Search

Distinct electrophysiological phenotypes are exhibited by biological cells that have differentiated into particular cell types. The usual approach when simulating the cardiac electrophysiology of tissue that includes different cell types is to model the different cell types as occupying spatially distinct yet coupled regions. Instead, we model the electrophysiology of well-mixed cells by using homogenisation to derive an extension to the commonly used monodomain or bidomain equations. These new equations permit spatial variations in the distribution of the different subtypes of cells and will reduce the computational demands of solving the governing equations. We validate the homogenisation computationally, and then use the new model to explain some experimental observations from stem cell-derived cardiomyocyte monolayers.

Objective: A flexible, efficient, and verifiable pacemaker cell model is essential to the design of real-time virtual hearts that can be used for closed-loop validation of cardiac devices. A new parametric model of pacemaker action potential is developed to address this need., Methods: The action potential phases are modeled using hybrid automaton with one piecewise-linear continuous variable. The model can capture rate-dependent dynamics, such as action potential duration restitution, conduction velocity restitution, and overdrive suppression by incorporating nonlinear update functions. Simulated dynamics of the model compared well with previous models and clinical data., Conclusion: The results show that the parametric model can reproduce the electrophysiological dynamics of a variety of pacemaker cells, such as sinoatrial node, atrioventricular node, and the His-Purkinje system, under varying cardiac conditions., Significance: This is an important contribution toward closed-loop validation of cardiac devices using real-time heart models.

Models of electrical activity in cardiac cells have become important research tools as they can provide a quantitative description of detailed and integrative physiology. However, cardiac cell models have many parameters, and how uncertainties in these parameters affect the model output is difficult to assess without undertaking large numbers of model runs. In this study we show that a surrogate statistical model of a cardiac cell model (the Luo-Rudy 1991 model) can be built using Gaussian process (GP) emulators. Using this approach we examined how eight outputs describing the action potential shape and action potential duration restitution depend on six inputs, which we selected to be the maximum conductances in the Luo-Rudy 1991 model. We found that the GP emulators could be fitted to a small number of model runs, and behaved as would be expected based on the underlying physiology that the model represents. We have shown that an emulator approach is a powerful tool for uncertainty and sensitivity analysis in cardiac cell models.

Impulse propagation in biological tissues is known to be modulated by structural heterogeneity. In cardiac muscle, improved understanding on how this heterogeneity influences electrical spread is key to advancing our interpretation of dispersion of repolarization. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a means of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, analysed against in vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies relevant characteristics of cardiac electrical propagation at tissue level. These include conduction effects on action potential (AP) morphology, the shortening of AP duration along the activation pathway and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media.

Numerical integration of mathematical models of heart cell electrophysiology provides an important computational tool for studying cardiac arrhythmias, but the abundance of available models complicates selecting an appropriate model. We study the behavior of two recently published models of human ventricular action potentials, the Grandi-Pasqualini-Bers (GPB) and the O'Hara-Virág-Varró-Rudy (OVVR) models, and compare the results with four previously published models and with available experimental and clinical data. We find the shapes and durations of action potentials and calcium transients differ between the GPB and OVVR models, as do the magnitudes and rate-dependent properties of transmembrane currents and the calcium transient. Differences also occur in the steady-state and S1-S2 action potential duration and conduction velocity restitution curves, including a maximum conduction velocity for the OVVR model roughly half that of the GPB model and well below clinical values. Between single cells and tissue, both models exhibit differences in properties, including maximum upstroke velocity, action potential amplitude, and minimum diastolic interval. Compared to experimental data, action potential durations for the GPB and OVVR models agree fairly well (although OVVR epicardial action potentials are shorter), but maximum slopes of steady-state restitution curves are smaller. Although studies show alternans in normal hearts, it occurs only in the OVVR model, and only for a narrow range of cycle lengths. We find initiated spiral waves do not progress to sustained breakup for either model. The dominant spiral wave period of the GPB model falls within clinically relevant values for ventricular tachycardia (VT), but for the OVVR model, the dominant period is longer than periods associated with VT. Our results should facilitate choosing a model to match properties of interest in human cardiac tissue and to replicate arrhythmia behavior more closely. Furthermore, by indicating areas where existing models disagree, our findings suggest avenues for further experimental work.

