Journal of Nonlinear Science

, Volume 2, Issue 2, pp 135–182

The transition from bursting to continuous spiking in excitable membrane models

  • D. Terman

DOI: 10.1007/BF02429854

Cite this article as:
Terman, D. J Nonlinear Sci (1992) 2: 135. doi:10.1007/BF02429854


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 oscillationsexcitable membranesFibonacci dynamics

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