Neural codes via homological invariants of polarized neural ideals

Abstract

For a neural code $\mathcal{C}\subseteq {\mathbb{F}}_2^n$, polarizing the canonical form generators of the neural ideal $J_{\mathcal{C}}$ yields a squarefree monomial ideal $\mathcal{P}(J_{\mathcal{C}})\subset k[x_1,\dots ,x_n,y_1,\dots ,y_n]$, the polarized neural ideal, and an associated simplicial complex ${\Delta }_{\mathcal{C}}$, the polar complex. We study the graded invariants $\mathrm{pd}(\mathcal{P}(J_{\mathcal{C}}))$ and $\mathrm{reg}(\mathcal{P}(J_{\mathcal{C}}))$ via the topology of ${\Delta }_{\mathcal{C}}$, showing that simple geometric features of the Hamming cube ${\mathbb{F}}_2^n$ (with Hamming distance) organize their extremal behavior. We prove $\mathrm{reg}(\mathcal{P}(J_{\mathcal{C}}))\le 2n-1$, with equality precisely when $\mathcal{C}$ is obtained from ${\mathbb{F}}_2^n$ by deleting an antipodal pair. Using connectedness properties of induced subcomplexes of ${\Delta }_{\mathcal{C}}$, we obtain $\mathrm{pd}(\mathcal{P}(J_{\mathcal{C}}))\le 2n-3$, and we give an explicit family of codes attaining equality, each consisting of antipodal pairs. At the opposite end, we identify the cube geometry behind the smallest values: $\mathrm{reg}(\mathcal{P}(J_{\mathcal{C}}))=1$ forces $\mathcal{C}$ to be a coordinate subcube of ${\mathbb{F}}_2^n$, while $\mathrm{pd}(\mathcal{P}(J_{\mathcal{C}}))=0$ forces $\mathcal{C}$ to be the complement of one. Finally, we construct families realizing large regions of the $(\mathrm{pd},\mathrm{reg})$-plot for fixed $n$.

29 pages, 4 figures