Coupled Iterated Map Models of Action Potential Dynamics in a One-dimensional Cable of Cardiac Cells

Low-dimensional iterated map models have been widely used to study action potential dynamics in isolated cardiac cells. Coupled iterated map models have also been widely used to investigate action potential propagation dynamics in one-dimensional (1D) coupled cardiac cells, however, these models are usually empirical and not carefully validated. In this study, we first developed two coupled iterated map models which are the standard forms of diffusively coupled maps and overcome the limitations of the previous models. We then determined the coupling strength and space constant by quantitatively comparing the 1D action potential duration profile from the coupled cardiac cell model described by differential equations with that of the coupled iterated map models. To further validate the coupled iterated map models, we compared the stability conditions of the spatially uniform state of the coupled iterated maps and those of the 1D ionic model and showed that the coupled iterated map model could well recapitulate the stability conditions, i.e., the spatially uniform state is stable unless the state is chaotic. Finally, we combined conduction into the developed coupled iterated map model to study the effects of coupling strength on wave stabilities and showed that the diffusive coupling between cardiac cells tends to suppress instabilities during reentry in a 1D ring and the onset of discordant alternans in a periodically paced 1D cable.     

Map-based model of the cardiac action potential

A simple computationally efficient model which is capable of replicating the basic features of cardiac cell action potential is proposed. The model is a four-dimensional map and demonstrates good correspondence with real cardiac cells. Various regimes of cardiac activity, which can be reproduced by the proposed model, are shown. Bifurcation mechanisms of these regimes transitions are explained using phase space analysis. The dynamics of 1D and 2D lattices of coupled maps which model the behavior of electrically connected cells is discussed in the context of synchronization theory.     

Period-doubling instability and memory in cardiac tissue

Theoretical studies have indicated that alternans (period-doubling instability) of action potential duration is associated with a restitution relation with a slope >or=1. However, recent experimental findings suggest that the slope of the restitution relation is not necessarily predictive of alternans. Here, we compared a return map memory model to action potential data from an ionic model and found that the memory model reproduced dynamics that could not be explained by a unidimensional restitution relation. Using linear stability analysis, we determined the onset of the alternans in the memory model and confirmed that the slope of the restitution curve was not predictive.     

Hysteresis and bistability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells

The transmembrane potential of a single quiescent cell isolated from rabbit ventricular muscle was recorded using a suction electrode in whole-cell recording mode. The cell was then driven with a periodic train of current pulses injected into the cell through the same recording electrode. When the interpulse interval or basic cycle length (BCL) was sufficiently long, 1:1 rhythm resulted, with each stimulus pulse producing an action potential. Gradual decrease in BCL invariably resulted in loss of 1:1 synchronization at some point. When the pulse amplitude was set to a fixed low level and BCL gradually decreased, N+1:N rhythms (N>/=2) reminiscent of clinically observed Wenckebach rhythms were seen. Further decrease in BCL then yielded a 2:1 rhythm. In contrast, when the pulse amplitude was set to a fixed high level, a period-doubled 2:2 rhythm resembling alternans rhythm was seen before a 2:1 rhythm occurred. With the pulse amplitude set to an intermediate level (i.e., to a level between those at which Wenckebach and alternans rhythms were seen), there was a direct transition from 1:1 to 2:1 rhythm as the BCL was decreased: Wenckebach and alternans rhythms were not seen. When at that point the BCL was increased, the transition back to 1:1 rhythm occurred at a longer BCL than that at which the {1:1-->2:1} transition had initially occurred, demonstrating hysteresis. With the BCL set to a value within the hysteresis range, injection of a single well-timed extrastimulus converted 1:1 rhythm into 2:1 rhythm or vice versa, providing incontrovertible evidence of bistability (the coexistence of two different periodic rhythms at a fixed set of stimulation parameters). Hysteresis between 1:1 and 2:1 rhythms was also seen when the stimulus amplitude, rather than the BCL, was changed. Simulations using numerical integration of an ionic model of a single ventricular cell formulated as a nonlinear system of differential equations provided results that were very similar to those found in the experiments. The steady-state action potential duration restitution curve, which is a plot of the duration of the action potential during 1:1 rhythm as a function of the recovery time or diastolic interval immediately preceding that action potential, was determined. Iteration of a finite-difference equation derived using the restitution curve predicted the direct {1:1<-->2:1} transition, as well as bistability, in both the experimental and modeling work. However, prediction of the action potential duration during 2:1 rhythm was not as accurate in the experiments as in the model. Finally, we point out a few implications of our findings for cardiac arrhythmias (e.g., Mobitz type II block, ischemic alternans). (c) 1999 American Institute of Physics.     

