  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.
