Time-dependent transients in an ionically based mathematical model of the canine atrial action potential

An electrodiffusive neuron-extracellular-glia model for exploring the genesis of slow potentials in the brain

Within the computational neuroscience community, there has been a focus on simulating the electrical activity of neurons, while other components of brain tissue, such as glia cells and the extracellular space, are often neglected. Standard models of extracellular potentials are based on a combination of multicompartmental models describing neural electrodynamics and volume conductor theory. Such models cannot be used to simulate the slow components of extracellular potentials, which depend on ion concentration dynamics, and the effect that this has on extracellular diffusion potentials and glial buffering currents. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine compartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for action potentials and dendritic calcium spikes in neurons, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. The edNEG model accounts for the concentration-dependent effects on ECS potentials that the standard models neglect. Using the edNEG model, we analyze these effects by splitting the extracellular potential into three components: one due to neural sink/source configurations, one due to glial sink/source configurations, and one due to extracellular diffusive currents. Through a series of simulations, we analyze the roles played by the various components and how they interact in generating the total slow potential. We conclude that the three components are of comparable magnitude and that the stimulus conditions determine which of the components that dominate.

The impact of wall thickness and curvature on wall stress in patient-specific electromechanical models of the left atrium

The left atrium (LA) has a complex anatomy with heterogeneous wall thickness and curvature. The anatomy plays an important role in determining local wall stress; however, the relative contribution of wall thickness and curvature in determining wall stress in the LA is unknown. We have developed electromechanical finite element (FE) models of the LA using patient-specific anatomical FE meshes with rule-based myofiber directions. The models of the LA were passively inflated to 10mmHg followed by simulation of the contraction phase of the atrial cardiac cycle. The FE models predicted maximum LA volumes of 156.5 mL, 99.3 mL and 83.4 mL and ejection fractions of 36.9%, 32.0% and 25.2%. The median wall thickness in the 3 cases was calculated as [Formula: see text] mm, [Formula: see text] mm, and [Formula: see text] mm. The median curvature was determined as [Formula: see text] [Formula: see text], [Formula: see text], and [Formula: see text]. Following passive inflation, the correlation of wall stress with the inverse of wall thickness and curvature was 0.55-0.62 and 0.20-0.25, respectively. At peak contraction, the correlation of wall stress with the inverse of wall thickness and curvature was 0.38-0.44 and 0.16-0.34, respectively. In the LA, the 1st principal Cauchy stress is more dependent on wall thickness than curvature during passive inflation and both correlations decrease during active contraction. This emphasizes the importance of including the heterogeneous wall thickness in electromechanical FE simulations of the LA. Overall, simulation results and sensitivity analyses show that in complex atrial anatomy it is unlikely that a simple anatomical-based law can be used to estimate local wall stress, demonstrating the importance of FE analyses.

An electrodiffusive, ion conserving Pinsky-Rinzel model with homeostatic mechanisms

In most neuronal models, ion concentrations are assumed to be constant, and effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials are not accounted for. Here, we present the electrodiffusive Pinsky-Rinzel (edPR) model, which we believe is the first multicompartmental neuron model that accounts for electrodiffusive ion concentration dynamics in a way that ensures a biophysically consistent relationship between ion concentrations, electrical charge, and electrical potentials in both the intra- and extracellular space. The edPR model is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron. Unlike the PR model, the edPR model includes homeostatic mechanisms and ion-specific leakage currents, and keeps track of all ion concentrations (Na⁺, K⁺, Ca²⁺, and Cl⁻), electrical potentials, and electrical conductivities in the intra- and extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity. For higher activity levels, or when homeostatic mechanisms are impaired, the homeostatic mechanisms fail in maintaining ion concentrations close to baseline, and the edPR model diverges from the PR model as it accounts for effects of concentration changes on neuronal firing. We envision that the edPR model will be useful for the field in three main ways. Firstly, as it relaxes commonly made modeling assumptions, the edPR model can be used to test the validity of these assumptions under various firing conditions, as we show here for a few selected cases. Secondly, the edPR model should supplement the PR model when simulating scenarios where ion concentrations are expected to vary over time. Thirdly, being applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy.

Neurons generate their electrical signals by letting ions pass through their membranes. Despite this fact, most models of neurons apply the simplifying assumption that ion concentrations remain effectively constant during neural activity. This assumption is often quite good, as neurons contain a set of homeostatic mechanisms that make sure that ion concentrations vary quite little under normal circumstances. However, under some conditions, these mechanisms can fail, and ion concentrations can vary quite dramatically. Standard models are thus not able to simulate such conditions. Here, we present what to our knowledge is the first multicompartmental neuron model that accounts for ion concentration variations in a way that ensures complete and consistent ion concentration and charge conservation. In this work, we use the model to explore under which activity conditions the ion concentration variations become important for predicting the neurodynamics. We expect the model to be of great value for the field of neuroscience, as it can be used to simulate a range of pathological conditions, such as spreading depression or epilepsy, which are associated with large changes in extracellular ion concentrations.

A compartmentalized mathematical model of mouse atrial myocytes

Various experimental mouse models are extensively used to research human diseases, including atrial fibrillation, the most common cardiac rhythm disorder. Despite this, there are no comprehensive mathematical models that describe the complex behavior of the action potential and [Ca²⁺]_i transients in mouse atrial myocytes. Here, we develop a novel compartmentalized mathematical model of mouse atrial myocytes that combines the action potential, [Ca²⁺]_i dynamics, and β-adrenergic signaling cascade for a subpopulation of right atrial myocytes with developed transverse-axial tubule system. The model consists of three compartments related to β-adrenergic signaling (caveolae, extracaveolae, and cytosol) and employs local control of Ca²⁺ release. It also simulates ionic mechanisms of action potential generation and describes atrial-specific Ca²⁺ handling as well as frequency dependences of the action potential and [Ca²⁺]_i transients. The model showed that the T-type Ca²⁺ current significantly affects the later stage of the action potential, with little effect on [Ca²⁺]_i transients. The block of the small-conductance Ca²⁺-activated K⁺ current leads to a prolongation of the action potential at high intracellular Ca²⁺. Simulation results obtained from the atrial model cells were compared with those from ventricular myocytes. The developed model represents a useful tool to study complex electrical properties in the mouse atria and could be applied to enhance the understanding of atrial physiology and arrhythmogenesis.NEW & NOTEWORTHY A new compartmentalized mathematical model of mouse right atrial myocytes was developed. The model simulated action potential and Ca²⁺ dynamics at baseline and after stimulation of the β-adrenergic signaling system. Simulations showed that the T-type Ca²⁺ current markedly prolonged the later stage of atrial action potential repolarization, with a minor effect on [Ca²⁺]_i transients. The small-conductance Ca²⁺-activated K⁺ current block resulted in prolongation of the action potential only at the relatively high intracellular Ca²⁺.

