Instability and Spatiotemporal Dynamics of Alternans in Paced Cardiac Tissue

Abstract

We derive an equation that governs the spatiotemporal dynamics of small amplitude alternans in paced cardiac tissue. We show that a pattern-forming linear instability leads to the spontaneous formation of stationary or traveling waves whose nodes divide the tissue into regions with opposite phase of oscillation of action potential duration. This instability is important because it creates dynamically a heterogeneous electrical substrate for the formation of conduction blocks and the induction of fibrillation if the tissue size exceeds a fraction of the pattern wavelength. We derive an analytical expression for this wavelength as a function of three basic length scales related to dispersion and intercellular electrical coupling. DOI: 10.1103/PhysRevLett.88.208101 PACS numbers: 87.19.Hh, 05.45. -a, 05.45.Gg, 89.75. -k It is well established that the duration of cardiac excitation can oscillate from beat to beat at sufficiently short pacing interval [1]. Pioneering studies by Nolasco and Dahlen [2] and Guevara et al. [3] have demonstrated that the generic sequence LSLS . . . of long and short action potential duration (APD), known as alternans, is a direct consequence of the restitution relationshipbetween the APD generated by the nth 1 1 stimulus, APD n11 , and the diastolic time interval DI n during which the tissue recovers its resting properties after the end of the previous ͑nth͒ action potential. If we denote the interval between the nth and nth 1 1 stimulus by T n , we must have DI n T n 2 APD n . Then, for a fixed period: T n t for all n, Eq. (1) yields the map APD n f͑t 2 APD n21 ͒, whose slope f 0 typically increases with decreasing period. If the slope of the restitution curve exceeds unity, the map undergoes a period doubling bifurcation to alternans.Over the last decade, the study of alternans [4 -11], and their control [12], has become a main focus of research because of the potentially crucial link of this dynamical instability with cardiac fibrillation [13]. However, there is presently no simple analytical understanding of how the bifurcation to alternans is manifested spatiotemporally in paced cardiac tissue. Analytical progress to date is limited to the one-dimensional circulation of electrical impulse in a ring of tissue [5 -7].In this Letter, we derive an equation that governs the spatiotemporal dynamics of alternans close to the onset of instability. This enables us to obtain a quantitative analytical understanding of the formation of recently observed complex patterns of APD oscillations that can promote fibrillation [8 -11]. A crucial feature of these patterns is that the APD oscillates with opposite phases in two (or more) spatially extended regions of tissue, i.e., with a sequence LSLS . . . in one region and SLSL . . . in the other. These "discordant alternans" have been observed experimentally in both two-dimensional [8] and linear strands [9] of cardiac tissue, as well as in ionic model simulations [9][10][11]. Moreover, they have been shown to lead to the formation of conduction bl...

Control of Electrical Alternans in Canine Cardiac Purkinje Fibers

“…More recently, Echebarria and Karma found in simulations that control from a single site was limited by an inability to influence alternans dynamics in tissue distant from the stimulating electrode, with control efficacy decreasing as pacing rate increased [17]. Furthermore, these authors showed analytically [17] that control failure from single-site pacing is a direct consequence of the wave nature of alternans elucidated in an earlier theoretical study of alternans [18]. In particular, they showed that the linear stability eigenmodes of the paced cable are governed by the standard one-dimensional Helmholtz equation with a spatial coupling term originating from the diffusive electrical coupling between cells and an additional spatially uniform external forcing imposed by the feedback control [17].…”

Supernormal conduction in cardiac tissue promotes concordant alternans and action potential bunching

“…Second, we demonstrate that concordant alternans (CA) is promoted and is observed in simulations of arbitrarily long paced one-dimensional cables. This phenomenon is theoretically explained by an amplitude equation approach [19]. A second important consequence is that SNC allows for period doubled spirals with straight defect lines (in contrast to spiraling defect lines in cardiac models with normal conduction [20]).…”

Dynamics of sustained reentry in a loop model with discrete gap junction resistances

“…Whether alternative approaches that have been proposed for the continuous loop would be more appropriate remains to be determined. [11,16,17] In any case, we are still far from a general low-dimensional model that could be applied in situations including a dynamic change of the intercellular coupling, as in [28,29,30].…”

“…[1,2,3,4] For the 1-D loop, most work has been done assuming the membrane to be a continuous and uniform cable with constant intracellular axial resistivity. [5,6,7,8,9,10,11,12,13,14,15] For different models representing the ionic properties of the membrane, propagation was found to change from stable period-1 propagation to quasi-periodic reentry when the length of the loop was reduced below a critical length. The quasi-periodic reentry was characterized by a spatial oscillation of the action potential duration as propagation proceeded around the loop.…”

“…[6,7,8,9,10] Different attempts were made to build simplified representations of the dynamics allowing analytical examina-tion of the nature of the bifurcation. [7,10,11,16,17,18] One of these approaches, which guides the present investigation, relies on an integral-delay model. [9,10] It is based on the assumption that both the speed of propagation and the action potential duration can be expressed as functions of the diastolic interval, which measures the recovery time from the end of the previous action potential.…”