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.