The neurogeometry of pinwheels as a sub-Riemannian contact structure

We present a geometrical model of the functional architecture of the primary Visual cortex (VI) and, more precisely, of its pinwheel structure. The problem is to understand from within how the internal unmanent geometry of the visual cortex Call produce the transcendent geometry of the external space. We use first the concept of blowing up to model V1 as a discrete approximation of a continuous fibration pi : R x P --> P with base space the space of the retina R and fiber the projective line P of the orientations of the plane. The core of the paper consists first in showing that the horizontal cortico-cortical connections of VI implement what the geometers call the contact structure of the fibration pi, and secondly in introducing an integrability condition and the integral curves associated with it. The paper develops then three applications: (i) to Field's, Hayes', and Hess' psychophysical concept of association field. (ii) to a variational model Of curved modal illusory contours (in the Spirit Of previous models due to Ullman. Horn. and Mumford), (iii) to Ermentrout's, Cowan's, Bressloff's, Golubitsky's models of visual hallucinations. (C) 2004 Elsevier Ltd. All rights reserved.