First passage time problems for diffusion processes have been
extensively investigated to model neuronal firing activity or
extinction times in population dynamics (see, for instance,
\cite{Ricciardi99}). In this paper we study the asymptotic
behavior of first passage times densities for a class of specially
confined temporally homogeneous diffusion processes in the presence
of an entrance or a reflecting boundary. The emphasis is on problems
of a rather mathematical nature, concerning the behavior of the first
passage time density and of its moments when the neuronal firing
threshold is in the neighborhood of the reflecting boundary, and when
it moves indefinitely away from it. Our asymptotic results are obtained
without need to determine beforehand the transition probability density
in the presence of entrance or reflecting boundaries; they depend,
instead, only on drift, infinitesimal variance, threshold and on the
entrance or the reflecting boundary of the process. Some evaluations of
moments of first passage time, in particular, mean and variance, are
performed by solving numerically, or analytically whenever possible,
Siegert's recursion equations \cite{Siegert51}, and by comparing the
results with those obtained through our approximate formulas. In the
case where the transition probability density is known, the goodness
of the obtained approximations can be verified. Such results appear
to be useful for neuronal modeling in the presence of reversal
potential especially to pinpoint the role of the involved parameters
in various models, some of which are the object of a somewhat detailed
analysis.