Aims: Despite recognition that understanding gross anatomy of the right atrium is important in the era of invasive electrophysiology, areas such as the right atrial appendage (RAA) wall or the vestibule are not fully appreciated. The aim of this study was to conduct an anatomical assessment focusing on these structures to gain further insights into electrophysiological procedures., Methods: Forty-four normal human hearts were examined macro- and microscopically., Results: Inside the RAA, two prominent muscle bundles; the crista terminalis (CT) and the sagittal bundle (SB) were identified. The medial wall at which the CT originated from and its surrounding area was the thickest part adjacent anteriorly to the thin aortic mound, suggesting non-uniform wall thickness of this area. Histological sections revealed that myocardial strands of the SB connected the CT and RAA tip, implying preferential anterior conduction of the sinus impulse. The vestibule had a thin myocardium with extensive fat covering along the epicardial side of the tricuspid annulus. The proximal portion of the right coronary artery (RCA) was relatively distant from the annulus, followed by gradual shortening of the distance from the endocardium to the RCA, which led to a very close relationship (<3.0 mm) at the inferior annulus., Conclusion: Non-uniform wall thickness and muscle fibre orientation in the RAA should be taken into consideration during lead/catheter positioning. The RCA proximity in the inferior portion of the vestibule and the deeper fatty plane of the anterior atrioventricular groove are important anatomical features relevant to accessory pathway ablation.

Various forms of heart disease are associated with remodeling of the heart muscle, which results in a perturbation of cell-to-cell electrical coupling. These perturbations may alter the trajectory of spiral wave drift in the heart muscle. We investigate the effect of spatially extended inhomogeneity of transverse cell coupling on the spiral wave trajectory using a simple active media model. The spiral wave was either attracted or repelled from the center of inhomogeneity as a function of cell excitability and gradient of the cell coupling. High levels of excitability resulted in an attraction of the wave to the center of inhomogeneity, whereas low levels resulted in an escape and termination of the spiral wave. The spiral wave drift velocity was related to the gradient of the coupling and the initial position of the wave. In a diseased heart, a region of altered transverse coupling corresponds with local gap junction remodeling that may be responsible for stabilization-destabilization of spiral waves and hence reflect potentially important targets in the treatment of heart arrhythmias.

Myocardial development is regulated by an elegantly choreographed ensemble of signaling events mediated by a multitude of intermediates that take a variety of forms. Cellular differentiation and maturation are a subset of vertically integrated processes that extend over several spatial and temporal scales to create a well-defined collective of cells that are able to function cooperatively and reliably at the organ level. Early efforts to understand the molecular mechanisms of cardiomyocyte fate determination focused primarily on genetic and chemical mediators of this process. However, increasing evidence suggests that mechanical interactions between the extracellular matrix (ECM) and cell surface receptors as well as physical interactions between neighboring cells play important roles in regulating the signaling pathways controlling the developmental processes of the heart. Interdisciplinary efforts have made it apparent that the influence of the ECM on cellular behavior occurs through a multitude of physical mechanisms, such as ECM boundary conditions, elasticity, and the propagation of mechanical signals to intracellular compartments, such as the nucleus. In addition to experimental studies, a number of mathematical models have been developed that attempt to capture the interplay between cells and their local microenvironment and the influence these interactions have on cellular self-assembly and functional behavior. Nevertheless, many questions remain unanswered concerning the mechanism through which physical interactions between cardiomyocytes and their environment are translated into biochemical cellular responses and how these signaling modalities can be utilized in vitro to fabricate myocardial tissue constructs from stem cell-derived cardiomyocytes that more faithfully represent their in vivo counterpart. These studies represent a broad effort to characterize biological form as a conduit for information transfer that spans the nanometer length scale of proteins to the meter length scale of the patient and may yield new insights into the contribution of mechanotransduction into heart development and disease.

In this review, we discuss recent studies on the mechanosensitive morphology and function of cardiomyocytes derived from embryos and neonates. For early cardiomyocytes cultured on substrates of various stiffnesses, contractile function as measured by force production, work output and calcium handling is optimized when the culture substrate stiffness mimics that of the tissue from which the cells were obtained. This optimal contractile function corresponds to changes in sarcomeric protein conformation and organization that promote contractile ability. In light of current models for myofibillogenesis, a recent mathematical model of striation and alignment on elastic substrates helps to illuminate how substrate stiffness modulates early myofibril formation and organization. During embryonic heart formation and maturation, cardiac tissue mechanics change dynamically. Experiments and models highlighted here have important implications for understanding cardiomyocyte differentiation and function in development and perhaps in regeneration processes.

