A variety of studies have now demonstrated that the basic 'wavelet-like' parameters of V1 neurons (e.g., orientation tuning, spatial frequency tuning etc.) provide an efficient means of representing natural images. 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.
Is there a single geometrical framework that can describe all of these non-linear properties? It is argued that existing non-linearities can be described by parameters relating to the curvature of the response surfaces. 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 iso-response surface). 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.
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