# Dynamic Behavior of a Paced Cardiac Fiber

#### Abstract

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 *η*, we estimate the number of beats *N *= *N*(*η,L*) required for the fiber to achieve approximate steady-state in the sense that spatial variation in the diastolic interval (DI) is bounded by *η*. 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.

*This paper has been withdrawn.*