Abstract
A variety of studies have now demonstrated that the basic parameters of V1 neurons (e.g., orientation tuning, spatial frequency tuning etc.) provide an efficient means of representing natural scenes. Much of this work has modeled visual neurons as an array of linear vectors. However, it is also well known that such neurons show a wide range of non-linear response properties. They include such effects as end-stopping, cross orientation inhibition, and other non classical surround effects, as well as invariance effects shown with complex cells, and facilitory effects demonstrated with contour integration. A number of theories have been put forward to explain each of these non-linearities. Possibly the most general theory is that of “contrast normalization”, but it is not a complete account, and provides no insights into the tiling of neurons needed to represent visual space. Here, we provide a single framework for describing these visual non-linearities. It is argued that existing non-linearities can be described by parameters relating to the curvature of the response manifold and parameters of response saturation. Each dimension of the neuron requires one generalized curvature parameter and is a function of the neighboring neurons. It is argued that all forms of invariance and generalization in neural response can be represented by negative curvature while hyper-selectivity and contrast normalization require positive curvature. However, for most neurons, the vast majority of dimensions show no curvature (flat response manifold). We show how these curvatures follow from an over-complete tiling of the response space while maintaining a particular form of independence. We believe that this notion of curvature is sufficient to describe a wide variety of sensory non-linearities including those at higher levels of the visual system.