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

Capillary oxygen regulates demand-supply coupling by triggering connexin40-mediated conduction: Rethinking the metabolic hypothesis

Coupling red blood cell (RBC) supply to O2 demand is an intricate process requiring O2 sensing, generation of a stimulus, and signal transduction that alters upstream arteriolar tone. Although actively debated, this process has been theorized to be induced by hypoxia and to involve activation of endothelial inwardly rectifying K+ channels (KIR) 2.1 by elevated extracellular K+ to trigger conducted hyperpolarization via connexin40 (Cx40) gap junctions to upstream resistors. This concept was tested in resting healthy skeletal muscle of Cx40-/- and endothelial KIR2.1-/- mice using state-of-the-art live animal imaging where the local tissue O2 environment was manipulated using a custom gas chamber. Second-by-second capillary RBC flow responses were recorded as O2 was altered. A stepwise drop in PO2 at the muscle surface increased RBC supply in capillaries of control animals while elevated O2 elicited the opposite response; capillaries were confirmed to express Cx40. The RBC flow responses were rapid and tightly coupled to O2; computer simulations did not support hypoxia as a driving factor. In contrast, RBC flow responses were significantly diminished in Cx40-/- mice. Endothelial KIR2.1-/- mice, on the other hand, reacted normally to O2 changes, even when the O2 challenge was targeted to a smaller area of tissue with fewer capillaries. Conclusively, microvascular O2 responses depend on coordinated electrical signaling via Cx40 gap junctions, and endothelial KIR2.1 channels do not initiate the event. These findings reconceptualize the paradigm of blood flow regulation in skeletal muscle and how O2 triggers this process in capillaries independent of extracellular K+.

The role of the Cx43/Cx45 gap junction voltage gating on wave propagation and arrhythmogenic activity in cardiac tissue

Gap junctions (GJs) formed of connexin (Cx) protein are the main conduits of electrical signals in the heart. Studies indicate that the transitional zone of the atrioventricular (AV) node contains heterotypic Cx43/Cx45 GJ channels which are highly sensitive to transjunctional voltage (Vj). To investigate the putative role of Vj gating of Cx43/Cx45 channels, we performed electrophysiological recordings in cell cultures and developed a novel mathematical/computational model which, for the first time, combines GJ channel Vj gating with a model of membrane excitability to simulate a spread of electrical pulses in 2D. Our simulation and electrophysiological data show that Vj transients during the spread of cardiac excitation can significantly affect the junctional conductance (gj) of Cx43/Cx45 GJs in a direction- and frequency-dependent manner. Subsequent simulation data indicate that such pulse-rate-dependent regulation of gj may have a physiological role in delaying impulse propagation through the AV node. We have also considered the putative role of the Cx43/Cx45 channel gating during pathological impulse propagation. Our simulation data show that Vj gating-induced changes in gj can cause the drift and subsequent termination of spiral waves of excitation. As a result, the development of fibrillation-like processes was significantly reduced in 2D clusters, which contained Vj-sensitive Cx43/Cx45 channels.

Role of cardiac mitofusins in cardiac conduction following simulated ischemia-reperfusion

Mitochondrial dysfunction induced by acute cardiac ischemia-reperfusion (IR), may increase susceptibility to arrhythmias by perturbing energetics, oxidative stress production and calcium homeostasis. Although changes in mitochondrial morphology are known to impact on mitochondrial function, their role in cardiac arrhythmogenesis is not known. To assess action potential duration (APD) in cardiomyocytes from the Mitofusins-1/2 (Mfn1/Mfn2)-double-knockout (Mfn-DKO) compared to wild-type (WT) mice, optical-electrophysiology was conducted. To measure conduction velocity (CV) in atrial and ventricular tissue from the Mfn-DKO and WT mice, at both baseline and following simulated acute IR, multi-electrode array (MEA) was employed. Intracellular localization of connexin-43 (Cx43) at baseline was evaluated by immunohistochemistry, while Cx-43 phosphorylation was assessed by Western-blotting. Mfn-DKO cardiomyocytes demonstrated an increased APD. At baseline, CV was significantly lower in the left ventricle of the Mfn-DKO mice. CV decreased with simulated-ischemia and returned to baseline levels during simulated-reperfusion in WT but not in atria of Mfn-DKO mice. Mfn-DKO hearts displayed increased Cx43 lateralization, although phosphorylation of Cx43 at Ser-368 did not differ. In summary, Mfn-DKO mice have increased APD and reduced CV at baseline and impaired alterations in CV following cardiac IR. These findings were associated with increased Cx43 lateralization, suggesting that the mitofusins may impact on post-MI cardiac-arrhythmogenesis.

Simulation of atrial fibrillation in a non-ohmic propagation model with dynamic gap junctions

Gap junctions exhibit nonlinear electrical properties that have been hypothesized to be relevant to arrhythmogenicity in a structurally remodeled tissue. Large-scale implementation of gap junction dynamics in 3D propagation models remains challenging. We aim to quantify the impact of nonlinear diffusion during episodes of arrhythmias simulated in a left atrial model. Homogenization of conduction properties in the presence of nonlinear gap junctions was performed by generalizing a previously developed mathematical framework. A monodomain model was solved in which conductivities were time-varying and depended on transjunctional potentials. Gap junction conductances were derived from a simplified Vogel-Weingart model with first-order gating and adjustable time constant. A bilayer interconnected cable model of the left atrium with 100  μm resolution was used. The diffusion matrix was recomputed at each time step according to the state of the gap junctions. Sinus rhythm and atrial fibrillation episodes were simulated in remodeled tissue substrates. Slow conduction was induced by reduced coupling and by diffuse or stringy fibrosis. Simulations starting from the same initial conditions were repeated with linear and nonlinear gap junctions. The discrepancy in activation times between the linear and nonlinear diffusion models was quantified. The results largely validated the linear approximation for conduction velocities >20 cm/s. In very slow conduction substrates, the discrepancy accumulated over time during atrial fibrillation, eventually leading to qualitative differences in propagation patterns, while keeping the descriptive statistics, such as cycle lengths, unchanged. The discrepancy growth rate was increased by impaired conduction, fibrosis, conduction heterogeneity, lateral uncoupling, fast gap junction time constant, and steeper action potential duration restitution.

The resting membrane potential of hSC-CM in a syncytium is more hyperpolarised than that of isolated cells

Human-induced pluripotent stem cell (hiPSC) and stem cell (hSC) derived cardiomyocytes (CM) are gaining popularity as in vitro model for cardiology and pharmacology studies. A remaining flaw of these cells, as shown by single-cell electrophysiological characterization, is a more depolarized resting membrane potential (RMP) compared to native CM. Most reports attribute this to a lower expression of the Kir2.1 potassium channel that generates the IK1 current. However, most RMP recordings are obtained from isolated hSC/hiPSC-CMs whereas in a more native setting these cells are interconnected with neighboring cells by connexin-based gap junctions, forming a syncytium. Hereby, these cells are electrically connected and the total pool of IK1 increases. Therefore, the input resistance (Ri) of interconnected cells is lower than that of isolated cells. During patch clamp experiments pipettes need to be well attached or sealed to the cell, which is reflected in the seal resistance (Rs), because a nonspecific ionic current can leak through this pipette-cell contact or seal and balance out small currents within the cell such as IK1. By recording the action potential of isolated hSC-CMs and that of hSC-CMs cultured in small monolayers, we show that the RMP of hSC-CMs in monolayer is approximately -20 mV more hyperpolarized compared to isolated cells. Accordingly, adding carbenoxolone, a connexin channel blocker, isolates the cell that is patch clamped from its neighboring cells of the monolayer and depolarizes the RMP. The presented data show that the recorded RMP of hSC-CMs in a syncytium is more negative than that determined from isolated hSC/hiPSC-CMs, most likely because the active pool of Kir2.1 channels increased.

