Interpreting the notion of simultaneous confinement in space and spatial frequency in terms of confinement, not variance, yields a rich class of intrinsically two-dimensional patterns

Jonathon Victor1,2,3 and B.W. Knight2,3

1Department of Neurology and Neuroscience, Weill Medical College of Cornell University
2The Rockefeller University
3Department of Biomathematical Sciences, Mount Sinai School of Medicine

Well-established theoretical reasons lead to consideration of functions that are limited in space and spatial frequency as the starting point for modeling receptive fields of visual neurons. Limited spatial spread economizes on connection length and allows for spatial localization. Limited spread in spatial frequency allows for analysis at a particular spatial scale. In the traditional view, spread is measured by variance. As Daugman (1985) showed via analogy with the uncertainty principle, the Gabor functions have the smallest joint spread in space and spatial frequency. While it is interesting to note that Gabor functions resemble receptive fields in V1, this view provides little insight into receptive fields in other visual areas, and also does not provide an indication of why typical V1 receptive fields have only a few lobes.

Here we present an alternative construction of the notion of "limited in space and spatial frequency," based on the concept of confinement. From this notion, a much richer set of functions, the 2-dimensional Hermite functions, emerge. These functions form a natural hierarchy. The first levels of this hierarchy contain functions that resemble Gabor functions with a small number of lobes, and thus resemble V1 receptive fields. Further down in the hierarchy are intrinsically two-dimensional functions, some of which resemble the non-Cartesian gratings, to which some V4 neurons respond preferentially (Gallant, 1996). In addition to their many interesting mathematical properties, the two-dimensional Hermite functions allow for efficient ("sparse") local synthesis of images.

While we make no claim that this view suffices to account for visual receptive field structure, we suggest that it provides a framework for a principled study of receptive fields, and that it is useful to think of receptive fields (along the V1-to-V2-to-V4 pathway) as not only expanding, but also increasing in their combined space-bandwidth aperture.