Bond graph modelling of the cardiac action potential: implications for drift and non-unique steady states

Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour. Bond graphs provide an explicit energy-based framework for modelling physical systems, which makes them well suited for examining conservation within electrophysiological models. We find that the charge conservation laws derived in previous studies are examples of the more general concept of a 'conserved moiety'. Conserved moieties explain model drift and non-unique steady states, generalizing the results from previous studies. The bond graph approach provides a rigorous method to check for drift and non-unique steady states in a wide range of cardiac action potential models, and can be extended to examine behaviours of other excitable systems.

Solving Inaccuracies in Anatomical Models for Electrocardiographic Inverse Problem Resolution by Maximizing Reconstruction Quality

Electrocardiographic Imaging has become an increasingly used technique for non-invasive diagnosis of cardiac arrhythmias, although the need for medical imaging technology to determine the anatomy hinders its introduction in the clinical practice. This paper explores the ability of a new metric based on the inverse reconstruction quality for the location and orientation of the atrial surface inside the torso. Body surface electrical signals from 31 realistic mathematical models and four AF patients were used to estimate the optimal position of the atria inside the torso. The curvature of the L-curve from the Tikhonov method, which was found to be related to the inverse reconstruction quality, was measured after application of deviations in atrial position and orientation. Independent deviations in the atrial position were solved by finding the maximal L-curve curvature with an error of 1.7 ± 2.4 mm in mathematical models and 9.1 ± 11.5 mm in patients. For the case of independent angular deviations, the error in location by using the L-curve was 5.8±7.1° in mathematical models and 12.4° ± 13.2° in patients. The ability of the L-curve curvature was tested also under superimposed uncertainties in the three axis of translation and in the three axis of rotation, and the error in location was of 2.3 ± 3.2 mm and 6.4° ± 7.1° in mathematical models, and 7.9±10.7 mm and 12.1°±15.5° in patients. The curvature of L-curve is a useful marker for the atrial position and would allow emending the inaccuracies in its location.

Complex restitution behavior and reentry in a cardiac tissue model for neonatal mice

Spatiotemporal dynamics in cardiac tissue emerging from the coupling of individual cardiomyocytes underlie the heart's normal rhythm as well as undesired and possibly life-threatening arrhythmias. While single cells and their transmembrane currents have been studied extensively, systematically investigating spatiotemporal dynamics is complicated by the nontrivial relationship between single-cell and emergent tissue properties. Mathematical models have been employed to bridge this gap and contribute to a deepened understanding of the onset, development, and termination of arrhythmias. However, no such tissue-level model currently exists for neonatal mice. Here, we build on a recent single-cell model of neonatal mouse cardiomyocytes by Wang and Sobie (Am. J. Physiol. Heart Circ. Physiol 294:H2565) to predict properties that are commonly used to gauge arrhythmogenicity of cardiac substrates. We modify the model to yield well-defined behavior for common experimental protocols and construct a spatially extended version to study emergent tissue dynamics. We find a complex action potential duration (APD) restitution behavior characterized by a nonmonotonic dependence on pacing frequency. Electrotonic coupling in tissue leads not only to changes in action potential morphology but can also induce spatially concordant and discordant alternans not observed in the single-cell model. In two-dimensional tissue, our results show that the model supports stable functional reentry, whose frequency is in good agreement with that observed in adult mice. Our results can be used to further constrain and validate the mathematical model of neonatal mouse cardiomyocytes with future experiments.

Atrial sources identification by causality analysis during atrial fibrillation

Ablation of electrical drivers during atrial fibrillation (AF) has been proved as an effective therapy to prevent recurrence of fibrillatory episodes. This study presents a new methodology based on causality analysis that is able to identify the hierarchical dominance of atrial areas driving AF. Realistic mathematical models of the atrial electrical activity during AF were used to assess the validity of our method. Identification of the dominant atrial propagation patterns was achieved by computing causal relations between multiple electrogram signals. The causal relationships between atrial areas during the fibrillatory processes were summarized into a recurrence map, highlighting the hierarchy and dominant areas. Recurrence maps computed from causality analysis allowed the identification of sites responsible for maintenance of the arrhythmia. These maps were able to locate the position of the atrial driver in fibrillatory processes with a single rotor, with 2 rotors or with several drivers. Additionally, the correspondence between the nodal values of the recurrence map and the distance to the rotor core has been established. Causal analysis consistently estimated propagation patterns and location of atrial drivers during AF. This methodology could guide ablation procedures in AF patients.

Myokit: A simple interface to cardiac cellular electrophysiology

Myokit is a new powerful and versatile software tool for modeling and simulation of cardiac cellular electrophysiology. Myokit consists of an easy-to-read modeling language, a graphical user interface, single and multi-cell simulation engines and a library of advanced analysis tools accessible through a Python interface. Models can be loaded from Myokit's native file format or imported from CellML. Model export is provided to C, MATLAB, CellML, CUDA and OpenCL. Patch-clamp data can be imported and used to estimate model parameters. In this paper, we review existing tools to simulate the cardiac cellular action potential to find that current tools do not cater specifically to model development and that there is a gap between easy-to-use but limited software and powerful tools that require strong programming skills from their users. We then describe Myokit's capabilities, focusing on its model description language, simulation engines and import/export facilities in detail. Using three examples, we show how Myokit can be used for clinically relevant investigations, multi-model testing and parameter estimation in Markov models, all with minimal programming effort from the user. This way, Myokit bridges a gap between performance, versatility and user-friendliness.

Unification of neuronal spikes, seizures, and spreading depression

The pathological phenomena of seizures and spreading depression have long been considered separate physiological events in the brain. By incorporating conservation of particles and charge, and accounting for the energy required to restore ionic gradients, we extend the classic Hodgkin-Huxley formalism to uncover a unification of neuronal membrane dynamics. By examining the dynamics as a function of potassium and oxygen, we now account for a wide range of neuronal activities, from spikes to seizures, spreading depression (whether high potassium or hypoxia induced), mixed seizure and spreading depression states, and the terminal anoxic "wave of death." Such a unified framework demonstrates that all of these dynamics lie along a continuum of the repertoire of the neuron membrane. Our results demonstrate that unified frameworks for neuronal dynamics are feasible, can be achieved using existing biological structures and universal physical conservation principles, and may be of substantial importance in enabling our understanding of brain activity and in the control of pathological states.

Effects of maximal sodium and potassium conductance on the stability of Hodgkin-Huxley model

Hodgkin-Huxley (HH) equation is the first cell computing model in the world and pioneered the use of model to study electrophysiological problems. The model consists of four differential equations which are based on the experimental data of ion channels. Maximal conductance is an important characteristic of different channels. In this study, mathematical method is used to investigate the importance of maximal sodium conductance g-Na and maximal potassium conductance g-K. Applying stability theory, and taking g-Na and g-K as variables, we analyze the stability and bifurcations of the model. Bifurcations are found when the variables change, and bifurcation points and boundary are also calculated. There is only one bifurcation point when g-Na is the variable, while there are two points when g-K is variable. The (g-Na,  g-K) plane is partitioned into two regions and the upper bifurcation boundary is similar to a line when both g-Na and g-K are variables. Numerical simulations illustrate the validity of the analysis. The results obtained could be helpful in studying relevant diseases caused by maximal conductance anomaly.