Coupling between cardiac cells-An important determinant of electrical impulse propagation and arrhythmogenesis

Cardiac arrhythmias are an important cause of sudden cardiac death-a devastating manifestation of many underlying causes, such as heart failure and ischemic heart disease leading to ventricular tachyarrhythmias and ventricular fibrillation, and atrial fibrillation causing cerebral embolism. Cardiac electrical propagation is a main factor in the initiation and maintenance of cardiac arrhythmias. In the heart, gap junctions are the basic unit at the cellular level that host intercellular low-resistance channels for the diffusion of ions and small regulatory molecules. The dual voltage clamp technique enabled the direct measurement of electrical conductance between cells and recording of single gap junction channel openings. The rapid turnover of gap junction channels at the intercalated disk implicates a highly dynamic process of trafficking and internalization of gap junction connexons. Recently, non-canonical roles of gap junction proteins have been discovered in mitochondria function, cytoskeletal organization, trafficking, and cardiac rescue. At the tissue level, we explain the concepts of linear propagation and safety factor based on the model of linear cellular structure. Working myocardium is adequately represented as a discontinuous cellular network characterized by cellular anisotropy and connective tissue heterogeneity. Electrical propagation in discontinuous cellular networks reflects an interplay of three main factors: cell-to-cell electrical coupling, flow of electrical charge through the ion channels, and the microscopic tissue structure. This review provides a state-of-the-art update of the cardiac gap junction channels and their role in cardiac electrical impulse propagation and highlights a combined approach of genetics, cell biology, and physics in modern cardiac electrophysiology.

Conductance heterogeneities induced by multistability in the dynamics of coupled cardiac gap junctions

In this paper, we study the propagation of the cardiac action potential in a one-dimensional fiber, where cells are electrically coupled through gap junctions (GJs). We consider gap junctional gate dynamics that depend on the intercellular potential. We find that different GJs in the tissue can end up in two different states: a low conducting state and a high conducting state. We first present evidence of the dynamical multistability that occurs by setting specific parameters of the GJ dynamics. Subsequently, we explain how the multistability is a direct consequence of the GJ stability problem by reducing the dynamical system's dimensions. The conductance dispersion usually occurs on a large time scale, i.e., thousands of heartbeats. The full cardiac model simulations are computationally demanding, and we derive a simplified model that allows for a reduction in the computational cost of four orders of magnitude. This simplified model reproduces nearly quantitatively the results provided by the original full model. We explain the discrepancies between the two models due to the simplified model's lack of spatial correlations. This simplified model provides a valuable tool to explore cardiac dynamics over very long time scales. That is highly relevant in studying diseases that develop on a large time scale compared to the basic heartbeat. As in the brain, plasticity and tissue remodeling are crucial parameters in determining the action potential wave propagation's stability.

Connexins in the Heart: Regulation, Function and Involvement in Cardiac Disease

Connexins are a family of transmembrane proteins that play a key role in cardiac physiology. Gap junctional channels put into contact the cytoplasms of connected cardiomyocytes, allowing the existence of electrical coupling. However, in addition to this fundamental role, connexins are also involved in cardiomyocyte death and survival. Thus, chemical coupling through gap junctions plays a key role in the spreading of injury between connected cells. Moreover, in addition to their involvement in cell-to-cell communication, mounting evidence indicates that connexins have additional gap junction-independent functions. Opening of unopposed hemichannels, located at the lateral surface of cardiomyocytes, may compromise cell homeostasis and may be involved in ischemia/reperfusion injury. In addition, connexins located at non-canonical cell structures, including mitochondria and the nucleus, have been demonstrated to be involved in cardioprotection and in regulation of cell growth and differentiation. In this review, we will provide, first, an overview on connexin biology, including their synthesis and degradation, their regulation and their interactions. Then, we will conduct an in-depth examination of the role of connexins in cardiac pathophysiology, including new findings regarding their involvement in myocardial ischemia/reperfusion injury, cardiac fibrosis, gene transcription or signaling regulation.

Mechano-Electric Coupling and Arrhythmogenic Current Generation in a Computational Model of Coupled Myocytes

A wide range of arrhythmogenic phenotypes have been associated with heterogeneous mechanical dyskinesis. Pro-arrhythmic effects are often associated with dysregulated intra-cellular calcium handling, especially via the development of intra- and inter-cellular calcium waves. Experimental evidence suggests that mechanical strain can contribute to the generation and maintenance of these calcium waves via a variety of mechano-electric coupling mechanisms. Most model studies of mechano-electric coupling mechanisms have been focused on mechano-sensitive ion channels, even though experimental studies have shown that intra- and inter-cellular calcium waves triggered by mechanical perturbations are likely to be more prevalent pro-arrhythmic mechanisms in the diseased heart. A one-dimensional strongly coupled computational model of electromechanics in rabbit ventricular cardiomyocytes showed that specific myocyte stretch sequences can modulate the susceptibility threshold for delayed after-depolarizations. In simulations of mechanically-triggered calcium waves in cardiomyocytes coupled to fibroblasts, susceptibility to calcium wave propagation was reduced as the current through the gap junction caused current drain from the myocytes. In 1D multi-cellular arrays coupled via gap junctions, mechanically-induced waves may contribute to synchronizing arrhythmogenic calcium waves and after-depolarizations.

Four-State Model for Simulating Kinetic and Steady-State Voltage-Dependent Gating of Gap Junctions

Gap junction (GJ) channels, formed of connexin (Cx) proteins, provide a direct pathway for metabolic and electrical cell-to-cell communication. These specialized channels are not just passive conduits for the passage of ions and metabolites but have been shown to gate robustly in response to transjunctional voltage, V_j, the voltage difference between two coupled cells. Voltage gating of GJs could play a physiological role, particularly in excitable cells, which can generate large transients in membrane potential during the propagation of action potentials. We present a mathematical/computational model of GJ channel voltage gating to assess properties of GJ channels that takes into account contingent gating of two series hemichannels and the distribution of V_j across each hemichannel. From electrophysiological recordings in cell cultures expressing Cx43 or Cx45, the principal isoforms expressed in cardiac tissue, various data sets were fitted simultaneously using global optimization. The results showed that the model is capable of describing both steady-state and kinetic properties of homotypic and heterotypic GJ channels composed of these Cxs. Moreover, mathematical analyses showed that the model can be simplified to a reversible two-state system and solved analytically using a rapid equilibrium assumption. Given that excitable cells are arranged in interconnected networks, the equilibrium assumption allows for a substantial reduction in computation time, which is useful in simulations of large clusters of coupled cells. Overall, this model can serve as a tool for the studying of GJ channel gating and its effects on the spread of excitation in networks of electrically coupled cells.

The control mechanisms of heart rate dynamics in a new heart rate nonlinear time series model

The control mechanisms and implications of heart rate variability (HRV) under the sympathetic (SNS) and parasympathetic nervous system (PNS) modulation remain poorly understood. Here, we establish the HR model/HRV responder using a nonlinear process derived from Newton's second law in stochastic self-restoring systems through dynamic analysis of physiological properties. We conduct model validation by testing, predictions, simulations, and sensitivity and time-scale analysis. We confirm that the outputs of the HRV responder can be accepted as the real data-generating process. Empirical studies show that the dynamic control mechanism of heart rate is a stable fixed point, rather than a strange attractor or transitions between a fixed point and a limit cycle; HR slope (amplitude) may depend on the ratio of cardiac disturbance or metabolic demand mean (standard deviation) to myocardial electrical resistance (PNS-SNS activity). For example, when metabolic demands remain unchanged, HR amplitude depends on PNS to SNS activity; when autonomic activity remains unchanged, HR amplitude during resting reflects basal metabolism. HR parameter alterations suggest that age-related decreased HRV, ultrareduced HRV in heart failure, and ultraelevated HRV in ST segment alterations refer to age-related decreased basal metabolism, impaired myocardial metabolism, and SNS hyperactivity triggered by myocardial ischemia, respectively.

