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The Transfer Functions of Cardiac Tissue during Stochastic Pacing
Abstract
The restitution properties of cardiac action potential duration (APD) and conduction velocity (CV) are important factors in arrhythmogenesis. They determine alternans, wavebreak, and the patterns of reentrant arrhythmias. We developed a novel approach to characterize restitution using transfer functions. Transfer functions relate an input and an output quantity in terms of gain and phase shift in the complex frequency domain. We derived an analytical expression for the transfer function of interbeat intervals (IBIs) during conduction from one site (input) to another site downstream (output). Transfer functions can be efficiently obtained using a stochastic pacing protocol. Using simulations of conduction and extracellular mapping of strands of neonatal rat ventricular myocytes, we show that transfer functions permit the quantification of APD and CV restitution slopes when it is difficult to measure APD directly. We find that the normally positive CV restitution slope attenuates IBI variations. In contrast, a negative CV restitution slope (induced by decreasing extracellular [K(+)]) amplifies IBI variations with a maximum at the frequency of alternans. Hence, it potentiates alternans and renders conduction unstable, even in the absence of APD restitution. Thus, stochastic pacing and transfer function analysis represent a powerful strategy to evaluate restitution and the stability of conduction.
... These discrepancies can be explained by the notion of ''memory'' [5,16,[19][20][21][22][23], reflecting the fact that APD depends not only on the previous DI, but on several previous DIs and APDs. In multicellular tissue, these discrepancies can also be explained by the fact that electrotonic interactions and a steep conduction velocity restitution relation can further affect the APD restitution slope at which alternans occurs by exerting important stabilizing or destabilizing effects [15,16,24]. ...
... In previous work [24], we introduced the concept of cardiac tissue as a ''filter'' transforming an input (e.g., a series of pacing intervals varying stochastically) into an output (e.g., the series of APDs or DIs). We examined the filter characteristics in the frequency domain in terms of gain and phase shift using the transfer functions between the series of pacing intervals and the series of APDs (H tRa ) and between the series of pacing intervals and the series of DIs (H tRd ), respectively. ...
... As a reference, we first computed these transfer functions for the classical first-order memoryless restitution function APD n = f(DI n21 ) [9] with a slope a = df/dDI at the operation point. These transfer functions are H tRa = a/(z+a) and H tRd = z/(z+a), with z = e 2pif , as can be deduced from Eqs. 12 and 15-19 and as we showed previously [24]. They are represented in Figure 5 A for a ranging from 0.1 to 0.9. ...
Alternans of cardiac action potential duration (APD) is a well-known arrhythmogenic mechanism which results from dynamical instabilities. The propensity to alternans is classically investigated by examining APD restitution and by deriving APD restitution slopes as predictive markers. However, experiments have shown that such markers are not always accurate for the prediction of alternans. Using a mathematical ventricular cell model known to exhibit unstable dynamics of both membrane potential and Ca²⁺ cycling, we demonstrate that an accurate marker can be obtained by pacing at cycle lengths (CLs) varying randomly around a basic CL (BCL) and by evaluating the transfer function between the time series of CLs and APDs using an autoregressive-moving-average (ARMA) model. The first pole of this transfer function corresponds to the eigenvalue (λ(alt)) of the dominant eigenmode of the cardiac system, which predicts that alternans occurs when λ(alt) ≤ -1. For different BCLs, control values of λ(alt) were obtained using eigenmode analysis and compared to the first pole of the transfer function estimated using ARMA model fitting in simulations of random pacing protocols. In all versions of the cell model, this pole provided an accurate estimation of λ(alt). Furthermore, during slow ramp decreases of BCL or simulated drug application, this approach predicted the onset of alternans by extrapolating the time course of the estimated λ(alt). In conclusion, stochastic pacing and ARMA model identification represents a novel approach to predict alternans without making any assumptions about its ionic mechanisms. It should therefore be applicable experimentally for any type of myocardial cell.
... The characteristic Equation (11) also provides the resonant modes of the electrical and calcium compartments. More complex filtering functions have been proposed previously [15,41]. ...
Cardiac EC coupling is triggered by rhythmic depolarizing current fronts originating from the sino-atrial node, and the way variability in rhythm is associated with variability in action potential duration (APD) and, in turn, in the variability of calcium transient amplitude (CTA) and contraction is a key determinant of beating stability. Sinusoidal-varying pacing rate is adopted here in order to establish whether APD and CTA oscillations, elicited in a human ventricular AP model (OR) under oscillatory pacing, are consistent with the dynamics of two coupled harmonic oscillators, e.g., a two-degree-of-freedom system of mass and springs (MS model). I show evidence that this is the case, and that the MS model, preliminarily fitted to OR behavior, retains key features of the physiological system, such as the dependence of APD and CTA oscillation amplitudes from average value and from beat-to-beat changes in pacing rate, and the phase relationship between them. The bi-directionality of coupling between APD and CTA makes it difficult to discriminate which one leads EC coupling dynamics under variable pacing. The MS model suggests that the calcium cycling, with its greater inertia chiefly determined by the SR calcium release, is the leading mechanism. I propose the present approach to also be relevant at the whole organ level, where the need of compact representations of electromechanical interaction, particularly in clinical practice, remains urgent.
... Through the kernel K(x), the gap junctional interface can be referred as a system whose input is the current i g (x) and generates an output u(x). Such convolution approach for modeling the cardiac dynamics has been implemented in computational studies [72][73][74]. Given that K(x) depends on the spatial variable x, it can be regarded as containing information about the spatial configuration of the gap junctions. ...
Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard diffusion mathematical equation, that considers the myocardium as a continuum. However, the modeling of structural properties and their influence on electrical propagation still reveal several limitations. In this paper, a model of cardiac electrical propagation is proposed based on complex order derivatives. By assuming that the myocardium has an underlying fractal process, the complex order dynamics emerges as an important modeling option. In this perspective, the real part of the order corresponds to the fractal dimension, while the imaginary part represents the log-periodic corrections of the fractal dimension. Indeed, the imaginary part in the derivative implies characteristic scales within the cardiac tissue. The analytical and numerical procedures for solving the related equation are presented. The sinus rhythm and persAF conditions are implemented using the Courtemanche formalism. The electrophysiological properties are measured and analyzed on different scales of observation. The results indicate that the complex order modulates the electrophysiology of the atrial system, through the variation of its real and imaginary parts. The combined effect of the two components yields a broad range of electrophysiological conditions. Therefore, the proposed model can be a useful tool for modeling electrical and structural properties during cardiac conduction.
... In the presence of memory or instabilities that do not originate from voltage, the APD restitution slope can no longer predict the onset of alternans. Using a stochastic pacing protocol and transfer function method [256,257], de Lange and colleagues can predict the onset of alternans in cardiac cells in the presence of memory or instabilities originating from intracellular Ca 2+ cycling. ...
In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.
... The visual similarity between the CV-restitution curve of figure 3(E) and that of figure 1(A) is striking: both have three distinctive regimes that have different slopes, either positive, negative or zero. This result was somewhat unexpected, since non-monotonic CV-restitution curves were found only when [K + ] o was abnormally low in previous experimental investigations by Kucera et al, who used similarly prepared cardiac cell cultures [15,16]. According to the schematic illustration of figure 1(D), a periodic pacing of an excitable medium, which supports non-monotonic CV restitution, in the negative slope regime may generate an unstable period-1 oscillatory wave train. ...
Conduction velocity (CV) restitution is a key property that characterizes any medium supporting traveling waves. It reflects not only the dynamics of the individual constituents but also the coupling mechanism that mediates their interaction. Recent studies have suggested that cardiac tissues, which have a non-monotonic CV-restitution property, can support alternans, a period-2 oscillatory response of periodically paced cardiac tissue. This study finds that single-hump, non-monotonic, CV-restitution curves are a common feature of in vitro cultures of rat cardiac cells. We also find that the Fenton–Karma model, one of the well-established mathematical models of cardiac tissue, supports a very similar non-monotonic CV restitution in a physiologically relevant parameter regime. Surprisingly, the mathematical model as well as the cell cultures support bistability and show cardiac memory that tends to work against the generation of an alternans. Bistability was realized by adopting two different stimulation protocols, 'S1S2', which produces a period-1 wave train, and 'alternans-pacing', which favors a concordant alternans. Thus, we conclude that the single-hump non-monotonicity in the CV-restitution curve is not sufficient to guarantee a cardiac alternans, since cardiac memory interferes and the way the system is paced matters.
... Similarly, delayed adaptation of action potential durations after abrupt changes in stimulation rates are well known (8). The intracellular mechanism underlying the repolarization dependency and thus the pattern of the QT/RR relationship are not completely understood and have been subject of both animal and modeling experiments of ionic currents (11,19,22). Why the QT/RR relationship and adaptation exhibits substantial intersubject differences with intrasubject stability (3,27) is not known. ...
