Two-dimensional spectral analysis of cortical receptive field profiles

Abstract

Most vision research embracing the spatial frequency paradigm has been conceptually and mathematically a one-dimensional analysis of two-dimensional mechanisms. Spatial vision models and the experiments sustaining them have generally treated spatial frequency as a one-dimensional variable, even though receptive fields and retinal images are two-dimensional and linear transform theory obliges any frequency analysis to preserve dimension. Four models of cortical receptive fields are introduced and studied here in 2D form, in order to illustrate the relationship between their excitatory/inhibitory spatial structure and their resulting 2D spectral properties. It emerges that only a very special analytic class of receptive fields possess independent tuning functions for spatial frequency and orientation; namely, those profiles whose two-dimensional Fourier Transforms are expressible as the separable product of a radial function and an angular function. Furthermore, only such receptive fields would have the same orientation tuning curve for single bars as for gratings. All classes lacking this property would describe cells responsive to different orientations for different spatial frequencies and vice versa; this is shown to be the case, for example, for the Hubel & Wiesel model of cortical orientation-tuned simple cells receiving inputs from an aligned row of center/surround LGN cells. When these results are considered in conjunction with psychophysical evidence for nonseparability of spatial frequency and orientation tuning properties within a “channel”, it becomes mandatory that future spatial vision research of the Fourier genre take on an explicitly two-dimensional character.