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On Chaotic Subthreshold Oscillations in a Simple Neuronal Model

Published online by Cambridge University Press:  09 June 2010

M. Zaks
M. Zaks*
Affiliation:
Institute of Physics, Humboldt University of Berlin, D-12489, Germany
*
* Corresponding author. E-mail: zaks@physik.hu-berlin.de
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Abstract

In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, a sequence of period-doubling bifurcations for small-scale oscillations precedes the transition into the spiking regime. For a wide range of values of the timescale separation parameter, this scenario is recovered numerically. Its relation to the singularly perturbed integrable system is discussed.

Keywords

slow-fast equationsperiod-doubling bifurcation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 1: Instability and patterns. Issue dedicated to the memory of A. Golovin , 2011 , pp. 149 - 162
Copyright
© EDP Sciences, 2010

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References

V. I. Arnold (Editor). Dynamical systems V: Bifurcation theory and catastrophe theory. Encyclopaedia of Mathematical Sciences. Springer. New York, Berlin, Heidelberg, 1999.
Brons, M., Krupa, M., Wechselberger, M.. Mixed mode oscillations due to the generalized canard phenomenon . Fields Institute Communications, 49 (2006), 39–63.Google Scholar
M. Brons, T. J. Kaper, H. G. Rotstein (Editors). Mixed Mode Oscillations: Experiment, Computation, and Analysis. Focus Issue of Chaos, 18 (2008).
Callot, J. L., Diener, F., Diener, M.. Problem of duck hunt . Compt. Rend. Acad. Sci., 286 (1978), 1059–1061. Google Scholar
Collet, P., Eckmann, J.-P., Koch, H.. On universality for area-preserving maps of the plane . Physica D, 3 (1981), 457–467.CrossRefGoogle Scholar
Eckhaus, W.. Relaxation oscillations including a standard chase on French ducks . Lect. Notes Math., 985 (1983), 449–494.CrossRefGoogle Scholar
Ermentrout, G. B.. Period doublings and possible chaos in neural models . SIAM J. Appl. Math., 44 (1984), 80–95.CrossRefGoogle Scholar
Feigenbaum, M. J.. Quantitative universality for a class of nonlinear transformations . J. Stat. Phys., 19 (1978), 25–52.CrossRefGoogle Scholar
Greene, J. M., MacKay, R. S., Vivaldi, F., Feigenbaum, M. J.. Universal behaviour in families of area-preserving maps . Physica D, 3 (1981), 468–486. CrossRefGoogle Scholar
J. Keener, J. Sneyd. Mathematical physiology. Springer, New York, 1998.
Milik, A., Szmolyan, P., Löffelmann, H., Gröller, E.. The geometry of mixed-mode oscillations in the 3d-autocatalator . Int. J. Bif. & Chaos, 8 (1998), 505–519.CrossRefGoogle Scholar
Rinzel, J.. Formal Classification of bursting mechanisms in excitable systems . Lecture Notes Biomathematics, 71 (1987) 267–281, Springer, New York. CrossRefGoogle Scholar
Rössler, O. E.. An equation for continuous chaos . Phys. Lett. A, 57 (1976), 397–398.CrossRefGoogle Scholar
Rotstein, H. G., Kuske, R.. Localized and asynchronous patterns via canards in coupled calcium oscillators . Physica D, 215 (2006), 46–61.CrossRefGoogle Scholar
Sailer, X., Zaks, M., Schimansky-Geier, L.. Collective dynamics in an ensemble of globally coupled FHN systems . Fluctuation & Noise Lett., 5 (2005), L299–L304.CrossRefGoogle Scholar
Verechtchaguina, T., Sokolov, I. M., Schimansky-Geier, L.. First passage time densities in non-Markovian models with subthreshold oscillations . Europhys. Lett., 73 (2006), 691–697.CrossRefGoogle Scholar
Wechselberger, M.. Existence and bifurcation of canards in R3 in the case of a folded node . SIAM J. Appl. Dyn. Sys., 4 (2005), 101–139.CrossRefGoogle Scholar
Zaks, M. A., Sailer, X., Schimansky-Geier, L., Neiman, A., Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems . Chaos, 15 (2005), 026117.CrossRefGoogle ScholarPubMed
Zisook, A. B.. Universal effects of dissipation in two-dimensional mappings . Phys. Rev. A, 24 (1981), 1640–1642.CrossRefGoogle Scholar