Conjectures regarding the nonlinear geometry of visual neurons

•Curvature in the response surface of a neuron describes many early non-linearities.•Endo-origin curvature describes invariance and tolerance.•Exo-origin curvature describes a number of non-classical effects.•Non-linearities in sparse coding networks are well described by this curvature. From the earliest stages of sensory processing, neurons show inherent non-linearities: the response to a complex stimulus is not a sum of the responses to a set of constituent basis stimuli. These non-linearities come in a number of forms and have been explained in terms of a number of functional goals. The family of spatial non-linearities have included interactions that occur both within and outside of the classical receptive field. They include, saturation, cross orientation inhibition, contrast normalization, end-stopping and a variety of non-classical effects. In addition, neurons show a number of facilitatory and invariance related effects such as those exhibited by complex cells (integration across position). Here, we describe an approach that attempts to explain many of the non-linearities under a single geometric framework. In line with Zetzsche and colleagues (e.g., Zetzsche et al., 1999) we propose that many of the principal non-linearities can be described by a geometry where the neural response space has a simple curvature. In this paper, we focus on the geometry that produces both increased selectivity (curving outward) and increased tolerance (curving inward). We demonstrate that overcomplete sparse coding with both low-dimensional synthetic data and high-dimensional natural scene data can result in curvature that is responsible for a variety of different known non-classical effects including end-stopping and gain control. We believe that this approach provides a more fundamental explanation of these non-linearities and does not require that one postulate a variety of explanations (e.g., that gain must be controlled or the ends of lines must be detected). In its standard form, sparse coding does not however, produce invariance/tolerance represented by inward curvature. We speculate on some of the requirements needed to produce such curvature.