The effects of discrete gap junction coupling on propagation in myocardium

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

Numerical methods for the detection of phase defect structures in excitable media

Electrical waves that rotate in the heart organize dangerous cardiac arrhythmias. Finding the region around which such rotation occurs is one of the most important practical questions for arrhythmia management. For many years, the main method for finding such regions was so-called phase mapping, in which a continuous phase was assigned to points in the heart based on their excitation status and defining the rotation region as a point of phase singularity. Recent analysis, however, showed that in many rotation regimes there exist phase discontinuities and the region of rotation must be defined not as a point of phase singularity, but as a phase defect line. In this paper, we use this novel methodology and perform a comparative study of three different phase definitions applied to in silico data and to experimental data obtained from optical voltage mapping experiments on monolayers of human atrial myocytes. We introduce new phase defect detection algorithms and compare them with those that appeared in literature already. We find that the phase definition is more important than the algorithm to identify sudden spatial phase variations. Sharp phase defect lines can be obtained from a phase derived from local activation times observed during one cycle of arrhythmia. Alternatively, similar quality can be obtained from a reparameterization of the classical phase obtained from observation of a single timeframe of transmembrane potential. We found that the phase defect line length was (35.9 ± 6.2)mm in the Fenton-Karma model and (4.01 ± 0.55)mm in cardiac human atrial myocyte monolayers. As local activation times are obtained during standard clinical cardiac mapping, the methods are also suitable to be applied to clinical datasets. All studied methods are publicly available and can be downloaded from an institutional web-server.

Calcium Signaling in the Photodamaged Skin: In Vivo Experiments and Mathematical Modeling

The epidermis forms an essential barrier against a variety of insults. The overall goal of this study was to shed light not only on the effects of accidental epidermal injury, but also on the mechanisms that support laser skin resurfacing with intra-epidermal focal laser-induced photodamage, a widespread medical practice used to treat a range of skin conditions. To this end, we selectively photodamaged a single keratinocyte with intense, focused and pulsed laser radiation, triggering Ca2+ waves in the epidermis of live anesthetized mice with ubiquitous expression of a genetically encoded Ca2+ indicator. Waves expanded radially and rapidly, reaching up to eight orders of bystander cells that remained activated for tens of minutes, without displaying oscillations of the cytosolic free Ca2+ concentration ([Formula: see text]). By combining in vivo pharmacological dissection with mathematical modeling, we demonstrate that Ca2+ wave propagation depended primarily on the release of ATP, a prime damage-associated molecular patterns (DAMPs), from the hit cell. Increments of the [Formula: see text] in bystander cells were chiefly due to Ca2+ release from the endoplasmic reticulum (ER), downstream of ATP binding to P2Y purinoceptors. ATP-dependent ATP release though connexin hemichannels (HCs) affected wave propagation at larger distances, where the extracellular ATP concentration was reduced by the combined effect of passive diffusion and hydrolysis due to the action of ectonucleotidases, whereas pannexin channels had no role. Bifurcation analysis suggests basal keratinocytes have too few P2Y receptors (P2YRs) and/or phospholipase C (PLC) to transduce elevated extracellular ATP levels into inositol trisphosphate (IP3) production rates sufficiently large to sustain [Formula: see text] oscillations.

Pinning and depinning: From periodic to chaotic and random media

We study the propagation of dissipative structures in inhomogeneous media with a focus on pinning and depinning transitions. We model spatial complexity in the medium as generated by dynamical systems. We are thus able to capture transitions from periodic to quasiperiodic, to homoclinic and heteroclinic, and to chaotic media. Depinning bifurcations exhibit universal laws depending on extreme value statistics that are encoded in the dimension of ergodic measures, only. A key condition limiting this approach bounds spatial Lyapunov exponents in terms of interface localization, and we explore the breakdown of smoothness and universality when this condition is violated and fluctuations in the medium occur on length scales shorter than a typical interface width.

Reciprocal Modulation of IK1-INa Extends Excitability in Cardiac Ventricular Cells

