  Motivated by the dynamics of resonant neurons we discuss the properties of
the first passage time (FPT) densities for nonmarkovian differentiable random
processes. We start from an exact expression for the FPT density in terms of an
infinite series of integrals over joint densities of level crossings, and
consider different approximations based on truncation or on approximate
summation of this series. Thus, the first few terms of the series give good
approximations for the FPT density on short times. For rapidly decaying
correlations the decoupling approximations perform well in the whole time
domain.
  As an example we consider resonate-and-fire neurons representing stochastic
underdamped or moderately damped harmonic oscillators driven by white Gaussian
or by Ornstein-Uhlenbeck noise. We show, that approximations reproduce all
qualitatively different structures of the FPT densities: from monomodal to
multimodal densities with decaying peaks. The approximations work for the
systems of whatever dimension and are especially effective for the processes
with narrow spectral density, exactly when markovian approximations fail.
