Singularities in Cortical Orientation and Direction Maps: Vortices, Strings, and Bubbles

Abstract

The maps of orientation tuning in areas V1 and V2 contain many singular points (‘vortices’) around which orientation rotates by ±π. By contrast, direction vortices (±2π rotation) appear in the motion processing area V5. These useful mappings of orientations/directions plus positions into two dimensions carry no evolutionary or developmental costs, since vortices are unavoidable and very robust structures in almost any 2D map of a rotationally periodic quantity in which smoothing competes with local noise. Vortices of opposite sign may annihilate or dissociate in pairs, depending on the noise level.

In many cases, the cell tuning has coupled orientation (π periodic) and direction (2π periodic) components. Interactions that favour alignment of both components can then produce additional singularity structures: Orientation-vortices must carry an odd number of ‘strings’, defined as the locus where the direction component inverts its sign. The strings can have tension, enabling them to bind opposite-sign vortices. In other parameter regimes, meandering strings may produce freely floating loops that enclose opposite direction ”bubbles” in areas of aligned orientation.