Pinwheel stability in a non-Euclidean model of pattern formation in the visual cortex

The structure of neural maps in the primary visual cortex arises from the problem of representing a high-dimensional stimulus manifold on an essentially two-dimensional piece of cortical tissue. In order to treat the problem theoretically, stimuli are usually represented by a set of features, such as centroid position, orientation, spatial frequency, phase etc. Inputs to the cortex are, however, activity distributions over afferent nerve fibers; i.e., they require, in principle, a description as high-dimensional vectors. We study the relation between high-dimensional maps, which can be assumed to rely on a Euclidean geometry, and low-dimensional feature maps, which need to be formulated in Riemannian space in order to represent high-dimensional maps to a good accuracy. We show numerically that the Riemannian framework allows for a suggestive explanation of the abundance of typical structural units ("pinwheels") in feature maps emerging in the course of the adaptation process from an initially unstructured state.