{"id":"https://openalex.org/W2474785193","doi":"https://doi.org/10.1109/tnnls.2016.2582512","title":"Fractional Hopfield Neural Networks: Fractional Dynamic Associative Recurrent Neural Networks","display_name":"Fractional Hopfield Neural Networks: Fractional Dynamic Associative Recurrent Neural Networks","publication_year":2016,"publication_date":"2016-07-14","ids":{"openalex":"https://openalex.org/W2474785193","doi":"https://doi.org/10.1109/tnnls.2016.2582512","mag":"2474785193","pmid":"https://pubmed.ncbi.nlm.nih.gov/27429451"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2016.2582512","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tnnls.2016.2582512","pdf_url":"https://ieeexplore.ieee.org/ielx7/5962385/8038919/07513418.pdf","source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://ieeexplore.ieee.org/ielx7/5962385/8038919/07513418.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101890639","display_name":"Yi\u2010Fei Pu","orcid":"https://orcid.org/0000-0003-2975-4976"},"institutions":[{"id":"https://openalex.org/I24185976","display_name":"Sichuan University","ror":"https://ror.org/011ashp19","country_code":"CN","type":"education","lineage":["https://openalex.org/I24185976"]},{"id":"https://openalex.org/I4210125143","display_name":"Chengdu University","ror":"https://ror.org/034z67559","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210125143"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yi-Fei Pu","raw_affiliation_strings":["College of Computer Science, Sichuan University, Chengdu, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer Science, Sichuan University, Chengdu, China","institution_ids":["https://openalex.org/I24185976","https://openalex.org/I4210125143"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100388188","display_name":"Yi Zhang","orcid":"https://orcid.org/0000-0002-5867-9322"},"institutions":[{"id":"https://openalex.org/I24185976","display_name":"Sichuan University","ror":"https://ror.org/011ashp19","country_code":"CN","type":"education","lineage":["https://openalex.org/I24185976"]},{"id":"https://openalex.org/I4210125143","display_name":"Chengdu University","ror":"https://ror.org/034z67559","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210125143"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang Yi","raw_affiliation_strings":["College of Computer Science, Sichuan University, Chengdu, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer Science, Sichuan University, Chengdu, China","institution_ids":["https://openalex.org/I24185976","https://openalex.org/I4210125143"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102009027","display_name":"Jiliu Zhou","orcid":"https://orcid.org/0000-0001-6864-9509"},"institutions":[{"id":"https://openalex.org/I24185976","display_name":"Sichuan University","ror":"https://ror.org/011ashp19","country_code":"CN","type":"education","lineage":["https://openalex.org/I24185976"]},{"id":"https://openalex.org/I4210125143","display_name":"Chengdu University","ror":"https://ror.org/034z67559","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210125143"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ji-Liu Zhou","raw_affiliation_strings":["College of Computer Science, Sichuan University, Chengdu, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer Science, Sichuan University, Chengdu, China","institution_ids":["https://openalex.org/I24185976","https://openalex.org/I4210125143"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":13.1669,"has_fulltext":true,"cited_by_count":122,"citation_normalized_percentile":{"value":0.98448124,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"28","issue":"10","first_page":"2319","last_page":"2333"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9997000098228455,"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":0.9997000098228455,"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.9941999912261963,"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"}},{"id":"https://openalex.org/T10244","display_name":"Chaos control and synchronization","score":0.9914000034332275,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.6945129036903381},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6582003831863403},{"id":"https://openalex.org/keywords/hopfield-network","display_name":"Hopfield network","score":0.6156336665153503},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5384146571159363},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.5098432898521423},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.4635777771472931},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.4535556435585022},{"id":"https://openalex.org/keywords/recurrent-neural-network","display_name":"Recurrent