On Different Mechanisms for Membrane Potential Bursting

  • John Rinzel
  • Young Seek Lee
Conference paper
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 66)


A number of mathematical models have been proposed to describe the electrical bursting activity of biological excitable membrane systems. Many of these models have been formulated for specific applications [4,5,14]. One of our goals has been to understand the basic underlying qualitative structure of these models and to distinguish, possibly different, classes of models for bursting. In this paper we contrast two examples which illustrate different mathematical mechanisms.


Manifold Arginine Resis Dura Congo 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, W. B., and J. A. Benson. 1985. The generation and modulation of endogenous rhythmicity in the Aplysia bursting pacemaker neurone R15. Prog. Biophys. Molec. Biol. 46:1–49.CrossRefGoogle Scholar
  2. 2.
    Atwater, I., C. M. Dawson, A. Scott, G. Eddiestone, E. Rojas. 1980. The nature of the oscillatory behavior in electrical activity for pancreatic β-cell. J. of Hormone and Metabolic Res., Suppl. 10:100–107.Google Scholar
  3. 3.
    Beigelman, P. M., B. Ribalet, and I. Atwater. 1977. Electrical activity of mouse pancreatic beta-cells II. Effects of glucose and arginine. J. Physiol., Paris 71:201–217.Google Scholar
  4. 4.
    Both, R., W. Finger, and R. A. Chaplain. 1976. Model predictions of the ionic mechanisms underlying the beating and bursting pacemaker characteristics of mulloscan neurons. Biiol. Cybernetics 23:1–11.CrossRefGoogle Scholar
  5. 5.
    Chay, T. R., and J. Keizer. 1983. Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J. 42:181–190.CrossRefGoogle Scholar
  6. 6.
    Chay, T. R., and J. Keizer. Theory of the effect of extracellular potassium on oscillations in the pancreatic β-cell. Biophys. J. (in press).Google Scholar
  7. 7.
    Doedel, E. J. 1981. AUTO: A program for the automatic bifurcation and analysis of autonomous systems, (Proc. 10th Manitoba Conf. on Num. Math. and Comput., Winnipeg, Canada), Cong. Num. 30:265–284.MathSciNetGoogle Scholar
  8. 8.
    Ermentrout, G. B., and N. Kopell. Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Applied Math. (in press).Google Scholar
  9. 9.
    FitzHugh, R. 1961. Impulses and physiological states in models of nerve membrane. Biophys. J. 1:445–466.CrossRefGoogle Scholar
  10. 10.
    Hindmarsh, J. L., and R. M. Rose. 1984. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B 221:87–102.CrossRefGoogle Scholar
  11. 11.
    Hodgkin, A. L., and A. F. Huxley. 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond) 117:500–544.Google Scholar
  12. 12.
    Honerkamp, J., G. Mutschler, and R. Seitz. 1985. Coupling of a slow and a fast oscillator can generate bursting. Bull. Math. Biol. 47:1–21.MATHMathSciNetGoogle Scholar
  13. 13.
    Nagumo, J. S., S. Arimoto, and S. Yoshizawa. 1962. An active pulse transmission line simulating nerve axon. Proc. IRE. 50:2061–2070.CrossRefGoogle Scholar
  14. 14.
    Plant, R. E., and M. Kim. 1976. Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys. J. 16:227–244.CrossRefGoogle Scholar
  15. 15.
    Plant, R. E. 1981. Bifurcation and resonance in a model for bursting nerve cells. J. Math. Biology 11:15–32.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Rinzel, J., and W. C. Troy. 1982. Bursting phenomena in a simplified Oregonator flow system model. J. Chem. Phys. 76:1775–1789.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Rinzel, J. Bursting oscillations in an excitable membrane model. In Proc. 8th Dundee Conf. on thé Theory of Ordinary and Partial Differential Equations (eds., B. D. Sleeman, R. J. Jarvis, and D. S. Jones). Springer-Verlag (in press).Google Scholar
  18. 18.
    Scott, A. M., I. Atwater, and E. Rojas. 1981. A method for the simultaneous measurement of insulin release and β-cell membrane potential in single mouse islet of Langerhans. Diabetologia 21:470–475.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • John Rinzel
    • 1
  • Young Seek Lee
    • 1
  1. 1.Mathematical Research Branch, NIADDKNational Institutes of HealthBethesadaUSA

Personalised recommendations