Curved feature metrics in models of visual cortex

We study the relation between maps of a high-dimensional stimulus manifold onto an essentially two-dimensional cortical area and low-dimensional maps of stimulus features such as centroid position, orientation, spatial frequency, etc. Whereas the former safely can be represented in a Euclidean space, the latter are shown to require a Riemannian metric in order to reach qualitatively similar stationary structures under a standard learning algorithm. We show that the non-Euclidean framework allows for a tentative explanation of the presence of the so-called “pinwheels” in feature maps and compare maps obtained numerically in the flat high-dimensional maps and in the curved low-dimensional case.