Experimental control of cardiac muscle alternans

We demonstrate that alternans in small pieces of in vitro paced bullfrog (Rana Catesbeiana) myocardium can be suppressed by making minute adjustments to the pacing period in response to real time measurements of the action potential duration. Control is possible over a large range of physiological conditions over many animals and the self-referencing control protocol can automatically adjust to changes in the pacing interval. Our results suggest the feasibility of developing low-energy methods for maintaining normal cardiac function.     

Mechanisms for initiation of cardiac discordant alternans

Electrical alternans, defined as a beat-to-beat change in the duration of the excited phase of cardiac cells, is among the known precursors of sudden cardiac death. It may appear as concordant (all the tissue presenting the same phase of oscillation) or discordant (with out-of-phase regions distributed among tissue). Spatially discordant alternans can lead to unidirectional block that initiates reentry and ventricular fibrillation. The role played by tissue heterogeneities and heart rate changes in their initiation remains, however, unclear. We study the mechanisms for initiation of spatially discordant alternans by numerical simulations of an ionic model spatially distributed in a one-dimensional cable and in an anatomical model of the rabbit heart. The effects of CV-restitution, ectopic beats, and the role of spatial gradients of electrical restitution properties are investigated. In homogeneous tissue, the origin of discordant alternans may be dynamical, through CV-restitution, or due to a localized change in the pacing period. We also find that a sudden change of stimulation rate can initiate discordant alternans in the presence of a spatial gradient of APD-restitution without necessitating CV-restitution. The mechanism of, and the conditions for, initiation are determined based on an iterated map analysis of beat to beat changes of APD. This analysis leads to the definition of a vulnerable window for initiation of discordant alternans. Moreover, the pattern of spatially discordant alternans is found to change slowly over several beats following initiation, as reflected in ECG recordings.     

Control of electrical alternans in canine cardiac purkinje fibers

Alternation in the duration of consecutive cardiac action potentials (electrical alternans) may precipitate conduction block and the onset of arrhythmias. Consequently, suppression of alternans using properly timed premature stimuli may be antiarrhythmic. To determine the extent to which alternans control can be achieved in cardiac tissue, isolated canine Purkinje fibers were paced from one end using a feedback control method. Spatially uniform control of alternans was possible when alternans amplitude was small. However, control became attenuated spatially as alternans amplitude increased. The amplitude variation along the cable was well described by a theoretically expected standing wave profile that corresponds to the first quantized mode of the one-dimensional Helmholtz equation. These results confirm the wavelike nature of alternans and may have important implications for their control using electrical stimuli.     

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.     

Instability and spatiotemporal dynamics of alternans in paced cardiac tissue

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.     

Unidirectional pinning and hysteresis of spatially discordant alternans in cardiac tissue

Spatially discordant alternans is a widely observed pattern of voltage and calcium signals in cardiac tissue that can precipitate lethal cardiac arrhythmia. Using spatially coupled iterative maps of the beat-to-beat dynamics, we explore this pattern's dynamics in the regime of a calcium-dominated period-doubling instability at the single-cell level. We find a novel nonlinear bifurcation associated with the formation of a discontinuous jump in the amplitude of calcium alternans at nodes separating discordant regions. We show that this jump unidirectionally pins nodes by preventing their motion away from the pacing site following a pacing rate decrease but permitting motion towards this site following a rate increase. This unidirectional pinning leads to strongly history-dependent node motion that is strongly arrhythmogenic.     

