Spectrum of a Linearized Amplitude Equation for Alternans in a Cardiac Fiber

Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suffer reversals of phase at various points along the fiber, a phenomenon called (spatially) discordant alternans. Either stationary or moving patterns are possible. Using a weak approximation, Echebarria and Karma proposed an equation to describe the spatiotemporal dynamics of small-amplitude alternans in a class of simple cardiac models, and they showed that an instability in this equation predicts the spontaneous formation of discordant alternans. To study the bifurcation, they computed the spectrum of the relevant linearized operator numerically, supplemented with partial analytical results. In the present paper we calculate this spectrum with purely analytical methods in two cases where a small parameter may be exploited: (i) small dispersion or (ii) a long fiber. From this analysis we estimate the parameter ranges in which the phase reversals of discordant alternans are stationary or moving.