Neural Codes and Neural ring endomorphisms

TL;DR

This paper explores the algebraic and topological properties of neural codes, proving a conjecture for certain classes, introducing new code classes, and analyzing neural ring endomorphisms.

Contribution

It proves a conjecture relating convex embedding dimensions, introduces Doublet maximal and Circulant code classes, and studies neural ring endomorphisms.

Findings

Proved the conjecture for specific classes of neural codes.

Characterized Doublet maximal codes as open convex iff max-intersection complete.

Counted neural ring endomorphisms for Circulant code subclasses.

Abstract

We investigate combinatorial, topological and algebraic properties of certain classes of neural codes. We look into a conjecture that states if the minimal \textit{open convex} embedding dimension of a neural code is two then its minimal \textit{convex} embedding dimension is also two. We prove the conjecture for two interesting classes of examples and provide a counterexample for the converse of the conjecture. We introduce a new class of neural codes, \textit{Doublet maximal}. We show that a Doublet maximal code is open convex if and only if it is max-intersection complete. We prove that surjective neural ring homomorphisms preserve max-intersection complete property. We introduce another class of neural codes, \textit{Circulant codes}. We give the count of neural ring endomorphisms for several sub-classes of this class.