Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells

Richard Bertram, Joël Tabak, Wondimu Teka, Theodore Vo, Martin Wechselberger

Research output: Chapter in Book/Report/Conference proceedingChapter (Book)Otherpeer-review


Dynamical systems theory provides a number of powerful tools for analyzing biological models, providing much more information than can be obtained from numerical simulation alone. In this chapter, we demonstrate how geometric singular perturbation analysis can be used to understand the dynamics of bursting in endocrine pituitary cells. This analysis technique, often called “fast/slow analysis,” takes advantage of the different time scales of the system of ordinary differential equations and formally separates it into fast and slow subsystems. A standard fast/slow analysis, with a single slow variable, is used to understand bursting in pituitary gonadotrophs. The bursting produced by pituitary lactotrophs, somatotrophs, and corticotrophs is more exotic, and requires a fast/slow analysis with two slow variables. It makes use of concepts such as canards, folded singularities, and mixed-mode oscillations. Although applied here to pituitary cells, the approach can and has been used to study mixed-mode oscillations in other systems, including neurons, intracellular calcium dynamics, and chemical systems. The electrical bursting pattern produced in pituitary cells differs fundamentally from bursting oscillations in neurons, and an understanding of the dynamics requires very different tools from those employed previously in the investigation of neuronal bursting. The chapter thus serves both as a case study for the application of recently-developed tools in geometric singular perturbation theory to an application in biology and a tutorial on how to use the tools.
Original languageEnglish
Title of host publicationMathematical Analysis of Complex Cellular Activity
EditorsChristopher K R T Jones, Björn Sandstede, Lai-Sang Young
Place of PublicationCham Switzerland
Number of pages52
ISBN (Electronic)9783319181141
ISBN (Print)9783319181134
Publication statusPublished - 2015
Externally publishedYes

Publication series

NameFrontiers in Applied Dynamical Systems: Reviews and Tutorials
ISSN (Print)2364-4532
ISSN (Electronic)2364-4931

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