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 $x_b$. 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 $x_b$. 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 monotonous first passage time distribution) fail.