{"id":"https://openalex.org/W2950320405","doi":"https://doi.org/10.48550/arxiv.1512.00897","title":"Pattern completion in symmetric threshold-linear networks","display_name":"Pattern completion in symmetric threshold-linear networks","publication_year":2015,"publication_date":"2015-12-02","ids":{"openalex":"https://openalex.org/W2950320405","doi":"https://doi.org/10.48550/arxiv.1512.00897","mag":"2950320405"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:1512.00897","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1512.00897","pdf_url":"https://arxiv.org/pdf/1512.00897","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1512.00897","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020021216","display_name":"Carina Curto","orcid":"https://orcid.org/0000-0001-9849-3592","observed_orcids":["https://orcid.org/0000-0001-9849-3592"]},"institutions":[{"id":"https://openalex.org/I130769515","display_name":"Pennsylvania State University","ror":"https://ror.org/04p491231","country_code":"US","type":"education","lineage":["https://openalex.org/I130769515"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Curto, Carina","raw_affiliation_strings":["Department of Mathematics , The Pennsylvania State University , University Park , PA 16802","Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , The Pennsylvania State University , University Park , PA 16802","institution_ids":["https://openalex.org/I130769515"]},{"raw_affiliation_string":"Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.#TAB#","institution_ids":["https://openalex.org/I130769515"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073186807","display_name":"Katherine Morrison","orcid":"https://orcid.org/0000-0002-8402-3212","observed_orcids":["https://orcid.org/0000-0002-8402-3212"]},"institutions":[{"id":"https://openalex.org/I130769515","display_name":"Pennsylvania State University","ror":"https://ror.org/04p491231","country_code":"US","type":"education","lineage":["https://openalex.org/I130769515"]},{"id":"https://openalex.org/I71966907","display_name":"University of Northern Colorado","ror":"https://ror.org/016bysn57","country_code":"US","type":"education","lineage":["https://openalex.org/I71966907"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Morrison, Katherine","raw_affiliation_strings":["Department of Mathematics , The Pennsylvania State University , University Park , PA 16802","School of Mathematical Sciences , University of Northern Colorado , Greeley , CO 80639","Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A., and School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , The Pennsylvania State University , University Park , PA 16802","institution_ids":["https://openalex.org/I130769515"]},{"raw_affiliation_string":"School of Mathematical Sciences , University of Northern Colorado , Greeley , CO 80639","institution_ids":["https://openalex.org/I71966907"]},{"raw_affiliation_string":"Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A., and School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639, U.S.A","institution_ids":["https://openalex.org/I130769515","https://openalex.org/I71966907"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10581","display_name":"Neural dynamics and brain function","score":0.8330000042915344,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10581","display_name":"Neural dynamics and brain function","score":0.8330000042915344,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.14640000462532043,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10448","display_name":"Memory and Neural Mechanisms","score":0.006000000052154064,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/symmetric-networks","display_name":"symmetric networks","score":0.7229999899864197},{"id":"https://openalex.org/keywords/pattern-completion","display_name":"pattern completion","score":0.847000002861023},{"id":"https://openalex.org/keywords/stable-fixed-points","display_name":"stable fixed points","score":0.9769999980926514},{"id":"https://openalex.org/keywords/maximal-cliques","display_name":"maximal cliques","score":0.8550000190734863},{"id":"https://openalex.org/keywords/error-correction","display_name":"error correction","score":0.9539999961853027},{"id":"https://openalex.org/keywords/recurrent-neural-networks","display_name":"recurrent neural networks","score":0.6340000033378601},{"id":"https://openalex.org/keywords/memory-encoding-and-retrieval","display_name":"memory encoding and retrieval","score":0.625},{"id":"https://openalex.org/keywords/distance-geometry","display_name":"distance geometry","score":0.9279999732971191}],"concepts":[{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.7786052823066711},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5887341499328613},{"id":"https://openalex.org/C136134351","wikidata":"https://www.wikidata.org/wiki/Q431937","display_name":"Antichain","level":3,"score":0.5556401014328003},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.49664169549942017},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46982312202453613},{"id":"https://openalex.org/C27333817","wikidata":"https://www.wikidata.org/wiki/Q6134913","display_name":"Least fixed