Analysis of a Non-linear Partial Difference Equation, and Its Application to Cardiac Dynamics

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

“…They are the same for tissue fibers as they are for loops; see, e.g. Stubna et al (2002). The last two terms in the definition of DI i;n in Eq.…”

“…For our purposes, we may think of APD as a cell's effective refractory period (ERP) during which no excitations, even strong ones, can elicit new action potentials. The restitution of APD, which can be experimentally measured or derived from ionic models, is the most extensively studied of restitution relations (Banville and Gray, 2002;Cain, 2007;Cain and Schaeffer, 2006;Cain et al, 2004;Cao et al, 1999;Chen et al, 2005;Cherry and Fenton, 2004;Cytrynbaum and Keener, 2002;Gilmour et al, 1999;Hall et al, 1999;Ito and Glass, 1992;Kalb et al, 2004;Koller et al, 1998Koller et al, , 2005Mitchell and Schaeffer, 2003;Nolasco and Dahlen, 1968;Qu et al, 1997;Sedaghat et al, 2005;Stubna et al, 2002;Ten Tusscher et al, 2006;Vinet, 2000;Watanabe et al, 2001;Watanabe and Koller, 2002). Any APD restitution function, whether experimentally measured or analytically derived can be used in this model.…”

“…For discussions of CV restitution functions see Banville and Gray (2002), Cao et al (1999), Chen et al (2005), Cytrynbaum and Keener (2002), Derksen et al (2003), Courtemanche et al (1993Courtemanche et al ( , 1996, Courtemanche and Vinet (2003), Frame and Simson (1988), Girouard et al (1996), Stubna et al (2002), Ten Tusscher et al (2006 and Watanabe et al (2001). For our simulations we use the following expression These numbers are obtained using Eq.…”

“…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.…”

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

“…The concepts of restitution and dispersion are well known and have been widely studied in various contexts, both theoretical and experimental; see, e.g., [2], [3], [5], [6], [7], [10], [13], [14], [15], [16], [24], [25], [26]. In this paper we reexamine the dynamical system in [14], limiting our focus to dynamics of sustained reentry.…”