  Neural or cortical fields are continuous assemblies of mesoscopic models,
also called neural masses, of neural populations that are fundamental in the
modeling of macroscopic parts of the brain. Neural fields are described by
nonlinear integro-differential equations. The solutions of these equations
represent the state of activity of these populations when submitted to inputs
from neighbouring brain areas. Understanding the properties of these solutions
is essential in advancing our understanding of the brain. In this paper we
study the dependency of the stationary solutions of the neural fields equations
with respect to the stiffness of the nonlinearity and the contrast of the
external inputs. This is done by using degree theory and bifurcation theory in
the context of functional, in particular infinite dimensional, spaces. The
joint use of these two theories allows us to make new detailed predictions
about the global and local behaviours of the solutions. We also provide a
generic finite dimensional approximation of these equations which allows us to
study in great details two models. The first model is a neural mass model of a
cortical hypercolumn of orientation sensitive neurons, the ring model. The
second model is a general neural field model where the spatial connectivity
isdescribed by heterogeneous Gaussian-like functions.
