Condition for alternans and its control in a two-dimensional mapping model of paced cardiac dynamics.

We investigate a two-dimensional mapping model of a paced, isolated cardiac cell that relates the duration of the action potential to the two preceding diastolic intervals as well as the preceding action potential duration. The model displays rate-dependent restitution and hence memory. We derive a criterion for the stability of the 1:1 response pattern displayed by the model. This criterion can be written in terms of experimentally measured quantities-the slopes of restitution curves obtained via different pacing protocols. In addition, we analyze the two-dimensional mapping model in the presence of closed-loop feedback control. The control is initiated by making small adjustments to the pacing interval in order to suppress alternans and stabilize the 1:1 pattern. We find that the domain of control does not depend on the functional form of the map, and, in the general case, is characterized by a combination of the slopes. We show that the gain gamma necessary to establish control may vary significantly depending on the value of the slope of the so-called standard restitution curve (herein denoted as S12), but that the product gammaS12 stays approximately in the same range.

[1]  B B Lerman,et al.  Nonlinear-dynamical arrhythmia control in humans , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[2]  A. Karma.  Electrical alternans and spiral wave breakup in cardiac tissue. , 1994, Chaos.

[3]  Friedberg Ck.  Computers in cardiology. , 1970 .

[4]  R. Gilmour,et al.  Biphasic restitution of action potential duration and complex dynamics in ventricular myocardium. , 1995, Circulation research.

[5]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[6]  F. Fenton,et al.  Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. , 2002, Chaos.

[7]  D. G. Schaeffer,et al.  Analysis of the Fenton-Karma model through an approximation by a one-dimensional map. , 2002, Chaos.

[8]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[9]  B. R. Jewell,et al.  A study of the factors responsible for rate‐dependent shortening of the action potential in mammalian ventricular muscle. , 1978, The Journal of physiology.

[10]  Richard A Gray,et al.  Effect of Action Potential Duration and Conduction Velocity Restitution and Their Spatial Dispersion on Alternans and the Stability of Arrhythmias , 2002, Journal of cardiovascular electrophysiology.

[11]  Daniel J Gauthier,et al.  Experimental control of cardiac muscle alternans. , 2002, Physical review letters.

[12]  R. Gilmour,et al.  Memory models for the electrical properties of local cardiac systems. , 1997, Journal of theoretical biology.

[13]  Eberhard Bodenschatz,et al.  Period-doubling instability and memory in cardiac tissue. , 2002, Physical review letters.

[14]  Daniel J. Gauthier,et al.  Prevalence of Rate-Dependent Behaviors in Cardiac Muscle , 1999 .

[15]  L. Glass,et al.  DYNAMIC CONTROL OF CARDIAC ALTERNANS , 1997 .

[16]  Harold M. Hastings,et al.  Memory in an Excitable Medium: A Mechanism for Spiral Wave Breakup in the Low-Excitability Limit , 1999 .

[17]  Blas Echebarria,et al.  Instability and spatiotemporal dynamics of alternans in paced cardiac tissue. , 2001, Physical review letters.

[18]  D. Chialvo,et al.  Electrical Restitution, Critical Mass, and the Riddle of Fibrillation , 1999, Journal of cardiovascular electrophysiology.

[19]  James P. Keener,et al.  Stability conditions for the traveling pulse: Modifying the restitution hypothesis. , 2002, Chaos.

[20]  Daniel J. Gauthier,et al.  Comment on ``Dynamic Control of Cardiac Alternans'' , 1997 .

[21]  Daniel J Gauthier,et al.  Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  R. Gilmour,et al.  Conduction block in one-dimensional heart fibers. , 2002, Physical review letters.

[23]  M. Franz,et al.  Cycle length dependence of human action potential duration in vivo. Effects of single extrastimuli, sudden sustained rate acceleration and deceleration, and different steady-state frequencies. , 1988, The Journal of clinical investigation.

[24]  F. Fenton,et al.  Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. , 1998, Chaos.

[25]  Flavio H. Fenton,et al.  SPATIOTEMPORAL CONTROL OF WAVE INSTABILITIES IN CARDIAC TISSUE , 1999 .

[26]  J Jalife,et al.  Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fibers. , 1990, Circulation research.