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

Braun, Wilhelm; Thul, Ruediger; Longtin, Andr�

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Authors

Wilhelm Braun

Andr� Longtin



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.

Citation

Braun, W., Thul, R., & Longtin, A. (in press). Evolution of moments and correlations in non-renewal escape-time processes. Physical Review E, 95, https://doi.org/10.1103/PhysRevE.95.052127

Journal Article Type Article
Acceptance Date Apr 27, 2017
Online Publication Date May 16, 2017
Deposit Date May 11, 2017
Publicly Available Date Mar 28, 2024
Journal Physical Review E
Print ISSN 2470-0045
Electronic ISSN 1550-2376
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 95
DOI https://doi.org/10.1103/PhysRevE.95.052127
Public URL https://nottingham-repository.worktribe.com/output/860484
Publisher URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE.95.052127

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