The inwardly rectifying potassium current (I_K1) and the fast inward sodium current (I_Na) are reciprocally modulated in mammalian ventricular myocytes. An increase in the expression of channels responsible for one of these two currents results in a corresponding increase in expression of the other. These currents are critical in the propagation of action potentials (AP) during the normal functioning of the heart. This study identifies a physiological role for I_K1–I_Na reciprocal modulation in ventricular fiber activation thresholds and conduction. Simulations of action potentials in single cells and propagating APs in cardiac fibers were carried out using an existing model of electrical activity in cardiac ventricular myocytes. The conductances, G_K1, of the inwardly rectifying potassium current, and G_Na, of the fast inward sodium current were modified independently and in tandem to simulate reciprocal modulation. In single cells, independent modulation of G_K1 alone resulted in changes in activation thresholds that were qualitatively similar to those for reciprocal G_K1–G_Na modulation and unlike those due to independent modulation of G_Na alone, indicating that G_K1 determines the cellular activation threshold. On the other hand, the variations in conduction velocity in cardiac cell fibers were similar for independent G_Na modulation and for tandem changes in G_K1–G_Na, suggesting that G_Na is primarily responsible for setting tissue AP conduction velocity. Conduction velocity dependence on G_K1–G_Na is significantly affected by the intercellular gap junction conductance. While the effects on the passive fiber space constant due to changes in both G_K1 and the intercellular gap junction conductance, G_gj, were in line with linear cable theory predictions, both conductances had surprisingly large effects on fiber activation thresholds. Independent modulation of G_K1 rendered cardiac fibers inexcitable at higher levels of G_K1 whereas tandem G_K1–G_Na changes allowed fibers to remain excitable at high G_K1 values. Reciprocal modulation of the inwardly rectifying potassium current and the fast inward sodium current may have a functional role in allowing cardiac tissue to remain excitable when I_K1 is upregulated.

Rate-dependent activation failure in isolated cardiac cells and tissue due to Na+ channel block

While it is well established that class-I antiarrhythmics block cardiac sodium channels, the mechanism of action of therapeutic levels of these drugs is not well understood. Using a combination of mathematical modeling and in vitro experiments, we studied the failure of activation of action potentials in single ventricular cells and in tissue caused by Na(+) channel block. Our computations of block and unblock of sodium channels by a theoretical class-Ib antiarrhythmic agent predict differences in the concentrations required to cause activation failure in single cells as opposed to multicellular preparations. We tested and confirmed these in silico predictions with in vitro experiments on isolated guinea-pig ventricular cells and papillary muscles stimulated at various rates (2-6.67 Hz) and exposed to various concentrations (5 × 10(-6) to 500 × 10(-6) mol/l) of lidocaine. The most salient result was that whereas large doses (5 × 10(-4) mol/l or higher) of lidocaine were required to inhibit action potentials temporarily in single cells, much lower doses (5 × 10(-6) mol/l), i.e., therapeutic levels, were sufficient to have the same effect in papillary muscles: a hundredfold difference. Our experimental results and mathematical analysis indicate that the syncytial nature of cardiac tissue explains the effects of clinically relevant doses of Na(+) channel blockers.

Nonlinear and Stochastic Dynamics in the Heart

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.

A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry

It is widely believed that the pulmonary veins (PVs) of the left atrium play the central role in the generation of anatomically induced atrial reentry but its mechanism has not been analytically explained. To understand this mechanism, a new analytic approach is proposed by adapting the geometric relative acceleration analysis from spacetime physics based on the hypothesis that a large relative acceleration can translate to a dramatic increase in the curvature of a wavefront and subsequently to conduction failure. By verifying the strong dependency of the propagational direction and the magnitude of anisotropy for conduction failure, this analytic method reveals that a unidirectional block can be generated by asymmetric propagation toward the PVs. This model is validated by computational tests in a T-shaped domain, computational simulations for three-dimensional atrial reentry and previous in-silico reports for anatomically induced atrial reentry.

Metastable dynamical patterns and their stabilization in arrays of bidirectionally coupled sigmoidal neurons

Transient patterns in a bistable ring of bidirectionally coupled sigmoidal neurons were studied. When the system had a pair of spatially uniform steady solutions, the instability of unstable spatially nonuniform steady solutions decreased exponentially with the number of neurons because of the symmetry of the system. As a result, transient spatially nonuniform patterns showed dynamical metastability: Their duration increased exponentially with the number of neurons and the duration of randomly generated patterns obeyed a power-law distribution. However, these metastable dynamical patterns were easily stabilized in the presence of small variations in coupling strength. Metastable rotating waves and their pinning in the presence of asymmetry in the direction of coupling and the disappearance of metastable dynamical patterns due to asymmetry in the output function of a neuron were also examined. Further, in a two-dimensional array of neurons with nearest-neighbor coupling, intrinsically one-dimensional patterns were dominant in transients, and self-excitation in these neurons affected the metastable dynamical patterns.