Triggered alternans in an ionic model of ischemic cardiac ventricular muscle

It has been known for several decades that electrical alternans occurs during myocardial ischemia in both clinical and experimental work. There are a few reports showing that this alternans can be triggered into existence by a premature ventricular contraction. Detriggering of alternans by a premature ventricular contraction, as well as pause-induced triggering and detriggering, have also been reported. We conduct a search for triggered alternans in an ionic model of ischemic ventricular muscle in which alternans has been described recently: a one-dimensional cable of length 3 cm, containing a central ischemic zone 1 cm long, with 1 cm segments of normal (i.e., nonischemic) tissue at each end. We use a modified form of the Luo-Rudy [Circ. Res. 68, 1501-1526 (1991)] ionic model to represent the ventricular tissue, modeling the effect of ischemia by raising the external potassium ion concentration ([K(+)](o)) in the central ischemic zone. As [K(+)](o) is increased at a fixed pacing cycle length of 400 ms, there is first a transition from 1:1 rhythm to alternans or 2:2 rhythm, and then a transition from 2:2 rhythm to 2:1 block. There is a range of [K(+)](o) over which there is coexistence of 1:1 and 2:2 rhythms, so that dropping a stimulus from the periodic drive train during 1:1 rhythm can result in the conversion of 1:1 to 2:2 rhythm. Within the bistable range, the reverse transition from 2:2 to 1:1 rhythm can be produced by injection of a well-timed extrastimulus. Using a stimulation protocol involving delivery of pre- and post-mature stimuli, we derive a one-dimensional map that captures the salient features of the results of the cable simulations, i.e., the {1:1-->2:2-->2:1} transitions with {1:1<-->2:2} bistability. This map uses a new index of the global activity in the cable, the normalized voltage integral. Finally, we put forth a simple piecewise linear map that replicates the {1:1<-->2:2} bistability observed in the cable simulations and in the normalized voltage integral map. (c) 2002 American Institute of Physics.     

Spatiotemporal dynamics of damped propagation in excitable cardiac tissue

Compared to steadily propagating waves (SPW), damped waves (DW), another solution to the nonlinear wave equation, are seldom studied. In cardiac tissue after electrical stimulation in an SPW wake, we observe DW with diminished amplitude and velocity that either gradually decrease as the DW dies, or exhibit a sharp amplitude increase after a delay to become an SPW. The cardiac DW-SPW transition is a key link in understanding defibrillation and stimulation close to the refractory period, and is ideal for a general study of DW dynamics.     

Spatiotemporal optical pulse control using microwaves

The Davey-Stewartson system allows us to describe the interaction between a spatiotemporal optical pulse and adequately matched microwaves. We show that the interaction can lead to the formation of a two-dimensional soliton which is robust in the sense that it occurs in a wide range of parameters of the incident optical pulse and microwaves, and of the material used.     

Spatiotemporal chaos control in two-wave driven systems

Spatiotemporal chaos control is considered by taking a one-dimensional driven/damped nonlinear drift-wave equation as a model. We apply an additional sinusoidal wave to suppress spatiotemporal chaos, and the system becomes a two-sinusoidal-wave driven system (the original driving wave with frequency ω and an additional controlling wave with frequency Ω). Numerical simulations show that when the frequency of the controlling wave is in the proper range, spatiotemporal chaos can be modified into a regular state where the amplitudes of all modes vary periodically with frequency Ω-ω while the phases of all modes evolve quasi-periodically with a running frequency Ω overlapped by a small modulation of frequency Ω-ω. The physical reason for this peculiar phenomenon is attributed to a frequency entrainment in the competition of the two external waves.     

Period-doubling bifurcation to alternans in paced cardiac tissue: crossover from smooth to border-collision characteristics

We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental techniques, we find a hybrid behavior: Very close to the bifurcation point, the dynamics is smooth, whereas further away it is border-collision-like. The essence of this behavior is captured by a model that exhibits what we call an unfolded border-collision bifurcation. This new model elucidates that, in an experiment, where only a limited number of data points can be measured, the smooth behavior of the bifurcation can easily be missed.     

Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements

Structured optical elements that control the spatial and temporal characteristics of femtosecond light pulses are analyzed and synthesized. We show that unique spatiotemporal effects can be attained based on the diffraction, refraction, and dispersive effects that appear in the femtosecond regime. We argue that the design requirements for ultrafast optics are beyond the achromatization considerations that are usually applied to incoherent illumination because of the need to consider coherent effects. Despite fundamental limitations in the space-time control of ultrashort pulses, we show the potential of this technique to improve simultaneously the spatial and the temporal resolution of a lens and to generate ultrafast pulse sequences.     

