Dissection of a model for neuronal parabolic bursting

John Rinzel, Young Seek Lee

Research output: Contribution to journalArticle

Abstract

We have obtained new insight into the mechanisms for bursting in a class of theoretical models. We study Plant's model [24] for Aplysia R-15 to illustrate our view of these so-called "parabolic" bursters, which are characterized by low spike frequency at the beginning and end of a burst. By identifying and analyzing the fast and slow processes we show how they interact mutually to generate spike activity and the slow wave which underlies the burst pattern. Our treatment is essentially the first step of a singular perturbation approach presented from a geometrical viewpoint and carried out numerically with AUTO [12]. We determine the solution sets (steady state and oscillatory) of the fast subsystem with the slow variables treated as parameters. These solutions form the slow manifold over which the slow dynamics then define a burst trajectory. During the silent phase of a burst, the solution trajectory lies approximately on the steady state branch of the slow manifold and during the active phase of spiking, the trajectory sweeps through the oscillation branch. The parabolic nature of bursting arises from the (degenerate) homoclinic transition between the oscillatory branch and the steady state branch. We show that, for some parameter values, the trajectory remains strictly on the steady state branch (to produce a resting steady state or a pure slow wave without spike activity) or strictly in the oscillatory branch (continuous spike activity without silent phases). Plant's model has two slow variables: a calcium conductance and the intracellular free calcium concentration, which activates a potassium conductance. We also show how bursting arises from an alternative mechanism in which calcium inactivates the calcium current and the potassium conductance is insensitive to calcium. These and other biophysical interpretations are discussed.

Original languageEnglish (US)
Pages (from-to)653-675
Number of pages23
JournalJournal of Mathematical Biology
Volume25
Issue number6
DOIs
StatePublished - Dec 1987

Fingerprint

Dissection
Bursting
Calcium
Branch
trajectories
Burst
Spike
calcium
inflorescences
Trajectories
Conductance
Trajectory
Slow Manifold
Potassium
potassium
Strictly
Aplysia
Model
Homoclinic
Singular Perturbation

Keywords

  • Bursting-Pacemaker
  • Neuronal modeling

ASJC Scopus subject areas

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

Cite this

Dissection of a model for neuronal parabolic bursting. / Rinzel, John; Lee, Young Seek.

In: Journal of Mathematical Biology, Vol. 25, No. 6, 12.1987, p. 653-675.

Research output: Contribution to journalArticle

Rinzel, John ; Lee, Young Seek. / Dissection of a model for neuronal parabolic bursting. In: Journal of Mathematical Biology. 1987 ; Vol. 25, No. 6. pp. 653-675.
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