Journal of Nonlinear Science

, Volume 2, Issue 2, pp 135–182 | Cite as

The transition from bursting to continuous spiking in excitable membrane models

  • D. Terman


Mathematical models for excitable membranes may exhibit bursting solutions, and, for different values of the parameters, the bursting solutions give way to continuous spiking. Numerical results have demonstrated that during the transition from bursting to continuous spiking, the system of equations may give rise to very complicated dynamics. The mathematical mechanism responsible for this dynamics is described. We prove that during the transition from bursting to continuous spiking the system must undergo a large number of bifurcations. After each bifurcation the system is increasingly chaotic in the sense that the maximal invariant set of a certain two-dimensional map is topologically equivalent to the shift on a larger set of symbols. The number of symbols is related to the Fibonacci numbers.

Key words

bursting oscillations excitable membranes Fibonacci dynamics 


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  1. 1.
    J. C. Alexander and D. Cai, On the dynamics of bursting systems, J. Math. Biol.29, 405 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    P. M. Beigelman, B Ribalet, and I. Atwater, Electrical activity of mouse pancreaticβ-cells II. Effects of glucose and arginine, J. Physiol. (Paris)73 (1977), 201–217.Google Scholar
  3. 3.
    T. R. Chay, Chaos in a three-variable model of an excitable cell, Phys. D. 16 (1985), 233–242.zbMATHCrossRefGoogle Scholar
  4. 4.
    T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J.42 (1983), 181–190.Google Scholar
  5. 5.
    T. R. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model, Biophys. J.47 (1985) 357–366.Google Scholar
  6. 6.
    B. Deng, Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal.21 (1990), 693–720.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J.21 (1971), 193–226.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Gukenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors Fields, Springer-Verlag, New York (1986).Google Scholar
  9. 9.
    A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London)117 (1952), 500–544.Google Scholar
  10. 10.
    H. P. Meissner, Electrical characteristics of the beta-cells in pancreatic islets. J. Physiol. (Paris)72 (1976), 757–767.Google Scholar
  11. 11.
    J. Moser,Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton (1973).zbMATHGoogle Scholar
  12. 12.
    C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J.35 (1981), 193–213.Google Scholar
  13. 13.
    J. Rinzel,Bursting oscillations in an excitable membrane model, inOrdinary and Partial Differential Equations (B. D. Sleeman and R. J. Jarvis, eds.), Springer-Verlag, New York (1985), 304–316.Google Scholar
  14. 14.
    J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, inMethods in Neuronal Modeling, From Synapses to Networks, (C. Koch and I. Seger, eds.), MIT Press, Cambridge, MA (1989), 135–169.Google Scholar
  15. 15.
    A. M. Scott, I. Atwater, and E. Rojas, A method for the simultaneous measurement of insulin release and beta-cell membrane potential in single mouse islet of Langerhans, Diabetologia21 (1981), 470–475.CrossRefGoogle Scholar
  16. 16.
    A. Sherman, J. Rinzel, and J. Keizer, Emergence of organized bursting in clusters of pancreaticβ-cells by channel sharing, Biophys. J.54 (1988), 411–425.CrossRefGoogle Scholar
  17. 17.
    D. Terman, Chaotic spikes arising from a model for bursting in excitable membranes, SIAM J. Appl. Math.51 (1991), 1418–1450.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Wiggins,Global Birfurcations and Chaos: Analytic Methods, Springer-Verlag, New York, Heidelberg, Berlin (1988).Google Scholar

Copyright information

© Springer-Verlag New York Inc 1992

Authors and Affiliations

  • D. Terman
    • 1
    • 2
  1. 1.Department of MathematicsOhio State UniversityColumbus
  2. 2.Mathematics Research Branch, NIDDKNational Institute of HealthBethesdaUSA

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