A theory of pattern formation in primary visual cortex (V1) is presented that takes
into account its crystalline–like structure. The cortex is partitioned into fundamental
domains or hypercolumns of a lattice describing the distribution of singularities or
pinwheels in the orientation preference map. Each hypercolumn is modelled as a
network of orientation and spatial frequency selective cells organized around a pair
of pinwheels, which are associated with high and low spatial frequency domains
respectively. The network topology of the hypercolumn is taken to be a sphere with
the pinwheels located at the poles of the sphere. Interactions between hypercolumns
are mediated by anisotropic long range lateral connections that link cells with similar
feature preferences. Using weakly nonlinear analysis we investigate the spontaneous
formation of cortical activity patterns through the simultaneous breaking of an
internal O(3) symmetry and a discrete lattice symmetry. The resulting patterns are
characterized by states in which each hypercolumn exhibits a tuned response to both
orientation and spatial frequency, and the distribution of optimal responses across
hypercolumns is doubly periodic or quasi–periodic with respect to the underlying
lattice