Mechanisms of ventricular arrhythmias: a dynamical systems-based perspective

Defining the cellular electrophysiological mechanisms for ventricular tachyarrhythmias is difficult, given the wide array of potential mechanisms, ranging from abnormal automaticity to various types of reentry and kk activity. The degree of difficulty is increased further by the fact that any particular mechanism may be influenced by the evolving ionic and anatomic environments associated with many forms of heart disease. Consequently, static measures of a single electrophysiological characteristic are unlikely to be useful in establishing mechanisms. Rather, the dynamics of the electrophysiological triggers and substrates that predispose to arrhythmia development need to be considered. Moreover, the dynamics need to be considered in the context of a system, one that displays certain predictable behaviors, but also one that may contain seemingly stochastic elements. It also is essential to recognize that even the predictable behaviors of this complex nonlinear system are subject to small changes in the state of the system at any given time. Here we briefly review some of the short-, medium-, and long-term alterations of the electrophysiological substrate that accompany myocardial disease and their potential impact on the initiation and maintenance of ventricular arrhythmias. We also provide examples of cases in which small changes in the electrophysiological substrate can result in rather large differences in arrhythmia outcome. These results suggest that an interrogation of cardiac electrical dynamics is required to provide a meaningful assessment of the immediate risk for arrhythmia development and for evaluating the effects of putative antiarrhythmic interventions.

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

Continuous description of lattice discreteness effects in front propagation

Models describing microscopic or mesoscopic phenomena in physics are inherently discrete, where the lattice spacing between fundamental components, such as in the case of atomic sites, is a fundamental physical parameter. The effect of spatial discreteness over front propagation phenomenon in an overdamped one-dimensional periodic lattice is studied. We show here that the study of front propagation leads in a discrete description to different conclusions that in the case of its, respectively, continuous description, and also that the results of the discrete model, can be inferred by effective continuous equations with a supplementary spatially periodic term that we have denominated Peierls-Nabarro drift, which describes the bifurcation diagram of the front speed, the appearance of particle-type solutions and their snaking bifurcation diagram. Numerical simulations of the discrete equation show quite good agreement with the phenomenological description.

Spiral wave breakup in excitable media with an inhomogeneity of conduction anisotropy

Many conditions remodel the heart muscle such that it results in a perturbation of cells coupling. The effect of this perturbation on the stability of the spiral waves of electrochemical activity is not clear. We used the FitzHugh-Nagumo model of an excitable medium to model the conduction of the activation waves in a two-dimensional system with inhomogeneous anisotropy level. Inhomogeneity of the anisotropy level was modeled by adding Gaussian noise to diffusion coefficients corresponding with lateral coupling of the cells. Low noise levels resulted in a stable propagation of the spiral wave. For large noise level conduction was not possible due to insufficient coupling in direction perpendicular to fibers. For intermediate noise intensities, the initial wave broke up into several independent spiral waves or waves circulating around conduction obstacles. At an optimal noise intensity, the number of wavelets was maximized-a form of anti-coherent resonance was obtained. Our results suggest that the inhomogeneity of conduction anisotropy may promote wave breakup and hence play an important role in the initiation and perpetuation of the cardiac arrhythmias.

Fast calcium wave propagation mediated by electrically conducted excitation and boosted by CICR

We have investigated synchronization and propagation of calcium oscillations, mediated by gap junctional excitation transmission. For that purpose we used an experimentally based model of normal rat kidney (NRK) cells, electrically coupled in a one-dimensional configuration (linear strand). Fibroblasts such as NRK cells can form an excitable syncytium and generate spontaneous inositol 1,4,5-trisphosphate (IP3)-mediated intracellular calcium waves, which may spread over a monolayer culture in a coordinated fashion. An intracellular calcium oscillation in a pacemaker cell causes a membrane depolarization from within that cell via calcium-activated chloride channels, leading to an L-type calcium channel-based action potential (AP) in that cell. This AP is then transmitted to the electrically connected neighbor cell, and the calcium inflow during that transmitted AP triggers a calcium wave in that neighbor cell by opening of IP3receptor channels, causing calcium-induced calcium release (CICR). In this way the calcium wave of the pacemaker cell is rapidly propagated by the electrically transmitted AP. Propagation of APs in a strand of cells depends on the number of terminal pacemaker cells, the L-type calcium conductance of the cells, and the electrical coupling between the cells. Our results show that the coupling between IP3-mediated calcium oscillations and AP firing provides a robust mechanism for fast propagation of activity across a network of cells, which is representative for many other cell types such as gastrointestinal cells, urethral cells, and pacemaker cells in the heart.

