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Properties of graphs of neural codes

Suhith K N, Neha Gupta

TL;DR

The connectedness and completeness of CCG are preserved under surjective morphisms between neural codes defined by A. Jeffs, and it is proved that a code whose CCG is complete is open convex.

Abstract

A neural code on $ n $ neurons is a collection of subsets of the set $ [n]=\{1,2,\dots,n\} $. In this paper, we study some properties of graphs of neural codes. In particular, we study codeword containment graph (CCG) given by Chan et al. (SIAM J. on Dis. Math., 37(1):114-145,2017) and general relationship graph (GRG) given by Gross et al. (Adv. in App. Math., 95:65-95, 2018). We provide a sufficient condition for CCG to be connected. We also show that the connectedness and completeness of CCG are preserved under surjective morphisms between neural codes defined by A. Jeffs (SIAM J. on App. Alg. and Geo., 4(1):99-122,2020). Further, we show that if CCG of any neural code $\mathcal{C}$ is complete with $|\mathcal{C}|=m$, then $\mathcal{C} \cong \{\emptyset,1,12,\dots,123\cdots m\}$ as neural codes. We also prove that a code whose CCG is complete is open convex. Later, we show that if a code $\mathcal{C}$ with $|\mathcal{C}|>3$ has its CCG to be connected 2-regular then $|\mathcal{C}| $ is even. The GRG was defined only for degree two neural codes using the canonical forms of its neural ideal. We first define GRG for any neural code. Then, we show the behaviour of GRGs under the various elementary code maps. At last, we compare these two graphs for certain classes of codes and see their properties.

Properties of graphs of neural codes

TL;DR

The connectedness and completeness of CCG are preserved under surjective morphisms between neural codes defined by A. Jeffs, and it is proved that a code whose CCG is complete is open convex.

Abstract

A neural code on neurons is a collection of subsets of the set . In this paper, we study some properties of graphs of neural codes. In particular, we study codeword containment graph (CCG) given by Chan et al. (SIAM J. on Dis. Math., 37(1):114-145,2017) and general relationship graph (GRG) given by Gross et al. (Adv. in App. Math., 95:65-95, 2018). We provide a sufficient condition for CCG to be connected. We also show that the connectedness and completeness of CCG are preserved under surjective morphisms between neural codes defined by A. Jeffs (SIAM J. on App. Alg. and Geo., 4(1):99-122,2020). Further, we show that if CCG of any neural code is complete with , then as neural codes. We also prove that a code whose CCG is complete is open convex. Later, we show that if a code with has its CCG to be connected 2-regular then is even. The GRG was defined only for degree two neural codes using the canonical forms of its neural ideal. We first define GRG for any neural code. Then, we show the behaviour of GRGs under the various elementary code maps. At last, we compare these two graphs for certain classes of codes and see their properties.
Paper Structure (17 sections, 22 theorems, 6 equations, 10 figures)

This paper contains 17 sections, 22 theorems, 6 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Neural ring homomorphisms
  4. Morphisms
  5. Codeword containment graph (CCG)
  6. Connectedness of CCG
  7. Complete CCG and morphisms
  8. 2-Regular CCG
  9. Canonical forms and their behavior under elementary code maps
  10. Neural ideal and its canonical form
  11. Algorithm to generate canonical form of neural ideals
  12. Behaviour of canonical forms under elementary code maps
  13. General relationship graph (GRG)
  14. Behaviour of general relationship graph under elementary code maps
  15. Comparing GRG and CCG for certain codes
  16. ...and 2 more sections

Key Result

Theorem 2.2

curto2020neural A map $\phi:\mathcal{R}_{\mathcal{D}}\rightarrow \mathcal{R}_{\mathcal{C}}$ is a neural ring homomorphism if and only if $q_\phi$ is a composition of the following code maps: 1. Permutation, 2. Adding a trivial neuron (or deleting a trivial neuron), 3. Duplication of a neuron (or del

Figures (10)

  • Figure 1: Codeword containment graphs
  • Figure 2: $G_\mathcal{C}$ when $\mathcal{C}=\{\emptyset,1,2,123,4\}$
  • Figure 3:
  • Figure 4: Complete codeword containment graph
  • Figure 5: 2-regular codeword containment graph
  • ...and 5 more figures

Theorems & Definitions (62)

  • Definition 2.1: Neural ring homomorphism
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Definition 3.1: Codeword containment graph (CCG)
  • Example 3.2
  • Remark 3.3
  • ...and 52 more