The transition from bursting to continuous spiking in excitable membrane models
- 306 Downloads
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 wordsbursting oscillations excitable membranes Fibonacci dynamics
Unable to display preview. Download preview PDF.
- 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
- 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.T. R. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model, Biophys. J.47 (1985) 357–366.Google Scholar
- 8.J. Gukenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors Fields, Springer-Verlag, New York (1986).Google Scholar
- 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.H. P. Meissner, Electrical characteristics of the beta-cells in pancreatic islets. J. Physiol. (Paris)72 (1976), 757–767.Google Scholar
- 12.C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J.35 (1981), 193–213.Google Scholar
- 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.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
- 18.S. Wiggins,Global Birfurcations and Chaos: Analytic Methods, Springer-Verlag, New York, Heidelberg, Berlin (1988).Google Scholar