Non-ohmic tissue conduction in cardiac electrophysiology: Upscaling the non-linear voltage-dependent conductance of gap junctions

Gap junctions are key mediators of intercellular communication in cardiac tissue, and their function is vital to sustaining normal cardiac electrical activity. Conduction through gap junctions strongly depends on the hemichannel arrangement and transjunctional voltage, rendering the intercellular conductance highly non-Ohmic, particularly under steady-state regimes of conduction. Despite this marked non-linear behavior, current tissue-level models of cardiac conduction are rooted in the assumption that gap-junctions conductance is constant (Ohmic), which results in inaccurate predictions of electrical propagation, particularly in the low junctional-coupling regime observed under pathological conditions. In this work, we present a novel non-Ohmic homogenization model (NOHM) of cardiac conduction that is suitable to tissue-scale simulations. Using non-linear homogenization theory, we develop a conductivity model that seamlessly upscales the voltage-dependent conductance of gap junctions, without the need of explicitly modeling gap junctions. The NOHM model allows for the simulation of electrical propagation in tissue-level cardiac domains that accurately resemble that of cell-based microscopic models for a wide range of junctional coupling scenarios, recovering key conduction features at a fraction of the computational complexity. A unique feature of the NOHM model is the possibility of upscaling the response of non-symmetric gap-junction conductance distributions, which result in conduction velocities that strongly depend on the direction of propagation, thus allowing to model the normal and retrograde conduction observed in certain regions of the heart. We envision that the NOHM model will enable organ-level simulations that are informed by sub- and inter-cellular mechanisms, delivering an accurate and predictive in-silico tool for understanding the heart function. Codes are available for download at https://github.com/dehurtado/NonOhmicConduction.

The heart relies on the propagation of electrical impulses that are mediated gap junctions, whose conduction properties vary depending on the transjunctional voltage. Despite this non-linear feature, current mathematical models assume that cardiac tissue behaves like an Ohmic (linear) material, thus delivering inaccurate results when simulated in a computer. Here we present a novel mathematical multiscale model that explicitly includes the non-Ohmic response of gap junctions in its predictions. Our results show that the proposed model recovers important conduction features modulated by gap junctions at a fraction of the computational complexity. This contribution represents an important step towards constructing computer models of a whole heart that can predict organ-level behavior in reasonable computing times.

Properties of cardiac conduction in a cell-based computational model

The conduction of electrical signals through cardiac tissue is essential for maintaining the function of the heart, and conduction abnormalities are known to potentially lead to life-threatening arrhythmias. The properties of cardiac conduction have therefore been the topic of intense study for decades, but a number of questions related to the mechanisms of conduction still remain unresolved. In this paper, we demonstrate how the so-called EMI model may be used to study some of these open questions. In the EMI model, the extracellular space, the cell membrane, the intracellular space and the cell connections are all represented as separate parts of the computational domain, and the model therefore allows for study of local properties that are hard to represent in the classical homogenized bidomain or monodomain models commonly used to study cardiac conduction. We conclude that a non-uniform sodium channel distribution increases the conduction velocity and decreases the time delays over gap junctions of reduced coupling in the EMI model simulations. We also present a theoretical optimal cell length with respect to conduction velocity and consider the possibility of ephaptic coupling (i.e. cell-to-cell coupling through the extracellular potential) acting as an alternative or supporting mechanism to gap junction coupling. We conclude that for a non-uniform distribution of sodium channels and a sufficiently small intercellular distance, ephaptic coupling can influence the dynamics of the sodium channels and potentially provide cell-to-cell coupling when the gap junction connection is absent.

The electrochemical wave traversing the heart during every beat is essential for cardiac pumping function and supply of blood to the body. Understanding the stability of this wave is crucial to understanding how lethal arrhythmias are generated. Despite this importance, our knowledge of the physical determinants of wave propagation are still evolving. One particular challenge has been the lack of accurate mathematical models of conduction at the cellular level. Because cardiac muscle is an electrical syncytium, in which direct charge transfer between cells drives wave propagation, classical bidomain and monodomain tissue models employ a homogenized approximation of this process. This approximation is not valid at the length scale of single cells, and prevents any analysis of how cellular structures impact cardiac conduction. Instead, so-called microdomain models must be used for these questions. Here we utilize a recently developed modelling framework that is well suited to represent small collections of cells. By applying this framework, we show that concentration of sodium channels at the longitudinal borders of myocytes accelerates cardiac conduction. We also demonstrate that when juxtaposed cells are sufficiently close, this non-uniform distribution induces large ephaptic currents, which contribute to intercellular coupling.

Ephaptic coupling rescues conduction failure in weakly coupled cardiac tissue with voltage-gated gap junctions

Electrical conduction in cardiac tissue is usually considered to be primarily facilitated by gap junctions, providing a pathway between the intracellular spaces of neighboring cells. However, recent studies have highlighted the role of coupling via extracellular electric fields, also known as ephaptic coupling, particularly in the setting of reduced gap junction expression. Further, in the setting of reduced gap junctional coupling, voltage-dependent gating of gap junctions, an oft-neglected biophysical property in computational studies, produces a positive feedback that promotes conduction failure. We hypothesized that ephaptic coupling can break the positive feedback loop and rescue conduction failure in weakly coupled cardiac tissue. In a computational tissue model incorporating voltage-gated gap junctions and ephaptic coupling, we demonstrate that ephaptic coupling can rescue conduction failure in weakly coupled tissue. Further, ephaptic coupling increased conduction velocity in weakly coupled tissue, and importantly, reduced the minimum gap junctional coupling necessary for conduction, most prominently at fast pacing rates. Finally, we find that, although neglecting gap junction voltage-gating results in negligible differences in well coupled tissue, more significant differences occur in weakly coupled tissue, greatly underestimating the minimal gap junctional coupling that can maintain conduction. Our study suggests that ephaptic coupling plays a conduction-preserving role, particularly at rapid heart rates.

Illuminating Myocyte-Fibroblast Homotypic and Heterotypic Gap Junction Dynamics Using Dynamic Clamp

Fibroblasts play a significant role in the development of electrical and mechanical dysfunction of the heart; however, the underlying mechanisms are only partially understood. One widely studied mechanism suggests that fibroblasts produce excess extracellular matrix, resulting in collagenous septa that slow propagation, cause zig-zag conduction paths, and decouple cardiomyocytes, resulting in a substrate for cardiac arrhythmia. An emerging mechanism suggests that fibroblasts promote arrhythmogenesis through direct electrical interactions with cardiomyocytes via gap junction (GJ) channels. In the heart, three major connexin (Cx) isoforms, Cx40, Cx43, and Cx45, form GJ channels in cell-type-specific combinations. Because each Cx is characterized by a unique time- and transjunctional voltage-dependent profile, we investigated whether the electrophysiological contributions of fibroblasts would vary with the specific composition of the myocyte-fibroblast (M-F) GJ channel. Due to the challenges of systematically modifying Cxs in vitro, we coupled native cardiomyocytes with in silico fibroblast and GJ channel electrophysiology models using the dynamic-clamp technique. We found that there is a reduction in the early peak of the junctional current during the upstroke of the action potential (AP) due to GJ channel gating. However, effects on the cardiomyocyte AP morphology were similar regardless of the specific type of GJ channel (homotypic Cx43 and Cx45, and heterotypic Cx43/Cx45 and Cx45/Cx43). To illuminate effects at the tissue level, we performed multiscale simulations of M-F coupling. First, we developed a cell-specific model of our dynamic-clamp experiments and investigated changes in the underlying membrane currents during M-F coupling. Second, we performed two-dimensional tissue sheet simulations of cardiac fibrosis and incorporated GJ channels in a cell type-specific manner. We determined that although GJ channel gating reduces junctional current, it does not significantly alter conduction velocity during cardiac fibrosis relative to static GJ coupling. These findings shed more light on the complex electrophysiological interplay between cardiac fibroblasts and myocytes.

