{"id":"https://openalex.org/W2097671547","doi":"https://doi.org/10.1109/tnn.2008.2009535","title":"Analysis of Continuous Attractors for 2-D Linear Threshold Neural Networks","display_name":"Analysis of Continuous Attractors for 2-D Linear Threshold Neural Networks","publication_year":2008,"publication_date":"2008-12-10","ids":{"openalex":"https://openalex.org/W2097671547","doi":"https://doi.org/10.1109/tnn.2008.2009535","mag":"2097671547","pmid":"https://pubmed.ncbi.nlm.nih.gov/19129036"},"language":"en","primary_location":{"id":"doi:10.1109/tnn.2008.2009535","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2008.2009535","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"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","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5040299913","display_name":"Lan Zou","orcid":"https://orcid.org/0000-0003-2443-0817"},"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"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lan Zou","raw_affiliation_strings":["Yangtze Center of Mathematics and the Department of Mathematics, Sichuan University, Chengdu 610064, China. lanzou@yahoo.com.cn","Dept. of Math., Sichuan Univ., Chengdu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yangtze Center of Mathematics and the Department of Mathematics, Sichuan University, Chengdu 610064, China. lanzou@yahoo.com.cn","institution_ids":["https://openalex.org/I24185976"]},{"raw_affiliation_string":"Dept. of Math., Sichuan Univ., Chengdu","institution_ids":["https://openalex.org/I24185976"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057282687","display_name":"Huajin Tang","orcid":"https://orcid.org/0000-0002-9857-1543"},"institutions":[{"id":"https://openalex.org/I115228651","display_name":"Agency for Science, Technology and Research","ror":"https://ror.org/036wvzt09","country_code":"SG","type":"government","lineage":["https://openalex.org/I115228651"]},{"id":"https://openalex.org/I3005327000","display_name":"Institute for Infocomm Research","ror":"https://ror.org/053rfa017","country_code":"SG","type":"facility","lineage":["https://openalex.org/I115228651","https://openalex.org/I3005327000","https://openalex.org/I91275662"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Huajin Tang","raw_affiliation_strings":["Institute of Infocomm Research, Agency for Science, Technology and Research, Singapore","Institute for Infocomm Research, Agency for Science Technology & Research, Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Infocomm Research, Agency for Science, Technology and Research, Singapore","institution_ids":["https://openalex.org/I115228651","https://openalex.org/I3005327000"]},{"raw_affiliation_string":"Institute for Infocomm Research, Agency for Science Technology & Research, Singapore","institution_ids":["https://openalex.org/I3005327000"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025285243","display_name":"Kay Chen Tan","orcid":"https://orcid.org/0000-0002-6802-2463"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Kay Chen Tan","raw_affiliation_strings":["Department of Electrical and Computer Engineering, National University of Singapore, Singapore","Dept. of Electr. & Comput. Eng., Nat. Univ. of Singapore, Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, National University of Singapore, Singapore","institution_ids":["https://openalex.org/I165932596"]},{"raw_affiliation_string":"Dept. of Electr. & Comput. Eng., Nat. Univ. of Singapore, Singapore","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019414664","display_name":"Weinian Zhang","orcid":"https://orcid.org/0000-0002-4566-499X"},"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"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Weinian Zhang","raw_affiliation_strings":["Yangtze Center of Mathematics and the Department of Mathematics, Sichuan University, Chengdu, China","Dept. of Math., Sichuan Univ., Chengdu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yangtze Center of Mathematics and the Department of Mathematics, Sichuan University, Chengdu, China","institution_ids":["https://openalex.org/I24185976"]},{"raw_affiliation_string":"Dept. of Math., Sichuan Univ., Chengdu","institution_ids":["https://openalex.org/I24185976"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2834,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.58350122,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"20","issue":"1","first_page":"175","last_page":"180"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10581","display_name":"Neural dynamics and brain function","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9986000061035156,"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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.9955000281333923,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.8727915287017822},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.7283324003219604},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.657753586769104},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.6530804634094238},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5348431468009949},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.4669932425022125},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4519405961036682},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3789862096309662},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3620623052120209},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2610885500907898},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.22590923309326172},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.191258043050766},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1678796410560608},