  We study a Fokker-Planck equation modelling the firing rates of two
interacting populations of neurons. This model arises in computational
neuroscience when considering, for example, bistable visual perception problems
and is based on a stochastic Wilson-Cowan system of differential equations. In
a previous work, the slow-fast behavior of the solution of the Fokker-Planck
equation has been highlighted. Our aim is to demonstrate that the complexity of
the model can be drastically reduced using this slow-fast structure. In fact,
we can derive a one-dimensional Fokker-Planck equation that describes the
evolution of the solution along the so-called slow manifold. This permits to
have a direct efficient determination of the equilibrium state and its
effective potential, and thus to investigate its dependencies with respect to
various parameters of the model. It also allows to obtain information about the
time escaping behavior. The results obtained for the reduced 1D equation are
validated with those of the original 2D equation both for equilibrium and
transient behavior.
