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

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 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 (x) over dot. 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.