Pattern Formation in the Primary Visual Cortex Martin Golubitsky Department of Mathematics University of Houston Twenty years ago Ermentrout and Cowan observated that drug induced visual hallucinations could be modeled by spontaneous pattern formation in the primary visual cortex. They assumed that the drug uniformly excites the cortex causing nonuniform patterns in the activity variable of cortical neurons. Kluver subdivided hallucination patterns into four classes, called "form constants", and the Ermentrout/Cowan theory produced only some of these form constants. During the past two decades experiments on the visual cortex have shown an unexpected structure. First, the neurons in the visual cortex appear to fire in response to the orientation of boundaries or contrast edges in the visual field. Thus, it is more appropriate to model the visual cortex as R^2xS^1 rather than R^2. Second, cortical neurons appear to be connected in two different ways, locally and globally, which changes the symmetry in model equations. We discuss how these changes affect the bifurcation analysis and lead to the recovery of Kluver's form constants. Perhaps surprisingly, the kind of pattern formation that occurs in the visual cortex depends on the fact that patterns are given in terms of line fields rather than level contours of functions, as is typically the case in reaction-diffusion systems. This kind of pattern formation may be expected in other fields, such as liquid crystals. This work is joint with Paul Bressloff, Jack Cowan, Peter Thomas, and Matthew Wiener.