# Applied Math Colloquium

### Bifurcation Theory and the Dynamics of Cardiac Arrhythmias

### Abstract

Mathematical models of cardiac arrhythmias come in two distinct forms. High-dimensional, biophysically detailed models give us realism; they talk about biophysical quantities that can be experimentally and clinically altered. But there is another critical kind of modeling: Low-dimensional modeling tries to isolate the essential dynamical phenomena responsible for a qualitative type of solution. It gives us deeper insights into causal mechanisms. The most important technique of low-dimensional modeling is bifurcation theory. A bifurcation is a qualitative change in the solution to a differential equation (ODE or PDE), as a key parameter is varied. Identification of these key parameters then becomes the central task, because it is by acting through them that we can produce or prevent the qualitative phenomenon. For example, consider the onset of Ventricular Fibrillation. We will show that VF arises as a series of bifurcations from simple reentry to complex spatio- temporally chaotic behavior, and that manipulations of the key parameters can prevent its onset. A second type of arrhythmia is the onset of the pathological voltage oscillations that are known as early after depolarizations (EADs). EADs are thought to be important triggers of arrhythmia in heart failure, Long QT syndromes and other cardiac conditions. We will show that the onset of these pathological oscillations is a bifurcation at the cellular level, and that manipulation of its key parameters can produce and prevent EADs.