Electrical Waves in a One-Dimensional Model of Cardiac Tissue

The electrical dynamics in the heart is modeled by a two-component PDE. Using geometric singular perturbation theory, it is shown that a traveling pulse solution, which corresponds to a single heartbeat, exists. One key aspect of the proof involves tracking the solution near a point on the slow manifold that is not normally hyperbolic. This is achieved by desingularizing the vector eld using a blow-up technique. This feature is relevant because it distinguishes cardiac impulses from, for example, nerve impulses. Stability of the pulse is also shown, by computing the zeros of the Evans function. Although the spectrum of one of the fast components is only marginally stable, due to essential spectrum that accumulates at the origin, it is shown that the spectrum of the full pulse consists of an isolated eigenvalue at zero and essential spectrum that is bounded away from the imaginary axis. Thus, this model provides an example in a biological application reminiscent of a previously observed mathematical phenomenon: that connecting an unstable { in this case marginally stable { front and back can produce a stable pulse. Finally, remarks are made regarding the existence and stability of spatially periodic pulses, corresponding to successive heartbeats, and their relationship with alternans, irregular action potentials that have been linked with arrhythmia.

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