Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. APPLIED DYNAMICAL SYSTEMS c  2013 Society for Industrial and Applied Mathematics Vol. 12, No. 2, pp. 789–830 Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting ∗ Theodore Vo † , Richard Bertram ‡ , and Martin Wechselberger † Abstract. Pseudo-plateau bursting is a type of oscillatory waveform associated with mixed mode dynamics in slow/fast systems and commonly found in neural bursting models. In a recent model for the electrical activity and calcium signaling in a pituitary lactotroph, two types of pseudo-plateau bursts were discovered: one in which the calcium drives the bursts and another in which the calcium simply follows them. Multiple methods from dynamical systems theory have been used to understand the bursting. The classic 2-timescale approach treats the calcium concentration as a slowly varying parameter and considers a parametrized family of fast subsystems. A more novel and successful 2-timescale approach divides the system so that there is only one fast variable and shows that the bursting arises from canard dynamics. Both methods can be effective analytic tools, but there has been little justification for one approach over the other. In this work, we use the lactotroph model to demonstrate that the two analysis techniques are different unfoldings of a 3-timescale system. We show that elementary applications of geometric singular perturbation theory in the 2-timescale and 3-timescale methods provide us with substantial predictive power. We use that predictive power to explain the transient and long-term dynamics of the pituitary lactotroph model. Key words. geometric singular perturbation theory, multiple timescales, mixed mode oscillations, bursting AMS subject classifications. 37N25, 34E17, 92B25, 37M05, 37G35 DOI. 10.1137/120892842 1. Introduction. Bursting is a type of complex oscillatory waveform commonly seen in the electrical activity of nerve and endocrine cells [34, 18, 42, 28, 43, 44]. Characterized by alternating periods of fast spiking in the active (depolarized) phase and quiescence during which the cell is repolarized, bursts are typically more efficient than spikes in evoking hormone and neurotransmitter release [33, 48]. Characteristics of the burst pattern such as frequency and duration determine how much calcium enters the cell, which in turn determines the level of hormone secretion [59]. One particular type of bursting that has been the focus of recent modeling efforts is pseudo-plateau bursting, which features small amplitude oscillations or spikes in the active phase superimposed on relaxation type oscillations [51, 47, 39, 56]. Pseudo-plateau bursts are distinguished from plateau bursts, which feature large amplitude fast spiking in the active phase [40, 4, 32, 57]. Using geometric singular perturbation theory [19, 26], the authors showed that the pseudo- ∗ Received by the editors September 9, 2012; accepted for publication (in revised form) by H. Osinga March 21, 2013; published electronically May 21, 2013. http://www.siam.org/journals/siads/12-2/89284.html † School of Mathematics and Statistics, University of Sydney, Sydney NSW, Australia (theodore.vo@sydney.edu.au, wm@maths.usyd.edu.au). The first author’s research was partially supported by an A.E. and F.A.Q. Stephens Scholarship and a Philipp Hofflin International Research Scholarship (University of Sydney). The third author’s research was partially supported by the Australian Research Council and the Marsden Fund, New Zealand. ‡ Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL 32306 (bertram@math.fsu.edu). This author’s research was supported by NSF grant DMS 1220063. 789 Downloaded 06/11/13 to 128.186.103.176. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php