Afferent geometry in the primate visual cortex and the generation of neuronal trigger features

Abstract

In previous work, it was suggested that the sequence regularity property of cortical neurons could be accounted for if the local geometric structure of the cortex were a recapituation of the global complex logarithmic structure of the retinotopic mapping. This model is developed in detail: the excitatory and inhibitory structure of cortical receptive fields may be approximated by a complex logarithmic local geometry, coupled with an intra-cortical lateral inhibition operator which may flow unidirectionally yet still create "rotating" receptive field structure. The direction of intra-cortical lateral inhibition follows the borders of cortical ocular dominance columns, which are the approximate images under the global complex logarithmic mapping, of exponentially spaced, horizontal straight lines in the visual field. Two different topological structures are discussed for the local cortical manifold. The binocular trigger features of cortical neurons follow from the same geometric model, and the ratio of binocular to monocular cortical cells is related to the size and shape of cortical dendritic tree's by an application of integral geometry. Recent results in optical pattern recognition are cited to suggest that the rotation and size invariant properties of the cortical map are essential to any cross-correlational basis for stereopsis. Finally, a meromorphic function is presented which is both locally and globally complex logarithmic in its structure, and therefore represents the model presented in this and previous papers in a concise mathematical form. This function is closely related to the description of a Karman vortex pattern, in fluid mechanics, and leads to the suggestion that the boundary conditions of layer IV of the cortex (i.e. periodic ocular dominance columns) are causally related to the existence of sequence regularity in the cortex. The developmental implications of this statement are that the specification of neural connections in the cortex may follow directly, both locally and globally, from the detailed nature of the cortical boundary conditions (i.e. anatomy), coupled with general physico-mathematical considerations of continuity and differentiability in the neural fiber flow.