point","level":5,"score":0.4377838373184204},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.427212119102478},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4226991832256317},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.41203612089157104},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37205570936203003},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3162655234336853},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.2294081151485443},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.10177525877952576},{"id":"https://openalex.org/C28642324","wikidata":"https://www.wikidata.org/wiki/Q656176","display_name":"Schauder fixed point theorem","level":4,"score":0.08887147903442383},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C206929604","wikidata":"https://www.wikidata.org/wiki/Q530152","display_name":"Picard\u2013Lindel\u00f6f theorem","level":3,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:1512.00897","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1512.00897","pdf_url":"https://arxiv.org/pdf/1512.00897","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.1512.00897","doi":"https://doi.org/10.48550/arxiv.1512.00897","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1512.00897","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1512.00897","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1512.00897","pdf_url":"https://arxiv.org/pdf/1512.00897","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1965448315","display_name":"Theory of threshold-linear networks and combinatorial neural codes.","funder_award_id":"1516881","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5060290107","display_name":"Memory encoding in spatially structured networks: dynamics, discrete geometry & topology","funder_award_id":"1225666","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7216168265","display_name":"Memory encoding in spatially structured networks: dynamics, discrete geometry & topology","funder_award_id":"1537228","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2950320405.pdf","grobid_xml":"https://content.openalex.org/works/W2950320405.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W30129623","https://openalex.org/W178022270","https://openalex.org/W2032987831","https://openalex.org/W2052515926","https://openalex.org/W2108902061","https://openalex.org/W2112246162","https://openalex.org/W2115761837","https://openalex.org/W2123745704","https://openalex.org/W2127780167","https://openalex.org/W2128084896","https://openalex.org/W2147103442","https://openalex.org/W2610857016"],"related_works":["https://openalex.org/W178022270","https://openalex.org/W2147103442","https://openalex.org/W2972304349","https://openalex.org/W2115761837","https://openalex.org/W7096967096","https://openalex.org/W2795469741","https://openalex.org/W2079222338","https://openalex.org/W2072617874","https://openalex.org/W4414740232","https://openalex.org/W2098411549"],"abstract_inverted_index":{"Threshold-linear":[0],"networks":[1,21,50,112],"are":[2,125],"a":[3,83,92,106,132,137,151],"common":[4],"class":[5],"of":[6,47,61,67,73,85,91,102,136,163,188,196],"firing":[7],"rate":[8],"models":[9],"that":[10,143],"describe":[11],"recurrent":[12],"interactions":[13],"among":[14],"neurons.":[15,69],"Unlike":[16],"their":[17,178],"linear":[18],"counterparts,":[19],"these":[20],"generically":[22],"possess":[23],"multiple":[24],"stable":[25,44,93,107,154],"fixed":[26,45,62,94,108,155],"points":[27,46,63,156],"(steady":[28],"states),":[29],"making":[30],"them":[31],"viable":[32],"candidates":[33],"for":[34,119,144,172,180],"memory":[35],"encoding":[36],"and":[37,55,176,183],"retrieval.":[38],"In":[39,70],"this":[40],"work,":[41],"we":[42,76,148,168],"characterize":[43],"general":[48],"threshold-linear":[49,111,200],"with":[51],"constant":[52],"external":[53],"drive,":[54],"discover":[56],"constraints":[57],"on":[58,193],"the":[59,71,78,89,123,160,194],"co-existence":[60],"involving":[64],"different":[65],"subsets":[66],"active":[68],"case":[72],"symmetric":[74],"networks,":[75,201],"prove":[77],"following":[79],"antichain":[80],"property:":[81],"if":[82],"set":[84],"neurons":[86],"$\u03c4$":[87,103],"is":[88],"support":[90,105],"point,":[95],"then":[96],"no":[97],"proper":[98],"subset":[99,133],"or":[100,134],"superset":[101,135],"can":[104,149],"point.":[109],"Symmetric":[110],"thus":[113],"appear":[114],"to":[115,128,159,205],"be":[116],"well":[117],"suited":[118],"pattern":[120,184],"completion,":[121],"since":[122],"dynamics":[124],"guaranteed":[126],"not":[127],"get":[129],"\"stuck\"":[130],"in":[131,199],"stored":[138],"pattern.":[139],"We":[140],"also":[141],"show":[142],"any":[145],"graph":[146],"G,":[147],"construct":[150],"network":[152,170],"whose":[153],"correspond":[157],"precisely":[158],"maximal":[161],"cliques":[162],"G.":[164],"As":[165],"an":[166],"application,":[167],"design":[169],"decoders":[171],"place":[173],"field":[174],"codes,":[175],"demonstrate":[177],"efficacy":[179],"error":[181],"correction":[182],"completion.":[185],"The":[186],"proofs":[187],"our":[189],"main":[190],"results":[191],"build":[192],"theory":[195],"permitted":[197],"sets":[198],"including":[202],"recently-developed":[203],"connections":[204],"classical":[206],"distance":[207],"geometry.":[208]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2019-06-27T00:00:00"}