  The first passage time density of a diffusion process to a time varying
threshold is of primary interest in different fields. Here we consider a
Brownian motion in presence of an exponentially decaying threshold to model the
neuronal spiking activity. Since analytical expressions of the first passage
time density are not available, we propose to approximate the curved boundary
by means of a continuous two-piecewise linear threshold. Explicit expressions
for the first passage time density towards the new boundary are provided. Then
we introduce different approximating linear threshold and describe how to
choose the optimal one minimizing the distance to the curved boundary and hence
the error in the corresponding passage time density. Theoretical means,
variances and coefficients of variation given by our method are then compared
with empirical quantities from simulated data as well as other firing
statistics derived under the assumption of a small amplitude of the
time-dependent change in the threshold. Finally maximum likelihood and moment
estimators of the parameters of the Wiener process are also proposed and
compared on simulated data.