neural network","score":0.45026659965515137},{"id":"https://openalex.org/keywords/bidirectional-associative-memory","display_name":"Bidirectional associative memory","score":0.4493924081325531},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4246992766857147},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4070868492126465},{"id":"https://openalex.org/keywords/content-addressable-memory","display_name":"Content-addressable memory","score":0.36230307817459106},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3458279073238373},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.32270729541778564},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.23266971111297607},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.22073808312416077},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11652901768684387},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.07891732454299927}],"concepts":[{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.6945129036903381},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6582003831863403},{"id":"https://openalex.org/C46421273","wikidata":"https://www.wikidata.org/wiki/Q1407668","display_name":"Hopfield network","level":3,"score":0.6156336665153503},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5384146571159363},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.5098432898521423},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.4635777771472931},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.4535556435585022},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.45026659965515137},{"id":"https://openalex.org/C120620853","wikidata":"https://www.wikidata.org/wiki/Q506355","display_name":"Bidirectional associative memory","level":4,"score":0.4493924081325531},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4246992766857147},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4070868492126465},{"id":"https://openalex.org/C53442348","wikidata":"https://www.wikidata.org/wiki/Q745101","display_name":"Content-addressable memory","level":3,"score":0.36230307817459106},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3458279073238373},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32270729541778564},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.23266971111297607},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.22073808312416077},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11652901768684387},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.07891732454299927},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","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/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2016.2582512","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tnnls.2016.2582512","pdf_url":"https://ieeexplore.ieee.org/ielx7/5962385/8038919/07513418.pdf","source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},{"id":"pmid:27429451","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/27429451","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":"IEEE transactions on neural networks and learning systems","raw_type":null}],"best_oa_location":{"id":"doi:10.1109/tnnls.2016.2582512","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tnnls.2016.2582512","pdf_url":"https://ieeexplore.ieee.org/ielx7/5962385/8038919/07513418.pdf","source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7506217808","display_name":"\u5206\u6570\u9636Hopfield\u795e\u7ecf\u7f51\u7edc\u7684\u7406\u8bba\u7814\u7a76","funder_award_id":"61571312","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2474785193.pdf","grobid_xml":"https://content.openalex.org/works/W2474785193.grobid-xml"},"referenced_works_count":62,"referenced_works":["https://openalex.org/W22297218","https://openalex.org/W121642006","https://openalex.org/W572373117","https://openalex.org/W1538983967","https://openalex.org/W1597286183","https://openalex.org/W1605182429","https://openalex.org/W1841080724","https://openalex.org/W1966976574","https://openalex.org/W1970484278","https://openalex.org/W1971873204","https://openalex.org/W1983345770","https://openalex.org/W1984606190","https://openalex.org/W1985438383","https://openalex.org/W1989957303","https://openalex.org/W1992172913","https://openalex.org/W2000829143","https://openalex.org/W2010526455","https://openalex.org/W2013777