  We introduce a generalized Langevin model system for different non-Markovian
effects in the well and barrier regions of a potential, and use it to
numerically study the dependence of the barrier-crossing time. In the
appropriate limits, our model interpolates between the theoretical
barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak
et al., which for a single barrier memory time can differ by several orders of
magnitude. Our model furthermore allows to test an analytic rate theory for
space-inhomogeneous memory, which disagrees with our numerical results in the
long well-memory regime. In this regime, we find that short barrier memory
decreases the barrier-crossing time as compared to long barrier memory. This is
in contrast with the short well-memory regime, where both our numerical results
and GH theory predict an acceleration of the barrier crossing time with
increasing barrier memory time. Both effects, the `Markovian-barrier
acceleration' and GH `non-Markovian-barrier acceleration' can be understood
from a committor analysis. Our model combines finite relaxation times of
orthogonal degrees of freedom with a space-inhomogeneous coupling to such
degrees, and represents a step towards more realistic modeling of physical
reaction coordinates.