Dynamics of sustained reentry in a loop model with discrete gap junction resistances

The dynamics of reentry is studied in a one-dimensional loop of model cardiac cells with discrete intercellular gap junction resistance (R). Each cell is represented by a continuous cable with ionic current given by a modified Beeler-Reuter formulation. For R below a limiting value, propagation is found to change from period-1 to quasiperiodic (QP) at a critical loop length (L(crit)) that decreases with R. Quasiperiodic reentry exists from L(crit) to a minimum length (L(min)), which also shortens with R. The decrease of L(crit) (R) is not a simple scaling, but the bifurcation can still be predicted from the slope of the restitution curve giving the duration of the action potential as a function of the diastolic interval. However, the shape of the restitution curve changes with R. An increase of R does not seem to increase the number of possible QP solutions since, as in the continuous cable, only two QP modes of propagation were found despite an extensive search through alternative initial conditions.

Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory, and conduction velocity restitution effects

We examine the utility of the action potential (AP) duration (APD) restitution curve slope in predicting the onset of electrical alternans when electrotonic and memory effects are considered. We develop and use two ionic cell models without memory that have the same restitution curve with slope >1 but different AP shapes and, therefore, different electrotonic effects. We also study a third cell model that incorporates short-term memory of previous cycle lengths, so that it has a family of S1-S2 restitution curves as well as a dynamic restitution curve with slope >1. Our results indicate that both electrotonic and memory effects can suppress alternans, even when the APD restitution curve is steep. In the absence of memory, electrotonic currents related to the shape of the AP, as well as conduction velocity restitution, can affect how alternans develops in tissue and, in some cases, can prevent its induction entirely, even when isolated cells exhibit alternans. When short-term memory is included, alternans may not occur in isolated cells, despite a steep APD restitution curve, and may or may not occur in tissue, depending on conduction velocity restitution. We show for the first time that electrotonic and memory effects can prevent conduction blocks and stabilize reentrant waves in two and three dimensions. Thus we find that the slope of the APD restitution curve alone does not always well predict the onset of alternans and that incorporating electrotonic and memory effects may provide a more useful alternans criterion. A Data Supplement containing movies and JAVA applets is available online at http://ajpheart.physiology.org/cgi/content/full/00747.2003/DC1 .

Propagating waves in one-dimensional discrete networks of coupled units

We investigate the behavior of discrete systems on a one-dimensional lattice composed of localized units interacting with each other through nonlocal, nonlinear reactive dynamics. In the presence of second-order and third-order steps coupling two or three neighboring sites, respectively, we observe, for appropriate initial conditions, the propagation of waves which subsist in the absence of mass transfer by diffusion. For the case of the third-order (bistable) model, a counterintuitive effect is also observed, whereby the homogeneously less stable state invades the more stable one under certain conditions. In the limit of a continuous space the dynamics of these networks is described by a generic evolution equation, from which some analytical predictions can be extracted. The relevance of this mode of information transmission in spatially extended systems of interest in physical chemistry and biology is discussed.

Simulation of waves in calcium models with 3D spherical geometry

Waves of calcium ions are present in fertilized eggs of many species. Models for pulse and tidal wave propagation have usually been studied in one or two spatial coordinates only. We examine in three spatial coordinates some established models, based on Ca(2+)-induced Ca(2+)-release from both (assumed) continuously or heterogeneously distributed stores of endoplasmic reticulum (ER) through channels activated by inositol triphosphate (IP(3)). With continuous IP(3) distribution decreasing radially towards the interior, we obtain concave pulse shapes for waves penetrating the interior. Concave waves are also recorded in systems with ER confined to distributions of small spheres (microdomains) inside the cell, which we simulate for front waves (tides) in bistable systems.

Termination of reentry in an inhomogeneous ring of model cardiac cells

Reentrant waves propagating in a ring or annulus of excitable media are a model of the basic mechanism underlying a major class of irregular cardiac rhythms known as anatomical reentry. Such reentrant waves are terminated by rapid electrical stimulation (pacing) from an implantable device. Because the mechanisms of such termination are poorly understood, we study pacing of anatomical reentry in a one-dimensional ring of model cardiac cells. For realistic off-circuit pacing, our model-independent results suggest that circuit inhomogeneities, and the electrophysiological dynamical changes they introduce, may be essential for terminating reentry in some cases.

Dipolar effects on soliton dynamics on a discrete ferromagnetic chain

The contributions of dipole-dipole interactions to the dynamics of solitons on a one-dimensional discrete easy-plane Heisenberg ferromagnet, in which the biquadratic exchange interactions are taken into account in addition to the Zeeman energy, the uniaxial anisotropy, and the exchange energy, are studied numerically. The results of a numerical simulation of the dynamics of a single soliton, as well as collision between a soliton-antisoliton pair, indicated that the energy-velocity curves for the solitons in the ferromagnetic chain present the signature of five different branches corresponding to different types of nonlinear elementary excitations in the chain.