Modern perspectives on numerical modeling of cardiac pacemaker cell

Cardiac pacemaking is a complex phenomenon that is still not completely understood. Together with experimental studies, numerical modeling has been traditionally used to acquire mechanistic insights in this research area. This review summarizes the present state of numerical modeling of the cardiac pacemaker, including approaches to resolve present paradoxes and controversies. Specifically we discuss the requirement for realistic modeling to consider symmetrical importance of both intracellular and cell membrane processes (within a recent "coupled-clock" theory). Promising future developments of the complex pacemaker system models include the introduction of local calcium control, mitochondria function, and biochemical regulation of protein phosphorylation and cAMP production. Modern numerical and theoretical methods such as multi-parameter sensitivity analyses within extended populations of models and bifurcation analyses are also important for the definition of the most realistic parameters that describe a robust, yet simultaneously flexible operation of the coupled-clock pacemaker cell system. The systems approach to exploring cardiac pacemaker function will guide development of new therapies such as biological pacemakers for treating insufficient cardiac pacemaker function that becomes especially prevalent with advancing age.

Bifurcations, sustained oscillations and torus bursting involving ionic concentrations dynamics in a canine atrial cell model

Atrial fibrillation is a disorganization of the electrical propagation in the atria often initiated by ectopic beats. This spontaneous activity might be associated with the appearance of sustained oscillations in some portion of the tissue. Adrenergic stress and specific gene polymorphisms known to promote atrial fibrillation are notably related to calcium and potassium channel conductances. We performed codimension-one and two bifurcation analysis along these conductances in an ionic canine atrial myocyte model. Two Hopf bifurcations were found, related to two distinct mechanisms: (1) a fast calcium gating-driven oscillator, and (2) a slow concentration-driven oscillator. These two mechanisms interact through a double Hopf bifurcation (HH) in a neighborhood of which a torus (Neimark-Sacker) bifurcation leads to bursting. A complex codimension-two theoretical scenario was identified around HH, through systematic comparison with the attractors found numerically. The concentration oscillator was further decomposed to reveal the minimal oscillating subnetwork, in which the Na(+)/Ca(2+) exchanger plays a prominent role.

Calcium dynamics underlying the myogenic response of the renal afferent arteriole

The renal afferent arteriole reacts to an elevation in blood pressure with an increase in muscle tone and a decrease in luminal diameter. This effect, known as the myogenic response, is believed to stabilize glomerular filtration and to protect the glomerulus from systolic blood pressure increases, especially in hypertension. To study the mechanisms underlying the myogenic response, we developed a mathematical model of intracellular Ca(2+) signaling in an afferent arteriole smooth muscle cell. The model represents detailed transmembrane ionic transport, intracellular Ca(2+) dynamics, the kinetics of myosin light chain phosphorylation, and the mechanical behavior of the cell. It assumes that the myogenic response is initiated by pressure-induced changes in the activity of nonselective cation channels. Our model predicts spontaneous vasomotion at physiological luminal pressures and KCl- and diltiazem-induced diameter changes comparable to experimental findings. The time-periodic oscillations stem from the dynamic exchange of Ca(2+) between the cytosol and the sarcoplasmic reticulum, coupled to the stimulation of Ca(2+)-activated potassium (KCa) and chloride (ClCa) channels, and the modulation of voltage-activated L-type channels; blocking sarco/endoplasmic reticulum Ca(2+) pumps, ryanodine receptors (RyR), KCa, ClCa, or L-type channels abolishes these oscillations. Our results indicate that the profile of the myogenic response is also strongly dependent on the conductance of ClCa and L-type channels, as well as the activity of plasmalemmal Ca(2+) pumps. Furthermore, inhibition of KCa is not necessary to induce myogenic contraction. Lastly, our model suggests that the kinetic behavior of L-type channels results in myogenic kinetics that are substantially faster during constriction than during dilation, consistent with in vitro observations (Loutzenhiser R, Bidani A, Chilton L. Circ. Res. 90: 1316-1324, 2002).

The force-frequency relationship: insights from mathematical modeling

The force-frequency relationship has intrigued researchers since its discovery by Bowditch in 1871. Many attempts have been made to construct mathematical descriptions of this phenomenon, beginning with the simple formulation of Koch-Wesser and Blinks in 1963 to the most sophisticated ones of today. This property of cardiac muscle is amplified by β-adrenergic stimulation, and, in a coordinated way, the neurohumoral state alters both frequency (acting on the sinoatrial node) as well as force generation (modifying ventricular myocytes). This synchronized tuning is needed to meet new metabolic demands. Cardiac modelers have already linked mechanical and electrical activity in their formulations and showed how those activities feedback on each other. However, now it is necessary to include neurological control to have a complete description of heart performance, especially when changes in frequency are involved. Study of arrhythmias (or antiarrhythmic drugs) based on mathematical models should incorporate this effect to make useful predictions or point out potential pharmaceutical targets.

Computational modeling of the human atrial anatomy and electrophysiology

This review article gives a comprehensive survey of the progress made in computational modeling of the human atria during the last 10 years. Modeling the anatomy has emerged from simple "peanut"-like structures to very detailed models including atrial wall and fiber direction. Electrophysiological models started with just two cellular models in 1998. Today, five models exist considering e.g. details of intracellular compartments and atrial heterogeneity. On the pathological side, modeling atrial remodeling and fibrotic tissue are the other important aspects. The bridge to data that are measured in the catheter laboratory and on the body surface (ECG) is under construction. Every measurement can be used either for model personalization or for validation. Potential clinical applications are briefly outlined and future research perspectives are suggested.

Steady-state solutions of cell volume in a cardiac myocyte model elaborated for membrane excitation, ion homeostasis and Ca2+ dynamics

The cell volume continuously changes in response to varying physiological conditions, and mechanisms underlying volume regulation have been investigated in both experimental and theoretical studies. Here, general formulations concerning cell volume change are presented in the context of developing a comprehensive cell model which takes Ca(2+) dynamics into account. Explicit formulas for charge conservation and steady-state volumes of the cytosol and endoplasmic reticulum (ER) are derived in terms of membrane potential, amount of ions, Ca(2+)-bound buffer molecules, and initial cellular conditions. The formulations were applied to a ventricular myocyte model which has plasma-membrane Ca(2+) currents with dynamic gating mechanisms, Ca(2+)-buffering reactions with diffusive and non-diffusive buffer proteins, and Ca(2+) uptake into or release from the sarcoplasmic reticulum (SR) accompanied by compensatory cationic or anionic currents through the SR membrane. Time-dependent volume changes in cardiac myocytes induced by varying extracellular osmolarity or by action potential generation were successfully simulated by the novel formulations. Through application of bifurcation analysis, the existence and uniqueness of steady-state solutions of the cell volume were validated, and contributions of individual ion channels and transporters to the steady-state volume were systematically analyzed. The new formulas are consistent with previous fundamental theory derived from simple models of minimum compositions. The new formulations may be useful for examination of the relationship between cell function and volume change in other cell types.

