  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.
