# Dresden 2006 – wissenschaftliches Programm

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# DY: Dynamik und Statistische Physik

## DY 24: Brownian Motion and Kinetic Theory I

### DY 24.4: Vortrag

### Dienstag, 28. März 2006, 15:15–15:30, H\"UL 186

**Effective Approximations of First Passage Time Distributions of Non-Markovian Processes** — •Tatjana Verechtchaguina, Igor M. Sokolov, and Lutz Schimansky-Geier — Institut für Physik, Humboildt Universität zu Berlin, 12489 Berlin

Motivated by the dynamics of resonant neurons we discuss the properties of the first passage time (FPT) densities for nonmarkovian differentiable random processes. We start from an exact expression for the FPT density in terms of an infinite series of integrals over joint densities of level crossings, and consider different approximations based on truncation or on approximate summation of this series. Thus, the first few terms of the series give good approximations for the FPT density on short times. For rapidly decaying correlations the decoupling approximations perform well in the whole time domain.

As an example we consider resonate-and-fire neurons representing stochastic underdamped or moderately damped harmonic oscillators driven by white Gaussian or by Ornstein-Uhlenbeck noise. We show, that approximations reproduce all qualitative different structures of the FPT densities: from monomodal to multimodal densities with decaying peaks. The approximations work for the systems of whatever dimension and are especially effective for the processes with narrow spectral density, exactly when markovian approximations fail.