  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.