Electrical alternans and spiral wave breakup in cardiac tissue

This paper reports the results of a theoretical investigation of spiral wave breakup in model equations of action potential propagation in cardiac tissue. A general formulation of these equations is described in which arbitrary experimentally determined restitution and dispersion curves can in principle be fitted. Spiral wave behavior is studied in two-dimension as a function of a parameter Re which controls the steepness of the restitution curve at short diastolic intervals. Spiral breakup is found to occur when the minimum period T(min), below which a periodically stimulated tissue exhibits alternans in action potential duration, exceeds by a finite amount the spiral rotation period T(S). At this point, oscillations in action potential duration are of sufficiently large amplitude to cause a spontaneous conduction block to form along the wavefront. The latter occurs closer to the initiation point of reentry (spiral tip) with increasing steepness and, hence, in smaller tissue sizes. Spiral breakup leads to a spatially disorganized wave activity which is always transient, except for tissues larger than some minimum size and within a very narrow range of Re which increases with dispersion.     

Understanding cardiac alternans: a piecewise linear modeling framework

Cardiac alternans is a beat-to-beat alternation in action potential duration (APD) and intracellular calcium (Ca(2+)) cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca(2+) and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard [Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 (2006)]. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca(2+) cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca(2+) transport, the model supports spatially varying patterns of alternans. We analyze the onset of this instability with a generalization of the master stability approach to accommodate the nonsmooth nature of our system.     

Control of spiral waves and turbulent states in a cardiac model by travelling—wave perturbations

We propose a travelling-wave perturbation method to control the spatiotemporal dynamics in a cardiac model.It is numerically demonstrated that the method can successfully suppress the wave instability(alternans in action potential duration) in the one-dimensional case and converty spiral waves and turbulent states to the normal travelling wave states in the two-dimensional case.An experimental scheme is suggested which may provide a new design for a cardiac defibrillator.     

基于心脏腔式结构的心电图元胞自动机建模

建立了包含心房肌、心室肌、房室腔、室间隔并考虑心室肌分层结构的心电图元胞自动机模型.利用所建立的模型,仿真了电信号在心脏中的传导,计算了正常和缺血情况下的场点电势走势.数值结果表明:正常情况下,模拟所得的场点电势呈现与标准心电图一致的P波、QRS波群、T波和J波;在心内膜下肌细胞缺血情况下,出现T波倒置的现象;在心外膜下肌细胞缺血情况下,T波变得高耸;在透壁缺血情况下,T波提前形成,QT间期缩短.将正常和异常情况下的场点电势走势与临床结果进行了对比,并分析了其形成与持续机制.研究结果可为准确阐明心电图与心肌细胞电活动之间的关系、探讨心电图的产生与持续机制提供参考.     

Conduction block in one-dimensional heart fibers

We present a nonlinear dynamical systems analysis of the transition to conduction block in one-dimensional cardiac fibers. We study a simple model of wave propagation in heart tissue that depends only on the recovery of action potential duration and conduction velocity. If the recovery function has slope >or=1 and the velocity recovery function is nonconstant, rapid activation causes dynamical heterogeneity and finally conduction block away from the activation site. This dynamical mechanism may play a role in the initiation and breakup of spiral waves in excitable media.     

Mechanisms of stochastic phase locking

Periodically driven nonlinear oscillators can exhibit a form of phase locking in which a well-defined feature of the motion occurs near a preferred phase of the stimulus, but a random number of stimulus cycles are skipped between its occurrences. This feature may be an action potential, or another crossing by a state variable of some specific value. This behavior can also occur when no apparent external periodic forcing is present. The phase preference is then measured with respect to a time scale internal to the system. Models of these behaviors are briefly reviewed, and new mechanisms are presented that involve the coupling of noise to the equations of motion. Our study investigates such stochastic phase locking near bifurcations commonly present in models of biological oscillators: (1) a supercritical and (2) a subcritical Hopf bifurcation, and, under autonomous conditions, near (3) a saddle-node bifurcation, and (4) chaotic behavior. Our results complement previous studies of aperiodic phase locking in which noise perturbs deterministic phase-locked motion. In our study however, we emphasize how noise can induce a stochastic phase-locked motion that does not have a similar deterministic counterpart. Although our study focuses on models of excitable and bursting neurons, our results are applicable to other oscillators, such as those discussed in the respiratory and cardiac literatures. (c) 1995 American Institute of Physics.     

