Convexity of Neural Codes with Four Maximal Codewords
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912665869221888 |
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| author | Ahmed, Saber Crepeau, Natasha Flores, Gisel Isekenegbe, Osiano Perez, Deanna Shiu, Anne |
| author_facet | Ahmed, Saber Crepeau, Natasha Flores, Gisel Isekenegbe, Osiano Perez, Deanna Shiu, Anne |
| contents | Place cells are neurons that act as biological position sensors, associated with and firing in response to regions of an environment to situate an organism in space. These associations are recorded in (combinatorial) neural codes, motivating the following mathematical question: Which neural codes are generated by a collection of convex open sets in Euclidean space? Giusti and Itskov showed that a necessary condition for convexity is the absence of ``local obstructions." This necessary condition is, in fact, sufficient for certain families of codes. One such family consists of all codes with up to three maximal codewords. In this article, we investigate codes with four maximal codewords, showing that for many such codes, convexity is characterized by the absence of local obstructions, whereas for other such codes, convexity is characterized by the absence of local obstructions and a second type of obstruction, a ``wheel". Key to our analysis is a case-by-case investigation based on the nerve complex of the set of maximal codewords of a neural code. Up to symmetry, there are 20 possible nerves; and our results fully characterize convexity in 15 of the 20 cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convexity of Neural Codes with Four Maximal Codewords Ahmed, Saber Crepeau, Natasha Flores, Gisel Isekenegbe, Osiano Perez, Deanna Shiu, Anne Combinatorics Neurons and Cognition 13F55, 52A20, 92C20 Place cells are neurons that act as biological position sensors, associated with and firing in response to regions of an environment to situate an organism in space. These associations are recorded in (combinatorial) neural codes, motivating the following mathematical question: Which neural codes are generated by a collection of convex open sets in Euclidean space? Giusti and Itskov showed that a necessary condition for convexity is the absence of ``local obstructions." This necessary condition is, in fact, sufficient for certain families of codes. One such family consists of all codes with up to three maximal codewords. In this article, we investigate codes with four maximal codewords, showing that for many such codes, convexity is characterized by the absence of local obstructions, whereas for other such codes, convexity is characterized by the absence of local obstructions and a second type of obstruction, a ``wheel". Key to our analysis is a case-by-case investigation based on the nerve complex of the set of maximal codewords of a neural code. Up to symmetry, there are 20 possible nerves; and our results fully characterize convexity in 15 of the 20 cases. |
| title | Convexity of Neural Codes with Four Maximal Codewords |
| topic | Combinatorics Neurons and Cognition 13F55, 52A20, 92C20 |
| url | https://arxiv.org/abs/2510.20323 |