Hyperbolic Planforms in Relation to Visual Edges and Textures Perception

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in previous work to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.Author Summary: Our visual perception of the world is remarkably stable despite the fact that we move our gaze and body. This must be the effect of the neuronal organization of the visual areas of our brains that succeed in maintaining in our consciouness a representation that seems to be protected from brutal variations. We propose a theory to account for an invariance that pertains to such image features as edges and textures. It is based on the simple assumption that the spatial variations of the image intensity, its derivatives, are extracted and represented in some visual brain areas by populations of neurons that excite and inhibit each other according to the values of these derivatives. Geometric transformations of the retinal image, caused say by eye movements, affect these derivatives. Assuming that their representations are invariant to these transformations, we predict the appearance of specific patterns of activity which we call hyperbolic planforms. It is surprising that the geometry that emerges from our investigations is not the usual Euclidean geometry but the much less familiar hyperbolic, non-Euclidean, geometry. We also propose some preliminary ideas for putting our theory to the test by actual measurements of brain activity.