Convex Neural Codes in Dimension 1

Rosen, Zvi; X., Zhang, Yan

doi:10.48550/arxiv.1702.06907

Cited by 6 publications

(9 citation statements)

References 14 publications

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“…This conjecture is related to a result of Rosen and Zhang. Rosen and Zhang characterized codes that are realizable by open convex sets in dimension 1 [7]. We speculate that these codes are precisely the codes with minimal convex embedding dimension 1 (Conjecture 1).…”

Section: Resultsmentioning

confidence: 89%

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Every binary code can be realized by convex sets

Muthiah

2018

Advances in Applied Mathematics

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Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by convex sets when there is no restriction on whether the sets are all open or closed. We achieve this by constructing a convex realization for an arbitrary code with k nonempty codewords in R k−1 . This result justifies the usual restriction of the definition of convex neural codes to include only those that can be realized by receptive fields that are all either open convex or closed convex. We also show that the dimension of our construction cannot in general be lowered.

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Section: Resultsmentioning

confidence: 89%

“…We introduce the following notation: [6]. Rosen and Zhang characterized codes that are realizable by open convex sets in dimension 1 [7]. Less work has been done investigating convex codes without regard to openness or closedness because these codes are not directly motivated from the receptive fields of place cells.…”

Section: Introductionmentioning

confidence: 99%

Every binary code can be realized by convex sets

Muthiah

2018

Advances in Applied Mathematics

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“…Theorem 3.4 states that convex codes with up to three maximal codewords have minimal embedding dimension one or two. To distinguish between these two possible dimensions, we refer the reader to the classification of 1-dimensional codes due to Rosen and Yan [13].…”

Section: Resultsmentioning

confidence: 99%

Neural Codes With Three Maximal Codewords: Convexity and Minimal Embedding Dimension

Johnston¹,

Shiu²,

Clare³

2020

Preprint

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Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous work has focused on addressing the question: how can we tell when a neural code is convex? Giusti and Itskov identified a local obstruction and proved that convex neural codes have no local obstructions. The converse is true for codes on up to four neurons, but false in general. Nevertheless, we prove that this converse holds for codes with up to three maximal codewords, and moreover the minimal embedding dimension of such codes is at most two.

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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%