In spite of its practical importance, the processes of defibrillation still remain obscure. Most theoretical approaches have been based on linear models [Knisley et al., 1994; Sepulveda et al., 1989; Krassowska et al., 1990], while some numerics of nonlinear models [Cartee, 1992] and theoretical studies with simplified models have been attempted [Pumir & Krinsky, 1996]. The reason for this lack of progress is the combination of nonlinearity and the hierarchical multi-timescale structure of biophysical excitation equation with the necessity for a representation of the complicated spatial structure for every cell. We have overcome these problems by applying a series of well known methods, to fullfill nonlinear averaging; the result is the simplified models of (19) and (23). This reduction of an infinite dimensional system to an ordinary differential system may be of value in a range of applications of nonlinear science.
These simplified models have been verified numerically and allow us to use biophysically detailed excitation equations, and so we are now in a position to provide a quantitative, theoretical explanation for the effects of changes in parameters in the excitation equations on the defibrillation threshold, and to design optimal defibrillation pulse parameters. Experimental techniques now exist [Zhou et al., 1995] for testing such quantitative descriptions of the mechanisms of defibrillation.