{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T09:17:12Z","timestamp":1774689432576,"version":"3.50.1"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2018,6]]},"abstract":"<jats:p> The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gr\u00f6bner basis with respect to that monomial order. How are these two types of generating sets \u2014 canonical forms and Gr\u00f6bner bases \u2014 related? Our main result states that if the canonical form of a neural ideal is a Gr\u00f6bner basis, then it is the universal Gr\u00f6bner basis (that is, the union of all reduced Gr\u00f6bner bases). Furthermore, we prove that this situation \u2014 when the canonical form is a Gr\u00f6bner basis \u2014 occurs precisely when the universal Gr\u00f6bner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Gr\u00f6bner basis? (2) When the universal Gr\u00f6bner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice. <\/jats:p>","DOI":"10.1142\/s0218196718500261","type":"journal-article","created":{"date-parts":[[2018,3,19]],"date-time":"2018-03-19T04:46:12Z","timestamp":1521434772000},"page":"553-571","source":"Crossref","is-referenced-by-count":11,"title":["Gr\u00f6bner bases of neural ideals"],"prefix":"10.1142","volume":"28","author":[{"given":"Rebecca","family":"Garcia","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX 77341-2206, USA"}]},{"given":"Luis David Garc\u00eda","family":"Puente","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX 77341-2206, USA"}]},{"given":"Ryan","family":"Kruse","sequence":"additional","affiliation":[{"name":"Mathematics Department, Central College, Pella, IA 50219, USA"}]},{"given":"Jessica","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bard College, Annandale, NY 12504, USA"}]},{"given":"Dane","family":"Miyata","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA"}]},{"given":"Ethan","family":"Petersen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA"}]},{"given":"Kaitlyn","family":"Phillipson","sequence":"additional","affiliation":[{"name":"Department of Mathematics, St. Edwards University, Austin, TX 78704-6489, USA"}]},{"given":"Anne","family":"Shiu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Texas A&amp;M University, College Station, TX 77843, USA"}]}],"member":"219","published-online":{"date-parts":[[2018,6,28]]},"reference":[{"key":"S0218196718500261BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2006.10.002"},{"key":"S0218196718500261BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0913-3"},{"key":"S0218196718500261BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2008.02.017"},{"key":"S0218196718500261BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-16721-3"},{"key":"S0218196718500261BIB006","doi-asserted-by":"publisher","DOI":"10.1137\/16M1073170"},{"key":"S0218196718500261BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/s11538-013-9860-3"},{"key":"S0218196718500261BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s10801-013-0450-0"},{"key":"S0218196718500261BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2017.10.002"},{"key":"S0218196718500261BIB012","volume-title":"Computational Commutative Algebra 1","author":"Kreuzer M.","year":"2008"},{"key":"S0218196718500261BIB013","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2016.11.006"},{"key":"S0218196718500261BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(88)80042-7"},{"key":"S0218196718500261BIB015","doi-asserted-by":"publisher","DOI":"10.1016\/0006-8993(71)90358-1"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196718500261","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T15:45:45Z","timestamp":1565106345000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196718500261"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6]]},"references-count":12,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2018,6,28]]},"published-print":{"date-parts":[[2018,6]]}},"alternative-id":["10.1142\/S0218196718500261"],"URL":"https:\/\/doi.org\/10.1142\/s0218196718500261","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6]]}}}