Islands of spatially discordant APD alternans underlie arrhythmogenesis by promoting electrotonic dyssynchrony in models of fibrotic rat ventricular myocardium

Fibrosis and altered gap junctional coupling are key features of ventricular remodelling and are associated with abnormal electrical impulse generation and propagation. Such abnormalities predispose to reentrant electrical activity in the heart. In the absence of tissue heterogeneity, high-frequency impulse generation can also induce dynamic electrical instabilities leading to reentrant arrhythmias. However, because of the complexity and stochastic nature of such arrhythmias, the combined effects of tissue heterogeneity and dynamical instabilities in these arrhythmias have not been explored in detail. Here, arrhythmogenesis was studied using in vitro and in silico monolayer models of neonatal rat ventricular tissue with 30% randomly distributed cardiac myofibroblasts and systematically lowered intercellular coupling achieved in vitro through graded knockdown of connexin43 expression. Arrhythmia incidence and complexity increased with decreasing intercellular coupling efficiency. This coincided with the onset of a specialized type of spatially discordant action potential duration alternans characterized by island-like areas of opposite alternans phase, which positively correlated with the degree of connexinx43 knockdown and arrhythmia complexity. At higher myofibroblast densities, more of these islands were formed and reentrant arrhythmias were more easily induced. This is the first study exploring the combinatorial effects of myocardial fibrosis and dynamic electrical instabilities on reentrant arrhythmia initiation and complexity.

Computational Approaches to Understanding the Role of Fibroblast-Myocyte Interactions in Cardiac Arrhythmogenesis

The adult heart is composed of a dense network of cardiomyocytes surrounded by nonmyocytes, the most abundant of which are cardiac fibroblasts. Several cardiac diseases, such as myocardial infarction or dilated cardiomyopathy, are associated with an increased density of fibroblasts, that is, fibrosis. Fibroblasts play a significant role in the development of electrical and mechanical dysfunction of the heart; however the underlying mechanisms are only partially understood. One widely studied mechanism suggests that fibroblasts produce excess extracellular matrix, resulting in collagenous septa. These collagenous septa slow propagation, cause zig-zag conduction paths, and decouple cardiomyocytes resulting in a substrate for arrhythmia. Another emerging mechanism suggests that fibroblasts promote arrhythmogenesis through direct electrical interactions with cardiomyocytes via gap junctions. Due to the challenges of investigating fibroblast-myocyte coupling in native cardiac tissue, computational modeling and in vitro experiments have facilitated the investigation into the mechanisms underlying fibroblast-mediated changes in cardiomyocyte action potential morphology, conduction velocity, spontaneous excitability, and vulnerability to reentry. In this paper, we summarize the major findings of the existing computational studies investigating the implications of fibroblast-myocyte interactions in the normal and diseased heart. We then present investigations from our group into the potential role of voltage-dependent gap junctions in fibroblast-myocyte interactions.

A brief history of tissue models for cardiac electrophysiology

The last four decades have produced a number of significant advances in the developments of computer models to simulate and investigate the electrical activity of cardiac tissue. The tissue descriptions that underlie these simulations have been built from a combination of clever insight and careful comparison with measured data at multiple scales. Tissue models have not only led to greater insights into the mechanisms of life-threatening arrhythmias but have been used to engineer new therapies to treat the consequences of cardiac disease. This paper is a look back at the early years in the cardiac modeling and the challenges facing the field as models move toward the clinic.

Stem cells can form gap junctions with cardiac myocytes and exert pro-arrhythmic effects

Stem cell therapy has been suggested to be a promising option for regeneration of injured myocardium, for example following a myocardial infarction. For clinical use cell-based therapies have to be safe and applicable and are aimed to renovate the architecture of the heart. Yet for functional and coordinated activity synchronized with the host myocardium stem cells have to be capable of forming electrical connections with resident cardiomyocytes. In this paper we discuss whether stem cells are capable of establishing functional electrotonic connections with cardiomyocytes and whether these may generate a risk for arrhythmias. Application of stem cells in the clinical setting with outcomes concerning arrhythmogenic safety and future perspectives will also briefly be touched upon.

Regulation of gap junctions by nitric oxide influences the generation of arrhythmias resulting from acute ischemia and reperfusion in vivo

Myocardial ischemia resulting from sudden occlusion of a coronary artery is one of the major causes in the appearance of severe, often life-threatening ventricular arrhythmias. Although the underlying mechanisms of these acute arrhythmias are many and varied, there is no doubt that uncoupling of gap junctions (GJs) play an important role especially in arrhythmias that are generated during phase Ib, and often terminate in sudden cardiac death. In the past decades considerable efforts have been made to explore mechanisms which regulate the function of GJs, and to find new approaches for protection against arrhythmias through the modulation of GJs. These investigations led to the development of GJ openers and inhibitors. The pharmacological modulation of GJs, however, resulted in conflicting results. It is still not clear whether opening or closing of GJs would be advantageous for the ischemic myocardium. Both maneuvers can result in protection, depending on the models, endpoints and the time of opening and closing of GJs. Furthermore, although there is substantial evidence that preconditioning decreases or delays the uncoupling of GJs, the precise mechanisms by which this attains have not yet been elucidated. In our own studies in anesthetized dogs preconditioning suppressed the ischemia and reperfusion-induced ventricular arrhythmias, and this protection was associated with the preservation of GJ function, manifested in less marked changes in electrical impedance, as well as in the maintenance of GJ permeability and phosphorylation of connexin43. Since we have substantial previous evidence that nitric oxide (NO) is an important trigger and mediator of the preconditioning-induced antiarrhythmic protection, we hypothesized that NO, among its several effects, may lead to this protection by influencing cardiac GJs. The hypotheses and theories relating to the pharmacological modulation of GJs will be discussed with particular attention to the role of NO.

Nonlinear gap junctions enable long-distance propagation of pulsating calcium waves in astrocyte networks

A new paradigm has recently emerged in brain science whereby communications between glial cells and neuron-glia interactions should be considered together with neurons and their networks to understand higher brain functions. In particular, astrocytes, the main type of glial cells in the cortex, have been shown to communicate with neurons and with each other. They are thought to form a gap-junction-coupled syncytium supporting cell-cell communication via propagating Ca²⁺ waves. An identified mode of propagation is based on cytoplasm-to-cytoplasm transport of inositol trisphosphate (IP₃) through gap junctions that locally trigger Ca²⁺ pulses via IP₃-dependent Ca²⁺-induced Ca²⁺ release. It is, however, currently unknown whether this intracellular route is able to support the propagation of long-distance regenerative Ca²⁺ waves or is restricted to short-distance signaling. Furthermore, the influence of the intracellular signaling dynamics on intercellular propagation remains to be understood. In this work, we propose a model of the gap-junctional route for intercellular Ca²⁺ wave propagation in astrocytes. Our model yields two major predictions. First, we show that long-distance regenerative signaling requires nonlinear coupling in the gap junctions. Second, we show that even with nonlinear gap junctions, long-distance regenerative signaling is favored when the internal Ca²⁺ dynamics implements frequency modulation-encoding oscillations with pulsating dynamics, while amplitude modulation-encoding dynamics tends to restrict the propagation range. As a result, spatially heterogeneous molecular properties and/or weak couplings are shown to give rise to rich spatiotemporal dynamics that support complex propagation behaviors. These results shed new light on the mechanisms implicated in the propagation of Ca²⁺ waves across astrocytes and the precise conditions under which glial cells may participate in information processing in the brain.

In recent years, the focus of Cellular Neuroscience has progressively stopped only being on neurons but started to include glial cells as well. Indeed, astrocytes, the main type of glial cells in the cortex, dynamically modulate neuron excitability and control the flow of information across synapses. Moreover, astrocytes have been shown to communicate with each other over long distances using calcium waves. These waves spread from cell to cell via molecular gates called gap junctions, which connect neighboring astrocytes. In this work, we used a computer model to question what biophysical mechanisms could support long-distance propagation of Ca²⁺ wave signaling. The model shows that the coupling function of the gap junction must be non-linear and include a threshold. This prediction is largely unexpected, as gap junctions are classically considered to implement linear functions. Recent experimental observations, however, suggest their operation could actually be more complex, in agreement with our prediction. The model also shows that the distance traveled by waves depends on characteristics of the internal astrocyte dynamics. In particular, long-distance propagation is facilitated when internal calcium oscillations are in their frequency-modulation encoding mode and are pulsating. Hence, this work provides testable experimental predictions to decipher long-distance communication between astrocytes.