Enhanced self-termination of re-entrant arrhythmias as a pharmacological strategy for antiarrhythmic action

Ventricular tachycardia and fibrillation are potentially lethal cardiac arrhythmias generated by high frequency, irregular spatio-temporal electrical activity. Re-entrant propagation has been demonstrated as a mechanism generating these arrhythmias in computational and in vitro animal models of these arrhythmias. Re-entry can be idealised in homogenous isotropic virtual cardiac tissues as spiral and scroll wave solutions of reaction-diffusion equations. A spiral wave in a bounded medium can be terminated if its core reaches a boundary. Ventricular tachyarrhythmias in patients are sometimes observed to spontaneously self-terminate. One possible mechanism for self-termination of a spiral wave is meander of its core to an inexcitable boundary. We have previously proposed the hypothesis that the spatial extent of meander of a re-entrant wave in the heart can be directly related to its probability of self-termination, and so inversely related to its lethality. Meander in two-dimensional virtual ventricular tissues based on the Oxsoft family of cell models, with membrane excitation parameters simulating the inherited long Q-T syndromes has been shown to be consistent with this hypothesis: the largest meander is seen in the syndrome with the lowest probability of death per arrhythmic episode. Here we extend our previous results to virtual tissues based on the Luo-Rudy family of models. Consistent with our hypothesis, for both families of models, whose different ionic mechanisms produce different patterns of meander, the LQT virtual tissue with the larger meander simulates the syndrome with the lower probability of death per episode. Further, we search the parameter space of the repolarizing currents to find their conductance parameter values that give increased meander of spiral waves. These parameters may provide targets for antiarrhythmic drugs designed to act by increasing the likelihood of self-termination of re-entrant arrhythmias. (c) 2002 American Institute of Physics.

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

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

Pulse propagation sustained by noise in arrays of bistable electronic circuits

One-dimensional arrays of nonlinear electronic circuits are shown to support propagation of pulses when operating in a locally bistable regime, provided the circuits are under the influence of a global noise. These external random fluctuations are applied to the parameter that controls the transition between bistable and monostable dynamics in the individual circuits. As a result, propagating fronts become destabilized in the presence of noise, and the system self-organizes to allow the transmission of pulses. The phenomenon is also observed in weakly coupled arrays, when propagation failure arises in the absence of noise.

Continuum approach to discreteness

We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls-Nabarro barriers and Bloch oscillations). We analyze the role of such methods in providing an effective "homogenization" of spatially discrete, as well as of heterogeneous continuum equations. Finally, we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations, as well as to compare them with other semicontinuum approximations.

Traveling kinks in discrete media: exact solution in a piecewise linear model

We present an exact integral representation of a traveling kink solution in a reaction-diffusion equation with a piecewise linear reaction function, complementing existence proofs and numerical observations of such solutions in discrete excitable media. The kink speed is determined through a matching condition, and is worked out explicitly in two limiting situations: the pinning limit, and the opposite limit of infinitely fast kink. Results on the pinning limit agree with those in a recent paper by Fath [Physica D 116, 176 (1998)]. The model includes a "recovery parameter" for a possible extension to a discrete FitzHugh-Nagumo-type system.

Heterogeneous versus discrete mapping problem

We propose a method for mapping a spatially discrete problem, stemming from the spatial discretization of a parabolic or hyperbolic partial differential equation of gradient type, to a heterogeneous one with certain comparable dynamical features pertaining, in particular, to coherent structures. We focus the analysis on a (1+1)-dimensional phi(4) model and confirm the theoretical predictions numerically. We also discuss possible generalizations of the method and the ensuing qualitative analogies between heterogeneous and discrete systems and their dynamics.

Influence of dynamic gap junction resistance on impulse propagation in ventricular myocardium: a computer simulation study

The gap junction connecting cardiac myocytes is voltage and time dependent. This simulation study investigated the effects of dynamic gap junctions on both the shape and conduction velocity of a propagating action potential. The dynamic gap junction model is based on that described by Vogel and Weingart (J. Physiol. (Lond.). 1998, 510:177-189) for the voltage- and time-dependent conductance changes measured in cell pairs. The model assumes that the conductive gap junction channels have four conformational states. The gap junction model was used to couple 300 cells in a linear strand with membrane dynamics of the cells defined by the Luo-Rudy I model. The results show that, when the cells are tightly coupled (6700 channels), little change occurs in the gap junction resistance during propagation. Thus, for tight coupling, there are negligible differences in the waveshape and propagation velocity when comparing the dynamic and static gap junction representations. For poor coupling (85 channels), the gap junction resistance increases 33 MOmega during propagation. This transient change in resistance resulted in increased transjunctional conduction delays, changes in action potential upstroke, and block of conduction at a lower junction resting resistance relative to a static gap junction model. The results suggest that the dynamics of the gap junction enhance cellular decoupling as a possible protective mechanism of isolating injured cells from their neighbors.

