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

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

Received: 27 September 2005
Accepted: 5 January 2006

Abstract

Motivated by the dynamics of resonant neurons we consider a differentiable, non-Markovian random process and particularly the time after which it will reach a certain level x_b 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 . 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 x_b. 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.