Bistable dynamics of cardiac cell models coupled by dynamic gap junctions linked to cardiac memory

In an earlier study, we suggested that adaptive gap junctions (GJs) might be a basis of cardiac memory, a phenomenon which refers to persistent electrophysiological response of the heart to external pacing. Later, it was also shown that the proposed mechanism of adaptation of GJs is consistent with known electrophysiology of GJs. In the present article, we show that a pair of cardiac cell models coupled by dynamic, voltage-sensitive GJs exhibits bistable behavior under certain conditions. Three kinds of cell pairs are considered: (1) a Noble-Noble cell pair that represents adjacent cells in Purkinje network, (2) a pair of DiFranceso-Noble cells that represents adjacent SA nodal cells, and (3) a model of Noble cell coupled to Luo-Rudy cell model, which represents an interacting pair of a Purkinje fiber and a ventricular myocyte. Bistability is demonstrated in all the three cases. We suggest that this bistability might be an underlying factor behind cardiac memory. Focused analysis of a pair of Noble cell models showed that bistability is obtained only when the properties of GJs "match" with the properties of the pair of cells that is coupled by the GJs. This novel notion of match between GJs and cardiac cell types might give an insight into specialized distributions of various connexin proteins in cardiac tissue.

Monotone dynamics of two cells dynamically coupled by a voltage-dependent gap junction

We introduce and analyse a simple model for two non-excitable cells that are dynamically coupled by a gap junction, a plaque of aqueous channels that electrically couple the cells. The gap junction channels have a low and high conductance state, and the transition rates between these states are voltage-dependent. We show that the number and stability of steady states of the system has a simple relationship with the determinant of the Jacobian matrix. For the case that channel opening rates decrease with increasing trans-junctional voltage, and closing rates increase with increasing trans-junctional voltage, we show that the system is monotone, with tridiagonal Jacobian matrix, and hence every initial condition evolves to a steady state, but that there may be multiple steady states.

Early afterdepolarisations and ventricular arrhythmias in cardiac tissue: a computational study

Afterdepolarisations are associated with arrhythmias in the heart, but are difficult to study experimentally. In this study we used a simplified computational model of 1D and 2D cardiac ventricular tissue, where we could control the size of the region generating afterdepolarisations, as well as the properties of the afterdepolarisation waveform. Provided the size of the afterdepolarisation region was greater than around 1 mm, propagating extrasystoles were produced in both 1D and 2D. The number of extrasystoles produced depended on the amplitude, period, and duration of the oscillatory EAD waveform. In 2D, re-entry was also initiated for specific combinations of EAD amplitude, period, and duration, with the afterdepolarisation region acting as a common pathway. The main finding from this modelling study is therefore that afterdepolarisations can act as potent sources of propagating extrasystoles, as well as a source of re-entrant activation.

Enhancement of ventricular gap-junction coupling by rotigaptide

AIMS

Rotigaptide is proposed to exert its anti-arrhythmic effects by improving myocardial gap-junction communication. To directly investigate the mechanisms of rotigaptide action, we treated cultured neonatal murine ventricular cardiomyocytes with clinical pharmacological doses of rotigaptide and directly determined its effects on gap-junctional currents.

METHODS AND RESULTS

Neonatal murine ventricular cardiomyocytes were enzymatically isolated and cultured for 1-4 days. Primary culture cell pairs were subjected to dual whole cell patch-clamp procedures to directly measure gap-junctional currents (I(j)) and voltage (V(j)). Rotigaptide (0-350 nM) was applied overnight or acutely perfused into 35 mm culture dishes. Rotigaptide (35-100 nM) acutely and chronically increased the resting gap-junction conductance (g(j)), and normalized steady-state minimum g(j) (G(min)) by 5-20%. Higher concentrations produced a diminishing response, which mimics the observed therapeutic efficacy of the drug. The inactivation kinetics was similarly slowed in a therapeutic concentration-dependent manner without affecting the V(j) dependence of inactivation or recovery. The effects of 0-100 nM rotigaptide on ventricular g(j) during cardiac action potential propagation were accurately modelled by computer simulations which demonstrate that clinically effective concentrations of rotigaptide can partially reverse conduction slowing due to decreases in g(j) and inactivation.

CONCLUSION

These results demonstrate that therapeutic concentrations of rotigaptide increase the resting gap-junction conductance and reduce the magnitude and kinetics of steady-state inactivation in a concentration-dependent manner. Rotigaptide may be effective in treating re-entrant forms of cardiac arrhythmias by improving conduction and preventing the formation of re-entrant circuits in partially uncoupled myocardium.

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.

Electrical and structural remodeling in left ventricular hypertrophy-a substrate for a decrease in QRS voltage?

Electrical remodeling in advanced stages of cardiovascular diseases creates a substrate for triggering and maintenance of arrhythmias. The electrical remodeling is a continuous process initiated already in the early stages of cardiological pathology. The aim of this opinion article was to discuss the changes in electrical properties of myocardium in left ventricular hypertrophy (LVH), with special focus on its early stage, as well as their possible reflection in the QRS amplitude of the electrocardiogram. It critically appraises the classical hypothesis related to the QRS voltage changes in LVH. The hypothesis of the relative voltage deficit is discussed in the context of supporting evidence from clinical studies, animal experiments, and simulation studies. The underlying determinants of electrical impulse propagation which may explain discrepancies between "normal" ECG findings and increased left ventricular size/mass in LVH are reviewed.

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.

Transverse propagation in an expanded PSpice model for cardiac muscle with gap-junction ion channels

Transverse propagation was previously found to occur in a two-dimensional model of cardiac muscle using the PSpice software program for electronic circuit design and analysis. Longitudinal propagation within each chain, and transverse propagation between parallel chains, occurred even when there were no gap-junction (g-j) channels inserted between the simulated myocardial cells either longitudinally or transversely. In those studies, there were pronounced edge (boundary) effects and end-effects even within single chains. Transverse velocity increased with increase in model size. The present study was performed to examine boundary effects on transverse propagation velocity when the length of the chains was held constant at 10 cells and the number of parallel chains was varied from 3 to 5, to 7, to 10, and to 20. The number of g-j channels was either zero, both longitudinally and transversely (0/0), or 100/100. Some experiments were also made at 100/0, 1/1, and 10/10. Transverse velocity and overall velocity (both longitudinal and transverse components) was calculated from the measured total propagation time (TPT), i.e., the elapsed time between when the first action potential (AP) and the last AP crossed the zero potential level. The transverse g-j channels were placed only at the ends of each chain, such that propagation would occur in a zigzag pattern. Electrical stimulation was applied intracellularly between cells A1 and A2. It was found that, with no g-j channels (0/0), overall velocity increased almost linearly when more and more chains were placed in parallel. In contrast, with many g-j channels (100/100), there was a much flatter relationship between overall velocity and number of parallel chains. The difference in velocities with 0/0 channels and 100/100 channels was reduced as the number of chains was increased. In conclusion, edges have important effects on propagation velocity (overall and transverse) in cardiac muscle simulations.

Dynamic behavior of gap junctions in each cardiac cycle: a novel view on the electrical coupling of normal cardiocytes

Two main mechanisms have been suggested for the propagation of action potentials in cardiac muscle cells: (1) the free flow of local circuit current through gap junctions and (2) the effect of electrical field. Different evidences confirm each of two mechanisms. We think that gap junctions are not continuously open in a normal heart cycle; instead, they open and close intermittently. In other words, gap junction has a dynamic behavior in each cardiac cycle, managing different routes of propagation in the diverse moments of normal cycle. Gap junctions could be open in phases 0, 1, 3, 4 and close in phase 2 (plateau) of action potential. Whenever gap junction is open, conduction can be fulfilled rapidly by current flow and whenever it is closed, the electrical field will be the main route of propagation. When the prejunctional cell is in the peak of action potential(AP), gap junction is closed and the postjunctional cell should use the electrical field to be stimulated. Then, when the prejunctional cell comes to the end of AP, the gap junction opens and current will potentiate the rising phase of AP in the postjunctional cell. Moreover, this process causes accumulation of calcium in the postjunctional cell near phase 2. We believe that our hypothesis on the mechanism of cardiac action potential propagation may have exciting advantages. This novel view on gap junction dynamic behavior may be useful for better exploitation of drugs or designing new remedies in arrhythmias. We also hypothesize that in conditions as cardiac failure, in which cardiac contractility is diminished and increasing intracellular calcium concentration is needed, gap junction closing drugs may be effective. It is worth noting that future clinical studies are needed to validate these predictions.

