Abstract
The theoretical description of nonrenewal stochastic systems is a challenge. Analytical results are often not available or can be obtained only 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 nonrenewal 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 nontrivial time scale 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 time-scale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allow for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCCs) and discuss the challenge of reliably computing the SCCs, 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 nonrenewal dynamics in a wide range of stochastic systems exhibiting short- and long-range correlations.
Highlights
A general property of diverse systems, ranging from superconducting quantum interference devices (SQUIDs) [1], to lasers [2] to excitable cells [36] is that time intervals between specic events are not statistically independent
Even if these approximations allow for some insight into the parameter dependence of e.g. serial correlations and can be used to understand experimental data, as exemplied in [9, 1316] in the context of excitable systems, it is desirable to understand the statistics of model systems without making simplifying assumptions
We show that our methods reproduce known stationary analytical results for the serial correlation coecients (SCC) when we consider the perfect integrate-and-re model with single exponential adaptation in a parameter regime where we have large negative correlations, demonstrating that our methods are sound, but SCC calculations are very sensitive to numerical inaccuracies
Summary
A general property of diverse systems, ranging from superconducting quantum interference devices (SQUIDs) [1], to lasers [2] to excitable cells [36] is that time intervals between specic events are not statistically independent. Given that the OUP is the basis for integrate-and-re (IF) neuron models, which are among the most popular neuron descriptions [29], we refer to events as spikes and to s(t) as a time-dependent adaptation current in the present study. In the non-renewal case we are studying here, subsequent ring times will in general not have the same distribution as T1. The values of the peak adaptation current after the kth ring are dened for k ≥ 1 as k s(0k) = s(t−) + κ : t = Ti ,. The knowledge of these distributions is key to understanding the non-renewal dynamics, as they form a hidden Markov model of the underlying non-Markovian dynamics [15, 16, 32]. This gives rise to the iFPT approach which we explain
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