The topology of defibrillation

We describe how Art Winfree's ideas about phase singularities can be used to understand the response of cardiac tissue with a random preexisting pattern of reentrant waves (fibrillation) to a large brief current stimulus. This discussion is organized around spatial dimension, beginning with a discussion of reentry on a periodic ring, followed by reentry in a two-dimensional planar domain (spiral waves), and ending with consideration of three-dimensional reentrant patterns (scroll waves). In all cases, we show how reentrant activity is changed by the application of a shock, describing conditions under which defibrillation is successful or not. Using topological arguments we draw the general conclusion that with a generic placement of stimulating electrodes, large-scale virtual electrodes do not give an adequate explanation for the mechanism of defibrillation.