Effect of transverse gap-junction channels on transverse propagation in an enlarged PSpice model of cardiac muscle

BACKGROUND

In previous PSpice modeling studies of simulated action potentials (APs) in parallel chains of cardiac muscle, it was found that transverse propagation could occur between adjacent chains in the absence of gap-junction (gj) channels, presumably by the electric field (EF) generated in the narrow interstitial space between the chains. Transverse propagation was sometimes erratic, the more distal chains firing out of order.

METHODS

In the present study, the propagation of complete APs was studied in a 2-dimensional network of 100 cardiac muscle cells (10 x 10 model). Various numbers of gj-channels (assumed to be 100 pS each) were inserted across the junctions between the longitudinal cells of each chain and between adjacent chains (only at the end cells of each chain). The shunt resistance produced by the gj-channels (Rgj) was varied from 100,000 M omega (0 gj-channels) to 1,000 M omega (10 channels), 100 M omega (100 channels) and 10 M omega (1,000 channels). Total propagation time (TPT) was measured as the difference between the times when the AP rising phase of the first cell (cell # A1) and the last cell (in the J chain) crossed 0 mV. When there were no gj-channels, the excitation was transmitted between cells by the EF, i.e., the negative potential generated in the narrow junctional clefts (e.g., 100 angstroms) when the prejunctional membrane fired an AP. For the EF mechanism to work, the prejunctional membrane must fire a fraction of a millisecond before the adjacent surface membrane. When there were many gj-channels (e.g., 100 or 1,000), the excitation was transmitted by local-circuit current flow from one cell to the next through these channels.

RESULTS

TPT was measured as a function of four different numbers of transverse gj-channels, namely 0, 10, 100 and 1,000, and four different numbers of longitudinal gj-channels, namely 0, 10, 100 and 1,000. Thus, 16 different measurements were made. It was found that increasing the number of transverse channels had no effect on TPT when the number of longitudinal channels was low (i.e., 0 or 10). In contrast, when the number of longitudinal gj-channels was high (e.g., 100 or 1,000), then increasing the number of transverse channels decreased TPT markedly.

CONCLUSION

Thus, complete APs could propagate along a network of 100 cardiac muscle cells even when no gj-channels were present between the cells. Insertion of transverse gj-channels greatly speeded propagation through the 10 x 10 network when there were also many longitudinal gj-channels.

Modeling the Calcium Gate of Cardiac Gap Junction Channel

We addressed the question how Ca2+ transients affect gap junction conductance (Gj) during action potential (AP) propagation by constructing a dynamic gap junction model coupled with a cardiac cell model. The kinetics of the Ca2+ gate was determined based on published experimental findings that the Hill coefficient for the [Ca2+]i-Gj relationship ranges from 3 to 4, indicating multiple ion bindings. It is also suggested that the closure of the Ca2+ gate follows a single exponential time course. After adjusting the model parameters, a two-state (open-closed) model, assuming simultaneous ion bindings, well described both the single exponential decay and the [Ca2+]i-Gj relationship. Using this gap junction model, 30 cardiac cell models were electrically connected in a one-dimensional cable. However, Gj decreased in a cumulative manner by the repetitive Ca2+ transients, and a conduction block was observed. We found that a reopening of the Ca2+ gate is possible only by assuming a sequential ion binding with one rate limiting step in a multistate model. In this model, the gating time constant (T) has a bell-shaped dependence on [Ca2+]i, with a peak around the half-maximal concentration of [Ca2+]i. Here we propose a five-state model including four open states and one closed state, which allows normal AP propagation; namely, the Gj is decreased -15% by a single Ca2+ transient, but well recovers to the control level during diastole. Under the Ca(2+)-overload condition, however, the conduction velocity is indeed decreased as demonstrated experimentally. This new gap junction model may also be useful in simulations of the ventricular arrhythmia.

Action potential repolarization enabled by Ca++ channel deactivation in PSpice simulation of smooth muscle propagation

BACKGROUND

Previously, only the rising phase of the action potential (AP) in cardiac muscle and smooth muscle could be simulated due to the instability of PSpice upon insertion of a second black box (BB) into the K+ leg of the basic membrane unit. This restriction was acceptable because only the transmission of excitation from one cell to the next was investigated.

METHODS

In the current work, the repolarization of the AP was accomplished by inserting a second BB into the Ca++ leg of the basic membrane unit. Repolarization of the AP was produced, not through an activation of the K+ channel conductance, but rather through a mimicking of the deactivation of the Ca++ channel conductance. Propagation of complete APs was studied in a chain (strand) of 10 smooth muscle cells, in which various numbers of gap-junction (gj) channels (assumed to be 100 pS each) were inserted across the cell junctions.

RESULTS

The shunt resistance across the junctions produced by the gj-channels (Rgj) was varied from 100,000 MOmega (0 gj-channels) to 10,000 MOmega (1 gj-channel), to 1,000 MOmega (10 channels), to 100 MOmega (100 channels), to 10 MOmega (1000 channels), and to 1.0 MOmega (10,000 channels). Velocity of propagation (theta, in cm/sec) was calculated from the measured total propagation time (TPT, the time difference between when the AP rising phase of the first cell and the last cell crossed -20 mV), assuming a constant cell length of 200 microm. When there were no gj-channels, or only one, the transmission of excitation between cells was produced by the electric field (EF), i.e., the negative junctional cleft potential, that is generated in the narrow junctional clefts (e.g., 100 A) when the prejunctional membrane fires an AP (a fraction of a millisecond before the adjacent surface membrane). There were significant end-effects at the termination of the strand, such that the last cell (cell #10) failed to fire, or fired after a prolonged delay. This end-effect was abolished when the strand termination resistance (Rbt) was increased from 1.0 KOmega to 600 MOmega. When there were 1000 or 10,000 gj-channels, the transmission of excitation was produced by local-circuit current flow from one cell to the next through the gj-channels.

DISCUSSION

In summary, it is now possible to simulate complete APs in smooth muscle cells that could propagate along a single chain of 10 cells, even when there were no gj-channels between the cells.

Repolarization of the action potential enabled by Na+ channel deactivation in PSpice simulation of cardiac muscle propagation

BACKGROUND

In previous studies on propagation of simulated action potentials (APs) in cardiac muscle using PSpice modeling, we reported that a second black-box (BB) could not be inserted into the K+ leg of the basic membrane unit because that caused the PSpice program to become very unstable. Therefore, only the rising phase of the APs could be simulated. This restriction was acceptable since only the mechanism of transmission of excitation from one cell to the next was being investigated.

METHODS AND RESULTS

We have now been able to repolarize the AP by inserting a second BB into the Na+ leg of the basic units. This second BB effectively mimicked deactivation of the Na+ channel conductance. This produced repolarization of the AP, not by activation of K+ conductance, but by deactivation of the Na+ conductance. The propagation of complete APs was studied in a chain (strand) of 10 cardiac muscle cells, in which various numbers of gap-junction (gj) channels (assumed to be 100 pS each) were inserted across the cell junctions. The shunt resistance across the junctions produced by the gj-channels (Rgj) was varied from 100,000 M? (0 gj-channels) to 10,000 M? (1 gj-channel), to 1,000 M? (10 channels), to 100 M? (100 channels), and 10 M? (1000 channels). The velocity of propagation (theta, in cm/s) was calculated from the measured total propagation time (TPT, the time difference between when the AP rising phase of the first cell and the last cell crossed -20 mV, assuming a cell length of 150 microm. When there were no gj-channels, or only a few, the transmission of excitation between cells was produced by the electric field (EF), i.e. the negative junctional cleft potential, that is generated in the narrow junctional clefts (e.g. 100 A) when the prejunctional membrane fires an AP. When there were many gj-channels (e.g. 1000 or 10,000), the transmission of excitation was produced by local-circuit current flow from one cell to the next through the gj-channels.