Data of large clinical study were used to investigate how much are the QT/RR patterns in healthy subjects curved and whether these curvatures differ between females and males. Day-time drug-free 12-lead Holter recordings were repeated 4 times in each of 176 female and 176 male healthy subjects aged 32.7±9.1 years. In each of the subjects, up to 1440 carefully verified QT interval measurements were obtained with QT/RR hysteresis corrected RR intervals. Individual subject data were used to fit the regression equations QT=χ+(δ/γ)(1-RR(γ))+ε where QT and RR are in seconds, ε are normally distributed zero centred errors, and the parameter γ provides the lowest regression residual. δ and γ represent QT/RR slope and curvature, respectively. Bootstrap technique showed intra-subject reproducibility of QT/RR slopes and curvatures. In females and males, the QT/RR curvatures were 0.544±0.661 and 0.797±0.706, respectively (p=0.0006). The corresponding QT/RR slopes were 0.158±0.030 and 0.139±0.023, respectively (p<0.0001). The QT/RR curvatures were related to QT/RR slopes but to neither individually corrected mean QTc intervals nor the individual QT/RR hysteresis profiles. The individual heart rate correction formula derived from the curvilinear regression provided significantly lower intra-subject variability of QTc interval than individual optimisation of linear or log-linear QT/RR heart rate corrections. QT/RR curvature can be reliable measured and expressed numerically. Corresponding heart rate correction formula provides more compact data than the previously proposed approaches. There are substantial sex differences in QT/RR patterns. Females have the QT/RR pattern not only steeper than males but also more curved.
Sudden changes in pacing cycle length are frequently associated with repolarization abnormalities initiating cardiac arrhythmias, and physiologists have long been interested in measuring the likelihood of these events before their manifestation. A marker of repolarization stability has been found in the electrical restitution (ER), the response of the ventricular action potential duration to a pre- or post-mature stimulation, graphically represented by the so-called ER curve. According to the restitution hypothesis (ERH), the slope of this curve provides a quantitative discrimination between stable repolarization and proneness to arrhythmias. ER has been studied at the body surface, whole organ, and tissue level, and ERH has soon become a key reference point in theoretical, clinical, and pharmacological studies concerning arrhythmia development, and, despite criticisms, it is still widely adopted. The ionic mechanism of ER and cellular applications of ERH are covered in the present review. The main criticism on ERH concerns its dependence from the way ER is measured. Over the years, in fact, several different experimental protocols have been established to measure ER, which are also described in this article. In reviewing the state-of-the art on cardiac cellular ER, I have introduced a notation specifying protocols and graphical representations, with the aim of unifying a sometime confusing nomenclature, and providing a physiological tool, better defined in its scope and limitations, to meet the growing expectations of clinical and pharmacological research.
Cardiomyocyte cultures exhibit spontaneous electrical and contractile activity, as in a natural cardiac pacemaker. In such preparations, beat rate variability exhibits features similar to those of heart rate variability in vivo. Mechanical deformations and forces feed back on the electrical properties of cardiomyocytes, but it is not fully elucidated how this mechano‐electrical interplay affects beating variability in such preparations. Using stretchable microelectrode arrays, we assessed the effects of the myosin inhibitor blebbistatin and the non‐selective stretch‐activated channel blocker streptomycin on beating variability and on the response of neonatal or fetal murine ventricular cell cultures against deformation. Spontaneous electrical activity was recorded without stretch and upon predefined deformation protocols (5% uniaxial and 2% equibiaxial strain, applied repeatedly for 1 min every 3 min). Without stretch, spontaneous activity originated from the edge of the preparations, and its site of origin switched frequently in a complex manner across the cultures. Blebbistatin did not change mean beat rate, but it decreased the spatial complexity of spontaneous activity. In contrast, streptomycin did not exert any manifest effects. During the deformation protocols, beat rate increased transiently upon stretch but, paradoxically, also upon release. Blebbistatin attenuated the response to stretch, whereas this response was not affected by streptomycin. Therefore, our data support the notion that in a spontaneously firing network of cardiomyocytes, active force generation, rather than stretch‐activated channels, is involved mechanistically in the complexity of the spatiotemporal patterns of spontaneous activity and in the stretch‐induced acceleration of beating. image
Key points
Monolayer cultures of cardiac cells exhibit spontaneous electrical and contractile activity, as in a natural cardiac pacemaker. Beating variability in these preparations recapitulates the power‐law behaviour of heart rate variability in vivo. However, the effects of mechano‐electrical feedback on beating variability are not yet fully understood.
Using stretchable microelectrode arrays, we examined the effects of the contraction uncoupler blebbistatin and the non‐specific stretch‐activated channel blocker streptomycin on beating variability and on stretch‐induced changes of beat rate.
Without stretch, blebbistatin decreased the spatial complexity of beating variability, whereas streptomycin had no effects.
Both stretch and release increased beat rate transiently; blebbistatin attenuated the increase of beat rate upon stretch, whereas streptomycin had no effects.
Active force generation contributes to the complexity of spatiotemporal patterns of beating variability and to the increase of beat rate upon mechanical deformation. Our study contributes to the understanding of how mechano‐electrical feedback influences heart rate variability.
The stability of different modes of heart-rate dynamics was studied in patients who had experienced a myocardial infarction. The stability of the time series of RR intervals was estimated according to the critical point of the transition of heart-rate dynamics from the linear to the chaotic mode. The point of transition to the chaotic mode was identified by occurrence of atrial extrasystoles. It was shown that the transition of heart-rate dynamics from the linear to the nonlinear mode in postinfarction patients occurs in a jump, in agreement with the theoretical results obtained previously by mathematical modeling.
Fibrotic myocardial remodeling is typically accompanied by the appearance of myofibroblasts (MFBs). In vitro, MFBs were shown to slow conduction and precipitate ectopic activity following gap junctional coupling to cardiomyocytes (CMCs). To gain further mechanistic insights into this arrhythmogenic MFB-CMC crosstalk, we performed numerical simulations in cell-based high-resolution two-dimensional tissue models that replicated experimental conditions. Cell dimensions were determined using confocal microscopy of single and co-cultured neonatal rat ventricular CMCs and MFBs. Conduction was investigated as a function of MFB density in three distinct cellular tissue architectures: CMC strands with endogenous MFBs, CMC strands with coating MFBs of two different sizes, and CMC strands with MFB inserts. Simulations were performed to identify individual contributions of heterocellular gap junctional coupling and of the specific electrical phenotype of MFBs. With increasing MFB density, both endogenous and coating MFBs slowed conduction. At MFB densities of 5–30%, conduction slowing was most pronounced in strands with endogenous MFBs due to the MFB-dependent increase in axial resistance. At MFB densities >40%, very slow conduction and spontaneous activity was primarily due to MFB-induced CMC depolarization. Coating MFBs caused non-uniformities of resting membrane potential, which were more prominent with large than with small MFBs. In simulations of MFB inserts connecting two CMC strands, conduction delays increased with increasing insert lengths and block appeared for inserts >1.2 mm. Thus, electrophysiological properties of engineered CMC-MFB co-cultures depend on MFB density, MFB size and their specific positioning in respect to CMCs. These factors may influence conduction characteristics in the heterocellular myocardium.
Spatially discordant alternans (DA) of action potential durations (APD) is thought to be more pro-arrhythmic than concordant alternans. Super normal conduction (SNC) has been reported to suppress formation of DA. An increase in conduction velocity (CV) as activation rate increases, i.e., a negative CV restitution, is widely considered as hallmark of SNC. Our aim in this study is to show that it is not an increase in CV for faster rates that prevents formation of DA, rather, it is the ratio of the CV for the short relative to the long activation that is critical in DA suppression. To illustrate this subtlety, we simulated this phenomenon using two approaches; (1) by using the standard, i.e., S1S2 protocol to quantify restitution and disabling the slow inactivation gate j of the sodium current (INa), and (2) by using the dynamic, i.e., S1S1 protocol for quantification of restitution and increasing INa at different cycle lengths (CL). Even though both approaches produced similar CV restitution curves, DA was suppressed only during the first approach, where the CV of the short of the long-short action potential (AP) pattern was selectively increased. These results show that negative CV restitution, which is considered characteristic of SNC, per se, is not causal in suppressing DA, rather, the critical factor is a change in the ratio of the velocities of the short and the long APs.
Alternans, a beat-to-beat alternation in the cardiac
action potential duration (APD), is a dynamical instability linked with the initiation of arrhythmias and sudden cardiac death, and arises via a period-doubling bifurcation when myocytes are stimulated at fast rates. In this study, we analyze the stability of a propagating electrical wave in a one-dimensional cardiac myocyte model in response to an arrhythmogenic rhythm known as alternate pacing. Using a discrete-time kinematic model and complex frequency (Z) domain analysis, we derive analytical expressions to predict phase reversals and spatial discordance in the interbeat interval (IBI) and APD, which, importantly, cannot be predicted with a model that neglects the influence of cell coupling on repolarization. We identify key dimensionless parameters that determine the transition from spatial concordance to discordance. Finally, we show that the theoretical predictions agree closely with numerical simulations of an ionic myocyte model, over a wide range of parameters, including variable IBI, altered ionic current gating, and reduced cell coupling. We demonstrate a novel approach to predict instability in cardiac tissue during alternate pacing and further illustrate how this approach can be generalized to more detail models of myocyte dynamics.
