Classification of Open and Closed Convex Codes on Five Neurons

TL;DR

This paper classifies all neural codes with five neurons as convex or non-convex, exploring their representation by open and closed convex sets, and provides a minimal example of a code that is open but not closed convex.

Contribution

It provides a complete classification of five-neuron neural codes regarding convexity and introduces the first known minimal example of an open convex code that is not closed convex.

Findings

All five-neuron codes are classified as convex or non-convex.

Identifies codes representable by open but not closed convex sets.

Provides a minimal example of an open but not closed convex code.

Abstract

Neural codes, represented as collections of binary strings, encode neural activity and show relationships among stimuli. Certain neurons, called place cells, have been shown experimentally to fire in convex regions in space. A natural question to ask is: Which neural codes can arise as intersection patterns of convex sets? While past research has established several criteria, complete conditions for convexity are not yet known for codes with more than four neurons. We classify all neural codes with five neurons as convex/non-convex codes. Furthermore, we investigate which of these codes can be represented by open versus closed convex sets. Interestingly, we find a code which is an open but not closed convex code and demonstrate a minimal example for this phenomenon.