Covering Relations in the Poset of Combinatorial Neural Codes

A combinatorial neural code is a subset of the power set $2^{[n]}$ on $[n]=\{1,\dots, n\}$, in which each $1\leq i\leq n$ represents a neuron and each element (codeword) represents the co-firing event of some neurons. Consider a space $X\subseteq\mathbb{R}^d$, simulating an animal's environment, and a collection $\mathcal{U}=\{U_1,\dots,U_n\}$ of open subsets of $X$. Each $U_i\subseteq X$ simulates a place field which is a specific region where a place cell $i$ is active. Then, the code of $\mathcal{U}$ in $X$ is defined as $\text{code}(\mathcal{U},X)=\left\{\sigma\subseteq[n]\bigg|\bigcap_{i\in\sigma} U_i\setminus\bigcup_{j\notin\sigma}U_j\neq\varnothing\right\}$. If a neural code $\mathcal{C}=\text{code}(\mathcal{U},X)$ for some $X$ and $\mathcal{U}$, we say $\mathcal{C}$ has a realization of open subsets of some space $X$. Although every combinatorial neural code obviously has a realization by some open subsets, determining whether it has a realization by some open convex subsets remains unsolved. Many studies attempted to tackle this decision problem, but only partial results were achieved. In fact, a previous study showed that the decision problem of convex neural codes is NP-hard. Furthermore, the authors of this study conjectured that every convex neural code can be realized as a minor of a neural code arising from a representable oriented matroid, which can lead to an equivalence between convex and polytope convex neural codes. Even though this conjecture has been confirmed in dimension two, its validity in higher dimensions is still unknown. To advance the investigation of this conjecture, we provide a complete characterization of the covering relations within the poset $\mathbf{P_{Code}}$ of neural codes.