Amplitude equation approach to spatiotemporal dynamics of cardiac   alternans
  Amplitude equations are derived that describe the spatiotemporal dynamics of cardiac alternans during periodic pacing of one- and two-dimensional homogeneous tissue and one-dimensional anatomical reentry in a ring of homogeneous tissue. These equations provide a simple physical understanding of arrhythmogenic patterns of period-doubling oscillations of action potential duration with a spatially varying phase and amplitude as well as explicit quantitative predictions that can be compared to ionic model simulations or experiments. The form of the equations is expected to be valid for a large class of ionic models but the coefficients are only derived analytically for a two-variable ionic model and calculated numerically for the original Noble model of Purkinje fiber action potential.In paced tissue, the main result is the existence of a linear instability that produces a periodic pattern of discordant alternans. Moreover, the patterns of alternans can be either stationary, with fixed nodes, or travelling, with moving nodes and hence quasiperiodic oscillations of action potential duration, depending on the relative strength of the destabilizing effect of CV-restitution and the stabilizing effect of diffusive coupling. In both the paced geometries and the ring, the onset of alternans is different in tissue than for a paced isolated cell. The implications of these results for alternans dynamics during two-dimensional reentry are briefly discussed. 
Blas Echebarria, Alain Karma