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.     

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.     

Alternans and the influence of ionic channel modifications: Cardiac three-dimensional simulations and one-dimensional numerical bifurcation analysis

Cardiac propagation is investigated by simulations using a realistic three-dimensional (3D) geometry including muscle fiber orientation of the ventricles of a rabbit heart and the modified Beeler-Reuter ionic model. Electrical excitation is introduced by a periodic pacing of the lower septum. Depending on the pacing frequency, qualitatively different dynamics are observed, namely, normal heart beat, T-wave alternans, and 2:1 conduction block at small, intermediate, and large pacing frequencies, respectively. In a second step, we performed a numerical stability and bifurcation analysis of a pulse propagating in a one-dimensional (1D) ring of cardiac tissue. The precise onset of the alternans instability is obtained from computer-assisted linear stability analysis of the pulse and computation of the associated spectrum. The critical frequency at the onset of alternans and the profiles of the membrane potential agree well with the ones obtained in the 3D simulations. Next, we computed changes in the wave profiles and in the onset of alternans for the Beeler-Reuter model with modifications of the sodium, calcium, and potassium channels, respectively. For this purpose, we employ the method of numerical bifurcation and stability analysis. While blocking of calcium channels has a stabilizing effect, blocked sodium or potassium channels lead to the occurrence of alternans at lower pacing frequencies. The findings regarding channel blocking are verified within three-dimensional simulations. Altogether, we have found T-wave alternans and conduction block in 3D simulations of a realistic rabbit heart geometry. The onset of alternans has been analyzed by numerical bifurcation and stability analysis of 1D wave trains. By comparing the results of the two approaches, we find that alternans is not strongly influenced by ingredients such as 3D geometry and propagation anisotropy, but depends mostly on the frequency of pacing (frequency of subsequent action potentials). In addition, we have introduced numerical bifurcation and stability analysis as a tool into heart modeling and demonstrated its efficiency in scanning a large set of parameters in the case of models with reduced conductivity. Bifurcation analysis also provides an accurate test for analytical theories of alternans as is demonstrated for the case of the restitution hypothesis.     

Spatiotemporal control of cardiac alternans

Electrical alternans are believed to be linked to the onset of life-threatening ventricular arrhythmias and sudden cardiac death. Recent studies have shown that alternans can be suppressed temporally by dynamic feedback control of the pacing interval. Here we investigate theoretically whether control can suppress alternans both temporally and spatially in homogeneous tissue paced at a single site. We first carry out ionic model simulations in a one-dimensional cable geometry which show that control is only effective up to a maximum cable length that decreases sharply away from the alternans bifurcation point. We then explain this finding by a linear stability analysis of an amplitude equation that describes the spatiotemporal evolution of alternans. This analysis reveals that control failure above a critical cable length is caused by the formation of standing wave patterns of alternans that are eigenfunctions of a forced Helmholtz equation, and therefore remarkably analogous to sound harmonics in an open pipe. We discuss the implications of these results for using control to suppress alternans in the human ventricles as well as to probe fundamental aspects of alternans morphogenesis. (c) 2002 American Institute of Physics.     

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.     

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.     

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.     

Bifurcation analysis of a normal form for excitable media: are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for traveling waves in one-dimensional excitable media. The normal form that has been recently proposed on phenomenological grounds is given in the form of a differential delay equation. The normal form exhibits a symmetry-preserving Hopf bifurcation that may coalesce with a saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans, which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.     

Effective action and vacuum instability in the chiral σ-model and the standard electroweak model

We develop a partial expansion of the effective action from fermion integration relevant for spatially small configurations of the meson fields in the chiral σ-model. It is shown that this leads (in this semiclassical approximation) to a vacuum instability. For weak Yukawa coupling, when the semiclassical calculation is valid, this would indicate an instability of the Weinberg-Salam model, implying that the latter is only an effective theory. We give an estimate of the approximate length scale at which this occurs.     

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.     

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.     

Moment equations for a spatially extended system of two competing species

The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulations of the time discrete version of LV equations, that is coupled map lattice (CML) model, we conclude that the anticorrelated oscillations of the species densities are strictly related to non-overlapping spatial patterns.     

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.     

Formation mechanism of banded structure during directional solidification of Sn–Cd peritectic alloy under convection condition

Directional solidification experiments of Sn-0.75 wt%Cd and Sn-1.6 wt%Cd peritectic alloys have been conducted under convection condition to investigate the formation mechanism of banded structure. Many types of banded structure were obtained, which cannot be interpreted by the Karma’s model. The reason for this conflict is that there are many banded structure formation mechanisms such as abundant nucleation, regrowth, fast radial cellular growth and radial competitive growth under convection condition, but the Karma’s model only considers the abundant nucleation and ignores other mechanisms. The analyses showed that these formation mechanisms changed along with an increase in alloy composition. Based on these analyses, a simple modified banding window, which considered these different formation mechanisms, has been presented. Compared with the banding window defined by the Karma’s model, this modified banding window contained it and could predict different banded structure formations under convection condition appropriately.     

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.     

Phase resetting effects for robust cycles between chaotic sets

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a R?ssler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.     

激光等离子体相互作用中超热电子输运过程的蒙特卡罗模拟

利用蒙特卡罗（Ｍｏｎｔｅ－Ｃａｒｌｏ）方法模拟了激光等离子体相互作用中所产生的超热电子在固体物质中的输运过程，模拟运算采用连续慢化（ＣＳＤＡ）和玻恩（Ｂｏｒｎ）近似的单一散射模型。文中给出了具有单能，束状分布的超热电子在双层固体靶中产生的Ｋα射线强度随表面靶层厚度的变化曲线。     

有扩散不稳定性的四阶反应-扩散系统的空间周期结构

本文利用分叉理论研究一个含参数的有扩散不稳定性的四阶反应-扩散系统的空间周期结构问题,其中扩散项服从Cahn-Hillard广义扩散定律。首先通过稳定性和奇异性的分析得到从空间均匀定态产生空间周期定态的判别准则,然后利用奇异摄动方法获得这个空间周期定态的二阶近似解。本文的理论结果很好地符合文献[3]中的数值结果。 关键词：