Analysis of an autonomous phase model for neuronal parabolic bursting.

S. M. Baer, J. Rinzel, H. Carrillo

Research output: Contribution to journalArticle

Abstract

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol 25, 653-675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steady-state assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.

Original languageEnglish (US)
Pages (from-to)309-333
Number of pages25
JournalJournal of Mathematical Biology
Volume33
Issue number3
StatePublished - 1995

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Nerve Tissue
Nonlinear Dynamics
Bursting
Dissection
Membranes
Burst
Model
Phase Plane
Bistability
Alternation
Structure-function
Nerve
Averaging
Activation
inactivation
nerve tissue
Membrane
Chemical activation
Tissue
Demonstrate

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Mathematics (miscellaneous)

Cite this

Analysis of an autonomous phase model for neuronal parabolic bursting. / Baer, S. M.; Rinzel, J.; Carrillo, H.

In: Journal of Mathematical Biology, Vol. 33, No. 3, 1995, p. 309-333.

Research output: Contribution to journalArticle

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