Neural ring homomorphism preserves mandatory sets required for open convexity

It has been studied by Curto et al. (SIAM J. on App. Alg. and Geom., 1(1) : 222 \unicode{x2013} 238, 2017) that a neural code that has an open convex realization does not have any local obstruction relative to the neural code. Further, a neural code $\mathcal{C}$ has no local obstructions if and only if it contains the set of mandatory codewords, $\mathcal{C}_{\min}(\Delta),$ which depends only on the simplicial complex $\Delta=\Delta(\mathcal{C})$. Thus if $\mathcal{C} \not \supseteq \mathcal{C}_{\min}(\Delta)$, then $\mathcal{C}$ cannot be open convex. However, the problem of constructing $\mathcal{C}_{\min}(\Delta)$ for any given code $\mathcal{C}$ is undecidable. There is yet another way to capture the local obstructions via the homological mandatory set, $\mathcal{M}_H(\Delta).$ The significance of $\mathcal{M}_H(\Delta)$ for a given code $\mathcal{C}$ is that $\mathcal{M}_H(\Delta) \subseteq \mathcal{C}_{\min}(\Delta)$ and so $\mathcal{C}$ will have local obstructions if $\mathcal{C}\not\supseteq\mathcal{M}_H(\Delta).$ In this paper we study the affect on the sets $\mathcal{C}_{\min}(\Delta)$ and $\mathcal{M}_H(\Delta)$ under the action of various surjective elementary code maps. Further, we study the relationship between Stanley-Reisner rings of the simplicial complexes associated with neural codes of the elementary code maps. Moreover, using this relationship, we give an alternative proof to show that $\mathcal{M}_H(\Delta)$ is preserved under the elementary code maps

Properties of graphs of neural codes

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

Neural category

A neural code on $n$ neurons is a collection of subsets of the set $[n]=\{1,2,\dots,n\}$. Curto et al. \cite{curto2013neural} associated a ring $\mathcal{R}_{\mathcal{C}}$ (neural ring) to a neural code $\mathcal{C}$. A special class of ring homomorphisms between two neural rings, called neural ring homomorphism, was introduced by Curto and Youngs \cite{curto2020neural}. The main work in this paper comprises constructing two categories. First is the $\mathfrak{C}$ category, a subcategory of SETS consisting of neural codes and code maps. Second is the neural category $\mathfrak{N}$, a subcategory of \textit{Rngs} consisting of neural rings and neural ring homomorphisms. Then, the rest of the paper characterizes the properties of these two categories like initial and final objects, products, coproducts, limits, etc. Also, we show that these two categories are in dual equivalence

Neural Codes and Neural ring endomorphisms

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.Comment: 22 pages, 6 figures and 9 reference