Reentrant waves in excitable media

This dissertation presents a study of instabilities in the propagation of excitation pulses within spatially-distributed models of cardiac reentry. In one-dimensional closed rings, I study the onset of oscillations in the dynamics of circulating pulses as the ring length is decreased. In two-dimensional sheets, I analyze the spontaneous breakup of rotating spiral waves. In both cases, numerical results illustrating the instability phenomena are obtained using simulations of a partial differential equation (PDE) that models cardiac electrical activity using the Beeler-Reuter (BR) equations. The properties of the PDE model are summarized using the restitution and dispersion curves. The restitution curve gives the dependence of the pulse duration on the recovery time, defined as the elapsed time between the onset of an excitation pulse and the end of the previous excitation pulse. The dispersion curve gives the dependence of the pulse speed on the recovery time. I use these two properties to construct simplified models aimed at capturing the essence of the instabilities observed in the PDE. On the ring, I derive an integral-delay equation for the evolution of the recovery time as a function of the distance along the ring that incorporates the restitution and the dispersion curves. Numerical simulations and bifurcation analysis of the delay equation explain and predict the dynamics of the PDE. In two-dimensions, I extend early work that presented the first clear demonstration of spiral wave breakup in a reasonable discretization of a continuous PDE model of cardiac propagation. Spiral breakup can be observed in the BR model, depending on the value of a parameter controlling the duration of the electrical pulses. I study the appearance of spiral wavebreaks and relate it to the change in restitution properties of the BR equations as the parameter is varied. Finally, the effects of restitution and dispersion in two dimensions are examined in a discrete space/continuous time model of cardiac propagation. Results about the dependence of the propagation speed on the excitation threshold and on the excitation front curvature are obtained analytically. Inclusion of restitution relations derived from the BR equations into this simple model can give rise to spiral wavebreaks.