By method of computer modeling we investigated a stability of different regimes of heart rate dynamics in relation to the change in the atrioventricular conduction parameters: refractory period, minimum atrioventricular delay and curvatures of delay functions in sinoatrial and atrioventricular nodes. It is shown that curvatures of delay functions in sinoatrial and atrioventricular nodes have the most significant influence on the stability of different regimes of heart rate dynamics. The minimum delays in sinoatrial and atrioventricular nodes have smaller impact. The parameters determining the refractory periods of sinoatrial and atrioventricular nodes are least significant in terms of stability of heart rate dynamics.
Conduction velocity (CV) and CV restitution are important substrate parameters for understanding atrial arrhythmias. The aim of this work is to (i) present a simple but feasible method to measure CV restitution in-vivo using standard circular catheters, and (ii) validate its feasibility with data measured during incremental pacing. From five patients undergoing catheter ablation, we analyzed eight datasets from sinus rhythm and incremental pacing sequences. Every wavefront was measured with a circular catheter and the electrograms were analyzed with a cosine-fit method that calculated the local CV. For each pacing cycle length, the mean local CV was determined. Furthermore, changes in global CV were estimated from the time delay between pacing stimulus and wavefront arrival. Comparing local and global CV between pacing at 500 and 300 ms, we found significant changes in seven of eight pacing sequences. On average, local CV decreased by 20 ± 15% and global CV by 17 ± 13%. The method allows for in-vivo measurements of absolute CV and CV restitution during standard clinical procedures. Such data may provide valuable insights into mechanisms of atrial arrhythmias. This is important both for improving cardiac models and also for clinical applications, such as characterizing arrhythmogenic substrates during sinus rhythm.
Supernormal conduction (SNC) in excitable cardiac tissue refers to an increase of pulse (or action potential) velocity with decreasing distance to the preceding pulse. Here we employ a simple ionic model to study the effect of SNC on the propagation of action potentials (APs) and the phenomenology of alternans in excitable cardiac tissue. We use bifurcation analysis and simulations to study attraction between propagating APs caused by SNC that leads to AP pairs and bunching. It is shown that SNC stabilizes concordant alternans in arbitrarily long paced one-dimensional cables. As a consequence, spiral waves in two-dimensional tissue simulations exhibit straight nodal lines for SNC in contrast to spiraling ones in the case of normal conduction.
Rotating spiral waves appear ubiquitously in a wide range of nonlinear systems, and they play important roles in many biological phenomena. Recently, unusual spiral waves, which support period-2 dynamics, have been found in several different systems including cardiac tissues as well as nonlinear chemical reaction-diffusion systems. They are potentially significant as an intermediate dynamic state linking regularly rotating period-1 spiral waves to complex dynamic states such as cardiac fibrillations; for example, it is intrinsic of period-2 spiral waves to have "line defects" and their instability can lead to a spatiotemporal chaos. Previous mathematical models regarding period-2 spiral waves are mostly based on a coupled system of period-2 oscillators, but these are inappropriate for the description of a large class of systems that are composed of (nonoscillatory) excitable elements--a good example being the heart. In this paper we hypothesize that excitable media, which support a nonmonotonic conduction velocity dispersion relation, can sustain period-2 oscillatory spiral waves. We explicitly demonstrate that the new mechanism can create period-2 spirals by computer simulations on a simple mathematical model describing spiral wave front dynamics.
Cardiac restitution is an important factor in arrhythmogenesis. Steep positive action potential duration and conduction velocity (CV) restitution slopes promote alternans and reentrant arrhythmias. We examined the consequences of supernormal conduction (characterized by a negative CV restitution slope) on patterns of conduction and alternans in strands of Luo-Rudy model cells and in cultured cardiac cell strands. Interbeat intervals (IBIs) were analyzed as a function of distance during S1S2 protocols and during pacing at alternating cycle lengths. Supernormal conduction was induced by decreasing [K(+)](o). In control [K(+)](o) simulations, S1S2 intervals converged toward basic cycle length with a length constant determined by both CV and the CV restitution slope. During alternant pacing, the amplitude of IBI alternans converged with a shorter length constant, determined also by the action potential duration restitution slope. In contrast, during supernormal conduction, S1S2 intervals and the amplitude of alternans diverged. This amplification (resonance) led to phase-locked or more complex alternans patterns, and then to distal conduction block. The convergence/divergence of IBIs was verified in the cultured strands, in which naturally occurring tissue heterogeneities resulted in prominent discontinuities of the spatial IBI profiles. We conclude that supernormal conduction potentiates alternans and spatial analysis of IBIs represents a powerful method to locate tissue heterogeneities.
The electrophysiology of cardiac tissue is altered during acute myocardial ischemia, making the tissue less excitable but, nonetheless, more susceptible to tachyarrythmias which frequently degenerate to fibrillation within several seconds. The transition from tachycardia to fibrillation is associated with the breakup of spiral waves into multiple offspring, and has been linked to steep restitution \(slope>1\) of action potential duration (APD). However, restitution curves become so flat during ischemia that this mechanism does not apply. We found that when the response of APD to recent activations is included in a model of excitable media, spiral breakup can occur even when the slope in APD restitutions is
Ventricular fibrillation is the leading cause of sudden cardiac death. In fibrillation, fragmented electrical waves meander erratically through the heart muscle, creating disordered and ineffective contraction. Theoretical and computer studies, as well as recent experimental evidence, have suggested that fibrillation is created and sustained by the property of restitution of the cardiac action potential duration (that is, its dependence on the previous diastolic interval). The restitution hypothesis states that steeply sloped restitution curves create unstable wave propagation that results in wave break, the event that is necessary for fibrillation. Here we present experimental evidence supporting this idea. In particular, we identify the action of the drug bretylium as a prototype for the future development of effective restitution-based antifibrillatory agents. We show that bretylium acts in accord with the restitution hypothesis: by flattening restitution curves, it prevents wave break and thus prevents fibrillation. It even converts existing fibrillation, either to a periodic state (ventricular tachycardia, which is much more easily controlled) or to quiescent healthy tissue.
We have investigated the role of wave-front curvature on propagation by following the wave front that was diffracted through a narrow isthmus created in a two-dimensional ionic model (Luo-Rudy) of ventricular muscle and in a thin (0.5-mm) sheet of sheep ventricular epicardial muscle. The electrical activity in the experimental preparations was imaged by using a high-resolution video camera that monitored the changes in fluorescence of the potentiometric dye di-4-ANEPPS on the surface of the tissue. Isthmuses were created both parallel and perpendicular to the fiber orientation. In both numerical and biological experiments, when a planar wave front reached the isthmus, it was diffracted to an elliptical wave front whose pronounced curvature was very similar to that of a wave front initiated by point stimulation. In addition, the velocity of propagation was reduced in relation to that of the original planar wave. Furthermore, as shown by the numerical results, wave-front curvature changed as a function of the distance from the isthmus. Such changes in local curvature were accompanied by corresponding changes in velocity of propagation. In the model, the critical isthmus width was 200 microns for longitudinal propagation and 600 microns for transverse propagation of a single planar wave initiated proximal to the isthmus. In the experiments, propagation depended on the width of the isthmus for a fixed stimulation frequency. Propagation through an isthmus of fixed width was rate dependent both along and across fibers. Thus, the critical isthmus width for propagation was estimated in both directions for different frequencies of stimulation. In the longitudinal direction, for cycle lengths between 200 and 500 milliseconds, the critical width was < 1 mm; for 150 milliseconds, it was estimated to be between 1.3 and 2 mm; and for the maximum frequency of stimulation (117 +/- 15 milliseconds), it was > 2.5 mm. In the transverse direction, critical width was between 1.78 and 2.32 mm for a basic cycle length of 200 milliseconds. It increased to values between 2.46 and 3.53 mm for a basic cycle length of 150 milliseconds. The overall results demonstrate that the curvature of the wave front plays an important role in propagation in two-dimensional cardiac muscle and that changes in curvature may cause slow conduction or block.
The contribution of cumulative changes in action potential duration (memory) to complex cellular electrophysiological behavior was investigated in canine cardiac Purkinje fibers. Complex behavior induced during constant pacing was caused by reciprocal interactions between the time to full repolarization (TFR), where TFR = response duration + latency, and the diastolic interval (DI). The relationship between TFR and the preceding DI during complex behavior differed from that obtained using a standard restitution protocol. In particular, higher-order periodicities and chaos were produced in fibers in which the restitution curve lacked the prerequisites for such behavior. To investigate whether shifts in the restitution curve might be expected during rapid pacing, the relationship between TFR of a test response (TFR(n + 1)) and the immediately preceding response (TFR(n)) was determined. For any fixed DI(n), reduction of TFR(n) from 240 to 130 ms was accompanied by a corresponding reduction of TFR(n + 1), whereas as TFR(n) was reduced further to 120 ms, TFR(n + 1) increased. Because of the dependence of TFR(n + 1) on TFR(n) (memory) and on the preceding DI(n) (restitution), the slope of the low-dimensional relationship between TFR(n + 1) and DI(n) at a constant pacing cycle length depended on the slopes of the restitution and memory functions. These results suggest that rapid accumulation and dissipation of memory may contribute importantly to complex electrical behavior in cardiac tissue.
