  We study a simplified model of the representation of colors in the primate
primary cortical visual area V1. The model is described by an initial value
problem related to a Hammerstein equation. The solutions to this problem
represent the variation of the activity of populations of neurons in V1 as a
function of space and color. The two space variables describe the spatial
extent of the cortex while the two color variables describe the hue and the
saturation represented at every location in the cortex. We prove the
well-posedness of the initial value problem. We focus on its stationary, i.e.
independent of time, and periodic in space solutions. We show that the model
equation is equivariant with respect to the direct product G of the group of
the Euclidean transformations of the planar lattice determined by the spatial
periodicity and the group of color transformations, isomorphic to O(2), and
study the equivariant bifurcations of its stationary solutions when some
parameters in the model vary. Their variations may be caused by the consumption
of drugs and the bifurcated solutions may represent visual hallucinations in
space and color. Some of the bifurcated solutions can be determined by applying
the Equivariant Branching Lemma (EBL) by determining the axial subgroups of G .
These define bifurcated solutions which are invariant under the action of the
corresponding axial subgroup. We compute analytically these solutions and
illustrate them as color images. Using advanced methods of numerical
bifurcation analysis we then explore the persistence and stability of these
solutions when varying some parameters in the model. We conjecture that we can
rely on the EBL to predict the existence of patterns that survive in large
parameter domains but not to predict their stability. On our way we discover
the existence of spatially localized stable patterns through the phenomenon of
"snaking".
