THEORY OF PULSE INSTABILITIES IN ELECTROPHYSIOLOGICAL MODELS OF EXCITABLE TISSUES

We develop a theoretical framework which allows, starting from multi-variable electrophysiological PDE models, to make ab initio quantitative predictions of the instability threshold and nonlinear dynamics of a pulse propagating in a ring of excitable tissue. This framework is based on the reduction of the PDE models to a single-front free-boundary problem in which the fast membrane-current variables are eliminated and only the dynamics of the slow relevant ones are retained. The solution of this free-boundary problem is in tum well approximated by a simple discrete map whose dimension D is equal to the number of slow membrane-current variables. This framework is applied to the Noble (D = 1) and Beeler-Reuter model (D = 3) of cardiac tissue and found to yield results in reasonably good quantitative agreement with PDE simulations. For the Beeler-Reuter model, the minimum period of stable propagation is found to depend sensitively on the dynamics of the calcium channel in a way which, together with the two-dimensional numerical simulations of Courtemanche and Winfree (Int. J. Bifurcation and Chaos 1 (1991) 431), strongly suggests that spiral breakup in this model occurs as a result of electrical alternans.