Abstract
We consider an extended three-dimensional Bonhoeffer–van der Pol oscillator which generalises the planar FitzHugh–Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. (Phys Lett A 374(36):3745–3751, 2010) and by Freire and Gallas (Phys Lett A 375:1097–1103, 2011). Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. (J Appl Dyn Syst 7(2):361–420, 2008) as a canonical form for such systems. Based on results previously obtained in Krupa et al. (2008), we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer–van der Pol oscillator from the point of view of geometric singular perturbation theory, thus complementing the findings reported in Sekikawa et al. (2010). In particular, we specify in detail the mixed-mode patterns that are observed upon variation of a bifurcation parameter which is naturally obtained by combining two of the original parameters in the system, and we derive asymptotic estimates for the corresponding parameter intervals. We thereby also disprove a conjecture of Tu (SIAM J Appl Math 49(2): 331–343, 1989), where it was postulated that no stable periodic orbits of mixed-mode type can be observed in an equivalent extension of the Bonhoeffer–van der Pol equations.
Similar content being viewed by others
References
Brøns, M., Krupa, M., Wechselberger, M.: Mixed mode oscillations due to the generalized canard phenomenon. In: Bifurcation Theory and Spatio-Temporal Pattern Formation, vol. 49 of Fields Institute Communications, pp. 39–63. American Mathematical Society, Providence, RI, (2006)
Curtu, R., Rubin, J.: Interaction of canard and singular Hopf mechanisms in a neural model. SIAM J. Appl. Dyn. Syst. 10(4), 1443–1479 (2011)
Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H.M., Wechselberger, M.: Mixed-mode oscillations with multiple time scales. SIAM Rev. 54(2), 211–288 (2012)
Dumortier, F.: Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations. In: Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), vol. 408 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pp. 19–73. Kluwer Academic Publisher, Dordrecht (1993)
Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), 1–100 (1996) (With an appendix by Cheng Zhi Li)
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
Freire, J.G., Gallas, J.A.C.: Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer–van der Pol and the Fitzhugh-Nagumo models of excitable systems. Phys. Lett. A 375, 1097–1103 (2011)
Guckenheimer, J.: Singular Hopf bifurcation in systems with two slow variables. SIAM J. Appl. Dyn. Syst. 7(4), 1355–1377 (2008)
Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Computational Neuroscience. MIT Press, Cambridge, MA (2007)
Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems (Montecatini Terme, 1994) Lecture Notes in Mathematics, vol. 1609, pp. 44–118. Springer, Berlin, (1995)
Koper, M.T.M.: Bifurcations of mixed-mode oscillations in a three-variable autonomous van der Pol-Duffing model with a cross-shaped phase diagram. Phys. D 80(1–2), 72–94 (1995)
Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points–fold and canard points in two dimensions. SIAM J. Math. Anal 33(2), 286–314 (2001)
Krupa, M., Wechselberger, M.: Local analysis near a folded saddle-node singularity. J. Differ. Equ. 248(12), 2841–2888 (2010)
Krupa, M., Popović, N., Kopell, N.: Mixed-mode oscillations in three time-scale systems: a prototypical example. SIAM J. Appl. Dyn. Syst. 7(2), 361–420 (2008)
Krupa, M., Popović, N., Kopell, N., Rotstein, H.G.: Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron. Chaos 18(1), 015106 (2008)
Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. A K Peters Ltd., Wellesley, MA, (1997). Reprint of the original [Academic Press, New York, 1974]
Sekikawa, M., Inaba, N., Yoshinaga, T., Hikihara, T.: Period-doubling cascades of canards from the extended Bonhoeffer–van der Pol oscillator. Phys. Lett. A 374(36), 3745–3751 (2010)
Shimizu, K., Yuto, S., Sekikawa, M., Inaba, N.: Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation. Phys. D 241(18), 1518–1526 (2012)
Szmolyan, P., Wechselberger, M.: Canards in \(\mathbb{R}^3\). J. Differ. Equ. 177(2), 419–453 (2001)
Szmolyan, P., Wechselberger, M.: Relaxation oscillations in \(\mathbb{R}^3\). J. Differ. Equ. 200(1), 69–104 (2004)
Tu, S.T.: A phase-plane analysis of bursting in the three-dimensional Bonhoeffer–van der Pol equations. SIAM J. Appl. Math. 49(2), 331–343 (1989)
Wechselberger, M.: Existence and bifurcation of canards in \(\mathbb{R}^3\) in the case of a folded node. SIAM J. Appl. Dyn. Syst., 4(1), 101–139 (2005) (electronic)
Wilson, C.J., Callaway, J.C.: Coupled oscillator model of the dopaminergic neuron of the substantia nigra. J. Neurophysiol. 83(5), 3084–3100 (2000)
Acknowledgments
The authors’ research was supported by the Research Foundation Flanders (FWO) under grant number G.0939.10N. Moreover, E. K. acknowledges support from the Polish Ministry of Science and Higher Education.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Maesschalck, P., Kutafina, E. & Popović, N. Three Time-Scales In An Extended Bonhoeffer–Van Der Pol Oscillator. J Dyn Diff Equat 26, 955–987 (2014). https://doi.org/10.1007/s10884-014-9356-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-014-9356-3
Keywords
- Bonhoeffer–van der Pol oscillator
- Mixed-mode oscillations
- Canards
- Geometric singular perturbation theory
- Blow-up technique