Topological singularities in cortical orientation maps: the sign theorem correctly predicts orientation column patterns in primate striate cortex

Abstract

Optical imaging methods have revealed the spatial arrangement of orientation columns across striate cortex, usually summarized in terms of two measurements at each cortical location: (i) a ‘best’ stimulus orientation, corresponding to the stimulus orientation that elicits a maximal response, and (ii) the magnitude of the response to the best orientation. This mapping has been described as continuous except at a set of singular points (also termed ‘vortices’ or ‘pinwheels’. Although prior work has shown that vortex patterns qualitatively similar to those observed in visual area 17 of the Macaque cortex can be produced by either band-pass or low-pass filtering of random vector fields, there has been to date little further topological characterization of the structure of cortical vortex patterns. Nevertheless, much theoretical work has been done in other disciplines on mappings analogous to the cortical orientation map. In particular, a recent theorem in the optics literature, termed the sign principle, states that adjacent vortices on zero crossings of a phase (orientation) mapping must always alternate in sign. Using digitized samples of recently published optical recording data in monkey striate cortex, we show that the cortical orientation data does indeed possess 100% anti-correlation in vortex sign for nearest-neighbour vortices, as predicted by the sign theorem. This provides strong experimental support for the assumptions of continuity of cortical vortex maps which underly the sign theorem. Similar analysis predicts a lack of ‘higher-order’ vortices in the cortical orientation map, which is also found to be in agreement with optical imaging observations. It also follows from this work that cortical vortices must be created simultaneously in clockwise–anti-clockwise pairs. This suggests a possible basis for a modular (hyper-columnar) relationship among pairs of cortical vortices that originate at the same developmental time. In summary, this work indicates that primate visual cortex orientation column structure is best understood in the context of other ‘ordered continuous media’ (e.g. liquid He³, cholesteric liquid crystals, random optical phase maps, to name only a few) in which an order parameter (orientation in this case) is mapped to a physical space, and in which the topological properties of the mapping determine the observable regularities of the system. We also point out that these methods may well be applied to a variety of other cortical map systems which admit an ‘order parameter’, i.e. for which each cortical position is assigned a continuous stimulus value.