Spiral wave breakup is a proposed mechanism underlying the transition from ventricular tachycardia to fibrillation. We examined the importance of the restitution of action potential duration (APD) and of conduction velocity (CV) to the stability of spiral wave reentry in a two-dimensional sheet of simulated cardiac tissue. The Luo-Rudy ventricular action potential model was modified to eliminate its restitution properties, which are caused by deactivation or recovery from inactivation of K+, Ca2+, and Na+ currents (IK, ICa, and INa, respectively). In this model, we find that 1) restitution of ICa and INa are the main determinants of the steepness of APD restitution; 2) for promoting spiral breakup, the range of diastolic intervals over which the APD restitution slope is steep is more important than the maximum steepness; 3) CV restitution promotes spiral wave breakup independently of APD restitution; and 4) "defibrillation" of multiple spiral wave reentry is most effectively achieved by combining an antifibrillatory intervention based on altering restitution with an antitachycardia intervention. These findings suggest a novel paradigm for developing effective antiarrhythmic drugs.
We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The model displays rate-dependent restitution so that the dynamic and S1-S2 restitution curves are different, a manifestation of memory in the model. We derive a criterion for the stability of the 1:1 response pattern displayed by this model. It is found that the stability criterion depends on the slope of both the dynamic and S1-S2 restitution curves, and that the pattern can be stable even when the individual slopes are greater or less than one. We discuss the relation between the stability criterion and the slope of the constant-BCL restitution curve. The criterion can also be used to determine the bifurcation from the 1:1 response pattern to alternans. We demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol, thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model. We illustrate our results by considering a specific map recently derived from a three-current membrane model and find that the stability of the 1:1 pattern is accurately described by our criterion. In addition, a numerical experiment is performed using the three-current model to illustrate the application of the pacing protocol and the evaluation of the criterion.
Electrical restitution, relating action potential duration (APD) to diastolic interval (DI), was believed to determine the stability of heart rhythm. However, recent studies demonstrate that stability also depends on long-term APD changes caused by memory. This study presents a new method for investigation of rate- and memory-dependent aspects of restitution and for assessment of mapping models of APD.
Bullfrog ventricular myocardium was paced with a "perturbed downsweep protocol." Starting from a basic cycle length (BCL) of 1,000 ms, the tissue was paced until steady state was achieved, followed by single beats of longer and shorter cycle lengths. BCL was decreased by 50 to 100 ms and the process repeated. All APDs were plotted as a function of the preceding DI, which allowed simultaneous observation of dynamic, S1-S2, and two constant-BCL restitution curves in a "restitution portrait." Responses were classified as 1:1 (stimulus:response), transient 2:2, or persistent 2:2 (alternans) and were related to the slopes of the restitution curves. None of these slopes approached unity for the persistent 2:2 response, demonstrating that the traditional restitution condition does not predict alternans. The restitution portrait was used to evaluate three mapping models of APD. The models with no memory and with one-beat memory did not produce restitution portraits similar to the experimental one. A model with two-beat memory produced a qualitatively similar portrait.
The restitution portrait allows a more comprehensive assessment of cardiac dynamics than methods used to date. Further study of models with memory may result in a clinical criterion for electrical instability.
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.
Slow conduction and unidirectional conduction block (UCB) are key mechanisms of reentry. Following abrupt changes in heart rate, dynamic changes of conduction velocity (CV) and structurally determined UCB may critically influence arrhythmogenesis. Using patterned cultures of neonatal rat ventricular myocytes grown on microelectrode arrays, we investigated the dynamics of CV in linear strands and the behavior of UCB in tissue expansions following an abrupt decrease in pacing cycle length (CL). Ionic mechanisms underlying rate-dependent conduction changes were investigated using the Pandit-Clark-Giles-Demir model. In linear strands, CV gradually decreased upon a reduction of CL from 500 ms to 230-300 ms. In contrast, at very short CLs (110-220 ms), CV first decreased before increasing again. The simulations suggested that the initial conduction slowing resulted from gradually increasing action potential duration (APD), decreasing diastolic intervals, and increasing postrepolarization refractoriness, which impaired Na(+) current (I(Na)) recovery. Only at very short CLs did APD subsequently shorten again due to increasing Na(+)/K(+) pump current secondary to intracellular Na(+) accumulation, which caused recovery of CV. Across tissue expansions, the degree of UCB gradually increased at CLs of 250-390 ms, whereas at CLs of 180-240 ms, it first increased and subsequently decreased. In the simulations, reduction of inward currents caused by increasing intracellular Na(+) and Ca(2+) concentrations contributed to UCB progression, which was reversed by increasing Na(+)/K(+) pump activity. In conclusion, CV and UCB follow intricate dynamics upon an abrupt decrease in CL that are determined by the interplay among I(Na) recovery, postrepolarization refractoriness, APD changes, ion accumulation, and Na(+)/K(+) pump function.
1 Introduction.- 1.1 Motivation.- 1.2 Systems theory concepts in finite dimensions.- 1.3 Aims of this book.- 2 Semigroup Theory.- 2.1 Strongly continuous semigroups.- 2.2 Contraction and dual semigroups.- 2.3 Riesz-spectral operators.- 2.4 Delay equations.- 2.5 Invariant subspaces.- 2.6 Exercises.- 2.7 Notes and references.- 3 The Cauchy Problem.- 3.1 The abstract Cauchy problem.- 3.2 Perturbations and composite systems.- 3.3 Boundary control systems.- 3.4 Exercises.- 3.5 Notes and references.- 4 Inputs and Outputs.- 4.1 Controllability and observability.- 4.2 Tests for approximate controllability and observability.- 4.3 Input-output maps.- 4.4 Exercises.- 4.5 Notes and references.- 5 Stability, Stabilizability, and Detectability.- 5.1 Exponential stability.- 5.2 Exponential stabilizability and detectability.- 5.3 Compensator design.- 5.4 Exercises.- 5.5 Notes and references.- 6 Linear Quadratic Optimal Control.- 6.1 The problem on a finite-time interval.- 6.2 The problem on the infinite-time interval.- 6.3 Exercises.- 6.4 Notes and references.- 7 Frequency-Domain Descriptions.- 7.1 The Callier-Desoer class of scalar transfer functions.- 7.2 The multivariable extension.- 7.3 State-space interpretations.- 7.4 Exercises.- 7.5 Notes and references.- 8 Hankel Operators and the Nehari Problem.- 8.1 Frequency-domain formulation.- 8.2 Hankel operators in the time domain.- 8.3The Nehari extension problem for state linear systems.- 8.4 Exercises.- 8.5 Notes and references.- 9 Robust Finite-Dimensional Controller Synthesis.- 9.1 Closed-loop stability and coprime factorizations.- 9.2 Robust stabilization of uncertain systems.- 9.3 Robust stabilization under additive uncertainty.- 9.4 Robust stabilization under normalized left-coprime-factor uncertainty.- 9.5 Robustness in the presence of small delays.- 9.6 Exercises.- 9.7 Notes and references.- A. Mathematical Background.- A.1 Complex analysis.- A.2 Normed linear spaces.- A.2.1 General theory.- A.2.2 Hilbert spaces.- A.3 Operators on normed linear spaces.- A.3.1 General theory.- A.3.2 Operators on Hilbert spaces.- A.4 Spectral theory.- A.4.1 General spectral theory.- A.4.2 Spectral theory for compact normal operators.- A.5 Integration and differentiation theory.- A.5.1 Integration theory.- A.5.2 Differentiation theory.- A.6 Frequency-domain spaces.- A.6.1 Laplace and Fourier transforms.- A.6.2 Frequency-domain spaces.- A.6.3 The Hardy spaces.- A.7 Algebraic concepts.- A.7.1 General definitions.- A.7.2 Coprime factorizations over principal ideal domains.- A.7.3 Coprime factorizations over commutative integral domains.- References.- Notation.
This report presents a survey of methods for the Distributed Parameter System Identification Problem. Fundamental concepts such as ‘ System Identification ’ and ‘ Distributed Parameter Systems ’ are briefly summarized in order to make precise what kind of problem will be considered here. The various identification methods are grouped into three disjoint classes, namely : ‘ Direct Method ’, ‘ Reduction to a Lumped Parameter System ’ and ‘ Reduction to an Algebraic Equation ’. Under this classification we give a general survey of the main approaches to the problem of identification in distributed systems
The sections in this article are1The Problem2Background and Literature3Outline4Displaying the Basic Ideas: Arx Models and the Linear Least Squares Method5Model Structures I: Linear Models6Model Structures Ii: Nonlinear Black-Box Models7General Parameter Estimation Techniques8Special Estimation Techniques for Linear Black-Box Models9Data Quality10Model Validation and Model Selection11Back to Data: The Practical Side of Identification
A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n + 1) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The
simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and
computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum,
needed in statistical estimation problems.