Spatiotemporal chaos synchronization of an uncertain network based on sliding mode control

The sliding mode control method is used to study spatiotemporal chaos synchronization of an uncertain network.The method is extended from synchronization between two chaotic systems to the synchronization of complex network composed of N spatiotemporal chaotic systems.The sliding surface of the network and the control input are designed.Furthermore,the effectiveness of the method is analysed based on the stability theory.The Burgers equation with spatiotemporal chaos behavior is taken as an example to simulate the experiment.It is found that the synchronization performance of the network is very stable.     

Spatiotemporal stability of a femtosecond hard-x-ray undulator source studied by control of coherent optical phonons

We report on the temporal and spatial stability of the first tunable femtosecond undulator hard-x-ray source for ultrafast diffraction and absorption experiments. The 2.5-1 Angstrom output radiation is driven by an initial 50 fs laser pulse employing the laser-electron slicing technique. By using x-ray diffraction to probe laser-induced coherent optical phonons in bulk bismuth, we estimate an x-ray pulse duration of 140+/-30 fs FWHM with timing drifts below 30 fs rms measured over 5 days. Optical control of coherent lattice motion is demonstrated.     

Introduction: Mapping and control of complex cardiac arrhythmias

This paper serves as an introduction to the Focus Issue on mapping and control of complex cardiac arrhythmias. We first introduce basic concepts of cardiac electrophysiology and describe the main clinical methods being used to treat arrhythmia. We then provide a brief summary of the main themes contained in the articles in this Focus Issue. In recent years there have been important advances in the ability to map the spread of excitation in intact hearts and in laboratory settings. This work has been combined with simulations that use increasingly realistic geometry and physiology. Waves of excitation and contraction in the heart do not always propagate with constant velocity but are often subject to instabilities that may lead to fluctuations in velocity and cycle time. Such instabilities are often treated best in the context of simple one- or two-dimensional geometries. An understanding of the mechanisms of propagation and wave stability is leading to the implementation of different stimulation protocols in an effort to modify or eliminate abnormal rhythms. (c) 2002 American Institute of Physics.     

Distributed predictive control of spiral wave in cardiac excitable media

In this paper, we propose the distributed predictive control strategies of spiral wave in cardiac excitable media. The modified FitzHugh--Nagumo model was used to express the cardiac excitable media approximately. Based on the control-Lyapunov theory, we obtained the distributed control equation, which consists of a positive control-Lyapunov function and a positive cost function. Using the equation, we investigate two kinds of robust control strategies: the time-dependent distributed control strategy and the space-time dependent distributed control strategy. The feasibility of the strategies was demonstrated via an illustrative example, in which the spiral wave was prevented to occur, and the possibility for inducing ventricular fibrillation was eliminated. The strategies are helpful in designing various cardiac devices. Since the second strategy is more efficient and robust than the first one, and the response time in the second strategy is far less than that in the first one, the former is suitable for the quick-response control systems. In addition, our spatiotemporal control strategies, especially the second strategy, can be applied to other cardiac models, even to other reaction-diffusion systems.     

Spatiotemporal shaping of terahertz pulses

We report temporal shaping of few-cycle terahertz pulses, using a slit in a conductive screen as a high-pass filter. The filter's cutoff frequency was tuned by changing the width of the slit; the slope of the cutoff transition was altered by changing the thickness of the screen. We measured the transmission function of the filters, using large-aperture photoconducting antennas to create and detect the incident and transmitted electric field. Our experimental results were in excellent agreement with the performed finite-difference time-domain simulations of the propagation of the pulse through the slit. When the screen thickness was greater than the slit width, the filter was well modeled by a short, planar waveguide. Using a simple transfer function, we accurately describe the sharp cutoff and dispersion of such a filter.     

交替行为对螺旋波影响的数值模拟研究

在离散可激发介质Greenberg-Hasting模型中引入交替(alternans)行为,研究了交替行为对螺旋波的影响.数值结果表明:在适当选择参数下,交替对螺旋波有很大影响,例如交替导致螺旋波的形状振荡,形成呼吸螺旋波,交替使螺旋波漫游、漂移,甚至使螺旋波漫游出系统的边界,交替使螺旋波破碎形成小螺旋波、反靶波和时空混沌等,首次在均匀介质中观察到交替导致传导障碍,使螺旋波破碎和消失,并对发生这些现象的机理进行了分析. 关键词： 离散可激发介质 螺旋波 靶波 漫游