The presence of connective tissue as well as interstitial clefts forms a natural barrier to the electrical propagation in the heart. At a microscopic scale, such uncoupling structures change the pattern of the electrical conduction from uniform towards complex and may play a role in the genesis of cardiac arrhythmias. The anatomical diversity of conduction structures and their topology at a microscopic size scale is overwhelming for experimental techniques. Mathematical models have been often employed to study the behavior of the electrical propagation at a sub-cellular level. However, very fine and computationally expensive meshes are required to capture all microscopic details found in the cardiac tissue. In this work, we present a numerical technique based on the finite element method which allows to reproduce the effects of microscopic conduction barriers caused by the presence of uncoupling structures without actually resolving these structures in a high resolution mesh, thereby reducing the computational costs significantly.

This paper explores the conditions under which nonmyocytes, when electrically coupled to myocytes, act as a passive load during the depolarization phase. Using theoretical arguments and numerical simulations in a tissue incorporating fibroblasts, the passive load approximation is shown to be accurate at low coupling conductances (< 2 nS). In this case, the effect on conduction velocity can be expressed as a function of the elevation in resting potential and the coupling only.

The development of voltage-sensitive dyes has revolutionized cardiac electrophysiology and made optical imaging of cardiac electrical activity possible. Photon diffusion models coupled to electrical excitation models have been successful in qualitatively predicting the shape of the optical action potential and its dependence on subsurface electrical wave orientation. However, the accuracy of the diffusion equation in the visible range, especially for thin tissue preparations, remains unclear. Here, we compare diffusion and Monte Carlo (MC) based models and we investigate the role of tissue thickness. All computational results are compared to experimental data obtained from intact guinea pig hearts. We show that the subsurface volume contributing to the epi-fluorescence signal extends deeper in the tissue when using MC models, resulting in longer optical upstroke durations which are in better agreement with experiments. The optical upstroke morphology, however, strongly correlates to the subsurface propagation direction independent of the model and is consistent with our experimental observations.

Introduction: Most cardiac arrhythmias are associated with pathology-triggered ion channel remodeling. However, multicellular effects, for example, exaggerated anisotropy and altered cell-to-cell coupling, can also indirectly affect action potential morphology and electrical stability via changed electrotonus. These changes are particularly relevant in structural heart disease, including hypertrophy and infarction. Recent computational studies showed that electrotonus factors into stability by altering dynamic properties (restitution). We experimentally address the question of how cell alignment and connectivity alter tissue function and whether these effects depend on the direction of wave propagation., Methods and Results: We show that cardiac cell arrangement can alter electrical stability in an in vitro cardiac tissue model by mechanisms both dependent and independent of the direction of wave propagation, and local structural remodeling can be felt beyond a space constant. Notably, restitution of action potential duration (APD) and conduction velocity was significantly steepened in the direction of cell alignment. Furthermore, prolongation of APD and calcium transient duration was found in highly anisotropic cell networks, both for longitudinal and transverse propagation. This is in contrast to expected correlation between wave propagation direction and APD based on electrotonic effects only, but is consistent with our findings of increased cell size and secretion of atrial natriuretic factor, a hypertrophy marker, in the aligned structures., Conclusion: Our results show that anisotropic structure is a potent modulator of electrical stability via electrotonus and molecular signaling. Tissue alignment must be taken into account in experimental and computational models of arrhythmia generation and in designing effective treatment therapies.