A mathematical model of the membrane action potential of the mammalian ventricular cell is introduced. The model is based, whenever possible, on recent single-cell and single-channel data and incorporates the possibility of changing extracellular potassium concentration [K]o. The fast sodium current, INa, is characterized by fast upstroke velocity (Vmax = 400 V/sec) and slow recovery from inactivation. The time-independent potassium current, IK1, includes a negative-slope phase and displays significant crossover phenomenon as [K]o is varied. The time-dependent potassium current, IK, shows only a minimal degree of crossover. A novel potassium current that activates at plateau potentials is included in the model. The simulated action potential duplicates the experimentally observed effects of changes in [K]o on action potential duration and rest potential. Physiological simulations focus on the interaction between depolarization and repolarization (i.e., premature stimulation). Results demonstrate the importance of the slow recovery of INa in determining the response of the cell. Simulated responses to periodic stimulation include monotonic Wenckebach patterns and alternans at normal [K]o, whereas at low [K]o nonmonotonic Wenckebach periodicities, aperiodic patterns, and enhanced supernormal excitability that results in unstable responses ("chaotic activity") are observed. The results are consistent with recent experimental observations, and the model simulations relate these phenomena to the underlying ionic channel kinetics.
We derive an "eikonal-curvature" equation to describe the propagation of action potential wavefronts in myocardium. This equation is used to study the effects of fiber orientation on propagation in the myocardial wall. There are significant computational advantages to the use of an eikonal-curvature equation over a full ionic model of action potential spread. With this model, it is shown that the experimentally observed misalignment of spreading action potential "ellipses" from fiber orientation in level myocardial surfaces is adequately explained by the rotation of fiber orientation through the myocardial wall. Additionally, it is shown that apparently high propagation velocities on the epicardial and endocardial surfaces are the result of propagation into the midwall region and acceleration along midwall fibers before reemergence at an outer surface at a time preceding what could be accomplished with propagation along the surface alone.
Supernormality, which can be defined as greater than normal excitability during or immediately after action potential repolarization, has been observed in a variety of cardiac preparations. However, as yet, no description of the dynamics of tissue response to repetitive stimulation in the presence of supernormal or relatively supernormal excitability has appeared. Isolated sheep cardiac Purkinje fibers (2-5 mm in length) were superfused with Tyrode's solution and stimulated with depolarizing current pulses through a suction pipette. Recovery of excitability, restitution of the action potential duration, and response patterns were measured in each fiber for a wide range of current amplitudes and stimulation frequencies. When the potassium chloride concentration of the Tyrode's solution was decreased from 7 to 4 mM, the excitability recovery function consistently changed from monophasic ("normal") to triphasic ("supernormal'). During repetitive stimulation at increasing rates, normal preparations responded only with gradual changes in the activation ratio, expressed as periodic phase-locked responses (i.e., Wenckebach-like patterns, etc.). Supernormal preparations showed a nonmonotonic change in the activation ratio, as well as complex aperiodic response patterns. Numerical results from an analytical model gave a quantitative basis for the relation between nonmonotonicity in the excitability function and the development of complex rhythms in cardiac Purkinje fibers. Both our experimental and theoretical results indicate that the presence of supernormality and the slope of the action potential duration restitution curve at short diastolic intervals are responsible for the development of chaotic dynamics. Moreover, our results give an accurate description of the supernormality phenomenon, predict the behavior expected under such conditions, and provide insight about the role of membrane recovery in determining regular and irregular frequency-dependent rhythm and conduction disturbances.
It is well known that in most cardiac tissues an increase in rate results in a decrease of excitability and, eventually, conduction block. We used microelectrode techniques to evaluate the rate and time dependence of excitation latency in 27 isolated guinea pig papillary muscles (GPPM). Latency was measured as the interval between the stimulus onset and action potential upstroke. When the intensity of current was just suprathreshold, prolongation of the basic cycle length (BCL) from 300 to 1,000 ms produced an increase in latency or failure of excitation. Such behavior was observed with extracellular as well as intracellular stimulation. Rate-dependent changes in latency were maximal during the first 10-20 s following the rate change and reached a steady state in approximately 200 s. Application of premature beats revealed the presence of a "supernormal phase" in which latency abbreviated. Strength-interval and strength-duration curves demonstrated that changes in excitability accurately paralleled those observed in latency. Hence, supernormal excitability at the end of the phase 3 repolarization was consistently observed in all ventricular muscle experiments. Deceleration-induced decrease of excitability was attended by hyperpolarization, increase of action potential upstroke velocity (Vmax) and action potential amplitude, and decrease in membrane resistance. Our data suggest that paradoxical rate-related changes of excitability in GPPM are the result of changes in the passive membrane properties. Under conditions of depressed conductivity, this particular behavior may account for the occurrence of bradycardia-dependent block.
Elucidation of the mechanisms of cardiac conduction disturbances leading to reentry will require resolution of the details of multidimensional propagation at a microscopic size scale (less than 200 micron). In practice, this will necessitate the combined analysis of extracellular and transmembrane action potentials. The purpose of this paper is to demonstrate the relationships between the time derivatives of the extracellular waveforms and the underlying action potentials in the experimental analysis of anisotropic propagation at this small size scale, and apply these relationships to human atrial muscle at different ages. The extracellular waveforms and their derivatives changed from a smooth contour during transverse propagation in young preparations to complex polyphasic waveforms in the older preparations. The major problem was to estimate the size and location of small groups of fibers that generated the complex waveforms in the older preparations. We found dissimilarities in the derivatives that distinguished source (bundle) size from the distance of the source to the measurement site. The differences in the extracellular waveforms and their derivatives indicated that there was electrical uncoupling of the side-to-side connections between small groups of fibers with aging. These changes produced a prominent zigzag course of transverse propagation at a microscopic level which, in turn, accounted for the increased complexity of the waveforms. The waveform differences also correlated with the development of extensive collagenous septa that separated small groups of fibers. The electrophysiological consequence was an age-related decrease in the "effective" transverse conduction velocities to the range of the very slow conduction (less than 0.08 m/sec) which makes it possible for reentry to occur in small regions of cardiac muscle with normal cellular electrophysiological properties.
The electrophysiological characteristics of the period of supernormal excitability and supernormal conduction were investigated in the isolated canine His Purkinje system. Strength interval curves were determined as the minimum transmembrane current required to bring the impaled fiber to threshold potential following a conducted action potential. During the period of supernormal excitability, 17.0 ± 4.6% (SD) less current than that required during diastole were needed to reexcite fibers throughout the left and right bundle branch Purkinje system. A period of supernormal excitability was not found in the His bundle proximal to its pseudobifurcation or in ventricular muscle. The period of supernormal excitability was voltage dependent in the bundle branch Purkinje system; it began during phase 3 at full repolarization (88.8 ± 5.6 [SD] mv) and reached minimum current requirements at about 74.3±5.8 mv. Action potentials evoked during this period were conducted faster than they were during diastole. The maximum rates of depolarization of these supernormally conducted action potentials were not greatly depressed compared with control rates. A period of supernormal conduction was not observed in the His bundle. When the external potassium concentration was increased from 2.7 mM to 5.0 mM or 7.5 mM, both the supernormal period of excitability and the period of supernormal conduction were eliminated in Purkinje fibers. (19 references).
We have studied the three-dimensional arrangement of ventricular muscle cells and the associated extracellular connective tissue matrix in dog hearts. Four hearts were potassium-arrested, excised, and perfusion-fixed at zero transmural pressure. Full-thickness segments were cut from the right and left ventricular walls at a series of precisely located sites. Morphology was visualized macroscopically and with scanning electron microscopy in 1) transmural planes of section and 2) planes tangential to the epicardial surface. The appearance of all specimens was consistent with an ordered laminar arrangement of myocytes with extensive cleavage planes between muscle layers. These planes ran radially from endocardium toward epicardium in transmural section and coincided with the local muscle fiber orientation in tangential section. Stereological techniques were used to quantify aspects of this organization. There was no consistent variation in the cellular organization of muscle layers (48.4 +/- 20.4 microns thick and 4 +/- 2 myocytes across) transmurally or in different ventricular regions (23 sites in 6 segments), but there was significant transmural variation in the coupling between adjacent layers. The number of branches between layers decreased twofold from subepicardium to midwall, whereas the length distribution of perimysial collagen fibers connecting muscle layers was greatest in the midwall. We conclude that ventricular myocardium is not a uniformly branching continuum but a laminar hierarchy in which it is possible to identify three axes of material symmetry at any point.
Ventricular tachycardias occurring in the chronic phase of myocardial infarction are caused by reentry. Areas of slow conduction, facilitating reentry, are often found in the infarcted zone. The purpose of this study was to elucidate the mechanism of slow conduction in the chronic infarcted human heart.
Spread of activation was studied in infarcted papillary muscles from hearts of patients who underwent heart transplantation because of infarction. Recordings were carried out on 10 papillary muscles that were superfused in a tissue bath. High-resolution mapping was performed in areas revealing slow conduction. Activation delay between sites perpendicular to the fiber direction and 1.4 mm apart could be as long as 45 milliseconds. Analysis of activation times revealed that activation spread in tracts parallel to the fiber direction. Conduction velocity in the tracts was between 0.6 and 1 m/s. Although tracts were separated from each other over distances up to 8 mm, they often connected with each other at one or more sites, forming a complex network of connected tracts. In this network, wave fronts could travel perpendicular to the fiber direction. Separation of tracts was due to collagenous septa. At sites where tracts were interconnected, the collagenous barriers were interrupted.
