{"id":"https://openalex.org/W1967675726","doi":"https://doi.org/10.1007/s004220050022","title":"Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations II. Singularity theoretic approach and highly degenerate bifurcations","display_name":"Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations II. Singularity theoretic approach and highly degenerate bifurcations","publication_year":2000,"publication_date":"2000-02-09","ids":{"openalex":"https://openalex.org/W1967675726","doi":"https://doi.org/10.1007/s004220050022","mag":"1967675726","pmid":"https://pubmed.ncbi.nlm.nih.gov/10664109"},"language":"en","primary_location":{"id":"doi:10.1007/s004220050022","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s004220050022","pdf_url":null,"source":{"id":"https://openalex.org/S87357125","display_name":"Biological Cybernetics","issn_l":"0340-1200","issn":["0340-1200","1432-0770"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Biological Cybernetics","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/A5010996863","display_name":"Hidekazu Fukai","orcid":null},"institutions":[{"id":"https://openalex.org/I37596208","display_name":"Osaka University of Human Sciences","ror":"https://ror.org/05cp38y47","country_code":"JP","type":"education","lineage":["https://openalex.org/I37596208"]},{"id":"https://openalex.org/I98285908","display_name":"The University of Osaka","ror":"https://ror.org/035t8zc32","country_code":"JP","type":"education","lineage":["https://openalex.org/I98285908"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hidekazu Fukai","raw_affiliation_strings":["Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan","Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , JP"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan","institution_ids":["https://openalex.org/I37596208"]},{"raw_affiliation_string":"Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , JP","institution_ids":["https://openalex.org/I98285908"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007424237","display_name":"Taishin Nomura","orcid":"https://orcid.org/0000-0002-6545-2097"},"institutions":[{"id":"https://openalex.org/I98285908","display_name":"The University of Osaka","ror":"https://ror.org/035t8zc32","country_code":"JP","type":"education","lineage":["https://openalex.org/I98285908"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Taishin Nomura","raw_affiliation_strings":["Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , , , , , JP","Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , JP"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , , , , , JP","institution_ids":["https://openalex.org/I98285908"]},{"raw_affiliation_string":"Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , JP","institution_ids":["https://openalex.org/I98285908"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113724958","display_name":"Shinji Doi","orcid":null},"institutions":[{"id":"https://openalex.org/I98285908","display_name":"The University of Osaka","ror":"https://ror.org/035t8zc32","country_code":"JP","type":"education","lineage":["https://openalex.org/I98285908"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Shinji Doi","raw_affiliation_strings":["Department of Electrical Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka 560-0871, Japan, , , , , , JP","\u200aDepartment of Electrical Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka 560-0871, Japan, , JP"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka 560-0871, Japan, , , , , , JP","institution_ids":["https://openalex.org/I98285908"]},{"raw_affiliation_string":"\u200aDepartment of Electrical Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka 560-0871, Japan, , JP","institution_ids":["https://openalex.org/I98285908"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101532585","display_name":"Shunsuke Sato","orcid":"https://orcid.org/0000-0002-1378-4428"},"institutions":[{"id":"https://openalex.org/I98285908","display_name":"The University of Osaka","ror":"https://ror.org/035t8zc32","country_code":"JP","type":"education","lineage":["https://openalex.org/I98285908"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Shunsuke Sato","raw_affiliation_strings":["Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , , , , , JP","Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , JP"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , , , , , JP","institution_ids":["https://openalex.org/I98285908"]},{"raw_affiliation_string":"Division of Biophysical Engineering, Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan, , JP","institution_ids":["https://openalex.org/I98285908"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":3590,"currency":"USD","value_usd":3590},"apc_paid":null,"fwci":1.753,"has_fulltext":false,"cited_by_count":29,"citation_normalized_percentile":{"value":0.86669554,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"82","issue":"3","first_page":"223","last_page":"229"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9994999766349792,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.9045107364654541},{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.7180423140525818},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.685924768447876},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.6817952394485474},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.6608338356018066},{"id":"https://openalex.org/keywords/parameter-space","display_name":"Parameter space","score":0.6529155969619751},{"id":"https://openalex.org/keywords/singularity-theory","display_name":"Singularity theory","score":0.6074485182762146},{"id":"https://openalex.org/keywords/bifurcation-theory","display_name":"Bifurcation theory","score":0.5455827116966248},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5286434292793274},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5252856016159058},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5086989998817444},{"id":"https://openalex.org/keywords/hodgkin\u2013huxley-model","display_name":"Hodgkin\u2013Huxley