For prevention of lethal arrhythmias, patients at risk receive implantable cardioverter-defibrillators, which use high-frequency antitachycardia pacing (ATP) to convert tachycardias to a normal rhythm. One of the suggested ATP mechanisms involves paced-induced drift of rotating waves followed by their collision with the boundary of excitable tissue. This study provides direct experimental evidence of this mechanism. In monolayers of neonatal rat cardiomyocytes in which rotating waves of activity were initiated by premature stimuli, we used the Ca(2+)-sensitive indicator fluo 4 to observe propagating wave patterns. The interaction of the spiral tip with a paced wave was then monitored at a high spatial resolution. In the course of the experiments, we observed spiral wave pinning to local heterogeneities within the myocyte layer. High-frequency pacing led, in a majority of cases, to successful termination of spiral activity. Our data show that 1) stable spiral waves in cardiac monolayers tend to be pinned to local heterogeneities or areas of altered conduction, 2) overdrive pacing can shift a rotating wave from its original site, and 3) the wave break, formed as a result of interaction between the spiral tip and a paced wave front, moves by a paced-induced drift mechanism to an area where it may become unstable or collide with a boundary. The data were complemented by numerical simulations, which was used to further analyze experimentally observed behavior.

The rotating fiber orientation within the cardiac wall substantially affects the electrical propagation and can cause intra-myocardial cusp waves. Numerical simulations have shown that the cusps form in layers where propagation is perpendicular to the fiber orientation and lead to complex wave front morphologies. They can travel across layers and break through at the epi- or endocardial surfaces where they cause apparent accelerations of propagation. The validation of these results remains a major experimental challenge. In the present study, we investigate both computationally and experimentally how intramural cusp waves can be detected using optical imaging. Our simulations show that cusps alter the optical upstroke morphology and can be detected well before they reach the surface (up to 1 mm deep). Experiments in Langendorff-perfused guinea pig hearts are consistent with our numerical findings.

Cell coupling is considered to be important for cardiac action potential propagation and arrhythmogenesis. We carried out computer simulations to investigate the effects of stimulation strength and cell-to-cell coupling on action potential duration (APD) restitution, APD alternans, and stability of reentry in models of isolated cell, one-dimensional cable, and two-dimensional tissue. Phase I formulation of the Luo and Rudy action potential model was used. We found that stronger stimulation resulted in a shallower APD restitution curve and onset of APD alternans at a faster pacing rate. Reducing diffusive coupling between cells prolonged APD. Weaker diffusive currents along the direction of propagation steepened APD restitution and caused APD alternans to occur at a slower pacing rate in tissue. Diffusive current due to curvature changed APD but had little effect on APD restitution slope and onset of instability. Heterogeneous cell coupling caused APD inhomogeneities in space. Reduction in coupling strength either uniformly or randomly had little effect on the rotation period and stability of a reentry, but random cell decoupling slowed the rotation period and, thus, stabilized the reentry, preventing it from breaking up into multiple waves. Therefore, in addition to its effects on action potential conduction velocity, diffusive cell coupling also affects APD in a rate-dependent manner, causes electrophysiological heterogeneities, and thus modulates the dynamics of cardiac excitation. These effects are brought about by the modulation of ionic current activation and inactivation.

Structural and functional cardiac anisotropy varies with the development, location, and pathophysiology in the heart. The goal of this study was to design a cell culture model system in which the degree, change in fiber direction, and discontinuity of anisotropy can be controlled over centimeter-size length scales. Neonatal rat ventricular myocytes were cultured on fibronectin on 20-mm diameter circular cover slips. Structure-function relationships were assessed using immunostaining and optical mapping. Cell culture on microabraded cover slips yielded cell elongation and coalignment in the direction of abrasion, and uniform, macroscopically continuous, elliptical propagation with point stimulation. Coarser microabrasion (wider and deeper abrasion grooves) increased longitudinal (23.5 to 37.2 cm/s; r=0.66) and decreased transverse conduction velocity (18.1 to 9.2 cm/s; r=-0.84), which resulted in increased longitudinal-to-transverse velocity anisotropy ratios (1.3 to 3.7, n=61). A thin transition zone between adjacent uniformly anisotropic areas with 45 degrees or 90 degrees difference in fiber orientation acted as a secondary source during 2x threshold field stimulus. Cell culture on cover slips micropatterned with 12- or 25- micro m wide fibronectin lines and previously coated with decreasing concentrations of background fibronectin yielded transition from continuous to discontinuous anisotropic architecture with longitudinally oriented intercellular clefts, decreased transverse velocity (16.9 to 2.6 cm/s; r=-0.95), increased velocity anisotropy ratios (1.6 to 5.6, n=70), and decreased longitudinal velocity (36.4 to 14.6 cm/s; r=-0.85) for anisotropy ratios >3.5. Cultures of cardiac myocytes with controlled degree, uniformity and continuity of structural, and functional anisotropy may enable systematic 2-dimensional in vitro studies of macroscopic structure-related mechanisms of reentrant arrhythmias. The full text of this article is available at http://www.circresaha.org.

