Conjectures regarding the nonlinear geometry of visual neurons

Abstract

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.

Selectivity, hyperselectivity, and the tuning of V1 neurons

“…A second approach argues that nonlinearities provide an efficient sparse overcomplete code (e.g., Golden, Vilankar, Wu, & Field, 2016;Zhu & Rozell, 2013). Following from Zetzsche's work (e.g., Zetzsche, Krieger, & Wegmann, 1999;Zetzsche & Rohrbein, 2001), we have argued that these nonlinearities follow from a relatively simple curvature in the iso-response surfaces of these neurons (Golden et al, 2016). We argued that sparse coding produces this curvature to reduce the redundancy resulting from the nonorthogonal neurons that are produced by using overcomplete codes.…”

“…One approach argues that these nonlinearities serve to control the gain of a neuron (e.g., Pagan, Simoncelli, & Rust, 2016;Schwartz & Simoncelli, 2001;Tolhurst & Heeger, 1997). A second approach argues that nonlinearities provide an efficient sparse overcomplete code (e.g., Golden, Vilankar, Wu, & Field, 2016;Zhu & Rozell, 2013). Following from Zetzsche's work (e.g., Zetzsche, Krieger, & Wegmann, 1999;Zetzsche & Rohrbein, 2001), we have argued that these nonlinearities follow from a relatively simple curvature in the iso-response surfaces of these neurons (Golden et al, 2016).…”

“…Rather, we have two goals here. First, along the lines of Zetzsche (e.g., Zetzsche et al, 1999;Zetzsche & Nuding, 2005) and Golden et al (2016), we wish to emphasize that many of these nonlinearities can be described in terms of simple curvature of the response surfaces. Second, we argue that this curvature is responsible for a number of interesting behaviors that we believe provide deeper insights into the function of these nonlinearities.…”

“…We follow this introduction with the forms of curvature produced by various nonlinear models. Some of these models were introduced in our previous work (Golden et al, 2016). However, we extend this discussion to include cascaded linearnonlinear models (e.g., Pagan et al, 2016;Schwartz, Pillow, Rust, & Simoncelli, 2006).…”

Towards a state-space geometry of neural responses to natural scenes: A steady-state approach

“…This geometric distribution of responses can then be understood in accordance with the distribution of images that have been projected to different encoding spaces (such as those defined by visual filter outputs). This general approach has been used in theories of sparse coding [29] and the non-linear behavior of visual neurons [30][31]. By focusing on the full state-space geometry of the responses produced by an evoked potential (rather than simple features of the response), our experiments will show that it is possible to provide both a rational model of the signal as well as to provide an estimate of the information carried by that signal.…”