RESUMEN

Consider a typical experimental protocol in which one end of a one-dimensional fiber of cardiac tissue is periodically stimulated, or paced, resulting in a train of propagating action potentials. There is evidence that a sudden change in the pacing period can initiate abnormal cardiac rhythms. In this paper, we analyze how the fiber responds to such a change in a regime without arrhythmias. In particular, given a fiber length $L$ and a tolerance , we estimate the number of beats $N = N(\eta, L)$ required for the fiber to achieve approximate steady-state in the sense that spatial variation in the diastolic interval (DI) is bounded by . We track spatial DI variation using an infinite sequence of linear integral equations which we derive from a standard kinematic model of wave propagation. The integral equations can be solved in terms of generalized Laguerre polynomials. We then estimate $N$ by applying an asymptotic estimate for generalized Laguerre polynomials. We find that, for fiber lengths characteristic of cardiac tissue, it is often the case that $N$ effectively exhibits no dependence on $L$. More exactly, (i) there is a critical fiber length $L^{*}$ such that, if $L < L^{*}$, the convergence to steady-state is slowest at the pacing site, and (ii) often, $L^{*}$ is substantially larger than the diameter of the whole heart.