This article investigates the quantitative predictive capabilities of region-specific models by comparing experimental electrograms obtained from in vivo mapping of the ventricular free wall with those obtained through simulation of a region specific three-dimensional bidomain model that incorporates measured fiber orientations. Epicardial electrograms were recorded from canine left ventricles during and after unipolar pacing using a 528-channel electrode plaque. Fiber directions throughout the tissue were estimated from diffusion-weighted MRI and from pace mapping. Electrograms were computed in the bidomain model with experimentally derived properties during paced activations at the same spatiotemporal resolution as those recorded experimentally. Epicardial potentials from model and experiment were directly compared, and sensitivities of these comparisons to reference electrode location and to the choice of material properties were analyzed. The comparisons performed here demonstrate, that (1) the stimulus artifact can be used to estimate the in vivo myocardial fiber architecture, (2) the correlation between simulated and experimental electrograms decreases with increasing pacing depth, and (3) the quantitative comparisons between bidomain model and experimental data are sensitive to both the description of the fiber architecture, and the location of the unipolar reference electrode, but relatively insensitive to moderate changes in the bidomain conductivities.

It is known that branching strands of cardiac tissue can form a substrate for very slow conduction. The branches slow conduction by acting as current loads drawing depolarizing current from the main strand ("pull" effect). It has been suggested that, upon depolarization of the branches, they become current sources reinjecting current back into the strand, thus enhancing propagation safety ("push" effect). It was the aim of this study to verify this hypothesis and to assess the contribution of the push effect to propagation velocity and safety. Conduction was investigated in strands of Luo-Rudy dynamic model cells that branch from either a single branch point or from multiple successive branch points. In single-branching strands, blocking the push effect by not allowing current to flow retrogradely from the branches into the strand did not significantly increase the branching-induced local propagation delay. However, in multiple branching strands, blocking the push effect resulted in a significant slowing of overall conduction velocity or even in conduction failure. Furthermore, for certain slow velocities, the safety factor for propagation was higher when slow conduction was caused by branching tissue geometry than by reduced excitability without branching. Therefore, these results confirm the proposed "pull and push" mechanism of slow, but nevertheless robust, conduction in branching structures. Slow conduction based on this mechanism could occur in the atrioventricular node, where multiple branching is structurally present. It could also support reentrant excitation in diseased myocardium where the substrate is structurally complex.

A pacemaker cell pair model and the dynamic interaction between the two pacemaker cells is described in this paper. It is an extension of our single pacemaker cell model, in which we studied its response to repetitive external depolarization stimulations. This model is a simple model based on the two most important functional properties of the cardiac pacemaker cells: its intrinsic pacemaker cycle length, which is an 'internal' parameter of the cell, and the phase response curve (PRC), which is an 'overall collective' function. The PRC contains all the 'information' about the possible interactions of the pacemaker cell with the outside world (interaction with surrounding cells, external stimulus, etc.). First, we examined the properties and solutions of 1:1 synchronization between two pacemaker cells. We found that in order to achieve synchronization between two pacemaker cells, there should be limitations on the PRC parameters, which depend on the cells intrinsic cycle lengths. Next, we investigated the 2:1 entrainment state between two interacting pacemaker cells. We found that there is not necessarily a unique solution for this state as there was for the 1:1 state. Finally, we ran our computer model to investigate the properties of more complex patterns of entrainment between two pacemaker cells. As a result of our analytical study, we unveil two new important parameters, which are fully defined as a function of the PRC parameters: (1) the 'accelerator factor' which describes the tendency of a pair of interacting pacemaker cells to synchronize at a common cycle length, which is closer to the faster cycle of the pair; (2) the 'degree of coupling', which describes the range of the 1:1 synchronization and the 'strength' of the interaction between a pair of interacting pacemaker cells. Those two interaction parameters arise as helpful 'tools' for the understanding of synchronization and mutual entrainment mechanisms between pacemaker cells. Therefore, this study establishes the PRC as an important determinant and a useful approach for the understanding of the dynamic interaction of pacemaker cells among themselves and with the outside world.