How the Hodgkin-Huxley equations inspired the Cardiac Physiome Project

Early modelling of cardiac cells (1960-1980) was based on extensions of the Hodgkin-Huxley nerve axon equations with additional channels incorporated, but after 1980 it became clear that processes other than ion channel gating were also critical in generating electrical activity. This article reviews the development of models representing almost all cell types in the heart, many different species, and the software tools that have been created to facilitate the cardiac Physiome Project.

In silico optimization of atrial fibrillation-selective sodium channel blocker pharmacodynamics

Atrial fibrillation (AF) is the most common type of clinical arrhythmia. Currently available anti-AF drugs are limited by only moderate efficacy and an unfavorable safety profile. Thus, there is a recognized need for improved antiarrhythmic agents with actions that are selective for the fibrillating atrium. State-dependent Na(+)-channel blockade potentially allows for the development of drugs with maximal actions on fibrillating atrial tissue and minimal actions on ventricular tissue at resting heart rates. In this study, we applied a mathematical model of state-dependent Na(+)-channel blocking (class I antiarrhythmic drug) action, along with mathematical models of canine atrial and ventricular cardiomyocyte action potentials, AF, and ventricular proarrhythmia, to determine the relationship between their pharmacodynamic properties and atrial-selectivity, AF-selectivity (atrial Na(+)-channel block at AF rates versus ventricular block at resting rates), AF-termination effectiveness, and ventricular proarrhythmic properties. We found that drugs that target inactivated channels are AF-selective, whereas drugs that target activated channels are not. The most AF-selective drugs were associated with minimal ventricular proarrhythmic potential and terminated AF in 33% of simulations; slightly fewer AF-selective agents achieved termination rates of 100% with low ventricular proarrhythmic potential. Our results define properties associated with AF-selective actions of class-I antiarrhythmic drugs and support the idea that it may be possible to develop class I antiarrhythmic agents with optimized pharmacodynamic properties for AF treatment.

A mathematical model of action potentials of mouse sinoatrial node cells with molecular bases

Genetically modified mice are popular experimental models for studying the molecular bases and mechanisms of cardiac arrhythmia. A postgenome challenge is to classify the functional roles of genes in cardiac function. To unveil the functional role of various genetic isoforms of ion channels in generating cardiac pacemaking action potentials (APs), a mathematical model for spontaneous APs of mouse sinoatrial node (SAN) cells was developed. The model takes into account the biophysical properties of membrane ionic currents and intracellular mechanisms contributing to spontaneous mouse SAN APs. The model was validated by its ability to reproduce the physiological exceptionally short APs and high pacing rates of mouse SAN cells. The functional roles of individual membrane currents were evaluated by blocking their coding channels. The roles of intracellular Ca²⁺-handling mechanisms on cardiac pacemaking were also investigated in the model. The robustness of model pacemaking behavior was evaluated by means of one- and two-parameter analyses in wide parameter value ranges. This model provides a predictive tool for cellular level outcomes of electrophysiological experiments. It forms the basis for future model development and further studies into complex pacemaking mechanisms as more quantitative experimental data become available.

Differential effectiveness of pharmacological strategies to reveal dormant pulmonary vein conduction: a clinical-experimental correlation

BACKGROUND

Atrial fibrillation recurs in ∼30%-40% of patients after pulmonary vein (PV) isolation (PVI) procedures, often because of restored PV-left atrial (LA) conduction. Adenosine or isoproterenol are used clinically to reveal dormant PV conduction and guide additional ablation.

OBJECTIVE

The purpose of this study was to assess the differential efficacy of adenosine and/or isoproterenol in revealing dormant PV conduction.

METHODS

In 25 patients undergoing PVI, dormant conduction was assessed sequentially in response to intravenous adenosine, isoproterenol, and adenosine plus isoproterenol in 100 PVs. To study mechanisms, PVs were isolated by radiofrequency ablation in coronary-perfused canine LA-PV preparations. After PVI, resting membrane potential from PV cells was recorded before and after 1 mM adenosine, 1 μM isoproterenol, 1 μM isoproterenol plus 1 mM adenosine, or no drug (controls).

RESULTS

Clinical PVI was successful in all 100 PVs, with dormant conduction in 31. Sensitivity for dormant conduction was isoproterenol 10%; adenosine 87% (P <.001 vs. isoproterenol); and isoproterenol + adenosine 100% (P = .13 vs. adenosine). Dormant PV conduction in vitro was revealed with adenosine (53%) and adenosine + isoproterenol (60%) but not with isoproterenol alone or in controls (P <.01). Radiofrequency lesions producing PVI depolarized resting membrane potential, causing inexcitability. Postablation, resting membrane potential hyperpolarized after both adenosine and isoproterenol, but adenosine-induced changes were greater (9.1 ± 0.6 mV, vs. 3.8 ± 0.6 mV; P <0.001), with no significant additional effect when isoproterenol was added to adenosine.

CONCLUSION

Adenosine is superior to isoproterenol in revealing dormant PVs clinically and experimentally because of more effective adenosine-induced hyperpolarization. Adding isoproterenol to adenosine had no significant additional value.

Automaticity in acute ischemia: bifurcation analysis of a human ventricular model

Acute ischemia (restriction in blood supply to part of the heart as a result of myocardial infarction) induces major changes in the electrophysiological properties of the ventricular tissue. Extracellular potassium concentration ([K(o)(+)]) increases in the ischemic zone, leading to an elevation of the resting membrane potential that creates an "injury current" (I(S)) between the infarcted and the healthy zone. In addition, the lack of oxygen impairs the metabolic activity of the myocytes and decreases ATP production, thereby affecting ATP-sensitive potassium channels (I(Katp)). Frequent complications of myocardial infarction are tachycardia, fibrillation, and sudden cardiac death, but the mechanisms underlying their initiation are still debated. One hypothesis is that these arrhythmias may be triggered by abnormal automaticity. We investigated the effect of ischemia on myocyte automaticity by performing a comprehensive bifurcation analysis (fixed points, cycles, and their stability) of a human ventricular myocyte model [K. H. W. J. ten Tusscher and A. V. Panfilov, Am. J. Physiol. Heart Circ. Physiol. 291, H1088 (2006)] as a function of three ischemia-relevant parameters [K(o)(+)], I(S), and I(Katp). In this single-cell model, we found that automatic activity was possible only in the presence of an injury current. Changes in [K(o)(+)] and I(Katp) significantly altered the bifurcation structure of I(S), including the occurrence of early-after depolarization. The results provide a sound basis for studying higher-dimensional tissue structures representing an ischemic heart.

Mathematical models of canine right and left atria cardiomyocytes

The aim of this study is to build two mathematical models of canine ionic currents specific to right atria and left atria. The canine left atria mathematical model was firstly modified from the Ramirez-Nattel-Courtemanche (RNC) model using the recently available experimental data of ionic currents and was further developed based on our own experimental data. A model of right atria was then built by considering the differences between right atria and left atria. The two developed models well reproduced the experimental data on action potential morphology, the rate dependence, and action potential duration restitution. They are useful for investigating the mechanisms underlying the heterogeneity of canine regional action potentials and would help the simulation of whole heart excitation propagation and cardiac arrhythmia in the near future.

