Voltage Interval Mappings for an Elliptic Bursting Model

Chapter
First Online:
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 12)

Abstract

We employed Poincaré return mappings for a parameter interval to an exemplary elliptic bursting model, the FitzHugh–Nagumo–Rinzel model. Using the interval mappings, we were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally, mixed-mode oscillations and quiescence in the FitzHugh–Nagumo–Rinzel model. We illustrate the wealth of information, qualitative and quantitative, that was derived from the Poincaré mappings, for the neuronal models and for similar (electro) chemical systems.

Keywords

Lyapunov Exponent Hopf Bifurcation Homoclinic Orbit Topological Entropy Phase Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgments

This work was supported by the NSF grant DMS-1009591, the RFFI grant 11-01-00001, the RSF grant 14-41-00044 and by the MESRF grant 02.B.49.21.003.

References

  1. 1.
    Albahadily, F., Ringland, J., Schell, M.: Mixed-mode oscillations in an electrochemical system. I. A Farey sequence which does not occur on a torus. J. Chem. Phys. 90(2), 813–822 (1989)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Argoul, F., Roux, J.: Quasiperiodicity in chemistry: An experimental path in the neighbourhood of a codimension-two bifurcation. Phys. Lett. A 108(8), 426–430 (1985)CrossRefGoogle Scholar
  3. 3.
    Arnold, V., Afraimovich, V., Ilyashenko, Y., Shilnikov, L.: Bifurcation theory. In: Arnold, V. (ed.) Dynamical Systems. Encyclopaedia of Mathematical Sciences, vol. V. Springer, Berlin (1994)Google Scholar
  4. 4.
    Benes, N., Barry, A., Kaper, T., Kramer, M., Burke, J.: An elementary model of torus canards. Chaos 21, 023131 (2011)Google Scholar
  5. 5.
    Bertram, R., Butte, M., Kiemel, T., Sherman, A.: Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57(3), 413–439 (1995) (PM:7728115)CrossRefMATHGoogle Scholar
  6. 6.
    Channell, P., Cymbalyuk, G., Shilnikov, A.: Applications of the Poincaré mapping technique to analysis of neuronal dynamics. Neurocomputing 70, 10–12 (2007)CrossRefGoogle Scholar
  7. 7.
    Channell, P., Cymbalyuk, G., Shilnikov, A.: Origin of bursting through homoclinic spike adding in a neuron model. Phys. Rev. Lett. 98(13), Art. 134101 (2007) (PM:17501202)Google Scholar
  8. 8.
    Channell, P., Fuwape, I., Neiman, A., Shilnikov, A.: Variability of bursting patterns in a neuron model in the presence of noise. J. Comput. Neurosci. 27(3), 527–542 (2009). doi:10.1007/s10827-009-0167-1. http://dx.doi.org/10.1007/s10827-009-0167-1
  9. 9.
    Cymbalyuk, G., Shilnikov, A.: Coexistence of tonic spiking oscillations in a leech neuron model. J. Comput. Neurosci. 18(3), 255–263 (2005). doi:10.1007/s10827-005-0354-7. http://dx.doi.org/10.1007/s10827-005-0354-7
  10. 10.
    Doi, J., Kumagai, S.: Generation of very slow neuronal rhythms and chaos near the hopf bifurcation in single neuron models. J. Comput. Neurosci. 19(3), 325–356 (2005). doi:10.1007/s10827-005-2895-1. http://dx.doi.org/10.1007/s10827-005-2895-1
  11. 11.
    Gaspard, M.K.P., Sluyters, J.: Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle-focus in the indium/thiocyanate electrochemical oscillator. J. Chem. Phys. 97(11), 8250–8260 (1992)CrossRefGoogle Scholar
  12. 12.
    Gaspard, P., Wang, X.: Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium. J. Stat. Phys. 48(1/2), 151–199 (1987)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Gavrilov, N., Shilnikov, L.: On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Math. USSR-Sb. 17(3), 467–485 (1972)CrossRefGoogle Scholar
  14. 14.
    Glendinning, P., Hall, T.: Zeros of the kneading invariant and topological entropy for Lorenz maps. Nonlinearity 9, 999–1014 (1996)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Griffiths, R., Pernarowski, M.: Return map characterizations for a model of bursting with two slow variables. SIAM J. Appl. Math. 66(6), 1917–1948 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Guckenheimer, J.: Towards a global theory of singularly perturbed systems. In: Broer, H.W., van Gils, S.A., Hoveijn, F., Takens, F. (eds.) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol. 19, pp. 214–225 (1996)Google Scholar
  17. 17.
