In this paper we use symmetric bifurcation theory to analyze spontaneous pattern formation in
a neural field model of the visual cortex, which is defined on the product space R
2 × S
1
, where
r ∈ R
2
represents spatial position and θ ∈ S
1
represents the contrast polarity and orientation of an
edge. A contrast polarity selective neuron responds to an oriented edge only when the difference
in contrast on either side of the edge has a particular sign. Hence, we take the variable θ ∈ S
1
to
lie in the range [0, 2π] rather than [0, π]. Assuming the existence of a spatially periodic stationary,
θ-independent solution, we show how the full θ-selective model, with only isotropic connections,
can be reduced to a simpler neural field equation with the θ-independent solution acting as an
input. We then construct a bifurcation problem whereby the θ-independent solution can become
unstable due to the intrinsic θ-selective circuitry, and use symmetric bifurcation theory to analyze
the types of new solutions that can arise according to the equivariant branching lemma. The stability
of these bifurcating solutions is then determined using a Lyapunov–Schmidt reduction. We show
that one of the stable bifurcating solutions exhibits patterns of θ preference that coincide with
the spatial contrast gradient of the original activity pattern. Finally, we show that the gradient
solution persists in the presence of anisotropic perturbations. Given previous evidence that there are
correlations between stimulus-evoked and spontaneous cortical activity patterns, our work suggests
that the same network architecture might be capable of performing the above task in the presence
of an input.