Symmetries, Pattern Formation, and Geometric Visual Hallucinations

Marty Golubitsky^*)

Cullen Distinguished Professor,

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 some of these form constants.

Since Hubel and Wiesel's work on hypercolumns, experiments on the visual cortex have shown that neurons in the visual cortex react to the orientation of boundaries or contrast edges in the visual field. Thus, in continuum models, hypercolumns can be modeled as circles (of orientation preference) and the visual cortex can be modeled as a circle at each point in the plane rather than as a plane. Moreover, cortical neurons appear to be connected in two different ways, locally and laterally, which changes the symmetry in model equations. We discuss how these changes affect the bifurcation analysis and lead to the recovery of all of Kluver's form constants. We also show how weak anisotropy in lateral connection between hypercolumns again changes the symmetries of the cortex and thereby leads to hallucinations that appear to move in the visual field.

^*) This work is joint with Paul Bressloff, Jack Cowan, Peter Thomas, Matthew Wiener, Lie June Shiau, and Andrew Torok.