Mechanisms by which adenosine restores conduction in dormant canine pulmonary veins

BACKGROUND

Adenosine acutely reconnects pulmonary veins (PVs) after radiofrequency application, revealing "dormant conduction" and identifying PVs at risk of reconnection, but the underlying mechanisms are unknown.

METHODS AND RESULTS

Canine PV and left-atrial (LA) action potentials were recorded with standard microelectrodes and ionic currents with whole-cell patch clamp before and after adenosine perfusion. PVs were isolated with radiofrequency current application in coronary-perfused LA-PV preparations. Adenosine abbreviated action potential duration similarly in PV and LA but significantly hyperpolarized resting potential (by 3.9+/-0.5%; P<0.05) and increased dV/dt(max) (by 34+/-10%) only in PV. Increased dV/dt(max) was not due to direct effects on I(Na), which was reduced similarly by adenosine in LA and PV but correlated with resting-potential hyperpolarization (r=0.80). Adenosine induced larger inward rectifier K(+)current (I(KAdo)) in PV (eg, -2.28+/-0.04 pA/pF; -100 mV) versus LA (-1.28+/-0.16 pA/pF). Radiofrequency ablation isolated PVs by depolarizing resting potential to voltages positive to -60 mV. Adenosine restored conduction in 5 dormant PVs, which had significantly more negative resting potentials (-57+/-6 mV) versus nondormant (-46+/-5 mV, n=6; P<0.001) before adenosine. Adenosine hyperpolarized both, but more negative resting-potential values after adenosine in dormant PVs (-66+/-6 mV versus -56+/-6 mV in nondormant; P<0.001) were sufficient to restore excitability. Adenosine effects on resting potential and conduction reversed on washout. Spontaneous recovery of conduction occurring in dormant PVs after 30 to 60 minutes was predicted by the adenosine response.

CONCLUSIONS

Adenosine selectively hyperpolarizes canine PVs by increasing I(KAdo). PVs with dormant conduction show less radiofrequency-induced depolarization than nondormant veins, allowing adenosine-induced hyperpolarization to restore excitability by removing voltage-dependent I(Na) inactivation and explaining the restoration of conduction in dormant PVs.

Uniqueness and stability of action potential models during rest, pacing, and conduction using problem-solving environment

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca2+ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 1-5 above, an efficient scheme is developed for computing propagation in multicellular models.

Synergism of coupled subsarcolemmal Ca2+ clocks and sarcolemmal voltage clocks confers robust and flexible pacemaker function in a novel pacemaker cell model

Recent experimental studies have demonstrated that sinoatrial node cells (SANC) generate spontaneous, rhythmic, local subsarcolemmal Ca(2+) releases (Ca(2+) clock), which occur during late diastolic depolarization (DD) and interact with the classic sarcolemmal voltage oscillator (membrane clock) by activating Na(+)-Ca(2+) exchanger current (I(NCX)). This and other interactions between clocks, however, are not captured by existing essentially membrane-delimited cardiac pacemaker cell numerical models. Using wide-scale parametric analysis of classic formulations of membrane clock and Ca(2+) cycling, we have constructed and initially explored a prototype rabbit SANC model featuring both clocks. Our coupled oscillator system exhibits greater robustness and flexibility than membrane clock operating alone. Rhythmic spontaneous Ca(2+) releases of sarcoplasmic reticulum (SR)-based Ca(2+) clock ignite rhythmic action potentials via late DD I(NCX) over much broader ranges of membrane clock parameters [e.g., L-type Ca(2+) current (I(CaL)) and/or hyperpolarization-activated ("funny") current (I(f)) conductances]. The system Ca(2+) clock includes SR and sarcolemmal Ca(2+) fluxes, which optimize cell Ca(2+) balance to increase amplitudes of both SR Ca(2+) release and late DD I(NCX) as SR Ca(2+) pumping rate increases, resulting in a broad pacemaker rate modulation (1.8-4.6 Hz). In contrast, the rate modulation range via membrane clock parameters is substantially smaller when Ca(2+) clock is unchanged or lacking. When Ca(2+) clock is disabled, the system parametric space for fail-safe SANC operation considerably shrinks: without rhythmic late DD I(NCX) ignition signals membrane clock substantially slows, becomes dysrhythmic, or halts. In conclusion, the Ca(2+) clock is a new critical dimension in SANC function. A synergism of the coupled function of Ca(2+) and membrane clocks confers fail-safe SANC operation at greatly varying rates.

Effect of bundle branch block on cardiac output: a whole heart simulation study

The heart is an electrically controlled fluid pump which operates by mechanical contraction. Whole heart modelling is a computationally daunting task which must incorporate several subsystems: mechanical, electrical, and fluidic. Numerous feedback mechanisms on many levels, and operating at different scales, exist to finely control behaviour. Understanding these interactions is necessary to understand heart operation, as well as pathologies and therapies. A review of the components in such a model is given. The authors then present a framework for their electro-mechano-fluidic whole heart model based on cable methods. The model incorporates atria and ventricles, and has functioning valves with papillary muscles. The effect of altered propagation due to left and right bundle branch block on cardiac output is examined using the cable-based model. Results are compared to clinically observed phenomena. Good agreement was obtained, but tighter coupling of mechanical and electrical events is needed to fully account for behaviour. Cable-based models offer an alternative to continuum models.

Loading effect of fibroblast-myocyte coupling on resting potential, impulse propagation, and repolarization: insights from a microstructure model

The numerous nonmyocytes present within the myocardium may establish electrical connections with myocytes through gap junctions, formed naturally or as a result of a cell therapy. The strength of the coupling and its potential impact on action potential characteristics and conduction are not well understood. This study used computer simulation to investigate the load-induced electrophysiological consequences of the coupling of myocytes with fibroblasts, where the fibroblast resting potential, density, distribution, and coupling strength were varied. Conduction velocity (CV), upstroke velocity, and action potential duration (APD) were analyzed for longitudinal and transverse impulse propagation in a two-dimensional microstructure tissue model, developed to represent a monolayer culture of cardiac cells covered by a layer of fibroblasts. The results show that 1) at weak coupling (<0.25 nS), the myocyte resting potential was elevated, leading to CV up to 5% faster than control; 2) at intermediate coupling, the myocyte resting potential elevation saturated, whereas the current flowing from the myocyte to the fibroblast progressively slowed down both CV and upstroke velocity; 3) at strong couplings (>8 nS), all of the effects saturated; and 4) APD at 90% repolarization was usually prolonged by 0-20 ms (up to 60-80 ms for high fibroblast density and coupling) by the coupling to fibroblasts. The changes in APD depended on the fibroblast resting potential. This complex, coupling-dependent interaction of fibroblast and myocytes also has relevance to the integration of other nonmyocytes in the heart, such as those used in cellular therapies.

Computer modelling of the sinoatrial node

Over the past decades patch-clamp experiments have provided us with detailed information on the different types of ion channels that are present in the cardiac cell membrane. Sophisticated cardiac cell models based on these data can help us understand how the different types of ion channels act together to produce the cardiac action potential. In the field of biological pacemaker engineering, such models provide important instruments for the assessment of the functional implications of changes in density of specific ion channels aimed at producing stable pacemaker activity. In this review, an overview is given of the progress made in cardiac cell modelling, with particular emphasis on the development of sinoatrial (SA) nodal cell models. Also, attention is given to the increasing number of publicly available tools for non-experts in computer modelling to run cardiac cell models.