567","https://openalex.org/W2031749344","https://openalex.org/W2034948711","https://openalex.org/W2037580505","https://openalex.org/W2038237190","https://openalex.org/W2044266667","https://openalex.org/W2048322459","https://openalex.org/W2057506102","https://openalex.org/W2070758751","https://openalex.org/W2075132021","https://openalex.org/W2076749418","https://openalex.org/W2088187864","https://openalex.org/W2094438648","https://openalex.org/W2095425517","https://openalex.org/W2099624136","https://openalex.org/W2099914796","https://openalex.org/W2100885571","https://openalex.org/W2102227134","https://openalex.org/W2107815783","https://openalex.org/W2111926818","https://openalex.org/W2112246162","https://openalex.org/W2128084896","https://openalex.org/W2129649323","https://openalex.org/W2140710924","https://openalex.org/W2141404703","https://openalex.org/W2142742885","https://openalex.org/W2145722250","https://openalex.org/W2151015649","https://openalex.org/W2153662340","https://openalex.org/W2157570850","https://openalex.org/W2163664861","https://openalex.org/W2166672111","https://openalex.org/W2168653854","https://openalex.org/W2204277842","https://openalex.org/W2208596464","https://openalex.org/W2225065364","https://openalex.org/W2259227674","https://openalex.org/W2392229071","https://openalex.org/W2496913885","https://openalex.org/W2538187287","https://openalex.org/W2540809067","https://openalex.org/W3148872539","https://openalex.org/W4238663582","https://openalex.org/W4251731575","https://openalex.org/W4300223101"],"related_works":["https://openalex.org/W4321273133","https://openalex.org/W1536116856","https://openalex.org/W2109892802","https://openalex.org/W2769111744","https://openalex.org/W2147932521","https://openalex.org/W2144581752","https://openalex.org/W2260028708","https://openalex.org/W2050720944","https://openalex.org/W1560986979","https://openalex.org/W3087113781"],"abstract_inverted_index":{"This":[0],"paper":[1],"mainly":[2,26],"discusses":[3],"a":[4,85,172],"novel":[5,86],"conceptual":[6],"framework:":[7],"fractional":[8,17,41,79,89,114,176,217],"Hopfield":[9,64],"neural":[10,24,42,50,65],"networks":[11,43,66],"(FHNN).":[12],"As":[13],"is":[14,54,159,208],"commonly":[15],"known,":[16],"calculus":[18,90],"has":[19],"been":[20],"incorporated":[21],"into":[22],"artificial":[23],"networks,":[25],"because":[27],"of":[28,78,101,137,156,166,206],"its":[29,119,125,142,181,185,189,210],"long-term":[30],"memory":[31],"and":[32,44,71,112,122,135,139,174,191,223],"nonlocality.":[33],"Some":[34],"researchers":[35],"have":[36],"made":[37],"interesting":[38],"attempts":[39],"at":[40],"gained":[45],"competitive":[46],"advantages":[47],"over":[48],"integer-order":[49],"networks.":[51],"Therefore,":[52],"it":[53],"naturally":[55],"makes":[56],"one":[57],"ponder":[58],"how":[59,72],"to":[60,67,73,83,91,131,144,160,194,214],"generalize":[61],"the":[62,68,99,113,133,145,164,175,195,215,221],"first-order":[63],"fractional-order":[69],"ones,":[70],"implement":[74,92,96,107],"FHNN":[75,108,162,193,207,219],"by":[76,109,170],"means":[77],"calculus.":[80],"We":[81],"propose":[82,161],"introduce":[84],"mathematical":[87],"method:":[88],"FHNN.":[93],"First,":[94],"we":[95,106,128],"fractor":[97,111,173],"in":[98,163],"form":[100,165],"an":[102,167],"analog":[103,168],"circuit.":[104],"Second,":[105],"utilizing":[110,171],"steepest":[115,177],"descent":[116,178],"approach,":[117,179],"construct":[118,180],"Lyapunov":[120,182,186],"function,":[121,183],"further":[123,140],"analyze":[124,132,188],"attractors.":[126],"Third,":[127],"perform":[129],"experiments":[130],"stability":[134],"convergence":[136],"FHNN,":[138],"discuss":[141],"applications":[143],"defense":[146,196],"against":[147,197],"chip":[148,198],"cloning":[149,199],"attacks":[150,200],"for":[151,201],"anticounterfeiting.":[152,202],"The":[153],"main":[154],"contribution":[155],"our":[157],"work":[158],"circuit":[169],"prove":[184],"stability,":[187],"attractors,":[190],"apply":[192],"A":[203],"significant":[204],"advantage":[205],"that":[209],"attractors":[211],"essentially":[212],"relate":[213],"neuron's":[216],"order.":[218],"possesses":[220],"fractional-order-stability":[222],"fractional-order-sensitivity":[224],"characteristics.":[225]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":22},{"year":2022,"cited_by_count":17},{"year":2021,"cited_by_count":18},{"year":2020,"cited_by_count":17},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":11},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}