# Prize Spotlight: Theodore Vo, Richard Bertram, and Martin Wechselberger

Theodore Vo, Richard Bertram, and Martin Wechselberger received the SIAM Outstanding Paper Prize at the 2017 SIAM Annual Meeting. They were recognized for their paper, “Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting,” published in *SIAM Journal on Applied Dynamical Systems, Volume 12, Issue 2 (2013)*.

The SIAM Outstanding Paper Prizes are awarded annually to the authors of the three most outstanding papers, in the opinion of the selection committee, published in SIAM journals in the three calendar years preceding the year before the award year. Priority is given to papers that bring a fresh look at an existing field or that open up new areas of applied mathematics.

**Q:*** Why are you excited about winning the prize?*

**A:** We are honored to be in the company of past, current, and future prize winners. The prize is also recognition of a productive and enjoyable collaboration.

**Q:*** Could you tell us a bit about the research that won you the prize?*

**A:** Pseudo-plateau bursting (PPB) is a type of oscillatory waveform featuring small-amplitude oscillations on an elevated voltage plateau commonly observed in pituitary cells. In a recent three-time-scale model for the electrical activity and calcium signaling in a pituitary lactotroph, two types of PPB were discovered: one in which the calcium drives the bursts and another in which the calcium is slaved to them. Such three-time-scale systems had received little attention at the time at which the research was done and were typically treated as two-time-scale problems, which is the natural setting for geometric singular perturbation theory (GSPT). The presence of a third time-scale, however, means that there are various ways in which GSPT could be implemented.

The classic two-time-scale approach treated the calcium concentration as a slowly varying parameter and considered a parametrized family of fast subsystems. Bursts are classified according to the fast subsystem bifurcations (with respect to the calcium) involved in the initiation/termination of the active phase. The small-amplitude spikes in the PPB are understood as transient oscillations generated by unstable limit cycles emanating from a subcritical Hopf bifurcation in the fast subsystem. This classic approach explains the origin and properties of the calcium-induced PPB, such as resetting behavior and burst termination.

A more novel two-time-scale analysis divides the system so that the voltage is the only fast variable, and shows that the bursting arises from canard dynamics. Key organizing structures are the attracting and repelling invariant slow manifolds of the system. Transverse intersections of these slow manifolds (a.k.a canard solutions) shape the dynamics. Canard theory then provides the theoretical basis for understanding burst properties such as the transition from spiking to bursting, and how spike-adding bifurcations can occur. In particular, the canards explain the origin and properties of the calcium-conducting PPB.

Both slow/fast analysis techniques yield key insights into the dynamics of PPB, but there has been little justification for one approach over the other. In this research, we use the lactotroph model to demonstrate that the two analysis techniques are different unfoldings of a three-time-scale system. We compare the two-time-scale methods with a three-time-scale analysis and demonstrate the efficacy of each technique. We assert that the three-time-scale decomposition provides the best results asymptotically, independent of the model, as it inherits all of the geometric information contained in the different two-time-scale analyses. As a result, the three-time-scale decomposition provides a remarkable degree of control and predictive power.

**Q:*** What does your research mean to the public?*

**A:** Bursting electrical oscillations are the means through which endocrine cells release hormones. Understanding the mechanism of bursting in pituitary cells is thus necessary for an understanding of hormone secretion from pituitary cells, which secrete hormones involved in reproduction, stress, growth, and lactation. Our research provides a mathematical approach that will help one understand the dynamics of mathematical models of the different types of pituitary cells. This includes the ability to predict drug effects on the behavior of the cells.

**Q: ***W**hat does being a SIAM member mean to you?*

**A:** SIAM brings together mathematicians from all over the world who are interested in using mathematics to solve real world problems. Having such a community is an important unifying element for those of us who use mathematics to solve problems or provide insights in fields such as biology, physics, or chemistry. This is an inherently interdisciplinary endeavor, so it is good to have a society like SIAM to represent those of us who work not just within the discipline of mathematics, but between disciplines.