Criteria for the Convergence, Oscillation, and Bistability of Pulse Circulation in a Ring of Excitable Media

Abstract

A discrete model based on a nonlinear difference equation (equivalent to a coupled map lattice of high dimension) is used to study the dynamics of a circulating pulse in a ring of excitable media, such as cardiac cells. Based on the global and local properties of monotonic restitution and dispersion curves, criteria are obtained for the asymptotic stability of the unique steady state (pulse circulating at constant frequency) as well as for nonconvergent oscillatory behavior of all nonequilibrium trajectories (pulse circulating at variable frequency). We also demonstrate that in certain cases the system is bistable, where an asymptotically stable equilibrium coexists with stable oscillatory solutions.

Complex temporal patterns of spontaneous initiation and termination of reentry in a loop of cardiac tissue

“…In this way, a SITR pattern becomes complex when the crossing of a threshold causes the beat pattern to switch from one stable state to another that crosses a threshold some beats later. Further, it is known that bistability occurs when the length of the loop is within a certain range; see e.g., Sedaghat et al (2005) and Vinet (2000). Therefore, in our model complex SITR patterns are not seen unless the length of the loop is within that range, with all other physiological parameters in the model held fixed.…”

“…The relative simplicity of RC and their importance in clinical arrhythmogenesis have made them objects of frequent study (Cain, 2007;Chen et al, 2005;Cytrynbaum and Keener, 2002;Courtemanche et al, 1993Courtemanche et al, , 1996Courtemanche and Vinet, 2003;Frame and Simson, 1988;Ito and Glass, 1992;Keener, 2002;Mines, 1913;Nagai et al, 2000;Qu et al, 1997;Quan and Rudy, 1990;Sedaghat et al, 2005;Vinet, 2000). Studying spontaneous initiation and termination of reentry (SITR) patterns in the loop context is more focused and simpler than in the whole heart and can offer potentially useful insights into the clinically observed patterns of VT occurrences.…”

“…As is commonly done in ''restitution-based'' models involving loops or fibers, the physiological parameters of the loop are encoded in the restitution functions for action potential duration (APD) and conduction velocity (CV); see e.g. Cain (2007), Cain et al (2004), Chialvo et al (1990), Courtemanche et al (1993Courtemanche et al ( , 1996, Courtemanche and Vinet (2003), Fox et al (2002b), Ito and Glass (1992), Qu et al (1997), Sedaghat et al (2005), Stubna et al (2002) and Vinet (2000). The inclusion of a pacing mechanism is obviously necessary in a study of SITR patterns since reentry needs to be initiated repeatedly if terminations occur.…”

Dynamic Behavior of a Paced Cardiac Fiber

“…For m ≥ 1, we find that (28) Reversing the order of integration, the iterated integral (28) simplifies to a single integral (29) Combining (27) and (29) yields (30) where (31) It follows that (32) 2 We remark that the trivial estimate ||y n || ∞ = ||T n y 0 || ∞ ≤ ||T|| n ||y 0 || ∞ is too weak since ||T|| can exceed 1. This is especially true close to the onset of alternans (i.e., as α → 1 − ), in which case T is a contraction only for very short fiber lengths.…”

“…Therefore, it is often desirable to track the progress of propagating action potentials without regard to the structure of the voltage profile. Indeed, many recent studies [4,8,11,12,16,30,34] have employed kinematic models [19,27] of wave propagation in cardiac fibers.…”