Algebraic signatures of convex and non-convex codes

Carina Pamela Curto, Elizabeth Gross, Jack Jeffries, Katherine Morrison, Zvi Rosen, Anne Shiu, Nora Youngs

Research output: Contribution to journalArticle

Abstract

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we develop algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley–Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.

Original languageEnglish (US)
Pages (from-to)3919-3940
Number of pages22
JournalJournal of Pure and Applied Algebra
Volume223
Issue number9
DOIs
StatePublished - Sep 1 2019

Fingerprint

Signature
Receptive Field
Non-convexity
Neuroscience
Generating Set
Algebraic Methods
Complete Intersection
Binary Code
Canonical form
Open set
Convex Sets
Convexity
Euclidean space
Neuron
Arrangement
Intersection

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Curto, C. P., Gross, E., Jeffries, J., Morrison, K., Rosen, Z., Shiu, A., & Youngs, N. (2019). Algebraic signatures of convex and non-convex codes. Journal of Pure and Applied Algebra, 223(9), 3919-3940. https://doi.org/10.1016/j.jpaa.2018.12.012
Curto, Carina Pamela ; Gross, Elizabeth ; Jeffries, Jack ; Morrison, Katherine ; Rosen, Zvi ; Shiu, Anne ; Youngs, Nora. / Algebraic signatures of convex and non-convex codes. In: Journal of Pure and Applied Algebra. 2019 ; Vol. 223, No. 9. pp. 3919-3940.
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Curto, CP, Gross, E, Jeffries, J, Morrison, K, Rosen, Z, Shiu, A & Youngs, N 2019, 'Algebraic signatures of convex and non-convex codes', Journal of Pure and Applied Algebra, vol. 223, no. 9, pp. 3919-3940. https://doi.org/10.1016/j.jpaa.2018.12.012

Algebraic signatures of convex and non-convex codes. / Curto, Carina Pamela; Gross, Elizabeth; Jeffries, Jack; Morrison, Katherine; Rosen, Zvi; Shiu, Anne; Youngs, Nora.

In: Journal of Pure and Applied Algebra, Vol. 223, No. 9, 01.09.2019, p. 3919-3940.

Research output: Contribution to journalArticle

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Curto CP, Gross E, Jeffries J, Morrison K, Rosen Z, Shiu A et al. Algebraic signatures of convex and non-convex codes. Journal of Pure and Applied Algebra. 2019 Sep 1;223(9):3919-3940. https://doi.org/10.1016/j.jpaa.2018.12.012