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Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells

  • Richard BertramEmail author
  • Joël Tabak
  • Wondimu Teka
  • Theodore Vo
  • Martin Wechselberger
Part of the Frontiers in Applied Dynamical Systems: Reviews and Tutorials book series (FIADS, volume 1)

Abstract

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.

Keywords

Hopf Bifurcation Pituitary Cell Slow Manifold Fast Subsystem Critical Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported by NSF grants DMS 0917664 to RB, DMS 1220063 to RB and JT, and NIH grant DK 043200 to RB and JT.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Richard Bertram
    • 1
    Email author
  • Joël Tabak
    • 2
  • Wondimu Teka
    • 3
  • Theodore Vo
    • 4
  • Martin Wechselberger
    • 5
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of Mathematics and Biological ScienceFlorida State UniversityTallahasseeUSA
  3. 3.Department of MathematicsIndiana University – Purdue University IndianapolisIndianapolisUSA
  4. 4.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  5. 5.Department of MathematicsUniversity of SydneySydneyAustralia

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