Encoding binary neural codes in networks of threshold-linear neurons

TL;DR

This paper introduces a simple encoding rule for threshold-linear neural networks that effectively stores binary patterns, revealing geometric conditions for successful encoding and demonstrating applications to hippocampal place field codes.

Contribution

The study provides a precise characterization of pattern storage in threshold-linear networks using a new encoding rule and geometric analysis, connecting neural coding with convex and distance geometry.

Findings

Binary patterns are stored when excitatory connections are geometrically balanced.

Certain neural codes are 'natural' and can be learned from few patterns.

Networks can nearly exactly encode hippocampal place fields with minimal patterns.

Abstract

Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as "permitted sets" of the network. We introduce a simple Encoding Rule that selectively turns "on" synapses between neurons that co-appear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states ("on" or "off"), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the Encoding Rule -- including unintended "spurious" states -- and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced -- i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are "natural" in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks.