Nonlinear AND interactions between frequency components and the selective processing of intrinsically two-dimensional signals by cortical neurons

Abstract

The standard model of visual processing is based on the selective properties of linear spatial filters which are tuned to different orientations and radial frequencies. This standard model is well suited for the description of a wide range of phenomena in vision but it is not clear, whether the whole range of basic properties of early vision is entirely within the models explanatory scope. Here we suggest that there exists a basic selective processing property in early vision which is definitely outside the explanatory scope of the standard model: the selectivity for intrinsically 2D signals. This property has already been observed in the classical experiments of Hubel and Wiesel, and has more recently been found in more complex form in the extra-classical receptive field properties of various visual neurons. We show here that this selectivity cannot be described within the framework of linear spatial filtering because of reasons which lie at the heart of the theory o f linear systems: the restriction of such systems to OR- combinations of their intrinsically 1D eigenfunctions. We present a general nonlinear framework for the modeling of i2D-selective systems which is based on AND-like combinations of frequency components, and which is closely related to the Wiener-Volterra representation of nonlinear systems. To our knowledge, i2D-selectivity is the only non- standard property for which such a theoretical framework yet exists. The framework enables the combination of the nonlinear i2D-selectivity with other basic selectivities of visual neurons, for examples with simple and complex-like properties, and makes it thus possible, to construct models for the variety of neurophysiological observations on the i2D-selective processing in visual neurons. As an insight of general interest for the recent discussion on second-order properties in early vision, the framework reveals the existence of extended equivalence classes in which nonlinear schemes can have very dissimilar structural properties, and lead nevertheless to identical input-output relations. Finally, there is a close relation between i2D-selectivity and the higher-order statistical redundancies in natural images.