A single pacemaker cell model and its response to repetitive external depolarization stimulations is described in this paper. This model is a simple model based on the two most important functional properties of the cardiac pacemaker cells. The first property is the intrinsic pacemaker cycle length, which is an 'internal' parameter of the cell, describing the most important feature of a pacemaker cell. The second functional property is the phase response curve (PRC), which is an 'overall collective' function: it contains all the 'information' about the possible interactions of the pacemaker cell with the outside world (external stimulus, interaction with surrounding cells, etc.). This study demonstrates that by representing the pacemaker cell only by two fundamental features, and by applying a simple physical-mathematical model, a global picture of the system can be achieved, allowing us to explore qualitatively various physiological phenomena related to the pacemaker function. For example, we demonstrated that the PRC is a crucial parameter in the prediction of the entrainment phenomena of a single pacemaker cell in response to a periodic train of depolarization pulses. Specifically, the PRC permits a quantitative determination of the 1:1 synchronization range for a single pacemaker cell and an external depolarization pulse. Moreover, we show that the PRC can be used to represent the type of external stimulus applied to the pacemaker (e.g. depolarization pulse) and its intensity. Therefore, the PRC emerges as an important determinant and a useful 'tool' for the understanding of the dynamic interaction of pacemaker cells with the outside world. As a result of our simulations, we unveil a new important parameter: the 'degree of influence', which determines the range of 1:1 synchronization between an external depolarization pulse and a pacemaker cell. This interaction parameter is a direct function of the PRC parameters. It appears to be a helpful 'tool' for the understanding of synchronization and mutual entrainment mechanisms between the pacemaker cell and an external stimulus, and therefore it supports the basic importance of the PRC in the description and determination of these mechanisms.

This study introduces a simple mathematical model for a pacemaker cell affected by an external parasympathetic and/or sympathetic input. The model presented is based on the two most important functional properties of the cardiac pacemaker cells. The first property is the intrinsic pacemaker cycle length, an "internal" parameter of the cell. The second basic property is the phase response curve (PRC), a function which reflects the various interactions of the pacemaker cell with the outside world (i.e. interaction with surrounding cells, external stimulus). The vagal stimulus is simulated as affecting the pacemaker cycle length via a PRC, while the sympathetic input is expressed in the model as a continuous reduction in the pacemaker cycle length. When combined vagal and sympathetic activation is allowed, our model shows that autonomic systems are also capable of interacting. First, we studied the entrainment phenomena resulting from a repetitively applied vagal stimulus. Various complex patterns of dynamic interaction between the pacemaker cell and the vagal input were simulated. The PRC parameters appear to be an important factor in the prediction of the entrainment phenomena. Specifically, they permit a quantitative description of the limits of a 1:1 synchronization zone. Next, we apply this model to qualitatively investigate the phenomenon of "accentuated antagonism" between parasympathetic and sympathetic autonomic branches. We examined the various options for this interaction in regulating the pacemaker periodicity. Although this model is a simplified reflection of the biological system, we conclude that it can mimic many aspects of the dynamic autonomic control and of the possible interactions between vagal and sympathetic stimulation of a pacemaker cell.

Patients with severe heart failure are at high risk of sudden cardiac death. In the majority of these patients, sudden cardiac death is thought to be due to ventricular tachyarrhythmias. Alterations of the electric properties of single myocytes in heart failure may favor the occurrence of ventricular arrhythmias in these patients by inducing early or delayed afterdepolarizations. Mathematical models of the cellular action potential and its underlying ionic currents could help to elucidate possible arrhythmogenic mechanisms on a cellular level. In the present study, selected ionic currents based on human data are incorporated into a model of the ventricular action potential for the purpose of studying the cellular electrophysiological consequences of heart failure. Ionic currents that are not yet sufficiently characterized in human ventricular myocytes are adopted from the action potential model developed by Luo and Rudy (LR model). The main results obtained from this model are as follows: The action potential in ventricular myocytes from failing hearts is longer than in nonfailing control hearts. The major underlying mechanisms for this prolongation are the enhanced activity of the Na+-Ca2+ exchanger, the slowed diastolic decay of the [Ca2+]i transient, and the reduction of the inwardly rectifying K+ current and the Na+-K+ pump current in myocytes of failing hearts. Furthermore, the fast and slow components of the delayed rectifier K+ current (I(Kr) and I(Ks), respectively) are of utmost importance in determining repolarization of the human ventricular action potential. In contrast, the influence of the transient outward K+ current on APD is only small in both cell groups. Inhibition of I(Kr) promotes the development of early afterdepolarizations in failing, but not nonfailing, myocytes. Furthermore, spontaneous Ca2+ release from the sarcoplasmic reticulum triggers a premature action potential only in failing myocytes. This model of the ventricular action potential and its alterations in heart failure is intended to serve as a tool for investigating the effects of therapeutic interventions on the electric excitability of the human ventricular myocardium.

