Mixed mode oscillations as a mechanism for pseudo-plateau bursting
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We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called “pseudo-plateau bursting”. Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic β-cells.
KeywordsPseudo-plateau burst Pituitary cell Conductance-based model Mixed mode oscillation Canard Slow-fast system Bifurcation analysis
Financial support was provided by the National Institutes of Health grant DA 43200 and National Science Foundation grant DMS 0917664 to RB. MW thanks the Vienna University of Technology and the Mathematical Biosciences Institute at Ohio State University for sabbatical support.
- Benoit, E. (1983). Syst‘emes lents-rapides dans r3 et leur canards. Asterisque, 109–110, 159–191.Google Scholar
- Brons, M., Krupa, M., & Wechselberger, M. (2006). Mixed mode oscillations due to the generalized canard phenomenon. Fields Institute Communications, 49, 39–63.Google Scholar
- Ermentrout, G. B. (2002). Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students. Philadelphia: SIAM Books.Google Scholar
- Jones, C. K. R. T. (1995). Geometric singular perturbation theory. In R. Johnson (Ed.), Dynamical systems. Lecture notes in mathematics (Vol. 1609, pp. 44–120). New York: Springer.Google Scholar
- Rinzel, J. (1987). A formal classification of bursting mechanisms in excitable systems. In E. Teramoto, & M. Yamaguti (Eds.), Mathematical topics in population biology, morphogenesis and neurosciences. Lecture notes in biomathematics (Vol. 71, pp. 267–281). Berlin: Springer.Google Scholar
- Rinzel, J., & Ermentrout, G. B. (1998). Analysis of neural excitability and oscillations. In C. Koch, & I. Segev (Eds.), Methods in neuronal modeling: From synapses to networks (2nd ed., pp. 251–292). Cambridge: MIT.Google Scholar