{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T10:33:48Z","timestamp":1743071628341},"reference-count":12,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2000,6]]},"abstract":"<jats:p> This paper presents analytical criteria for local activity in reaction\u2013diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin\u2013Huxley CNN, which has cells defined by the equations of the cardiac Purkinje fiber model of morphogenesis that was first introduced in [Noble, 1962] to describe the long-lasting action and pace-maker potentials of the Purkinje fiber of the heart. The bifurcation diagrams of the Hodgkin\u2013Huxley CNN's supply a possible explanation for why a heart with a normal heart-rate may stop beating suddenly: The cell parameter of a normal heart is located in a locally active unstable domain and just nearby an edge of chaos. The membrane potential along a fiber is simulated in a Hodgkin\u2013Huxley CNN by a computer. As a second application, we present a smoothed Chua's circuit (SCC) CNN. The bifurcation diagrams of the SCC CNN's show that there does not exist a locally passive domain, and the edges of chaos corresponding to different fixed-cell parameters are significantly different. Our computer simulations show that oscillatory patterns, chaotic patterns, or divergent patterns may emerge if the selected cell parameters are located in locally active domains but nearby the edge of chaos. This research demonstrates once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells. <\/jats:p>","DOI":"10.1142\/s0218127400000876","type":"journal-article","created":{"date-parts":[[2002,8,24]],"date-time":"2002-08-24T21:40:19Z","timestamp":1030225219000},"page":"1295-1343","source":"Crossref","is-referenced-by-count":25,"title":["ANALYTICAL CRITERIA FOR LOCAL ACTIVITY OF REACTION\u2013DIFFUSION CNN WITH FOUR STATE VARIABLES AND APPLICATIONS TO THE HODGKIN\u2013HUXLEY EQUATION"],"prefix":"10.1142","volume":"10","author":[{"given":"LEQUAN","family":"MIN","sequence":"first","affiliation":[{"name":"Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"KENNETH R.","family":"CROUNSE","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LEON O.","family":"CHUA","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,5,2]]},"reference":[{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497001618"},{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.1109\/81.739186"},{"key":"p_4","doi-asserted-by":"publisher","DOI":"10.1109\/81.473564"},{"key":"p_6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000152"},{"key":"p_7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000899"},{"key":"p_8","doi-asserted-by":"publisher","DOI":"10.1142\/S021812749800187X"},{"key":"p_10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497001515"},{"key":"p_11","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1952.sp004717"},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1952.sp004718"},{"key":"p_13","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1952.sp004719"},{"key":"p_16","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127400000037"},{"key":"p_18","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1962.sp006849"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127400000876","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:43:49Z","timestamp":1565124229000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127400000876"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,6]]},"references-count":12,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2012,5,2]]},"published-print":{"date-parts":[[2000,6]]}},"alternative-id":["10.1142\/S0218127400000876"],"URL":"https:\/\/doi.org\/10.1142\/s0218127400000876","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,6]]}}}