CONCLUSION

We have now been able to simulate complete APs in cardiac muscle cells that could propagate along a single chain of 10 cells, even when there were no gj-channels between the cells.

Surface contact measurement of electrical cell uncoupling in the mouse heart during ischemia

Electrical cell uncoupling via gap junction closure is assumed to cause characteristic changes of the passive dielectric spectrum of ischemic heart tissue. In order to find an independent evidence for this assumption, we analysed heart tissue during ischemia, measured the open state of gap junctions by means of dye transfer and correlated this parameter with the time course of the dielectric permittivity. The hearts were pre-ischemically arrested by perfusion with Ringer solution containing 20 mmol/L of potassium (group KCL, n=10). This solution was also used with the addition of two gap junction blockers, either 3 mmol/L heptanol (group HEP, n=4) or 20 micromol/L palmitoleic acid (group PA, n=7). During subsequent ischemia at 21.0+/-0.5 degrees C, we monitored the passive dielectric permittivity spectrum and the spread of dye. After a sigmoidal increase the dielectric permittivity reached an upper plateau at 61+/-22 min of ischemia in KCL, at 45+/-7 min in PA, and already during perfusion at 2+/-1 min in group HEP. At the beginning of ischemia, dye migrated to neighbouring cells in groups KCL and PA but not in HEP. In KCL and PA, the intercellular diffusion of dye stopped after 64+/-26 and 40+/-11 min of ischemia, respectively. Our results suggest that the sigmoidal increase in dielectric permittivity and the reduction of dye diffusion depend on a common mechanism, namely gap junction closure.

Combined electric field and gap junctions on propagation of action potentials in cardiac muscle and smooth muscle in PSpice simulation

Propagation of action potentials in cardiac muscle and smooth muscle were simulated using the PSpice program. Excitation was transmitted from cell to cell along a strand of 6 cells (cardiac muscle) or 10 cells (smooth muscle) either not connected (control) or connected by low-resistance tunnels (gap-junction connexons). A significant negative cleft potential (V(jv) ) develops in the narrow junctional cleft when the pre-JM fires. V(jc) depolarizes the postjunctional membrane (post-JM) to threshold by a patch-clamp action. With few connecting tunnels, cell-to-cell transmission by the EF mechanism was facilitated. With many tunnels, propagation was dominated by the low-resistance mechanism, and propagation velocity (theta) became very fast and nonphysiological. In conclusion, when the 2 mechanisms for cell-to-cell transfer of excitation were combined, the two mechanisms facilitated each other in a synergistic manner. When there were many connecting tunnels, the tunnel mechanism was dominant.

Propagated repolarization of simulated action potentials in cardiac muscle and smooth muscle

BACKGROUND

Propagation of repolarization is a phenomenon that occurs in cardiac muscle. We wanted to test whether this phenomenon would also occur in our model of simulated action potentials (APs) of cardiac muscle (CM) and smooth muscle (SM) generated with the PSpice program.

METHODS

A linear chain of 5 cells was used, with intracellular stimulation of cell #1 for the antegrade propagation and of cell #5 for the retrograde propagation. The hyperpolarizing stimulus parameters applied for termination of the AP in cell #5 were varied over a wide range in order to generate strength / duration (S/D) curves. Because it was not possible to insert a second "black box" (voltage-controlled current source) into the basic units representing segments of excitable membrane that would allow the cells to respond to small hyperpolarizing voltages, gap-junction (g.j.) channels had to be inserted between the cells, represented by inserting a resistor (Rgj) across the four cell junctions.

RESULTS

Application of sufficient hyperpolarizing current to cell #5 to bring its membrane potential (Vm) to within the range of the sigmoidal curve of the Na+ conductance (CM) or Ca++ conductance (SM) terminated the AP in cell #5 in an all-or-none fashion. If there were no g.j. channels (Rgj = infinity), then only cell #5 repolarized to its stable resting potential (RP; -80 mV for CM and -55 mV for SM). The positive junctional cleft potential (VJC) produced only a small hyperpolarization of cell #4. However, if many g.j. channels were inserted, more hyperpolarizing current was required (for a constant duration) to repolarize cell #5, but repolarization then propagated into cells 4, 3, 2, and 1. When duration of the pulses was varied, a typical S/D curve, characteristic of excitable membranes, was produced. The chronaxie measured from the S/D curve was about 1.0 ms, similar to that obtained for muscle membranes.

CONCLUSIONS

These experiments demonstrate that normal antegrade propagation of excitation can occur in the complete absence of g.j. channels, and therefore no low-resistance pathways between cells, by the electric field (negative VJC) developed in the narrow junctional clefts. Because it was not possible to insert a second black-box into the basic units that would allow the cells to respond to small hyperpolarizing voltages, only cell #5 (the cell injected with hyperpolarizing pulses) repolarized in an all-or-none manner. But addition of many g.j. channels allowed repolarization to propagate in a retrograde direction over all 5 cells.

Gap-junction channels inhibit transverse propagation in cardiac muscle

The effect of adding many gap-junctions (g-j) channels between contiguous cells in a linear chain on transverse propagation between parallel chains was examined in a 5 × 5 model (5 parallel chains of 5 cells each) for cardiac muscle. The action potential upstrokes were simulated using the PSpice program for circuit analysis. Either a single cell was stimulated (cell A1) or the entire chain was stimulated simultaneously (A-chain). Transverse velocity was calculated from the total propagation time (TPT) from when the first AP crossed a V_m of -20 mV and the last AP crossed -20 mV. The number of g-j channels per junction was varied from zero to 100, 1,000 and 10,000 (R_gj of ∞, 100 MΩ, 10 MΩ, 1.0 MΩ, respectively). The longitudinal resistance of the interstitial fluid (ISF) space between the parallel chains (R_ol2) was varied between 200 KΩ (standard value) and 1.0, 5.0, and 10 MΩ. The higher the R_ol2 value, the tighter the packing of the chains. It was found that adding many g-j channels inhibited transverse propagation by blocking activation of all 5 chains, unless R_ol2 was greatly increased above the standard value of 200 KΩ. This was true for either method of stimulation. This was explained by, when there is strong longitudinal coupling between all 5 cells of a chain awaiting excitation, there must be more transfer energy (i.e., more current) to simultaneously excite all 5 cells of a chain.

Dynamic model for ventricular junctional conductance during the cardiac action potential

The ventricular action potential was applied to paired neonatal murine ventricular myocytes in the dual whole cell configuration. During peak action potential voltages >100 mV, junctional conductance (g(j)) declined by 50%. This transjunctional voltage (V(j))-dependent inactivation exhibited two time constants that became progressively faster with increasing V(j). G(j) returned to initial peak values during action potential repolarization and even exceeded peak g(j) values during the final 5% of repolarization. This facilitation of g(j) was observed <30 mV during linearly decreasing V(j) ramps. The same behavior was observed in ensemble averages of individual gap junction channels with unitary conductances of 100 pS or lower. Immunohistochemical fluorescent micrographs and immunoblots detect prominent amounts of connexin (Cx)43 and lesser amounts of Cx40 and Cx45 proteins in cultured ventricular myocytes. The time dependence of the g(j) curves and channel conductances are consistent with the properties of predominantly homomeric Cx43 gap junction channels. A mathematical model depicting two inactivation and two recovery phases accurately predicts the ventricular g(j) curves at different rates of stimulation and repolarization. Functional differences are apparent between ventricular myocytes and Cx43-transfected N2a cell gap junctions that may result from posttranslational modification. These observations suggest that gap junctions may play a role in the development of conduction block and the genesis and propagation of triggered arrhythmias under conditions of slowed conduction (<10 cm/s).

