{"id":"https://openalex.org/W2556205324","doi":"https://doi.org/10.1162/neco_a_00919","title":"Exponentially Long Orbits in Hopfield Neural Networks","display_name":"Exponentially Long Orbits in Hopfield Neural Networks","publication_year":2016,"publication_date":"2016-11-21","ids":{"openalex":"https://openalex.org/W2556205324","doi":"https://doi.org/10.1162/neco_a_00919","mag":"2556205324","pmid":"https://pubmed.ncbi.nlm.nih.gov/27870611"},"language":"en","primary_location":{"id":"doi:10.1162/neco_a_00919","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_00919","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://infoscience.epfl.ch/record/226670","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058895875","display_name":"Samuel P. Muscinelli","orcid":"https://orcid.org/0000-0002-5256-2289"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Samuel P. Muscinelli","raw_affiliation_strings":["School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), 1015 Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), 1015 Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003480477","display_name":"Wulfram Gerstner","orcid":"https://orcid.org/0000-0002-4344-2189"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Wulfram Gerstner","raw_affiliation_strings":["School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), 1015 Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), 1015 Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006129821","display_name":"Johanni Brea","orcid":"https://orcid.org/0000-0002-4636-0891"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Johanni Brea","raw_affiliation_strings":["School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), 1015 Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), 1015 Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5003480477","https://openalex.org/A5006129821","https://openalex.org/A5058895875"],"corresponding_institution_ids":["https://openalex.org/I5124864"],"apc_list":null,"apc_paid":null,"fwci":0.4285,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.79732766,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"29","issue":"2","first_page":"458","last_page":"484"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9994999766349792,"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/T10581","display_name":"Neural dynamics and brain function","score":0.9986000061035156,"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.9972000122070312,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopfield-network","display_name":"Hopfield network","score":0.6748160123825073},{"id":"https://openalex.org/keywords/cover","display_name":"Cover (algebra)","score":0.5955954790115356},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5856444239616394},{"id":"https://openalex.org/keywords/orbit","display_name":"Orbit (dynamics)","score":0.5557267665863037},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.548751175403595},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.522680938243866},{"id":"https://openalex.org/keywords/dynamical-system","display_name":"Dynamical system (definition)","score":0.4890225827693939},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4667022228240967},{"id":"https://openalex.org/keywords/state-space","display_name":"State space","score":0.4342576861381531},{"id":"https://openalex.org/keywords/noise","display_name":"Noise (video)","score":0.4195355474948883},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36287418007850647},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.329967737197876},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.25406789779663086},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.13180837035179138}],"concepts":[{"id":"https://openalex.org/C46421273","wikidata":"https://www.wikidata.org/wiki/Q1407668","display_name":"Hopfield network","level":3,"score":0.6748160123825073},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.5955954790115356},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5856444239616394},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.5557267665863037},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.548751175403595},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.522680938243866},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.4890225827693939},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4667022228240967},{"id":"https://openalex.org/C72434380","wikidata":"https://www.wikidata.org/wiki/Q230930","display_name":"State space","level":2,"score":0.4342576861381531},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.4195355474948883},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36287418007850647},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.329967737197876},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.25406789779663086},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.13180837035179138},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0}],"mesh":[{"descriptor_ui":"D000818","descriptor_name":"Animals","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000818","descriptor_name":"Animals","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000818","descriptor_name":"Animals","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D007858","descriptor_name":"Learning","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D007858","descriptor_name":"Learning","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D007858","descriptor_name":"Learning","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D008959","descriptor_name":"Models, Neurological","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008959","descriptor_name":"Models, Neurological","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008959","descriptor_name":"Models, Neurological","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D009434","descriptor_name":"Neural Pathways","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D009434","descriptor_name":"Neural Pathways","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D009434","descriptor_name":"Neural Pathways","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D009474","descriptor_name":"Neurons","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D009474","descriptor_name":"Neurons","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D009474","descriptor_name":"Neurons","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true}],"locations_count":4,"locations":[{"id":"doi:10.1162/neco_a_00919","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_00919","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},{"id":"pmid:27870611","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/27870611","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural computation","raw_type":null},{"id":"pmh:oai:infoscience.epfl.ch:226670","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/226670","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:infoscience.tind.io:226670","is_oa":true,"landing_page_url":"https://infoscience.epfl.ch/handle/20.500.14299/135588","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"research article"}],"best_oa_location":{"id":"pmh:oai:infoscience.epfl.ch:226670","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/226670","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W93405014","https://openalex.org/W1507575253","https://openalex.org/W1582051163","https://openalex.org/W1661597557","https://openalex.org/W1996739924","https://openalex.org/W2010937504","https://openalex.org/W2030450972","https://openalex.org/W2030805642","https://openalex.org/W2077554996","https://openalex.org/W2080792322","https://openalex.org/W2084259975","https://openalex.org/W2087514146","https://openalex.org/W2089097360","https://openalex.org/W2093002383","https://openalex.org/W2098868783","https://openalex.org/W2099231026","https://openalex.org/W2101222204","https://openalex.org/W2105398397","https://openalex.org/W2115648661","https://openalex.org/W2126209209","https://openalex.org/W2128084896","https://openalex.org/W2133671888","https://openalex.org/W2147896465","https://openalex.org/W2161739581","https://openalex.org/W2914624816","https://openalex.org/W3105747146","https://openalex.org/W4298069009"],"related_works":["https://openalex.org/W4250831671","https://openalex.org/W2230606337","https://openalex.org/W2089869689","https://openalex.org/W4388681848","https://openalex.org/W2078952919","https://openalex.org/W2945666329","https://openalex.org/W2617651062","https://openalex.org/W2330535098","https://openalex.org/W2377513064","https://openalex.org/W5179970"],"abstract_inverted_index":{"We":[0,41,100],"show":[1],"that":[2,15,33,46,69,91],"Hopfield":[3],"neural":[4],"networks":[5],"with":[6],"synchronous":[7],"dynamics":[8,105],"and":[9,55,65,81],"asymmetric":[10],"weights":[11,86],"admit":[12],"stable":[13],"orbits":[14,48,90],"form":[16],"sequences":[17,27],"of":[18,83],"maximal":[19,94],"length.":[20],"For":[21],"[Formula:":[22,30,52],"see":[23,31,53],"text]":[24],"units,":[25],"these":[26],"have":[28],"length":[29],"text];":[32],"is,":[34],"they":[35],"cover":[36],"the":[37,63,66,84,103],"full":[38],"state":[39],"space.":[40],"present":[42],"a":[43,58],"mathematical":[44],"proof":[45],"maximal-length":[47],"exist":[49],"for":[50],"all":[51],"text],":[54],"we":[56],"provide":[57],"method":[59],"to":[60,78,112],"construct":[61],"both":[62],"sequence":[64],"weight":[67],"matrix":[68],"allow":[70],"its":[71],"production.":[72],"The":[73],"orbit":[74],"is":[75],"relatively":[76],"robust":[77],"dynamical":[79],"noise,":[80],"perturbations":[82],"optimal":[85],"reveal":[87],"other":[88],"periodic":[89],"are":[92],"not":[93],"but":[95],"typically":[96],"still":[97],"very":[98],"long.":[99],"discuss":[101],"how":[102],"resulting":[104],"on":[106],"slow":[107],"time-scales":[108],"can":[109],"be":[110],"used":[111],"generate":[113],"desired":[114],"output":[115],"sequences.":[116]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
