Europhys. Lett.
Volume 73, Number 5, March 2006
Page(s) 691 - 697
Section General
Published online 18 January 2006
Europhys. Lett., 73 (5), pp. 691-697 (2006)
DOI: 10.1209/epl/i2005-10449-7

First passage time densities in non-Markovian models with subthreshold oscillations

T. Verechtchaguina, I. M. Sokolov and L. Schimansky-Geier

Institute for Physics, Humboldt-University at Berlin Newton Str. 15, D-12489 Berlin, Germany

received 27 September 2005; accepted in final form 5 January 2006
published online 18 January 2006

Motivated by the dynamics of resonant neurons we consider a differentiable, non-Markovian random process x(t) and particularly the time after which it will reach a certain level xb for the first time. The probability density of this first passage time is expressed as infinite series of integrals over joint probability densities of x and its velocity $\dot{x}$. Approximating higher-order terms of this series through the lower-order ones leads to closed expressions in the cases of vanishing and moderate correlations between subsequent crossings of xb. For a linear oscillator driven by white or coloured Gaussian noise, which models a resonant neuron, we show that these approximations reproduce the complex structures of the first passage time densities characteristic for the underdamped dynamics, where Markovian approximations (giving monomodal first passage time distribution) fail.

05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
02.50.Ey - Stochastic processes.

© EDP Sciences 2006