The initiation and propagation of the cardiac impulse depends on intrinsic properties of cells, geometrical arrangements, and intercellular coupling resistances. To address the issue of the interplay between these factors in a simple way, we have used a system, based on the van Capelle and Dürrer model, including a pacemaker and a non-pacemaker cell linked by an ohmic coupling resistance. The influence of asymmetrical cell sizes and electronic load was investigated by using numerical simulations and continuation-bifurcation techniques. The loading of a small pacemaker cell by a large non-pacemaker one (pacemaker: non-pacemaker size ratio = 0.3) was expressed as a pronounced early repolarization in the pacemaker cell and a quite long latency for the impulse propagation. Using coupling resistance as the continuation parameter, three behavioral zones were detected from low to high coupling resistance values: a zone of total quiescence (0:0), a zone of effective entertainment (1:1), and a zone of total block (1:0 pattern). At the boundary between 1:1 and 1:0 patterns, for relatively high coupling resistance values, a cascade of period doubling bifurcations emerged, corresponding to discrete changes of propagation patterns leading into irregular dynamics. Another route to irregular dynamics was also observed in the parameter space. The high sensitivity of the detected irregular dynamics to initial conditions and the positive value of the maximum Lyapunov exponent allowed us to identify these dynamics as being chaotic. Since neither intermediate block patterns nor irregular dynamics were observed with larger size ratios, we suggest that the interplay between resting membrane conductance of the non-pacemaker cell and intercellular coupling may bring about these rhythmic disturbances.

A two-dimensional anisotropic model of cardiac ventricular muscle was used to study the effects of discontinuities (barriers), such as dead cells or high-resistance areas, on longitudinal plane-wave propagation. Problems in propagation appear when long barriers become thicker and their spacing closer. Short barriers with large widths and small spacing also cause propagation disturbances and significant delays in their vicinity. If the plane wave front propagates through the barriers, the velocity returns to near normal within one-length constant away from the end of the barrier region. For a funnel-like structure, an opening of 13 cells should exist for longitudinal plane wave propagation. For smaller openings, the ratio of openings required for propagation to occur when traveling from a narrow to a wider area of tissue is proportional to the anisotropy ratio, which can cause unidirectional block. Tortuosity, created by spatial distribution of dead cell barriers, can facilitate propagation by changing the effective impedance the wave front sees, and can create multiple local delays, which may result in discrepancies when measuring propagation velocity.

A theoretical model of action potential propagation in a thick strand of cardiac muscle is presented. The calculation takes into account the anisotropic and syncytial properties of the tissue, the presence of the interstitial space, the effect of the surrounding tissue bath, and the variation of the potential both along the strand length and across the strand cross section. The bidomain model is used to represent the electrical properties of the tissue, and the Ebihara-Johnson model is used to represent the properties of the active sodium channels. The calculated wave front is curved, with the action potential at the surface of the strand leading that at the center. The rate of rise of the action potential and the time constant of the action potential foot vary with depth into the tissue. The velocity of the wave front is nearly independent of strand radius for radii greater than 0.5 mm. The conduction velocity decreases as the volume fraction of the interstitial space decreases. In the limit of tightly packed cells, an action potential propagates quickly over the surface of the strand; the bulk of the tissue is then excited by a slow inward wave front initiated on the surface. This model does not predict an increase in conduction velocity when cells are tightly packed, a hypothesis that has been proposed previously to explain the fast conduction velocity in Purkinje fibers of some species.