Quantitative techniques for steady-state calculation and dynamic integrated modelling of membrane potential and intracellular ion concentrations

The membrane potential (E(m)) is a fundamental cellular parameter that is primarily determined by the transmembrane permeabilities and concentration gradients of various ions. However, ion gradients are themselves profoundly influenced by E(m) due to its influence upon transmembrane ion fluxes and cell volume (V(c)). These interrelationships between E(m), V(c) and intracellular ion concentrations make computational modelling useful or necessary in order to guide experimentation and to achieve an integrated understanding of experimental data, particularly in complex, dynamic, multi-compartment systems such as skeletal and cardiac myocytes. A variety of quantitative techniques exist that may assist such understanding, from classical approaches such as the Goldman-Hodgkin-Katz equation and the Gibbs-Donnan equilibrium, to more recent "current-summing" models as exemplified by cardiac myocyte models including those of DiFrancesco & Noble, Luo & Rudy and Puglisi & Bers, or the "charge-difference" modelling technique of Fraser & Huang so far applied to skeletal muscle. In general, the classical approaches provide useful and important insights into the relationships between E(m), V(c) and intracellular ion concentrations at steady state, providing their core assumptions are fully understood, while the more recent techniques permit the modelling of changing values of E(m), V(c) and intracellular ion concentrations. The present work therefore reviews the various approaches that may be used to calculate E(m), V(c) and intracellular ion concentrations with the aim of establishing the requirements for an integrated model that can both simulate dynamic systems and recapitulate the key findings of classical techniques regarding the cellular steady state. At a time when the number of cellular models is increasing at an unprecedented rate, it is hoped that this article will provide a useful and critical analysis of the mathematical techniques fundamental to each of them.

Impact of Varying Ablation Patterns in a Simulation Model of Persistent Atrial Fibrillation

BACKGROUND

Several strategies of endovascular ablation with varying success rates and proarrhythmic effects have been proposed to treat persistent atrial fibrillation (AF). Evaluation of ablation patterns by computer simulation provides a tool for examination of its effectiveness and side effects.

METHODS AND RESULTS

A biophysical model of the human atria based on magnetic resonance imaging derived geometry and a membrane kinetics model was used. Uniform conduction properties were assigned to the monolayer surface representing the atria. After induction of AF by burst pacing, progressively broader ablation patterns were applied: (A) individual pulmonary vein isolation (PVI); (B) double ipsilateral PVI; (C) double PVI with a roofline; (D) double PVI with a lateral mitral isthmus line, and (E) double PVI with both linear lesions. In addition, the influence of incomplete linear lesions and dilated atria were simulated. The incidence of AF termination was found to increase from pattern (A) to (E). Atrial flutter rate increased with incomplete ablations and in dilated atria.

CONCLUSION

Computer simulation of various ablation patterns in persistent AF is feasible and can reproduce clinical results of catheter ablation. This model can be used to develop and simulate new ablation patterns and anticipate success rates and potential adverse effects.

Stretch-Sensitive KCNQ1Mutation

OBJECTIVES

This study sought to evaluate mutations in genes encoding the slow component of the cardiac delayed rectifier K+ current (I(Ks)) channel in familial atrial fibrillation (AF).

BACKGROUND

Although AF can have a genetic etiology, links between inherited gene defects and acquired factors such as atrial stretch have not been explored.

METHODS

Mutation screening of the KCNQ1, KCNE1, KCNE2, and KCNE3 genes was performed in 50 families with AF. The effects of mutant protein on cardiac I(Ks) activation were evaluated using electrophysiological studies and human atrial action potential modeling.

RESULTS

One missense KCNQ1 mutation, R14C, was identified in 1 family with a high prevalence of hypertension. Atrial fibrillation was present only in older individuals who had developed atrial dilation and who were genotype positive. Patch-clamp studies of wild-type or R14C KCNQ1 expressed with KCNE1 in CHO cells showed no statistically significant differences between wild-type and mutant channel kinetics at baseline, or after activation of adenylate cyclase with forskolin. After exposure to hypotonic solution to elicit cell swelling/stretch, mutant channels showed a marked increase in current, a leftward shift in the voltage dependence of activation, altered channel kinetics, and shortening of the modeled atrial action potential duration.

CONCLUSIONS

These data suggest that the R14C KCNQ1 mutation alone is insufficient to cause AF. Rather, we suggest a model in which a "second hit", such as an environmental factor like hypertension, which promotes atrial stretch and thereby unmasks an inherited defect in ion channel kinetics (the "first hit"), is required for AF to be manifested. Such a model would also account for the age-related increase in AF development.

Frequency-dependent effects of various IKr blockers on cardiac action potential duration in a human atrial model

Rapidly activating K(+) current (I(Kr)) blockers prolong action potential (AP) duration (APD) in a reverse-frequency-dependent manner and may induce arrhythmias, including torsade de pointes in the ventricle. The I(Kr) blocker dofetilide has been approved for treatment of atrial arrhythmias, including fibrillation. There are, however, a limited number of studies on the action of I(Kr) blockers on atrial AP. When we tested a mathematical model of the human atrial AP (M Courtemanche, RJ Ramirez, S Nattel. Am J Physiol Heart Circ Physiol 275: H301-H321, 1998) to examine the effects of dofetilide-type I(Kr) blockade, this model could not reproduce the reverse-frequency-dependent nature of I(Kr) blockade on atrial APD. We modified the model by introducing a slowly activating K(+) current activation parameter. As the slow time constant was increased, dofetilide-type blockade induced more prominent reverse-frequency-dependent APD prolongation. Using the modified model, we also examined the effects of two more types of I(Kr) blockade similar to those of quinidine and vesnarinone. Voltage- and time-dependent block of I(Kr) through the onset of inhibition by quinidine is much faster than by vesnarinone. When we incorporated the kinetics of the effects of these drugs on I(Kr) into the model, we found that quinidine-type blockade caused a reverse-frequency-dependent prolongation of APD that was similar to the effect of dofetilide-type blockade, whereas vesnarinone-type blockade did not. This finding coincides with experimental observations. The lack of the reverse frequency dependence in vesnarinone-type blockade was accounted for by the slow development of I(Kr) blockade at depolarized potentials. These results suggest that the voltage- and time-dependent nature of I(Kr) blockade by drugs may be critical for the phenotype of the drug effect on atrial AP.

