Abstract: In cardiac tissue, beat-to-beat alternation of action potential duration (APD) is a warning sign of potentially serious pathologies. When APD alternans is detected, it is desirable to coax the tissue back to a normal rhythm in which APD has little beat-to-beat variation. Mathematically, this is can be accomplished by applying feedback control to stabilize an unstable equilibrium near a periodic (or chaotic) orbit. Clinically, it is accomplished by applying well-timed electrical stimuli via a medical device such as a pacemaker. Such device intervention can be implemented in several ways, two of which are point stimulation and far-field pacing (FFP). In point stimulation, the device applies spatially localized stimuli through the tip of an electrode, whereas in FFP, large plate electrodes apply pulsed electric fields pulses across the entire heart. FFP creates "virtual" electrodes within the tissue by depolarizing or hyperpolarizing cells near the boundaries of non-conducting obstacles (e.g., dead tissue) and, if the field strength is strong enough, propagating action potentials can emanate from these obstacles. In this study, we analyze a particular feedback control algorithm (extended time-delay autosynchronization, ETDAS) for timing the stimuli in point stimulation, with the goal of controlling alternans in zero and one-dimensional samples of cardiac tissue (i.e., a single cell or a long fiber of cells joined end-to-end), as well as the use of ETDAS as a method for timing the stimuli applied during FFP. Previous theoretical and experimental studies have shown that special cases of ETDAS can terminate alternans in small, "zero-dimensional" patches of cardiac cells in which spatial extent is negligible; however, those special cases of ETDAS perform rather poorly in controlling the spatially discordant alternans in one-dimensional fibers. Here, we explore whether the added robustness of ETDAS can enlarge the spatial domain over which point stimulation can succeed, ultimately comparing our results with those obtained using FFP.

Brief Biography of the Speaker:
John W. Cain graduated from Duke University, Durham, NC, USA in 2005 with a Ph.D. in Mathematics. From 2005-2011, he served on the mathematics department faculty at Virginia Commonwealth University and as a Fellow of VCU's Center for the Study of Biological Complexity. In August 2011, Dr. Cain moved to the University of Richmond, where he is Associate Professor of Mathematics and Computer Science. His scholarly work lies at the interface of mathematics and medicine, and involves problems in cardiac electrophysiology, dynamics of biochemical reaction networks, and wound healing. Dr. Cain's research has been featured in interviews with Science, the American Mathematical Society, and in the Notices of the AMS (April 2011). In addition to his biomathematics research articles, he has co-authored two textbooks on differential equations, dynamical systems and bifurcations, both of which are available free-of-charge (by electronic request).