Theoretical and experimental study of two discrete coupled Nagumo chains

We analyze front wave (kink and antikink) propagation and pattern formation in a system composed of two coupled discrete Nagumo chains using analytical and numerical methods. In the case of homogeneous interaction among the chains, we show the possibility of the effective control on wave propagation. In addition, physical experiments on electrical chains confirm all theoretical behaviors.

Multipulses in discrete Hamiltonian nonlinear systems

In this work, the behavior of multipulses in discrete Hamiltonian nonlinear systems is investigated. The discrete nonlinear Schrödinger equation is used as the benchmark system for this study. A singular perturbation methodology as well as a variational approach are implemented in order to identify the dominant factors in the discrete problem. The results of the two methodologies are shown to coincide in assessing the interplay of discreteness and exponential tail-tail pulse interaction. They also allow one to understand why, contrary to what is believed for their continuum siblings, discrete systems can sustain (static) multipulse configurations, a conclusion that is subsequently verified by numerical experiment.

Propagation failure in discrete bistable reaction-diffusion systems: theory and experiments

Wave front propagation failure is investigated in discrete bistable reaction-diffusion systems. We present a theoretical approach including dissipative effects and leading to an analytical expression of the critical coupling beyond which front propagation can occur as a function of the nonlinearity threshold parameter. Our theoretical predictions are confirmed by numerical simulations and experimental results on an equivalent electrical diffusive lattice.

Universal scaling of wave propagation failure in arrays of coupled nonlinear cells

We study the onset of the propagation failure of wave fronts in systems of coupled cells. We introduce a new method to analyze the scaling of the critical external field at which fronts cease to propagate, as a function of intercellular coupling. We find the universal scaling of the field throughout the range of couplings and show that the field becomes exponentially small for large couplings. Our method is generic and applicable to a wide class of cellular dynamics in chemical, biological, and engineering systems. We confirm our results by direct numerical simulations.

Noise sustained propagation: local versus global noise

We expand on prior results on noise supported signal propagation in arrays of coupled bistable elements. We present and compare experimental and numerical results for kink propagation under the influence of local and global fluctuations. As demonstrated previously for local noise, an optimum range of global noise power exists for which the medium acts as a reliable transmission "channel." We discuss implications for propagation failure in a model of cardiac tissue, and present a general theoretical framework based on discrete kink statistics. Valid for generic bistable chains, the theory captures the essential features observed in our experiments and numerical simulations.

Avalanche of bifurcations and hysteresis in a model of cellular differentiation

Cellular differentiation in a developing organism is studied via a discrete bistable reaction-diffusion model. A system of undifferentiated cells is allowed to receive an inductive signal emanating from its environment. Depending on the form of the nonlinear reaction kinetics, this signal can trigger a series of bifurcations in the system. Differentiation starts at the surface where the signal is received and either cells change type up to a given distance or, under other conditions, the differentiation process propagates throughout the whole domain. When the signal diminishes, hysteresis is observed.

Comparison between the role of discontinuities in cardiac conduction and in a one-dimensional hardware model

In real electrophysiological experiments, irregularities in the extracellular excitation spread are believed to depend on cardiac tissue microstructure. An electronic hardware model was developed to analyze this dependence by placing some inhomogeneities (slow propagation areas) in the medium. The position of such inhomogeneities is correlated with abnormal delays and irregularities measured in signal propagation.

Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation

Wave propagation in ventricular muscle is rendered highly anisotropic by the intramural rotation of the fiber. This rotational anisotropy is especially important because it can produce a twist of electrical vortices, which measures the rate of rotation (in degree/mm) of activation wavefronts in successive planes perpendicular to a line of phase singularity, or filament. This twist can then significantly alter the dynamics of the filament. This paper explores this dynamics via numerical simulation. After a review of the literature, we present modeling tools that include: (i) a simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potential (Beeler-Reuter and others), and (ii) a semi-implicit algorithm for the fast solution of monodomain cable equations with rotational anisotropy. We then discuss selected results of a simulation study of vortex dynamics in a parallelepipedal slab of ventricular muscle of varying wall thickness (S) and fiber rotation rate (theta(z)). The main finding is that rotational anisotropy generates a sufficiently large twist to destabilize a single transmural filament and cause a transition to a wave turbulent state characterized by a high density of chaotically moving filaments. This instability is manifested by the propagation of localized disturbances along the filament and has no previously known analog in isotropic excitable media. These disturbances correspond to highly twisted and distorted regions of filament, or "twistons," that create vortex rings when colliding with the natural boundaries of the ventricle. Moreover, when sufficiently twisted, these rings expand and create additional filaments by further colliding with boundaries. This instability mechanism is distinct from the commonly invoked patchy failure or wave breakup that is not observed here during the initial instability. For modified Beeler-Reuter-like kinetics with stable reentry in two dimensions, decay into turbulence occurs in the left ventricle in about one second above a critical wall thickness in the range of 4-6 mm that matches experiment. However this decay is suppressed by uniformly decreasing excitability. Specific experiments to test these results, and a method to characterize the filament density during fibrillation are discussed. Results are contrasted with other mechanisms of fibrillation and future prospects are summarized. (c)1998 American Institute of Physics.

Syncytial integration by a network of coupled bipolar cells in the retina

A model system for syncytial integration is the outer vertebrate retina, where graded signals or electrotonic potentials interact laterally via gap junctions to form an integrated response that is relayed by chemical synapses to the next layer of interconnected cells. Morphological and physiological experiments confirm that bipolar cells form quasisyncytial lattices, and so this review will aim to address two important issues: the function of coupling in visual information processing and the construction of a robust mathematical model that can adequately simulate signal spread in the bipolar cell syncytium. It is shown that the role of coupling in bipolar cells differs from that associated in the presynaptic networks, namely, loss in spatial resolution in order to increase the signal-to-noise ratio. The intrinsic membrane properties of bipolar cells which give rise to voltage-dependent currents are inactive over the normal in vivo operating range of membrane potential and may be shunted as a direct result of electrotonic coupling, suppressing any possibility of action potential propagation in the bipolar cell syncytium. It is therefore speculated that the mechanisms underlying processing of information in bipolar networks are dependent on the structure of bipolar cells and in particular, on the presence of gap junctions. It is proposed that a three-dimensional model which incorporates the spatial properties of each bipolar cell in the network in the form of a leaky cable is the most likely model to simulate signal spread in the bipolar cell syncytium in vivo. This is because discrete network models represent each bipolar cell in the syncytium as isopotential units without any spatial structure, and thus are unable to reproduce the temporal characteristics of electrotonic potential spread within the central receptive field of bipolar cells.

Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: simulations in a three-dimensional bidomain model

INTRODUCTION

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.

METHODS AND RESULTS

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.

CONCLUSIONS

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.

Spatial phase pattern in one-dimensional arrays of limit cycle oscillators with discrete coupling

We present a model of limit cycle oscillators for collective oscillations in intracellular calcium concentration in cell communities. A phase-dependent discrete coupling between nearest neighbors is introduced into the model on the basis of the experimental observation that intercellular transmission of calcium or calcium mobilizing messenger is effected by gap junction and gap junctional permeability is affected by intracellular calcium concentration. The spatial phase pattern of several clusters in which oscillations are in phase is found with the phase-dependent discrete coupling.

Effect of an externally applied electric field on excitation propagation in the cardiac muscle

Classical theory of potential distribution in cardiac muscle (cable theory) postulates that all effects of electric field (internally or externally applied) should decay exponentially with a space constant of the order of the tissue space constant ( approximately 1 mm). Classical theory does not take into account the cellular structure of the heart. Here, we formulate a mathematical model of excitation propagation taking into account cellular gap junctions. Investigation of the model has shown that the classical description is correct on the macroscopic scale only. At microscopic scale, electric field is modulated with a spatial period equal to the cell size (Plonsey and Barr), with the zero average. A very important new feature found here is that this effect of electric field does not decay at arbitrary big distances from the electrode. It opens the new way to control the excitation propagation in the cardiac muscle. In particular, we show that electric field can modify the velocity of propagation of an impulse in cardiac tissue at arbitrary big distances from electrode. In 2-dimensions, it can make rotating waves drift. To test these predictions, experiments with cardiac preparations are proposed.

NASPE Young Investigator Awardee-1993. Computer model of the atrioventricular node predicts reentrant arrhythmias

INTRODUCTION

Following atrial premature beats, the AV node may exhibit sustained reentrant tachyarrhythmias, isolated echo beats, or discontinuities in the recovery curve (the plot of conduction time versus atrial cycle length). A computer model was used to examine the hypothesis that spatial variation of AV nodal passive electrical resistance may account for these phenomena.