Steady-state solutions in mathematical models of atrial cell electrophysiology and their stability

The steady states of the Fenton-Karma, the Courtemanche and the Nygren cell models were studied by determining the fixed points of the dynamical system describing their cell kinetics. The linear stability of the fixed points was investigated, as well as their response to external stimuli. Symbolic calculations were carried out as far as possible in order to prove the existence of these fixed points. In the Fenton-Karma model, a unique stable fixed point was found, namely the resting state. In contrast, the Courtemanche model had an infinite number of fixed points. A bifurcation diagram was constructed by classifying these fixed points according to a conservation law. Initial conditions were identified, for which the dynamical behavior of the cell was auto-oscillatory. In its original formulation, the Nygren model had no fixed point. After having restored charge conservation, the system was found to have an infinite number of fixed points, resulting in a bifurcation diagram similar to that of the Courtemanche model. The approach proposed in this paper assists in the exploration of the high-dimensional parameter space of the cell models and the identification of the conditions leading to spontaneous pacemaker activity.

Characterization of the in vivo and in vitro electrophysiological effects of the novel antiarrhythmic agent AZD7009 in atrial and ventricular tissue of the dog

This study evaluated the effects of the novel antiarrhythmic agent AZD7009 on atrial and ventricular repolarization and on the Na+-current system, using Vmax as an index. Anesthetized dogs were infused with AZD7009 or azimilide to produce three pseudo steady-state plasma concentrations in vivo. Microelectrode techniques were used to record action potentials and effective refractory period (ERP) in vitro. Whereas AZD7009 concentration-dependently increased atrial ERP (AERP, by 48 +/- 7 milliseconds maximum, P < 0.001 versus vehicle), the increases in ventricular ERP (VERP, 8 +/- 4 milliseconds) and QT interval (2 +/- 5.5 milliseconds) were small and not concentration-dependent. For azimilide, the AERP increase was less, whereas VERP and QT increases were substantially larger than with AZD7009. In vitro, AZD7009 concentration-dependently reduced Vmax and increased action potential duration (APD). ERP was increased through APD lengthening and post-repolarization refractoriness. The suppression of Vmax, but not APD prolongation, showed frequency-dependence. APD and ERP increases were more pronounced in atrial than ventricular tissue: in atria, 2 microM AZD7009 increased APD90 and ERP from 224 +/- 7 to 318 +/- 7 milliseconds and 241 +/- 7 milliseconds to 378 +/- 17 milliseconds; versus 257 +/- 5 to 283 +/- 7 milliseconds and 253 +/- 12 to 300 +/- 11 milliseconds respectively in ventricles. Thus, AZD7009 potently and predominantly increases atrial refractoriness in the dog, with actions mediated by combined effects on repolarization and the Na+-current system.

Substrate size as a determinant of fibrillatory activity maintenance in a mathematical model of canine atrium

Tissue size has been considered an important determinant of atrial fibrillation (AF), but recent work has questioned the critical size hypothesis. Here, we use a previously developed mathematical model of the two-dimensional canine atrium with realistic action potential, ionic, and conduction properties to address substrate size effects on the maintenance of fibrillatory activity. Cholinergic AF was simulated at different acetylcholine (ACh) concentrations ([ACh]) and distributions, with substrate area varied 11.1-fold. Automated phase singularity detection was used to facilitate the analysis of arrhythmic activity. The duration of activity induced by a single extrastimulus increased with increasing substrate dimensions. Two general mechanisms underlying activity were observed and were differentially affected by substrate size. For large mean [ACh], single primary rotors anchored in low-[ACh] zones maintained activity and substrate dimensions were not critical. At lower mean [ACh], extensive spiral wave meander prevented the emergence of single stable rotors. Prolonged activity was favored when substrate size permitted a sufficiently large number of simultaneous longer-lasting rotors that extinction of all was unlikely. Thus either single dominant rotor or multiple reentrant spiral generator mechanisms could maintain fibrillatory activity in this model and were differentially dependent on substrate size. These results speak to recent debates about the role in AF of single driver rotors versus multiple reentrant circuit mechanisms by suggesting that either may maintain fibrillatory atrial activity depending on atrial size and electrophysiological properties.

An ionic model for rhythmic activity in small clusters of embryonic chick ventricular cells

We recorded transmembrane potential in whole cell recording mode from small clusters (2-4 cells) of spontaneously beating 7-day embryonic chick ventricular cells after 1-3 days in culture and investigated effects of the blockers D-600, diltiazem, almokalant, and Ba2+. Electrical activity in small clusters is very different from that in reaggregates of several hundred embryonic chick ventricular cells, e.g., TTX-sensitive fast upstrokes in reaggregates vs. TTX-insensitive slow upstrokes in small clusters (maximum upstroke velocity approximately 100 V/s vs. approximately 10 V/s). On the basis of our voltage- and current-clamp results and data from the literature, we formulated a Hodgkin-Huxley-type ionic model for the electrical activity in these small clusters. The model contains a Ca2+ current (ICa), three K+ currents (IKs, IKr, and IK1), a background current, and a seal-leak current. ICa generates the slow upstroke, whereas IKs, IKr, and IK1 contribute to repolarization. All the currents contribute to spontaneous diastolic depolarization, e.g., removal of the seal-leak current increases the interbeat interval from 392 to 535 ms. The model replicates the spontaneous activity in the clusters as well as the experimental results of application of blockers. Bifurcation analysis and simulations with the model predict that annihilation and single-pulse triggering should occur with partial block of ICa. Embryonic chick ventricular cells have been used as an experimental model to investigate various aspects of spontaneous beating of cardiac cells, e.g., mutual synchronization, regularity of beating, and spontaneous initiation and termination of reentrant rhythms; our model allows investigation of these topics through numerical simulation.

Rate dependence and regulation of action potential and calcium transient in a canine cardiac ventricular cell model

BACKGROUND

Computational biology is a powerful tool for elucidating arrhythmogenic mechanisms at the cellular level, where complex interactions between ionic processes determine behavior. A novel theoretical model of the canine ventricular epicardial action potential and calcium cycling was developed and used to investigate ionic mechanisms underlying Ca2+ transient (CaT) and action potential duration (APD) rate dependence.

METHODS AND RESULTS

The Ca2+/calmodulin-dependent protein kinase (CaMKII) regulatory pathway was integrated into the model, which included a novel Ca2+-release formulation, Ca2+ subspace, dynamic chloride handling, and formulations for major ion currents based on canine ventricular data. Decreasing pacing cycle length from 8000 to 300 ms shortened APD primarily because of I(Ca(L)) reduction, with additional contributions from I(to1), I(NaK), and late I(Na). CaT amplitude increased as cycle length decreased from 8000 to 500 ms. This positive rate-dependent property depended on CaMKII activity.

CONCLUSIONS

CaMKII is an important determinant of the rate dependence of CaT but not of APD, which depends on ion-channel kinetics. The model of CaMKII regulation may serve as a paradigm for modeling effects of other regulatory pathways on cell function.

