Sparse Neural Codes and Convexity

TL;DR

This paper explores the geometric and combinatorial properties of neural codes, especially 2-sparse codes, using convex geometry to understand their realizability and embedding dimensions in low-dimensional spaces.

Contribution

It characterizes realizable 2-sparse neural codes via intersection-completeness and establishes bounds on their embedding dimensions in low-dimensional Euclidean spaces.

Findings

Intersection-completeness characterizes realizable 2-sparse codes.

Realizable 2-sparse codes have embedding dimension at most 3.

Realizations with closed sets are equivalent to those with open sets in 2D and 3D.

Abstract

Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in $\mathbb{R}^d$. Combinatorial objects known as \emph{neural codes} can then be used to extract features of the space covered by these convex regions. We apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets. We restrict our attention primarily to sparse codes in low dimensions. We find that intersection-completeness characterizes realizable $2$-sparse codes, and show that any realizable $2$-sparse code has embedding dimension at most $3$. Furthermore, we prove that in $\mathbb{R}^2$ and $\mathbb{R}^3$, realizations of $2$-sparse codes using closed sets are equivalent to those with open sets, and this allows us to provide some preliminary results on distinguishing which $2$-sparse codes have embedding dimension at most $2$.