{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T16:14:11Z","timestamp":1780762451582,"version":"3.54.1"},"reference-count":42,"publisher":"MIT Press - Journals","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2013,11]]},"abstract":"<jats:p> Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as \u201cpermitted sets\u201d of the network. We introduce a simple encoding rule that selectively turns \u201con\u201d synapses between neurons that coappear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states (\u201con\u201d or \u201coff\u201d), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the encoding rule, including unintended \u201cspurious\u201d states, and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced\u2014i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are natural in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks. <\/jats:p>","DOI":"10.1162\/neco_a_00504","type":"journal-article","created":{"date-parts":[[2013,7,29]],"date-time":"2013-07-29T17:48:52Z","timestamp":1375120132000},"page":"2858-2903","source":"Crossref","is-referenced-by-count":19,"title":["Encoding Binary Neural Codes in Networks of Threshold-Linear Neurons"],"prefix":"10.1162","volume":"25","author":[{"given":"Carina","family":"Curto","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anda","family":"Degeratu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Albert-Ludwig-Universit\u00e4t, Freiburg 79104, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vladimir","family":"Itskov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00337259"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623257"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623257"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.32.1007"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4916(87)90092-3"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780195100273.001.0001"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1016\/j.tins.2012.03.008"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/054"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.92.9.3844"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1126\/science.1195870"},{"key":"B12","volume-title":"Theory and applications of distance geometry.","author":"Blumenthal L. 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