model","score":0.45142096281051636},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.44959110021591187},{"id":"https://openalex.org/keywords/equilibrium-point","display_name":"Equilibrium point","score":0.4214057922363281},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.24454230070114136},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22680839896202087},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.119299978017807},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11653199791908264},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08517277240753174},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06721159815788269}],"concepts":[{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.9045107364654541},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.7180423140525818},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.685924768447876},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.6817952394485474},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.6608338356018066},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.6529155969619751},{"id":"https://openalex.org/C207769094","wikidata":"https://www.wikidata.org/wiki/Q4245771","display_name":"Singularity theory","level":3,"score":0.6074485182762146},{"id":"https://openalex.org/C85075877","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Bifurcation theory","level":4,"score":0.5455827116966248},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5286434292793274},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5252856016159058},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5086989998817444},{"id":"https://openalex.org/C101877181","wikidata":"https://www.wikidata.org/wiki/Q995215","display_name":"Hodgkin\u2013Huxley model","level":2,"score":0.45142096281051636},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.44959110021591187},{"id":"https://openalex.org/C94766913","wikidata":"https://www.wikidata.org/wiki/Q1530271","display_name":"Equilibrium point","level":3,"score":0.4214057922363281},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.24454230070114136},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22680839896202087},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.119299978017807},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11653199791908264},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08517277240753174},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06721159815788269},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0}],"mesh":[{"descriptor_ui":"D001369","descriptor_name":"Axons","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D001369","descriptor_name":"Axons","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D001369","descriptor_name":"Axons","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D008432","descriptor_name":"Mathematical Computing","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008432","descriptor_name":"Mathematical Computing","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008432","descriptor_name":"Mathematical Computing","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008433","descriptor_name":"Mathematics","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008433","descriptor_name":"Mathematics","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008433","descriptor_name":"Mathematics","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008564","descriptor_name":"Membrane Potentials","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D008564","descriptor_name":"Membrane Potentials","qualifier_ui":"Q000502","qualifier_name":"physiology","is_major_topic":false},{"descriptor_ui":"D008564","descriptor_name":"Membrane Potentials","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":true},{"descriptor_ui":"D008959","descriptor_name":"Models, Neurological","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008959","descriptor_name":"Models, Neurological","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D010507","descriptor_name":"Periodicity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D010507","descriptor_name":"Periodicity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D010507","descriptor_name":"Periodicity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false}],"locations_count":2,"locations":[{"id":"doi:10.1007/s004220050022","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s004220050022","pdf_url":null,"source":{"id":"https://openalex.org/S87357125","display_name":"Biological Cybernetics","issn_l":"0340-1200","issn":["0340-1200","1432-0770"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Biological Cybernetics","raw_type":"journal-article"},{"id":"pmid:10664109","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/10664109","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":"Biological cybernetics","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W176118634","https://openalex.org/W1965778005","https://openalex.org/W1965873706","https://openalex.org/W1985940938","https://openalex.org/W1986799785","https://openalex.org/W2009655055","https://openalex.org/W2027031319","https://openalex.org/W2089876099","https://openalex.org/W2090631087","https://openalex.org/W2094086293","https://openalex.org/W2565118972"],"related_works":["https://openalex.org/W1547025490","https://openalex.org/W2106448822","https://openalex.org/W2546487931","https://openalex.org/W3145971316","https://openalex.org/W2387129305","https://openalex.org/W2042777683","https://openalex.org/W2319243846","https://openalex.org/W1194905386","https://openalex.org/W1995298252","https://openalex.org/W1982223217"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2014,"cited_by_count":2}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}