Characterization of a hyperpolarization-activated time-dependent potassium current in canine cardiomyocytes from pulmonary vein myocardial sleeves and left atrium

Cardiomyocytes from the pulmonary vein sleeves (PVs) are known to play an important role in atrial fibrillation. PVs have been shown to exhibit time-dependent hyperpolarization-induced inward currents of uncertain nature. We observed a time-dependent K(+) current upon hyperpolarization of PV and left atrial (LA) cardiomyocytes (I(KH)) and characterized its biophysical and pharmacological properties. The activation time constant was weakly voltage dependent, ranging from 386 +/- 14 to 427 +/- 37 ms between -120 and -90 mV, and the half-activation voltage averaged -93 +/- 4 mV. I(KH) was larger in PV than LA cells (e.g. at -120 mV: -2.8 +/- 0.3 versus-1.9 +/- 0.2 pA pF(-1), respectively, P < 0.01). The reversal potential was approximately -84 mV with 5.4 mm[K(+)](o) and changed by 55.7 +/- 2.4 mV per decade [K(+)](o) change. I(KH) was exquisitely Ba(2+) sensitive, with a 50% inhibitory concentration (IC(50)) of 2.0 +/- 0.3 microm (versus 76.0 +/- 17.9 microm for instantaneous inward-rectifier current, P < 0.01), and showed similar Cs(+) sensitivity to instantaneous current. I(KH) was potently blocked by tertiapin-Q, a selective Kir3-subunit channel blocker (IC(50) 10.0 +/- 2.1 nm), was unaffected by atropine and was significantly increased by isoproterenol (isoprenaline), carbachol and the non-hydrolysable guanosine triphosphate analogue GTPgammaS. I(KH) activation by carbachol required GTP in the pipette and was prevented by pertussis toxin pretreatment. Tertiapin-Q delayed repolarization in atropine-exposed multicellular atrial preparations studied with standard microelectrodes (action potential duration pre- versus post-tertiapin-Q: 190.4 +/- 4.3 versus 234.2 +/- 9.9 ms, PV; 202.6 +/- 2.6 versus 242.7 +/- 6.2 ms, LA; 2 Hz, P < 0.05 each). Seven-day atrial tachypacing significantly increased I(KH) (e.g. at -120 mV in PV: from -2.8 +/- 0.3 to -4.5 +/- 0.5 pA pF(-1), P < 0.01). We conclude that I(KH) is a time-dependent, hyperpolarization-activated K(+) current that likely involves Kir3 subunits and appears to play a significant role in atrial physiology.

Cellular electrophysiology of canine pulmonary vein cardiomyocytes: action potential and ionic current properties

Pulmonary vein (PV) cardiomyocytes play an important role in atrial fibrillation; however, little is known about their specific cellular electrophysiological properties. We applied standard microelectrode recording and whole-cell patch-clamp to evaluate action potentials and ionic currents in canine PVs and left atrium (LA) free wall. Resting membrane potential (RMP) averaged -66 +/- 1 mV in PVs and -74 +/- 1 mV in LA (P < 0.0001) and action potential amplitude averaged 76 +/- 2 mV in PVs vs. 95 +/- 2 mV in LA (P < 0.0001). PVs had smaller maximum phase 0 upstroke velocity (Vmax: 98 +/- 9 vs. 259 +/- 16 V s(-1), P < 0.0001) and action potential duration (APD): e.g. at 2 Hz, APD to 90% repolarization in PVs was 84 % of LA (P < 0.05). Na+ current density under voltage-clamp conditions was similar in PV and LA, suggesting that smaller Vmax in PVs was due to reduced RMP. Inward rectifier current density in the PV cardiomyocytes was approximately 58% that in the LA, potentially accounting for the less negative RMP in PVs. Slow and rapid delayed rectifier currents were greater in the PV (by approximately 60 and approximately 50 %, respectively), whereas transient outward K+ current and L-type Ca2+ current were significantly smaller (by approximately 25 and approximately 30%, respectively). Na(+)-Ca(2+)-exchange (NCX) current and T-type Ca2+ current were not significantly different. In conclusion, PV cardiomyocytes have a discrete distribution of transmembrane ion currents associated with specific action potential properties, with potential implications for understanding PV electrical activity in cardiac arrhythmias.

Electrical remodeling of the epicardial border zone in the canine infarcted heart: a computational analysis

The density and kinetics of several ionic currents of cells isolated from the epicardial border zone of the infarcted heart (IZs) are markedly different from cells from the noninfarcted canine epicardium (NZs). To understand how these changes in channel function affect the action potential of the IZ cell as well as its response to antiarrhythmic agents, we developed a new ionic model of the action potential of a cell that survives in the infarct (IZ) and one of a normal epicardial cell (NZ) using formulations based on experimental measurements. The difference in action potential duration (APD) between NZ and IZ cells during steady-state stimulation (basic cycle length = 250 ms) was 6 ms (156 ms in NZ and 162 ms in IZ). However, because IZs exhibit postrepolarization refractoriness, the difference in the effective refractory period (ERP), calculated using a propagation model of a single fiber of 100 cells, was 43 ms (156 ms in NZ and 199 ms in IZ). Either an increase in L-type Ca(2+) current (to simulate the effects of BAY Y5959) or a decrease of both or either delayed rectifier currents (e.g., to simulate the effects of azimilide, sotalol, and chromanol) had significant effects on NZ ERP. In contrast, the effects of these agents in IZs were minor, in agreement with measurements in the in situ canine infarcted heart. Therefore 1) because IZs exhibit postrepolarization refractoriness, conclusions drawn from APD measurements cannot be extrapolated directly to ERPs; 2) ionic currents that are the major determinants of APD and the ERP in NZs are less important in IZs; and 3) differential effects of either BAY Y5959 or azimilide in NZs versus IZs are predicted to decrease ERP dispersion and in so doing prevent initiation of arrhythmias in a substrate of inhomogeneous APD/ERPs.

Cholinergic atrial fibrillation in a computer model of a two-dimensional sheet of canine atrial cells with realistic ionic properties

Classical concepts of atrial fibrillation (AF) have been rooted in Moe's multiple-wavelet hypothesis and simple cellular-automaton computer model. Recent experimental work has raised questions about the multiple-wavelet mechanism, suggesting a discrete "driver region" underlying AF. We reexplored the theoretical basis for AF with a 2-dimensional computer model of a 5x10-cm sheet of atrial cells with realistic ionic and coupling properties. Vagal actions were formulated based on patch-clamp studies of acetylcholine (ACh) effects. In control, a single extrastimulus resulted in a highly meandering unstable spiral wave. Simulated electrograms showed fibrillatory activity, with a dominant frequency (DF, 6.5 Hz) that correlated with the mean rate. Uniform ACh reduced core meander of the spiral wave by approximately 70% (as measured by the standard deviation of spiral-wave tip position) and accelerated the DF to 17.0 Hz. Simulated vagally induced refractoriness heterogeneity caused wavefront breakup as accelerated reentrant activity in regions of short refractoriness impinged on regions unable to respond in a 1:1 fashion because of longer refractoriness. In 7 simulations spanning the range of conditions giving sustained AF, 5 were maintained by single dominant spiral waves. On average, 3.0+/-1.3 wavelets were present (range, 1 to 7). Most wavelets were short-lived and did not contribute to AF maintenance. In contrast to predictions of the multiple-wavelet hypothesis, but in agreement with recent experimental evidence, our model indicates that AF can result from relatively stable primary spiral-wave generators and is significantly organized. Our results suggest that vagal AF may arise from ACh-induced stabilization of the primary spiral-wave generator and disorganization of the heterogeneous tissue response. The full text of this article is available at http://www.circresaha.org.