{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T19:36:44Z","timestamp":1760384204370},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"12","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2003,12]]},"abstract":"<jats:p> We analyze the asymptotic structure of the Hodgkin\u2013Huxley system of equations, in terms of the concepts of slow manifold and fast foliation, based on Tikhonov's theorem on asymptotics of solutions of slow\u2013fast systems of differential equations. We test Zeeman's conjecture that the jump onset\u2013slow return structure of the action potential in realistic equations of biological excitability may be due to a cusp singularity of the slow manifold with respect to the fast foliation. We find that although the cusp singularity can appear in such equations, the characteristic features in question cannot be reproduced within the Tikhonov scheme and require development of different asymptotic approaches. <\/jats:p>","DOI":"10.1142\/s0218127403008764","type":"journal-article","created":{"date-parts":[[2004,1,29]],"date-time":"2004-01-29T04:19:29Z","timestamp":1075349969000},"page":"3805-3825","source":"Crossref","is-referenced-by-count":23,"title":["THE ASYMPTOTIC STRUCTURE OF THE HODGKIN\u2013HUXLEY EQUATIONS"],"prefix":"10.1142","volume":"13","author":[{"given":"REBECCA","family":"SUCKLEY","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences,  University of Liverpool, Liverpool, L69 7ZL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"VADIM N.","family":"BIKTASHEV","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences,  University of Liverpool, Liverpool, L69 7ZL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"H2296","volume":"282","author":"Bernus O.","journal-title":"Am. J. Physiol."},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.89.168102"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127403008958"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.85.884"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(01)00129-1"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1063\/1.166311"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(61)86902-6"},{"key":"rf8","doi-asserted-by":"crossref","first-page":"27","DOI":"10.17323\/1609-4514-2001-1-1-27-47","volume":"1","author":"Guckenheimer J.","journal-title":"Moscow Math. J."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/S0079-6107(02)00006-8"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1952.sp004764"},{"key":"rf11","first-page":"575","volume":"31","author":"Tikhonov A. N.","journal-title":"Mat. Sbornik"},{"key":"rf12","first-page":"327","volume":"32","author":"Tyson J. J.","journal-title":"Physica"},{"key":"rf13","volume-title":"Differential Equations for the Heartbeat and Nerve Impulse","author":"Zeeman E. C.","year":"1972"},{"key":"rf14","volume-title":"Catastrophe Theory. Selected Papers 1972\u20131977","author":"Zeeman E. C.","year":"1977"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127403008764","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T11:12:50Z","timestamp":1565176370000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127403008764"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,12]]},"references-count":14,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2003,12]]}},"alternative-id":["10.1142\/S0218127403008764"],"URL":"https:\/\/doi.org\/10.1142\/s0218127403008764","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,12]]}}}