Slow conduction perpendicular to the fiber direction in infarcted myocardial tissue is caused by a "zigzag" course of activation at high speed. Activation proceeds along pathways lengthened by branching and merging bundles of surviving myocytes ensheathed by collagenous septa.
A modeling study is presented to explore the effects of tissue conductivity, fiber orientation, and presence of an adjoining extracellular volume conductor on electrical conduction in cardiac muscle. Simulated results are compared with those of classical in vitro experiments on superfused thin layer preparations and on whole hearts.
The tissue is modeled as a three-dimensional bidomain block adjoining an isotropic bath. In the thin layer model, the fibers are assumed parallel. In the thick block model, fiber rotation, curvature, and tipping are incorporated. Results from the thin layer model explain experimental observations that the rate of rise of the entire action potential upstroke is faster and the magnitude of the extracellular potential is smaller across fibers than along fibers in a uniformly propagating front. The simulation identified that this behavior only arises in tissue with unequal anisotropy in the two spaces and adjoining an extracellular bath. Simulated conduction and potential distributions in the thick block model are shown to well approximate experimental maps. The potentials are sensitive to changes in the fiber orientations. A slight 5 degrees tipping of intramural fibers out of the planes parallel to the epicardium and endocardium will lead to an asymmetry of the magnitudes of the positive regions. In addition, the introduction of fiber curvature leads to more realistic isochrone and extracellular potential distributions. The orientation of the central negative region of the extracellular potential is shown to be determined by the average of the fiber direction at the plane of pacing and the plane of recording.
The simulations demonstrate the sensitivity of spread of activation and potential time courses and distributions to the underlying electrical properties in both thick and thin slabs. The bidomain model is shown to be a useful representation of cardiac tissue for interpreting experimental data of activation.
A multicellular ventricular fiber model was used to determine mechanisms of slowed conduction and conduction failure during acute ischemia. We simulated the three major pathophysiological component conditions of acute ischemia: elevated [K+]o, acidosis, and anoxia. Elevated [K+]o was the major determinant of conduction, causing supernormal conduction, depressed conduction, and conduction block as [K+]o was gradually increased from 4.5 to 14.4 mmol/L. Only elevated [K+]o caused conduction failure when varied within the range reported for acute ischemia. Before block, depressed upstrokes consisted of two distinct components: the first to the fast Na+ current (INa) and the second to the L-type Ca2+ current (ICa(L)). Even in highly depressed conduction, excitability was maintained by INa, with conduction block occurring at 95% INa inactivation. However, because ICa(L) supported the later phase of the depressed upstroke, ICa(L) enhanced conduction and delayed block by increasing the electrotonic source current. At [K+]o = 18 mmol/L, slow action potentials generated by ICa(L) were obtained with 10% ICa(L) augmentation. However, in the presence of acidosis and anoxia, significantly larger (120%) ICa(L) augmentation was required. The depressant effect was due mostly to anoxic activation of outward ATP-sensitive K+ current, which counteracts inward ICa(L) and, by lowering the action potential amplitude, decreases the electrotonic current available to depolarize downstream cells. The simulations highlight the interactive nature of electrophysiological ischemic changes during propagation and demonstrate that both membrane changes and load factors (by downstream fiber) must be considered.
It is traditionally assumed that impulse propagation in cardiac muscle is determined by the combination of two factors: (1) the active properties of cardiac cell membranes and (2) the passive electrical characteristics of the network formed by cardiac cells. However, advances made recently in the theory of generic excitable media suggest that an additional factor-the geometry of excitation wavefronts -may play an important role. In particular, impulse propagation strongly depends on the wavefront curvature on a small spatial scale. In the heart, excitation wavefronts have pronounced curvatures in several situations including waves initiated by small electrodes, waves emerging from narrow tissue structures, and waves propagating around the sharp edges of anatomical obstacles or around a zone of functional conduction block during spiral wave rotation. In this short review we consider the theoretical background relating impulse propagation to wavefront curvature and we estimate the role of wavefront curvature in electrical stimulation, formation of conduction block, and the dynamic behavior of spiral waves.
In cardiac tissue, reduced membrane excitability and reduced gap junction coupling both slow conduction velocity of the action potential. However, the ionic mechanisms of slow conduction for the two conditions are very different. We explored, using a multicellular theoretical fiber, the ionic mechanisms and functional role of the fast sodium current, INa, and the L-type calcium current, ICa(L), during conduction slowing for the two fiber conditions. A safety factor for conduction (SF) was formulated and computed for each condition. Reduced excitability caused a lower SF as conduction velocity decreased. In contrast, reduced gap junction coupling caused a paradoxical increase in SF as conduction velocity decreased. The opposite effect of the two conditions on SF was reflected in the minimum attainable conduction velocity before failure: decreased excitability could reduce velocity to only one third of control (from 54 to 17 cm/s) before failure occurred, whereas decreased coupling could reduce velocity to as low as 0.26 cm/s before block. Under normal conditions and conditions of reduced excitability, ICa(L) had a minimal effect on SF and on conduction. However, ICa(L) played a major role in sustaining conduction when intercellular coupling was reduced. This phenomenon demonstrates that structural, nonmembrane factors can cause a switch of intrinsic membrane processes that support conduction. High intracellular calcium concentration, [Ca]i, lowered propagation safety and caused earlier block when intercellular coupling was reduced. [Ca]i affected conduction via calcium-dependent inactivation of ICa(L). The increase of safety factor during reduced coupling suggests a major involvement of uncoupling in stable slow conduction in infarcted myocardium, making microreentry possible. Reliance on ICa(L) for this type of conduction suggests ICa(L) as a possible target for antiarrhythmic drug therapy.
It was the aim of this study to characterize the spread of activation at the cellular level in cardiac tissue during conduction slowing, a key element of reentrant arrhythmias; therefore, activation patterns were assessed at high spatiotemporal resolution in narrow (70 to 80 microm) and wide (230 to 270 microm) linear strands of cultured neonatal rat ventricular myocytes, using multiple site optical recording of transmembrane voltage. Slow conduction was induced by graded elevation of [K+]o, by applying tetrodotoxin, or by exposing the preparations to the gap junctional uncouplers palmitoleic acid or 1-octanol. The main findings of the study are 4-fold: (1) gap junctional uncoupling reduced conduction velocity (range, 37 to 47 cm/s under control conditions) to a substantially larger extent before block (</=1 cm/s; ultra-slow conduction) than did a reduction of excitability (range, approximately 10 to 15 cm/s); (2) activation wavefronts during uncoupling meandered within the boundaries of the preparations, resulting in a pronounced additional slowing of conduction in wide cell strands; (3) at the cellular level, propagation during uncoupling-induced ultra-slow conduction was sustained by sequentially activated tissue patches, each of which consisted of a few cells being activated simultaneously; and (4) depending on the uncoupler used, maximal action potential upstroke velocities during ultra-slow conduction were either slightly (palmitoleic acid) or highly (1-octanol) depressed. Thus, depolarizing inward currents, the spatial pattern and degree of gap junctional coupling, and geometrical factors all contribute in a concerted manner to conduction slowing, which, at its extreme (0.25 cm/s measured over 1 mm), can reach values low enough to permit, theoretically, reentrant excitation to occur in minuscule areas of cardiac tissue (<1 mm2).
Sudden cardiac death resulting from ventricular fibrillation can be separated into 2 components: initiation of tachycardia and degeneration of tachycardia to fibrillation. Clinical drug studies such as CAST and SWORD demonstrated that focusing exclusively on the first component is inadequate as a therapeutic modality. The hope for developing effective pharmacological therapy rests on a comprehensive understanding of the second component, the transition from tachycardia to fibrillation. We summarize evidence that the transition from tachycardia to fibrillation is a transition to spatiotemporal chaos, with similarities to the quasiperiodic transition to chaos seen in fluid turbulence. In this scenario, chaos results from the interaction of multiple causally independent oscillatory motions. Simulations in 2-dimensional cardiac tissue suggest that the destabilizing oscillatory motions during spiral-wave reentry arise from restitution properties of action potential duration and conduction velocity. The process of spiral-wave breakup in simulated cardiac tissue predicts remarkably well the sequence by which tachycardia degenerates to fibrillation in real cardiac tissue. Modifying action potential duration and conduction velocity restitution characteristics can prevent spiral-wave breakup in simulated cardiac tissue, suggesting that drugs with similar effects in real cardiac tissue may have antifibrillatory efficacy (the Restitution Hypothesis). If valid for the real heart, the Restitution Hypothesis will support a new paradigm for antiarrhythmic drug classification, incorporating an antifibrillatory profile based on effects on cardiac restitution and the traditional antitachycardia profile (classes 1 through 4).
See article by Taggart et al. [50] (pages 454-462) in this issue.
Studies conducted over the past dozen years or so have demonstrated that ventricular myocardium is not homogeneous, as previously thought, but is comprised of at least three electrophysiologically distinct cell types: epicardial, M and endocardial cells (see [1,2] for reviews). These three cell types have also been shown to possess different pharmacologic profiles and to respond differently to a variety of pathophysiologic states [3–8].