    Guckenheimer, J.: Singular Hopf bifurcation in systems with two slow variables. SIAM J. Appl. Dyn. Syst. 7(4), 1355–1377 (2008)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Hudson, J., Marinko, D.: An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov–Zhabotinskii reaction. J. Chem. Phys. 71(4), 1600–1606 (1979)CrossRefGoogle Scholar
  19. 19.
    Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES 51, 137–173 (1980)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Koper, M., Gaspard, P.: Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle-focus in the indium/thiocyanate electricochemical oscillators. J. Chem. Phys. 97(11), 8250–8260 (1992)CrossRefGoogle Scholar
  21. 21.
    Kramer, M., Traub, R., Kopell, N.: New dynamics in cerebellar Purkinje cells: Torus canards. Phys. Rev. Lett. 101(6), Art. 068103 (2008)Google Scholar
  22. 22.
    Kuznetsov, A., Kuznetsov, S., Stankevich, N.: A simple autonomous quasiperiodic self-oscillator. Commun. Nonlinear Sci. Numer. Simul. 15, 1676–1681 (2010)CrossRefGoogle Scholar
  23. 23.
    Li, M.C., Malkin, M.: Smooth symmetric and Lorenz models for unimodal maps. Int. J. Bifurc. Chaos 13(11), 3353–3371 (2003)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Medvedev, G.: Transition to bursting via deterministic chaos. Phys. Rev. Lett. 97, Art. 048102 (2006)Google Scholar
  25. 25.
    Mira, C.: Chaotic Dynamics from the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism. World Scientific, Singapore (1987)CrossRefMATHGoogle Scholar
  26. 26.
    Mischenko, E., Kolesov, Y., Kolesov, A., Rozov, N.: Asymptotic Methods in Singularly Perturbed Systems. Monographs in Contemporary Mathematics. Consultants Bureau, New York (1994)CrossRefGoogle Scholar
  27. 27.
    Neiman, A., Dierkes, K., Lindner, B., Shilnikov, A.: Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells. J. Math. Neurosci. 1(11) (2011)Google Scholar
  28. 28.
    Neishtadt, A.I.: On delayed stability loss under dynamical bifurcations I. Differ. Equ. 23, 1385–1390 (1988)MathSciNetGoogle Scholar
  29. 29.
    Rinzel, J.: A formal classification of bursting mechanisms in excitable systems. In: Gleason, A.M. (ed.) Proceedings of the International Congress of Mathematics, pp. 1578–1593. AMS, Providence (1987)Google Scholar
  30. 30.
    Rinzel, J., Lee, Y.S.: Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25(6), 653–675 (1987)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Shilnikov, A.: On bifurcations of the Lorenz attractor in the Shimizu–Morioka model. Phys. D 62(1–4), 338–346 (1993)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Shilnikov, A., Kolomiets, M.: Methods of the qualitative theory for the Hindmarsh–Rose model: A case study. A tutorial. Int. J. Bifurc. Chaos 18(7), 1–32 (2008)MathSciNetGoogle Scholar
  33. 33.
    Shilnikov, A., Rulkov, N.: Origin of chaos in a two-dimensional map modelling spiking-bursting neural activity. Int. J. Bifurc. Chaos 13(11), 3325–3340 (2003)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Shilnikov, L., Shilnikov, A., Turaev, D., Chua, L.: Methods of Qualitative Theory in Nonlinear Dynamics, vols. 1 and 2. World Scientific, Singapore (1998, 2001)CrossRefMATHGoogle Scholar
  35. 35.
    Su, J., Rubin, J., Terman, D.: Effects of noise on elliptic bursters. Nonlinearity 17, 133–157 (2004)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Tikhonov, A.: On the dependence of solutions of differential equations from a small parameter. Mat. Sb. 22(64), 193–204 (1948)Google Scholar
  37. 37.
    Wojcik, J., Shilnikov, A.: Voltage interval mappings for activity transitions in neuron models for elliptic bursters. Phys. D 240, 1164–1180 (2011)CrossRefMATHGoogle Scholar
  38. 38.
    Zaks, M.: On chaotic subthreshold oscillations in a simple neuronal model. Math. Model. Nat. Phenom. 6(1), 1–14 (2011)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Zaks, M.A., Sailer, X., Schimansky-Geier, L., Neiman, A.B.: Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems. Chaos 15(2), Art. 26117 (2005). doi:10.1063/1.1886386. http://dx.doi.org/10.1063/1.1886386

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Applied Technology AssociatesAlbuquerqueUSA
  2. 2.Department of Computational Mathematics and CyberneticsLobachevsky State University of Nizhni NovgorodNizhni NovgorodRussia

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