Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T11:37:24.916Z Has data issue: false hasContentIssue false
  • English
  • Français

On Chaotic Subthreshold Oscillations in a Simple NeuronalModel

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
Get access

Abstract

In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, asequence of period-doubling bifurcations for small-scale oscillations precedes thetransition into the spiking regime. For a wide range of values of the timescale separationparameter, this scenario is recovered numerically. Its relation to the singularlyperturbed 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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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