The three cells types differ principally with respect to repolarization characteristics. Ventricular epicardial and M cells display action potentials with a prominent transient outward current ( I to)-mediated phase 1, giving rise to a notched appearance of the action potential. The absence of a prominent notch in the endocardium is a consequence of a much smaller I to. Similar regional differences in I to are found in canine, feline, rabbit, rat and human ventricular myocytes (see [1] for references). Recent studies also indicate that I to and the action potential notch are much larger in right vs. left ventricular epicardial [9] and M [10] cells. The transmural gradient in the amplitude of the I to-mediated action potential notch underlies the normal J wave or J point elevation in the ECG [11] and its accentuation, particularly in the right ventricle, contributes to the development of life-threatening arrhythmias in patients with the Brugada syndrome and various forms of idiopathic ventricular fibrillation [12,13]. The presence of a prominent I to in right ventricular epicardium has also been shown to sensitize this tissue to the effects of ischemia [14]. Accentuation of the action potential notch and eventual loss of the dome in right ventricular epicardium but not endocardium has been shown to contribute to ischemia-induced ST segment elevation [12].
The M cells are …
*Tel.: +1-315-735-2217; fax: +1-315-735-5648 ca{at}mmrl.edu
Ventricular fibrillation (VF) is the leading cause of sudden cardiac death. Yet, the mechanisms of VF remain elusive. Pixel-by-pixel spectral analysis of optical signals was carried out in video imaging experiments using a potentiometric dye in the Langendorff-perfused guinea pig heart. Dominant frequencies (peak with maximal power) were distributed throughout the ventricles in clearly demarcated domains. The fastest domain (25 to 32 Hz) was always on the anterior left ventricular (LV) wall and was shown to result from persistent rotor activity. Intermittent block and breakage of wavefronts at specific locations in the periphery of such rotors were responsible for the domain organization. Patch-clamping of ventricular myocytes from the LV and the right ventricle (RV) demonstrated an LV-to-RV drop in the amplitude of the outward component of the background rectifier current (I(B)). Computer simulations suggested that rotor stability in LV resulted from relatively small rectification of I(B) (presumably I(K1)), whereas instability, termination, and wavebreaks in RV were a consequence of strong rectification. This study provides new evidence in the isolated guinea pig heart that a persistent high-frequency rotor in the LV maintains VF, and that spatially distributed gradients in I(K1) density represent a robust ionic mechanism for rotor stabilization and wavefront fragmentation.
Computer simulations are used to characterize conduction of the cardiac action potential under the following conditions: 1. Acute myocardial ischemia, 2. Reduced membrane excitability, 3. Reduced intercellular coupling, and 4. Propagation through inhomogeneous tissue structures.
Spatially defined growth of cells in culture is a useful model for studies ranging from the characterization of cellular motility to the analysis of network behaviour in structurally defined ensembles of excitable cells. Current methodological approaches for obtaining patterned growth include sophisticated modifications of surface chemistry, stamping techniques and microfluidics. The implementation of most of these techniques requires the availability of highly specialized apparatus and some of the methods are specific for certain cell types and/or substrate materials. The goal of the present study was to develop a cell-patterning technique that can be implemented by any laboratory working with cell culture and that is highly adaptable in terms of cell types and substrate materials. The method is based on a photolithographic process that permits the patterned deposition of attachment factors of choice on surfaces previously coated with agar with a spatial resolution (maximal deviation from a straight line) of +/-3 micro m. Because agar efficiently prevents cell adhesion, patterned growth obtained with this technique displays virtually no off-pattern cell attachment. The method permitted the patterning of cardiomyocytes, fibroblasts and HeLa cells on either glass substrates or polymer-coated materials with a spatial resolution of a few micrometers.
Restitution of action potential duration (APD) is thought to be critical in activation instability. Although restitution is used to predict APD during sequential changes in diastolic interval (DI), currently used protocols to determine restitution do not use sequential changes in DI. We explored restitution using a new pacing protocol to change DI sequentially and independently of APD. Transmembrane potentials were recorded from right ventricular endocardial tissue isolated from six dogs. We used three patterns of DIs: oscillatory, to demonstrate differences in APDs depending on previous activation history; random, to minimize effects of previous activation history, each DI preceding an APD had an equal probability of being short or long; and linear, to compare restitution relationship obtained during sequential changes in DI with those obtained using currently used protocols; DIs mimicked those that resulted using currently used protocols, except that they changed in sequence. During oscillatory DIs, restitution showed bimodal trajectory similar to hysteresis. Decrease in APD during decreasing DIs was faster than increase in APD during increasing DIs. When effects of previous activation history were minimized, we observed that for a given DI there were multiple values of APD. Restitution relationship obtained during sequential changes in DI was shallower than those obtained using currently used protocols. Our results show that the new pacing protocol may permit direct evaluation of effects of memory on APD. Sequential and explicit control of DI suggests that use of a unimodal relationship to predict APD when DIs change in sequence may not be appropriate.
The dependence of action potential duration (APD) on the preceding diastolic interval (DI), i.e., restitution, has been purported to predict the development of alternans and reentrant arrhythmias. However, restitution depends on the history of activation (i.e., memory), and its relevance to arrhythmia induction and maintenance is unknown.
Using a dual-camera video imaging system, we recorded action potentials from thousands of sites on the surface of the isolated pig heart. A steady-state pacing (SSP) protocol was performed to generate the SSP APD restitution curve. During SSP, the minimum DI and APD were 57 +/- 6 ms and 107 +/- 6 ms, respectively. The restitution slope was >1 for DIs <85 +/- 5 ms; however, alternans were not observed. Abrupt decreases in cycle length (CL) resulted in a rapid (<5 beats) decrease in APD followed by a slower decrease to "steady state." DI, APD pairs for the initial beats following these rate changes were significantly above the SSP restitution curve. DI, APD pairs measured during sustained ventricular fibrillation clustered significantly below the SSP restitution curve, at significantly shorter APDs (57 +/- 4 ms) and DIs (49 +/- 6 ms) than could be achieved during SSP. In addition, abrupt increases in CL following SSP resulted in APDs significantly shorter than those predicted from the SSP restitution curve.
Our results indicate that the responses of APD and DI to sudden rate changes and during arrhythmias are not predicted by the SSP restitution relationship. Acute dynamics act to damp out the proarrhythmic oscillations predicted from the SSP restitution curve.
The restitution hypothesis proposes that adaptation of cardiac action potential duration (APD) to rate changes is a predictor of ventricular fibrillation (VF). Conventional restitution kinetics plots the APD of a premature beat as a function of the previous diastolic interval (DI), and VF vulnerability is related to how rapidly APD shortens with decreasing DI. However, APD depends not only on the previous DI but also on the history of previous APDs and DIs. For a comprehensive understanding of APD restitution, we developed a random stimulation protocol and curve fitted each APD with the previous DIs and APDs using multiple autoregressive analyses.
Guinea pig hearts (n = 5) were perfused and stained with di-4 ANEPPS to record optical APs from 252 sites. Activation and repolarization times were detected in real time from one pixel and hearts were stimulated at random DIs (range 0-50 or 0-100 ms). We found that the first, second, and third previous APDs and DIs are required to obtain the best curve fit, which provides the most significant feedback control to APD and up to six previous beats contributed to curve fits (R > 0.8). The coefficients relating the previous DI to APD increased systematically in going from apex to base reflecting the intrinsic gradient of APD across the epicardium.
Random restitution is more comprehensive than steady-state restitution, being based on random and dynamic DIs, and makes possible characterization of restitution in only 32 seconds to track changes in restitution during time-varying conditions such as ischemia/reperfusion.
Investigation of relationship between diastolic-interval (DI)-dependent restitution of action potential duration (APD) and alternans of APD has produced conflicting results. We used a novel pacing protocol to determine the role of restitution in alternans by minimizing changes in DI preceding each activation.
Transmembrane potentials were recorded from right ventricular endocardial tissue isolated from five dogs. We used three pacing sequences: (i) The tissue was paced at a constant DI for 100 beats. (ii) The DIs were changed randomly between two sequences of constant DI. (iii) Each constant DI trial was followed by constant cycle length trial where pacing cycle length was equal to average cycle length during previous constant DI trial.
Alternans of APD occurred even when DIs preceding each activation were invariant. Slopes of restitution during constant DI pacing were both negative and positive and were much larger than unity. Alternans amplitude during constant cycle length pacing was larger than during constant DI, 32.2 +/- 12.3 versus 7.5 +/- 2.8 msec, P < 0.01. Random perturbation of DI decreased alternans amplitude during constant DI pacing from 14.7 +/- 4.8 to 10.5 +/- 3.4 msec, P < 0.01.
Our results indicate that mechanism of repolarization alternans has restitution-dependent and restitution-independent components. However, our results also provide direct evidence that shows that DI-dependent restitution of APD is not a necessary mechanism for the alternans to exist. Ability to pace with explicit control of DI provides a novel approach to dissect mechanisms of alternans into restitution-dependent and restitution-independent effects.