A biophysical model of vertebrate olfactory epithelium and bulb exhibiting gap junction dependent odor-evoked spatiotemporal patterns of activity

This work describes a biophysical model of the initial stages of vertebrate olfactory system containing structures representing the olfactory epithelium and bulb. Its main novelty is the introduction of gap junctions connecting neurons both in the epithelium and bulb, and of biologically detailed dendrodendritic synapses between granule and mitral cells in the bulb. The model was used to simulate the effect of an odor presentation on the neural activity pattern in the epithelium and bulb. During the time for which an odor is presented with a constant concentration, there are spatiotemporal patterns in the epithelium and bulb generated by the couplings due to the gap junctions and/or dendrodendritic synapses. A study varying the strength of the gap junction coupling shows that the spatiotemporal patterns, both in the epithelium and bulb, are dependent of the coupling strength. It is also shown that the olfactory bulb's spatiotemporal pattern depends on the existence of the dendrodendritic connections between mitral and granule cells. If these spatiotemporal patterns really exist in the early processing stages of the olfactory system they may be used for odor coding and the gap junctions and dendrodendritic synapses might have a role on it.

Facilitating arrhythmia simulation: the method of quantitative cellular automata modeling and parallel running

Background

Many arrhythmias are triggered by abnormal electrical activity at the ionic channel and cell level, and then evolve spatio-temporally within the heart. To understand arrhythmias better and to diagnose them more precisely by their ECG waveforms, a whole-heart model is required to explore the association between the massively parallel activities at the channel/cell level and the integrative electrophysiological phenomena at organ level.

Methods

We have developed a method to build large-scale electrophysiological models by using extended cellular automata, and to run such models on a cluster of shared memory machines. We describe here the method, including the extension of a language-based cellular automaton to implement quantitative computing, the building of a whole-heart model with Visible Human Project data, the parallelization of the model on a cluster of shared memory computers with OpenMP and MPI hybrid programming, and a simulation algorithm that links cellular activity with the ECG.

Results

We demonstrate that electrical activities at channel, cell, and organ levels can be traced and captured conveniently in our extended cellular automaton system. Examples of some ECG waveforms simulated with a 2-D slice are given to support the ECG simulation algorithm. A performance evaluation of the 3-D model on a four-node cluster is also given.

Conclusions

Quantitative multicellular modeling with extended cellular automata is a highly efficient and widely applicable method to weave experimental data at different levels into computational models. This process can be used to investigate complex and collective biological activities that can be described neither by their governing differentiation equations nor by discrete parallel computation. Transparent cluster computing is a convenient and effective method to make time-consuming simulation feasible. Arrhythmias, as a typical case, can be effectively simulated with the methods described.

Action potential modulation of connexin40 gap junctional conductance

Connexin40 (Cx40) is abundantly expressed in the atrial myocardium, ventricular conduction system, and vascular endothelial and smooth muscle cells of the mammalian cardiovascular system. Rapid conduction through cardiac tissues depends on electrotonic transfer of the action potential between neighboring cells. To determine whether transjunctional voltages (Vj) elicited by an action potential can modulate conductance of Cx40 gap junctions, simulated myocardial action potentials were applied as voltage-clamp waveforms to Cx40 gap junctions expressed in mouse neuro2A (N2A) cells. Junctional currents resembled the action potential morphology but declined by >50% from peak to near-constant plateau values. Kinetics of Cx40 voltage gating were examined at peak voltages > or =100 mV, and decay time constants changed e-fold per 17.6 mV for Vj > +/-40 mV. Junctional conductance recovered during phase 3 repolarization and early diastole to initial values. These phasic changes in junctional conductance were due to rapid decay kinetics, increasing to tens of milliseconds at peak Vj of 130 mV, and the increase in the steady-state conductance curve as Vj returned toward 0 mV. Time-dependent conductance curves for Cx40 were modeled with one inactivation and two recovery Vj-dependent components. There was a temporal correlation between development of conduction delay or block and the inactivation phase of junctional conductance. Likewise, recovery of junctional conductance was coincident with recovery from refractoriness, suggesting that gap junctions may play a role in the genesis and propagation of cardiac arrhythmias.

Basic mechanisms of cardiac impulse propagation and associated arrhythmias

Propagation of excitation in the heart involves action potential (AP) generation by cardiac cells and its propagation in the multicellular tissue. AP conduction is the outcome of complex interactions between cellular electrical activity, electrical cell-to-cell communication, and the cardiac tissue structure. As shown in this review, strong interactions occur among these determinants of electrical impulse propagation. A special form of conduction that underlies many cardiac arrhythmias involves circulating excitation. In this situation, the curvature of the propagating excitation wavefront and the interaction of the wavefront with the repolarization tail of the preceding wave are additional important determinants of impulse propagation. This review attempts to synthesize results from computer simulations and experimental preparations to define mechanisms and biophysical principles that govern normal and abnormal conduction in the heart.

Transverse propagation of action potentials between parallel chains of cardiac muscle and smooth muscle cells in PSpice simulations

BACKGROUND

We previously examined transverse propagation of action potentials between 2 and 3 parallel chain of cardiac muscle cells (CMC) simulated using the PSpice program. The present study was done to examine transverse propagation between 5 parallel chains in an expanded model of CMC and smooth muscle cells (SMC).

METHODS

Excitation was transmitted from cell to cell along a strand of 5 cells not connected by low-resistance tunnels (gap-junction connexons). The entire surface membrane of each cell fired nearly simultaneously, and nearly all the propagation time was spent at the cell junctions, the junctional delay time being about 0.3-0.5 ms (CMC) or 0.8-1.6 ms (SMC). A negative cleft potential (Vjc) develops in the narrow junctional clefts, whose magnitude depends on the radial cleft resistance (Rjc), which depolarizes the postjunctional membrane (post-JM) to threshold. Propagation velocity (theta) increased with amplitude of Vjc. Therefore, one mechanism for the transfer of excitation from one cell to the next is by the electric field (EF) that is generated in the junctional cleft when the pre-JM fires. In the present study, 5 parallel stands of 5 cells each (5 x 5 model) were used.

RESULTS

With electrical stimulation of the first cell of the first strand (cell A1), propagation rapidly spread down that chain and then jumped to the second strand (B chain), followed by jumping to the third, fourth, and fifth strands (C, D, E chains). The rapidity by which the parallel chains became activated depended on the longitudinal resistance of the narrow extracellular cleft between the parallel strands (Rol2); the higher the Rol2 resistance, the faster the theta. The transverse resistance of the cleft (Ror2) had almost no effect. Increasing Rjc decreases the total propagation time (TPT) over the 25-cell network. When the first cell of the third strand (cell C1) was stimulated, propagation spread down the C chain and jumped to the other two strands (B and D) nearly simultaneously.

CONCLUSIONS

Transverse propagation of excitation occurred at multiple points along the chain as longitudinal propagation was occurring, causing the APs in the contiguous chains to become bunched up. Transverse propagation was more erratic and labile in SMC compared to CMC. Transverse transmission of excitation did not require low-resistance connections between the chains, but instead depended on the value of Rol2. The tighter the packing of the chains facilitated transverse propagation.

Regulation of connexin43 gap junctional conductance by ventricular action potentials

Transjunctional voltage regulates cardiac gap junctional conductance, but the kinetics of inactivation were considered too slow to affect cardiac action potential propagation. Connexin43 (Cx43) is abundantly expressed in the atrial and ventricular myocardium and the rapid ventricular conduction tissues (ie, His-Purkinje system) of the mammalian heart and is important to conduction through these cardiac tissues. The kinetics of Cx43 voltage gating were examined at peak action potential voltages using simulated ventricular myocardial action potential waveforms or pulse protocols exceeding 100-mV transjunctional potentials. Junctional current responses approximate the action potential morphology but conductance calculations reveal a 50% to 60% decline from peak to near constant plateau values. Junctional conductance recovers during phase 3 repolarization and early diastole to initial values. The bases for these transient changes in junctional conductance are the rapid decay kinetics in tens of milliseconds at peak transjunctional voltages (Vj) of 130 mV and the gradual increase in junctional conductance as Vj returns toward 0 mV. The decay time constants change e-fold per 22.1 mV above the half-inactivation voltage for Cx43 gap junctions of +/-58 mV. A realistic dynamic model for changes in junctional resistance between excitable and nonexcitable cells during cardiac action potential propagation was developed based on these findings. This dynamic model of cardiac gap junctions will further our understanding of the role gap junctions play in the genesis and propagation of cardiac arrhythmias. The full text of this article is available online at http://www.circresaha.org.