Hyperplane Neural Codes and the Polar Complex

Itskov, Vladimir; Kunin, Alex; Rosen, Zvi

doi:10.48550/arxiv.1801.02304

Cited by 4 publications

(5 citation statements)

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“…Itskov, Kunin, and Rosen showed that if C is a nondegenerate hyperplane code, the polar complex of C is shellable [9]. Thus, our result that polar complexes of inductively pierced codes are shellable motivated us to look for hyperplane realizations of inductively pierced codes.…”

Section: Introductionmentioning

confidence: 72%

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Geometry, combinatorics, and algebra of inductively pierced codes

Lienkaemper

2018

Preprint

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Convex neural codes are combinatorial structures describing the intersection pattern of a collection of convex sets. Inductively pierced codes are a particularly nice subclass of neural codes introduced in the information visualization literature by Stapleton et al. in 2011 and to the convex codes literature by Gross et al. in 2016.Here, we show that all inductively pierced codes are nondegenerate convex codes and nondegenerate hyperplane codes. In particular, we prove that a k-inductively pierced code on n neurons has a convex realization with balls in R k+1 and with half spaces in R n . We characterize the simplicial and polar complexes of inductively pierced codes, showing the simplicial complexes are disjoint unions of vertex decomposable clique complexes and that the polar complexes are shellable. In an earlier version of this preprint, we gave a flawed proof that toric ideals of k-inductively pierced codes have quadratic Gröbner bases under the term order induced by a shelling order of the polar complex of C. We now state this as a conjecture, inspired by computational evidence.

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Section: Introductionmentioning

confidence: 72%

“…Inductively pierced codes are hyperplane codes. Itskov, Kunin, and Rosen show that if C is a nondegenerate hyperplane code, Γ(C) is shellable [9]. Further, they showed all other known combinatorial characterizations of nondegenerate hyperplane codes follow from the shellability of Γ(C).…”

Section: 3mentioning

confidence: 97%

Geometry, combinatorics, and algebra of inductively pierced codes

Lienkaemper

2018

Preprint

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“…We define the region R 12 to contain the face-interior labeled by (12), and R 1 to contain the dashed lines of its boundary and the vertex (1) (so it is closed), which is why in Proposition 3.10 we must pass from the unions of regions R σ to their relative interiors.…”

Section: Locally Good Codes Are Good-cover Codesmentioning

confidence: 99%

Neural codes, decidability, and a new local obstruction to convexity

Chen¹,

Frick²,

Shiu³

2018

Preprint

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Given an intersection pattern of arbitrary sets in Euclidean space, is there an arrangement of convex open sets in Euclidean space that exhibits the same intersections? This question is combinatorial and topological in nature, but is motivated by neuroscience. Specifically, we are interested in a type of neuron called a place cell, which fires precisely when an organism is in a certain region, usually convex, called a place field. The earlier question, therefore, can be rephrased as follows: Which neural codes, that is, patterns of neural activity, can arise from a collection of convex open sets? To address this question, Giusti and Itskov proved that convex neural codes have no "local obstructions," which are defined via the topology of a code's simplicial complex. Codes without local obstructions are called locally good, because the obstruction precludes the code from encoding the intersections of open sets that form a good cover. In other words, every good-cover code is locally good. Here we prove the converse: Every locally good code is a good-cover code. We also prove that the good-cover decision problem is undecidable. Finally, we reveal a stronger type of local obstruction that prevents a code from being convex, and prove that the corresponding decision problem is NP-hard. Our proofs use combinatorial and topological methods.

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“…Our work fits into the literature on neural codes as follows. Like previous works, we are motivated by the question of convexity in neural codes [3,6,14,15,16,19,21], with a specific interest in using neural ideals to study convexity [5,7,8,10,11,17]. Also, our factor complexes are motivated by the closely related polar complexes introduced recently by Güntürkün et al [9] (see also [1,11]).…”

Section: Introductionmentioning

confidence: 99%