METHODS AND RESULTS

A computer model of a rectangular lattice of electrotonically linked elements whose ionic kinetics simulated nodal ionic flux was developed. The model showed that there exists a resistance value that minimizes the effective refractory period, because high resistance prevents depolarization of distal elements, while low resistance allows leakage of depolarizing current by electrotonic transmission, preventing activation of proximal elements. High resistances stabilized reentry by slowing conduction. Simulations incorporating equal resistance values between elements predicted increased AV nodal conduction times with increasing prematurity of atrial impulses. A model with a gradual change in resistance between fibers produced discontinuities and tachycardia, but not both simultaneously. Uniform anisotropy produced preferential transverse block, leading to echo beats and "fast-slow" tachycardia, but not recovery curve discontinuities. Nonuniform anisotropy could produce reentry, but tachycardia often occurred without discontinuities. Dividing the lattice into two electrotonically linked parallel pathways with different resistance values ("dual pathway model") predicted recovery curve discontinuities, echo beats, and tachycardia. At critical atrial cycle lengths, only the (high resistance) slow pathway conducted antegradely, while the fast pathway conducted retrogradely, to generate the typical "slow-fast" tachycardia. Responses of the dual pathway model to ablation were consistent with clinical data, including the previous observation of a decrease in fast pathway effective refractory period after slow pathway ablation.

CONCLUSION

Differences in passive electrical resistance of electronically linked dual pathways within the AV node may account for functional longitudinal dissociation, reentrant arrhythmias, and responses to catheter ablation therapy.

Generation of reentry in anisotropic myocardium

INTRODUCTION

We investigated numerically the effects of the rotation of fiber axis orientation through the myocardial wall on wave propagation.

METHODS AND RESULTS

We show that because of this rotation and inherent discrete properties of myocardium, a premature stimulus can create unidirectional conduction block leading to reentry.

CONCLUSION

The dynamics of the subsequent reentrant patterns are complicated by the presence of rotational anisotropy, as the center of reentry drifts, and the reentry terminates in finite time when it collides with the domain boundary.

Myocardial electrical propagation in patients with idiopathic dilated cardiomyopathy

Myocardial propagation may contribute to fatal arrhythmias in patients with idiopathic dilated cardiomyopathy (IDC). We examined this property in 15 patients with IDC undergoing cardiac transplantation and in 14 control subjects. An 8 x 8 array with electrodes 2 mm apart was used to determine the electrical activation sequence over a small region of the left ventricular surface. Tissue from the area beneath the electrode array was examined in the patients with IDC. The patients with IDC could be divided into three groups. Group I (n = 7) had activation patterns and estimates of longitudinal (theta L = 0.84 +/- 0.09 m/s) and transverse (theta T = 0.23 +/- 0.05 m/s) conduction velocities that were no different from controls (theta L = 0.80 +/- 0.08 m/s, theta T = 0.23 +/- 0.03 m/s). Group II (n = 4) had fractionated electrograms and disturbed transverse conduction with normal longitudinal activation, features characteristic of nonuniform anisotropic properties. Two of the control patients also had this pattern. Group III (n = 4) had fractionated potentials and severely disturbed transverse and longitudinal propagation. The amount of myocardial fibrosis correlated with the severity of abnormal propagation. We conclude that (a) severe contractile dysfunction is not necessarily accompanied by changes in propagation, and (b) nonuniform anisotropic propagation is present in a large proportion of patients with IDC and could underlie ventricular arrhythmias in this disorder.

A bidomain model with periodic intracellular junctions: a one-dimensional analysis

The classical bidomain model of cardiac tissue views the intracellular and extracellular (interstitial) spaces as two coupled but separate continua. In the present study, the classical bidomain model has been extended by introducing a periodic conductivity in the intracellular space to represent the junctional discontinuity between abutting myocytes. In this model the junctional region of a myocyte is represented in a way that permits variation of junction size and conductivity profile. Employing spectral techniques, a new method was developed for solving the coupled differential equations governing the intracellular and extracellular potentials in a tissue preparation of finite dimensions. Different spectral representations are used for the aperiodic intra- and extracellular potentials (finite Fourier integral transform) and for the periodic intracellular conductivity (Fourier series). As a first application of the method, the response of a 50-cell, single interior fiber to a defibrillating current is examined under steady-state conditions. Transmembrane as well as intra- and extracellular potential distributions along the fiber were calculated.

An eikonal-curvature equation for action potential propagation in myocardium

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.