Resetting and annihilation of reentrant activity in a model of a one-dimensional loop of ventricular tissue

Resetting and annihilation of reentrant activity by a single stimulus pulse (S1) or a pair (S1-S2) of coupled pulses are studied in a model of one-dimensional loop of cardiac tissue using a Beeler-Reuter-type ionic model. Different modes of reentry termination are described. The classical mode of termination by unidirectional block, in which a stimulus produces only a retrograde front that collides with the activation front of the reentry, can be obtained for both S1 and S1-S2 applied over a small vulnerable window. We demonstrate that another scenario of termination-that we term collision block-can also be induced by the S1-S2 protocol. This scenario is obtained over a much wider range of S1-S2 coupling intervals than the one leading to a unidirectional block. In the collision block, S1 produces a retrograde front, colliding with the activation front of the pre-existing reentry, and an antegrade front propagating in the same direction as the initial reentry. Then, S2 also produces an antegrade and a retrograde front. However, the propagation of these fronts in the spatial profile of repolarization left by S1 leads to a termination of the reentrant activity. More complex behaviors also occur in which the antegrade fronts produced by S1 and S2 both persist for several turns, displaying a growing alternation in action potential duration ("alternans amplification") that may lead to the termination of the reentrant activity. The hypothesis that both collision block and alternans amplification depend on the interaction between the action potential duration restitution curve and the recovery curve of conduction velocity is supported by the fact that the dynamical behaviors were reproduced using an integro-delay equation based on these two properties. We thus describe two new mechanisms (collision block and alternans amplification) whereby electrical stimulation can terminate reentrant activity. (c) 2002 American Institute of Physics.     

Complete dynamical analysis of myocardial cell exposed to magnetic flux

The heart is the essential, yet complex, component of the human cardiovascular system. In the past few decades, researchers have taken giant steps toward better understanding of the cardiac system and there have been proposed some mathematical models to describe the heart's function. In this paper, a new Fitzhugh-Nagumo neuron (FNN) model is proposed to model the electrical activity of the heart in which the effect of magnetic flux is considered. Magnetic field can greatly affect the heart's function. The dynamical analyses of the model, including quantitative assessment of the system's equilibria and its stability, phase portraits analysis, bifurcation and Lyapunov exponents analysis, and basin of attraction analysis, are carried out. In addition, a model of cardiac tissue is designed to study the electrical spatiotemporal activity of heart tissue under the electromagnetic effects. Our numerical simulations confirm that the electromagnetic excitation can change the normal rhythm of the heart. It can initiate the reentrant excitations leading to emergence of spiral seeds. This study highlights the role of electromagnetic induction in dynamical instability of the action potential duration, and thus the chaotic dynamics in the cardiac tissue.     

Scroll wave dynamics in a model of the heterogeneous heart

Scroll waves are found in physical, chemical and biological systems and underlie many significant processes including life-threatening cardiac arrhythmias. The theory of scroll waves predicts scroll wave dynamics should be substantially affected by heterogeneity of cardiac tissue together with other factors including shape and anisotropy. In this study, we used our recently developed analytical model of the human ventricle to identify effects of shape, anisotropy, and regional heterogeneity of myocardium on scroll wave dynamics. We found that the main effects of apical-base heterogeneity were an increased scroll wave drift velocity and a shift towards the region of maximum action potential duration. We also found that transmural heterogeneity does not substantially affect scroll wave dynamics and only in extreme cases changes the attractor position.     

The simplified FitzHugh-Nagumo model with action potential duration restitution: Effects on 2D wave propagation

A modification of the simplified FitzHugh-Nagumo (FN) equations is proposed for introducing a residual component of the slow variable, which determines the restitution of action potential duration (APD) also known as the interval-excitation duration relationship. The three-step-wise approximation of ε(E) which is widely used in current publications is replaced in a new model by a four-step approximation. This change is used for studying by computer simulation the effects of APD restitution properties independently of the APD and refractory period on 2D wave propagation in an isotropic matrix (made by 128 × 128 nodes). The method for fitting the model to the given experimental restitution data (obtained from myocardial cells) is presented. The computer simulations implemented on a massively parallel computer (Connection Machine) showed at least three important qualitative distinctions in behavior which demonstrate the effect of APD restitution: changes in the speed and wavelength of propagated waves with the period of stimulation, non-stationary propagation of spiral waves, and site-specific induction of spiral waves with premature stimulation not on the tail of the previous wave. Quantitative effects of differing restitution properties are expressed in the size and location of a window of vulnerability in 2D excitable media. These windows are characterized by the appearance of single and double spiral waves in response to premature stimulation applied inside the window. Thus the APD restitution incorporated in the FN model produces a significant effect on the formation and propagation of spiral waves.     