Previous models of cardiac cellular electrophysiology have been based largely on voltage-clamp measurements obtained from multicellular preparations and often combined data from different regions of the heart and a variety of species. We have developed a model of cardiac pacemaking based on a comprehensive set of voltage-clamp measurements obtained from single cells isolated from one specific tissue type, the bullfrog sinus venosus (SV). Consequently, sarcolemmal current densities and kinetics are not influenced by secondary phenomena associated with multicellular preparations, allowing us to realistically simulate processes thought to be important in pacemaking, including the Na(+)-K+ pump and Na(+)-Ca2+ exchanger. The membrane is surrounded extracellularly by a diffusion-limited space and intracellularly by a limited myoplasmic volume containing Ca2(+)-binding proteins (calmodulin, troponin). The model makes several predictions regarding mechanisms involved in pacing. 1) Primary pacemaking cannot be attributed to any single current but arises from both the lack of a background K+ current and a complex interaction between Ca2+, delayed-rectifier K+, and background leak currents. 2) Ca2+ current displays complex behavior and is important during repolarization. 3) Because of Ca2+ buffering by myoplasmic proteins, the Na(+)-Ca2+ exchanger current is small and has little influence on action potential repolarization but may modulate the maximum diastolic potential. 4) The Na(+)-K+ pump current does not play an active role in repolarization but is of sufficient size to modulate the rate of diastolic depolarization. 5) K+ accumulation and Ca2+ depletion may occur in the extracellular spaces but play no role in either the diastolic depolarization or repolarization of a single action potential. This model illustrates the importance of basing simulations on quantitative measurements of ionic currents in myocytes and of including both electrogenic transporter mechanisms and Ca2+ buffering by myoplasmic Ca2(+)-binding proteins.

We have investigated the oscillatory behavior of cardiac cellular elements simulated by two electrophysiological models: the van Capelle and Durrer (VCD) model and the sinoatrial node cell model of Yanagihara, Noma, and Irisawa (YNI). The VCD model behavior was examined systematically by using continuation-bifurcation analysis. Bifurcation diagrams were constructed as a function of Qit1, an intrinsic parameter of the model, which sets both maximum diastolic potential and depolarization threshold of the cell. The existence of stable high amplitude oscillations was evidenced between two Hopf bifurcation points (HB). Near each HB, a zone of bistability was detected. Close to the HB that corresponded to high values of Qit1, a high amplitude periodic stable state coexisted with a stable steady state. Close to the other HB, in a narrow range of lower Qit1 values, a relatively high amplitude periodic stable state coexisted with a low amplitude periodic stable state. There was no stable steady state in the latter bistability zone. Through the use of phase-plane representations and the determination of separatrices between the different attractor basins, we could deduce the conditions of timing, polarity, and strength needed for a pulse perturbation to send the system from one state to another and vice versa. The YNI model was analyzed by numerical simulation, and the oscillatory behavior of the sinoatrial node cell was explored while applying a depolarizing bias current of various strengths. Results were similar to those obtained from the VCD model in that there were two bistability regions for two different ranges of applied bias current. Depending on current intensity, annihilation of pacemaker activity could be achieved in both zones. However, the coexistence of two oscillatory stable states was never observed in the YNI model. From the behavioral similarities of these different models, we can conclude that bistabilities and annihilation phenomena can be found in transitional zones between quiescence and rhythmic activity.

The effects of the discrete cellular structure on propagation of electrical excitation in cardiac muscle were studied in a one-dimensional fiber model containing a periodic intercalated disk structure. Globally, the macroscopic velocity of propagation follows the behavior associated with propagation in a continuous tissue (except for high values of disk resistance). In addition, the computed spatial extracellular potential along the fiber is a smooth biphasic waveform and does not reflect the underlying discrete cellular structure of the tissue. Other results of the simulations demonstrate the discontinuous nature of propagation and the importance of the structure in arrhythmogenesis. Vmax displays a biphasic behavior as a function of increasing intercalated disk resistance. An initial "paradoxical" increase in Vmax (with a simultaneous decrease in conduction velocity) is followed by a decrease that leads to decremental propagation and conduction block. The time constant of the foot of the action potential (tau foot) increases monotonically with increasing intercalated disk resistance. An increase in the leakage current to extracellular space brings about a significant decrease in the action potential duration and a loss of the plateau. This major effect is accompanied by a relatively smaller decrease in conduction velocity. Collision of two activation wavefronts results in a significant (100%) increase in Vmax and a very small (0.6%) decrease in tau foot.