TY - JOUR
T1 - Algebraic signatures of convex and non-convex codes
AU - Curto, Carina
AU - Gross, Elizabeth
AU - Jeffries, Jack
AU - Morrison, Katherine
AU - Rosen, Zvi
AU - Shiu, Anne
AU - Youngs, Nora
N1 - Publisher Copyright:
© 2018 Elsevier B.V.
PY - 2019/9
Y1 - 2019/9
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85059170349&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85059170349&partnerID=8YFLogxK
U2 - 10.1016/j.jpaa.2018.12.012
DO - 10.1016/j.jpaa.2018.12.012
M3 - Article
C2 - 31534273
AN - SCOPUS:85059170349
SN - 0022-4049
VL - 223
SP - 3919
EP - 3940
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 9
ER -