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09601181745529175}],"concepts":[{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.8727915287017822},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.7283324003219604},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.657753586769104},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.6530804634094238},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5348431468009949},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.4669932425022125},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4519405961036682},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3789862096309662},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3620623052120209},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2610885500907898},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.22590923309326172},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.191258043050766},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1678796410560608},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09601181745529175},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","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":"D000818","descriptor_name":"Animals","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D005246","descriptor_name":"Feedback","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D005246","descriptor_name":"Feedback","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D005246","descriptor_name":"Feedback","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":"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":"D014793","descriptor_name":"Visual Cortex","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D014793","descriptor_name":"Visual Cortex","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D014793","descriptor_name":"Visual Cortex","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D016014","descriptor_name":"Linear Models","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D016014","descriptor_name":"Linear Models","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D016014","descriptor_name":"Linear Models","qualifier_ui":null,"qualifier_name":null,"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":5,"locations":[{"id":"doi:10.1109/tnn.2008.2009535","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2008.2009535","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"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","raw_type":"journal-article"},{"id":"pmid:19129036","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/19129036","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","raw_type":"Journal Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.667.2134","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.667.2134","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www1.i2r.a-star.edu.sg/%7Ehtang/Tang09.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.724.7983","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.724.7983","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www1.i2r.a-star.edu.sg/%7Ehtang/TNN09_ContinuousAttractors.pdf","raw_type":"text"},{"id":"pmh:oai:scholarbank.nus.edu.sg:10635/81962","is_oa":false,"landing_page_url":"http://scholarbank.nus.edu.sg/handle/10635/81962","pdf_url":null,"source":{"id":"https://openalex.org/S7407052290","display_name":"National University of Singapore","issn_l":null,"issn":null,"is_oa":false,"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":"Scopus","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1572063013","https://openalex.org/W1578196132","https://openalex.org/W1603053247","https://openalex.org/W1991233288","https://openalex.org/W1993545737","https://openalex.org/W2036958054","https://openalex.org/W2128747957","https://openalex.org/W2129481537","https://openalex.org/W2130837362","https://openalex.org/W2134114905","https://openalex.org/W2134683599","https://openalex.org/W2139411083","https://openalex.org/W2144789942","https://openalex.org/W2148830696","https://openalex.org/W2157518752","https://openalex.org/W2506699060","https://openalex.org/W3143844809"],"related_works":["https://openalex.org/W2051058708","https://openalex.org/W154868527","https://openalex.org/W1494268238","https://openalex.org/W1516574938","https://openalex.org/W1983207144","https://openalex.org/W2490706771","https://openalex.org/W1976468483","https://openalex.org/W2966150910","https://openalex.org/W4291418281","https://openalex.org/W2469703203"],"abstract_inverted_index":{"This":[0],"brief":[1],"investigates":[2],"continuous":[3],"attractors":[4],"of":[5,28,49,60],"the":[6,13,26,50,57,79,83,94],"well-developed":[7],"model":[8],"in":[9,34],"visual":[10,86],"cortex,":[11],"i.e.,":[12],"linear":[14],"threshold":[15],"(LT)":[16],"neural":[17],"networks,":[18],"based":[19],"on":[20,31,78],"a":[21,72],"parameterized":[22],"2-D":[23],"model.":[24],"On":[25],"basis":[27],"existing":[29],"results":[30,70],"nondegenerate":[32,61],"equilibria":[33,40,64],"mathematics,":[35],"we":[36],"further":[37],"discuss":[38],"degenerate":[39,63],"for":[41,75],"such":[42],"networks":[43],"and":[44,47,62,85],"present":[45],"properties":[46],"distributions":[48],"equilibria,":[51],"which":[52],"enables":[53],"us":[54],"to":[55,92],"draw":[56],"coexistence":[58],"conditions":[59],"(e.g.,":[65],"singular":[66],"lines).":[67],"Our":[68],"theoretical":[69,95],"provide":[71],"useful":[73],"framework":[74],"precise":[76],"tuning":[77],"network":[80],"parameters,":[81],"e.g.,":[82],"feedbacks":[84],"inputs.":[87],"Simulations":[88],"are":[89],"also":[90],"presented":[91],"illustrate":[93],"findings.":[96]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}