用晚钠电流终止心脏中的螺旋波和时空混沌

采用人类心脏模型研究了用晚钠电流控制二维心脏组织中的螺旋波和时空混沌,我们提出这样的控制策略来产生晚钠电流:让慢失活门变量j始终等于0.7,同时实时调节钠电流的快失活门变量h的阈值电压V_I,即先让阈值电压V_I经过T_1时间从71.55 mV均匀减少到50.55 mV,然后经过T_2时间再从50.55 mV均匀增加到71.55 mV,当阈值电压V_I回到71.55 mV,钠电流的快、慢失活门变量恢复正常变化.数值模拟结果表明:只要适当选择控制时间,不论心肌细胞是否存在自发的晚钠电流,控制产生的晚钠电流都可以有效抑制螺旋波和时空混沌,而且需要的晚钠电流都很小,且控制时间都很短,因为螺旋波和时空混沌消失主要是通过传导障碍消失,少数情况下时空混沌是通过转变为靶波消失.我们希望这种控制方法能为室颤控制提供新的思路.     

Study of atrial arrhythmias in a computer model based on magnetic resonance images of human atria

The maintenance of multiple wavelets appears to be a consistent feature of atrial fibrillation (AF). In this paper, we investigate possible mechanisms of initiation and perpetuation of multiple wavelets in a computer model of AF. We developed a simplified model of human atria that uses an ionic-based membrane model and whose geometry is derived from a segmented magnetic resonance imaging data set. The three-dimensional surface has a realistic size and includes obstacles corresponding to the location of major vessels and valves, but it does not take into account anisotropy. The main advantage of this approach is its ability to simulate long duration arrhythmias (up to 40 s). Clinically relevant initiation protocols, such as single-site burst pacing, were used. The dynamics of simulated AF were investigated in models with different action potential durations and restitution properties, controlled by the conductance of the slow inward current in a modified Luo-Rudy model. The simulation studies show that (1) single-site burst pacing protocol can be used to induce wave breaks even in tissue with uniform membrane properties, (2) the restitution-based wave breaks in an atrial model with realistic size and conduction velocities are transient, and (3) a significant reduction in action potential duration (even with apparently flat restitution) increases the duration of AF. (c) 2002 American Institute of Physics.     

Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity

It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. (c) 2002 American Institute of Physics.     

Filament instability and rotational tissue anisotropy: A numerical study using detailed cardiac models

The role of cardiac tissue anisotropy in the breakup of vortex filaments is studied using two detailed cardiac models. In the Beeler-Reuter model, modified to produce stable spiral waves in two dimensions, we find that anisotropy can destabilize a vortex filament in a parallelepipedal slab of tissue. The mechanisms of the instability are similar to the ones reported in previous work on a simplified cardiac model by Fenton and Karma [Chaos 8, 20 (1998)]. In the Luo-Rudy model, also modified to produce stable spiral waves in two dimensions, we find that anisotropy does not destabilize filaments. A possible explanation for this model-dependent behavior based on spiral tip trajectories is offered. (c) 2001 American Institute of Physics.     

Torus maps and the problem of a one-dimensional optical resonator with a quasiperiodically moving wall

We study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically (with d≥2 incommensurate frequencies). We show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d-dimensional torus that preserves a foliation by irrational straight lines. In particular, the Jacobian of this map has (d−1) eigenvalues equal to 1. We present rigorous and numerical results about several dynamical features of such maps. We also show how these dynamical features translate into properties for the field in the cavity. In particular, we show that when the torus map satisfies a KAM theorem—which happens for a Cantor set of positive measure of parameters—the energy of the electromagnetic field remains bounded. When the torus map is in a resonant region—which happens in open sets of parameters inside the gaps of the previous Cantor set—the energy grows exponentially.     

Restitution in mapping models with an arbitrary amount of memory

Restitution, the characteristic shortening of action potential duration (APD) with increased heart rate, has been studied extensively because of its purported link to the onset of fibrillation. Restitution is often represented in the form of mapping models where APD is a function of previous diastolic intervals (DIs) and/or APDs, A(n+1)=F(D(n),A(n),D(n-1),A(n-1),...), where A(n+1) is the APD following a DI given by D(n). The number of variables previous to D(n) determines the degree of memory in the mapping model. Recent experiments have shown that mapping models should contain at least three variables (D(n),A(n),D(n-1)) to reproduce a restitution portrait (RP) that is qualitatively similar to that seen experimentally, where the RP shows three different types of restitution curves (RCs) [dynamic, S1-S2, and constant-basic cycle length (BCL)] simultaneously. However, an interpretation of the different RCs has only been presented in detail for mapping models of one and two variables. Here we present an analysis of the different RCs in the RP for mapping models with an arbitrary amount of memory. We determine the number of variables necessary to represent the different RCs in the RP. We also present a graphical visualization of these RCs. Our analysis reveals that the dynamic and S1-S2 RCs reside on two-dimensional surfaces, and therefore provide limited information for mapping models with more than two variables. However, constant-BCL restitution is a feature of the RP that depends on higher dimensions and can possibly be used to determine a lower bound on the dimensionality of cardiac dynamics.