{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,21]],"date-time":"2026-06-21T23:05:04Z","timestamp":1782083104699,"version":"3.54.5"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2015,11,30]],"date-time":"2015-11-30T00:00:00Z","timestamp":1448841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CONICYT fellowship Beca Magister Nacional","award":["22110804"],"award-info":[{"award-number":["22110804"]}]},{"name":"FONDECYT project","award":["1120329"],"award-info":[{"award-number":["1120329"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Conductance-based (CB) models are a class of high dimensional dynamical systems derived from biophysical principles to describe in detail the electrical dynamics of single neurons. Despite the high dimensionality of these models, the dynamics observed for realistic parameter values is generically planar and can be minimally described by two equations. In this work, we derive the conditions to have a Bogdanov\u2013Takens (BT) bifurcation in CB models, and we argue that it is plausible that these conditions are verified for experimentally-sensible values of the parameters. We show numerically that the cubic BT normal form, a two-variable dynamical system, exhibits all of the diversity of bifurcations generically observed in single neuron models. We show that the Morris\u2013Lecar model is approximately equivalent to the cubic Bogdanov\u2013Takens normal form for realistic values of parameters. Furthermore, we explicitly calculate the quadratic coefficient of the BT normal form for a generic CB model, obtaining that by constraining the theoretical I-V curve\u2019s curvature to match experimental observations, the normal form appears to be naturally cubic. We propose the cubic BT normal form as a robust minimal model for single neuron dynamics that can be derived from biophysically-realistic CB models.<\/jats:p>","DOI":"10.3390\/e17127850","type":"journal-article","created":{"date-parts":[[2015,11,30]],"date-time":"2015-11-30T10:57:24Z","timestamp":1448881044000},"page":"7859-7874","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["The Bogdanov\u2013Takens Normal Form: A Minimal Model for Single Neuron Dynamics"],"prefix":"10.3390","volume":"17","author":[{"given":"Ulises","family":"Pereira","sequence":"first","affiliation":[{"name":"Department of Statistics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60615, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre","family":"Coullet","sequence":"additional","affiliation":[{"name":"INLN, UMR 7335 CNRS, Universite de Nice Sophia-Antipolis, 1361 routes des Lucioles, Sophia-Antipolis, 06560 Valbonne, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enrique","family":"Tirapegui","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, FCFM, Universidad de Chile, Casilla 487-3, Santiago 6511226, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,11,30]]},"reference":[{"key":"ref_1","unstructured":"Pereira, U. (2013). Toward a Universal Description for Single Neuron Dynamics. [Master\u2019s Thesis, Universidad de Chile]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0167-2789(87)90049-2","article-title":"A simple global characterization for normal forms of singular vector fields","volume":"29","author":"Elphick","year":"1987","journal-title":"Physica D"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Haragus, M., and Iooss, G. (2011). Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Springer. [1st ed.].","DOI":"10.1007\/978-0-85729-112-7"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ermentrout, G.B., and Terman, D.H. (2010). Mathematical Foundations of Neuroscience, Springer. [1st ed.].","DOI":"10.1007\/978-0-387-87708-2"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Carnevale, N.T., and Hines, M.L. (2006). The NEURON Book, Cambridge University Press.","DOI":"10.1017\/CBO9780511541612"},{"key":"ref_6","unstructured":"Dayan, P., and Abbott, L.F. (2001). Theoretical Neuroscience, The MIT Press."},{"key":"ref_7","unstructured":"Izhikevich, E.M. (2010). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ranjan, R., Khazen, G., Gambazzi, L., Ramaswamy, S., Hill, S.L., Schurmann, F., and Markram, H. (2011). Channelpedia: An integrative and interactive database for ion channels. Front. Neuroinform., 5.","DOI":"10.3389\/fninf.2011.00036"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1136\/jamia.1996.97084512","article-title":"ModelDB: An environment for running and storing computational models and their results applied to neuroscience","volume":"3","author":"Peterson","year":"1996","journal-title":"J. Am. Med. Inform. Assoc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1113\/jphysiol.1948.sp004260","article-title":"The local electric changes associated with repetitive action in a non-medullated axon","volume":"107","author":"Hodgkin","year":"1948","journal-title":"J. Physiol."},{"key":"ref_11","first-page":"251","article-title":"Analysis of neural excitability and oscillations","volume":"2","author":"Rinzel","year":"1998","journal-title":"Method. Neuron. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/BF02477753","article-title":"Mathematical models of threshold phenomena in the nerve membrane","volume":"17","author":"FitzHugh","year":"1955","journal-title":"Bull. Math. Biophys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Morris, C., and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J., 35.","DOI":"10.1016\/S0006-3495(81)84782-0"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/BF00197717","article-title":"Reduction of conductance-based neuron models","volume":"66","author":"Kepler","year":"1992","journal-title":"Biol. Cybern."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","article-title":"A quantitative description of membrane current and its application to conduction and excitation in nerve","volume":"117","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_16","unstructured":"Johnston, D., Wu, S.M.S., and Gray, R. (1995). Foundations of Cellular Neurophysiology, The MIT press."},{"key":"ref_17","unstructured":"Pereira, U., and Tirapegui, E. (2012, January 3\u20137). Una Ecuaci\u00f3n Universal Para la Din\u00e1mica Neuronal. Processdings of the XVII Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics, Santiago, Chile."},{"key":"ref_18","unstructured":"Pereira, U., and Tirapegui, E. (2012, January 21\u201323). Una Ecuaci\u00f3n Universal Para la Din\u00e1mica Neuronal. Proceedings of the XVIII Simposio Chileno de F\u00edsica, La Serena, Chile."},{"key":"ref_19","unstructured":"Hille, B. (2001). Ion Channels of Excitable Membranes, Sinauer."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1214\/aoms\/1177729893","article-title":"Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix","volume":"21","author":"Sherman","year":"1950","journal-title":"Ann. Math. Statist."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1016\/j.aml.2006.11.016","article-title":"Eigenvalues of rank-one updated matrices with some applications","volume":"20","author":"Ding","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kuznetsov, Y.A. (2004). Elements of Applied Bifurcation Theory, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"ref_23","unstructured":"Guckenheimer, J., and Holmes, P. (2002). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer. [7th ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3535","DOI":"10.1142\/S0218127405014209","article-title":"Practical computation of normal forms on center manifolds at degenerate Bogdanov\u2013Takens bifurcations","volume":"15","author":"Kuznetsov","year":"2005","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Coombes, S., and Bressloff, P.C. (2005). Bursting: The Genesis of Rhythm in the Nervous System, World Scientific. [1st ed.].","DOI":"10.1142\/9789812703231"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Kirst, C., Ammer, J., Felmy, F., Herz, A., and Stemmler, M. (2015). Fundamental Structure and Modulation of Neuronal Excitability: Synaptic Control of Coding, Resonance, and Network Synchronization. bioRxiv, 022475.","DOI":"10.1101\/022475"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/12\/7850\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:52:57Z","timestamp":1760215977000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/12\/7850"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,30]]},"references-count":26,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["e17127850"],"URL":"https:\/\/doi.org\/10.3390\/e17127850","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,11,30]]}}}