Structural remodeling of the myocardium associated with mechanical overload or cardiac infarction is accompanied by the appearance of myofibroblasts. These fibroblast-like cells express alpha-smooth muscle actin (alphaSMA) and have been shown to express connexins in tissues other than heart. The present study examined whether myofibroblasts of cardiac origin establish heterocellular gap junctional coupling with cardiomyocytes and whether ensuing electrotonic interactions affect impulse propagation. For this purpose, impulse conduction characteristics (conduction velocity [theta] and maximal upstroke velocity [dV/dtmax]) were assessed optically in cultured strands of cardiomyocytes, which were coated with fibroblasts of cardiac origin. Immunocytochemistry showed that cultured fibroblasts underwent a phenotype switch to alphaSMA-positive myofibroblasts that expressed connexin 43 and 45 both among themselves and at contact sites with cardiomyocytes. Myofibroblasts affected theta and dV/dtmax in a cell density-dependent manner; a gradual increase of myofibroblast-to-cardiomyocyte ratios up to 7:100 caused an increase of both theta and dV/dtmax, which was followed by a progressive decline at higher ratios. On full coverage of the strands with myofibroblasts (ratio >20:100), theta fell <200 mm/s. This biphasic dependence of theta and dV/dtmax on myofibroblast density is reminiscent of "supernormal conduction" and is explained by a myofibroblast density-dependent gradual depolarization of the cardiomyocyte strands from -78 mV to -50 mV as measured using microelectrode recordings. These findings suggest that myofibroblasts, apart from their role in structural remodeling, might contribute to arrhythmogenesis by direct electrotonic modulation of conduction and that prevention of their appearance might represent an antiarrhythmic therapeutic target.
Computer simulations and nonlinear dynamics have provided invaluable tools for illuminating the underlying mechanisms of cardiac arrhythmias. Here, we review how this approach has led to major insights into the mechanisms of spatially discordant alternans, a key arrhythmogenic factor predisposing the heart to re-entry and lethal arrhythmias. During spatially discordant alternans, the action potential duration (APD) alternates out of phase in different regions of the heart, markedly enhancing dispersion of refractoriness so that ectopic beats have a high probability of inducing reentry. We show how, at the cellular level, instabilities in membrane voltage (ie, steep APD restitution slope) and intracellular Ca (Cai) cycling dynamics cause APD and the Cai transient to alternate and how the characteristics of alternans are affected by different "modes" of the bidirectional coupling between voltage and Cai. We illustrate how, at the tissue level, additional factors, such as conduction velocity restitution and ectopic beats, promote spatially discordant alternans. These insights have illuminated the mechanistic basis underlying the clinical association of cardiac alternans (eg, T wave alternans) with arrhythmia risk, which may lead to novel therapeutic approaches to avert sudden cardiac death.
We used a mathematical model to investigate effects of repolarizing currents I(kr) and I(ks), calcium (Ca) current I(CaL), and Ca dynamics in network sarcoplasmic reticulum and junctional sarcoplasmic reticulum (JSR) on hysteresis in restitution of action potential duration. Enhanced I(kr) increased slope of restitution, hysteresis loop thickness, and delay between peaks of diastolic intervals and action potential duration. Increase in I(ks) decreased loop thickness and peak delay. Decrease in I(CaL) had effects similar to increasing I(kr), except slope of restitution decreased markedly. Uptake of Ca into the network sarcoplasmic reticulum had less effect on hysteresis than transfer of Ca into JSR. Faster transfer of Ca into JSR markedly decreased loop thickness and peak delay. Our results provide insight into mechanisms responsible for this newly identified property of restitution. Such information will be valuable in studies where modification of hysteresis is used to investigate its role in arrhythmogenesis.
Beat-to-beat alternation of the action potential duration (APD) in paced cardiac cells has been linked to the onset of lethal arrhythmias. Both experimental and theoretical studies have shown that alternans at the single cell level can be caused by unstable membrane voltage (V(m)) dynamics linked to steep APD-restitution, or unstable intracellular calcium (Ca) cycling linked to high sensitivity of Ca release from the sarcoplasmic reticulum on sarcoplasmic reticulum Ca load. Identifying which of these two mechanisms is the primary cause of cellular alternans, however, has remained difficult since Ca and V(m) are bidirectionally coupled. Here, we use numerical simulations of a physiologically detailed ionic model to show that the origin of alternans can be inferred by measuring the length scales over which APD and Ca(i) alternans reverse phase during spatially discordant alternans. The main conclusion is that these scales are comparable to a few millimeters and equal when alternans is driven by APD restitution, but differ markedly when alternans is driven predominantly by unstable Ca cycling. In the latter case, APD alternans still reverses phase on a millimeter tissue scale due to electrotonic coupling, while Ca alternans reverses phase on a submillimeter cellular scale. These results show that experimentally accessible measurements of Ca(i) and V(m) in cardiac tissue can be used to shed light on the cellular origin of alternans.
Heart rate variability (HRV) exhibits fluctuations characterized by a power law behavior of its power spectrum. The interpretation of this nonlinear HRV behavior, resulting from interactions between extracardiac regulatory mechanisms, could be clinically useful. However, the involvement of intrinsic variations of pacemaker rate in HRV has scarcely been investigated. We examined beating variability in spontaneously active incubating cultures of neonatal rat ventricular myocytes using microelectrode arrays. In networks of mathematical model pacemaker cells, we evaluated the variability induced by the stochastic gating of transmembrane currents and of calcium release channels and by the dynamic turnover of ion channels. In the cultures, spontaneous activity originated from a mobile focus. Both the beat-to-beat movement of the focus and beat rate variability exhibited a power law behavior. In the model networks, stochastic fluctuations in transmembrane currents and stochastic gating of calcium release channels did not reproduce the spatiotemporal patterns observed in vitro. In contrast, long-term correlations produced by the turnover of ion channels induced variability patterns with a power law behavior similar to those observed experimentally. Therefore, phenomena leading to long-term correlated variations in pacemaker cellular function may, in conjunction with extracardiac regulatory mechanisms, contribute to the nonlinear characteristics of HRV.
Adaptation of action potential duration (APD) to pacing cycle length (CL) has been previously characterized in isolated cardiomyocytes for sudden changes in constant CL and for pre-/postmature stimuli following constant pacing trains. However, random fluctuations characterize both physiological sinus rhythm (up to 10% of mean CL) and intrinsic beat-to-beat APD at constant pacing rate. We analysed the beat-to-beat sensitivity of each APD to the preceding CL during constant–sudden, random or linearly changing pacing trains in single patch clamped rat left ventricular myocytes, in the absence of the autonomic and electrotonic effects that modulate rate dependency in the intact heart. Beat-to-beat variability of APD at −60 mV (APD−60 mV), quantified as s.d. over 10-beat sequences, increased with corresponding mean APD. When measured as coefficient of variability (CV), APD−60 mV variability was inversely proportional to pacing frequency (from 1.2% at 5 Hz to 3.2% at 0.2 Hz). It was increased, at a basic CL (BCL) of 250 ms, by 55% by the L-type calcium current (ICaL) blocker nifedipine, and decreased by 23% by the transient-outward potassium current (Ito) blocker 4-aminopyridine. Variability of APD at BCL of 250 ms prevented the detection of random changes of CL smaller than ∼5%. Ten per cent random changes in CL were detected as a 40% increase in CV of APD and tended to correlate with it (r= 0.43). Block of ICaL depressed this correlation (r= 0.23), whereas block of Ito significantly increased it (r= 0.67); this was similar with linearly changing CL ramps (ranging ±10% and ±20% of 250 ms). We conclude that beat-to-beat APD variability, a major determinant of the propensity for development of arrhythmia in the heart, is present in isolated myocytes, where it is dependent on mean APD and pacing rate. Action potential duration shows a beat-to-beat positive correlation with preceding randomly/linearly changing CL, which can be pharmacologically modulated.
Reentry is a mechanism underlying numerous cardiac arrhythmias. During reentry, head-tail interactions of the action potential can cause cycle length (CL) oscillations and affect the stability of reentry. We developed a method based on a difference-delay equation to determine the slopes of the action potential duration and conduction velocity restitution functions, known to be major determinants of reentrant arrhythmogenesis, from the spatial period P and the decay length D of damped CL oscillations. Using this approach, we analyzed CL oscillations after the induction of reentry and the resetting of reentry with electrical stimuli in rings of cultured neonatal rat ventricular myocytes grown on microelectrode arrays and in corresponding simulations with the Luo-Rudy model. In the experiments, P was larger and D was smaller after resetting impulses compared to the induction of reentry, indicating that reentry became more stable. Both restitution slopes were smaller. Consistent with the experimental findings, resetting of simulated reentry caused oscillations with gradually increasing P, decreasing D, and decreasing restitution slopes. However, these parameters remained constant when ion concentrations were clamped, revealing that intracellular ion accumulation stabilizes reentry. Thus, the analysis of CL oscillations during reentry opens new perspectives to gain quantitative insight into action potential restitution.
Kras-sowska,etalTherestitutionportrait:anewmethodforinvestigating rate-dependent restitution
- S S Kalb
- H M Dobrovolny
- E G Tolkacheva
- S F Idriss
Kalb, S. S., H. M. Dobrovolny, E. G. Tolkacheva, S. F. Idriss, W. Kras-sowska,etal.2004.Therestitutionportrait:anewmethodforinvestigating rate-dependent restitution. J. Cardiovasc. Electrophysiol. 15:698–709