Algebraic signatures of convex and non-convex codes |
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Authors: | Carina Curto Elizabeth Gross Jack Jeffries Katherine Morrison Zvi Rosen Anne Shiu Nora Youngs |
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Institution: | 1. Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA;2. Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, HI 96822, USA;3. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA;4. School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639, USA;5. Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA;6. Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;7. Department of Mathematics and Statistics, Colby College, Waterville, ME 04901, USA |
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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. |
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Keywords: | 92 13 Neural coding Convex codes Neural ideal Local obstructions Simplicial complexes Links |
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