general numerical approach for studying non-renewal processes that overcomes limitations of analytical and Monte Carlo methods, applicable to complex neuronal models.

## Key findings

- The method accurately matches Monte Carlo simulations.
- Transition to stationarity depends on all system parameters.
- Serial correlation coefficients are sensitive to numerical inaccuracies.

## Abstract

The theoretical description of non-renewal stochastic systems is a challenge. Analytical results are often not available or can only be obtained under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad-hoc Monte--Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of non-renewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a non-trivial timescale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and timescale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allows for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCC), and discuss the challenge of reliably computing the SCC which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of non-renewal dynamics in a wide range of stochastic systems exhibiting short and long-range correlations.