Computer modelling of the sinoatrial node

Computational modeling of cardiac electrophysiology and arrhythmogenesis: toward clinical translation

The complexity of cardiac electrophysiology, involving dynamic changes in numerous components across multiple spatial (from ion channel to organ) and temporal (from milliseconds to days) scales, makes an intuitive or empirical analysis of cardiac arrhythmogenesis challenging. Multiscale mechanistic computational models of cardiac electrophysiology provide precise control over individual parameters, and their reproducibility enables a thorough assessment of arrhythmia mechanisms. This review provides a comprehensive analysis of models of cardiac electrophysiology and arrhythmias, from the single cell to the organ level, and how they can be leveraged to better understand rhythm disorders in cardiac disease and to improve heart patient care. Key issues related to model development based on experimental data are discussed, and major families of human cardiomyocyte models and their applications are highlighted. An overview of organ-level computational modeling of cardiac electrophysiology and its clinical applications in personalized arrhythmia risk assessment and patient-specific therapy of atrial and ventricular arrhythmias is provided. The advancements presented here highlight how patient-specific computational models of the heart reconstructed from patient data have achieved success in predicting risk of sudden cardiac death and guiding optimal treatments of heart rhythm disorders. Finally, an outlook toward potential future advances, including the combination of mechanistic modeling and machine learning/artificial intelligence, is provided. As the field of cardiology is embarking on a journey toward precision medicine, personalized modeling of the heart is expected to become a key technology to guide pharmaceutical therapy, deployment of devices, and surgical interventions.

The Impact of Potassium Dynamics on Cardiomyocyte Beating in Hemodialysis Treatment

Background: Observational studies of intermittent hemodialysis therapy have reported that the excess decrease in K+ concentration in plasma (KP) during treatment is associated with the destabilization of cardiac function. Elucidating the mechanism by which the decrease in KP impairs myocardial excitation is indispensable for a deeper understanding of prescription design. Methods: In this study, by using an electrophysiological mathematical model, we investigated the relationship between KP dynamics and cardiomyocyte excitability for the first time. Results: The excess decrease in KP during treatment destabilized cardiomyocyte excitability through the following events: (1) a decrease in KP led to the prolongation of the depolarization phase of ventricular cells due to the reduced potassium efflux rate of the Kr channel, temporarily enhancing contraction force; (2) an excess decrease in KP activated the transport of K+ and Na+ through the funny channel in sinoatrial nodal cells, disrupting automaticity; (3) the excess decrease in KP also resulted in a significant decrease in the resting membrane potential of ventricular cells, causing contractile dysfunction. Avoiding an excess decrease in KP during treatment contributed to the maintenance of cardiomyocyte excitability. Conclusions: The results of these mathematical analyses showed that it is necessary to implement personal prescription or optimal control of K+ concentration in dialysis fluid based on predialysis KP from the perspective of regulatory science in dialysis treatment.

Spectral Analysis of Heart Rate Variability in Time-Varying Conditions and in the Presence of Confounding Factors

The tools for spectrally analyzing heart rate variability (HRV) has in recent years grown considerably, with emphasis on the handling of time-varying conditions and confounding factors. Time-frequency analysis holds since long an important position in HRV analysis, however, this technique cannot alone handle a mean heart rate or a respiratory frequency which vary over time. Overlapping frequency bands represents another critical condition which needs to be dealt with to produce accurate spectral measurements. The present survey offers a comprehensive account of techniques designed to handle such conditions and factors by providing a brief description of the main principles of the different methods. Several methods derive from a mathematical/statistical model, suggesting that the model can be used to simulate data used for performance evaluation. The inclusion of a respiratory signal, whether measured or derived, is another feature of many recent methods, e.g., used to guide the decomposition of the HRV signal so that signals related as well as unrelated to respiration can be analyzed. It is concluded that the development of new approaches to handling time-varying scenarios are warranted, as is benchmarking of performance evaluated in technical as well as in physiological/clinical terms.

The virtual sinoatrial node: What did computational models tell us about cardiac pacemaking?

Since its discovery, the sinoatrial node (SAN) has represented a fascinating and complex matter of research. Despite over a century of discoveries, a full comprehension of pacemaking has still to be achieved. Experiments often produced conflicting evidence that was used either in support or against alternative theories, originating intense debates. In this context, mathematical descriptions of the phenomena underlying the heartbeat have grown in importance in the last decades since they helped in gaining insights where experimental evaluation could not reach. This review presents the most updated SAN computational models and discusses their contribution to our understanding of cardiac pacemaking. Electrophysiological, structural and pathological aspects - as well as the autonomic control over the SAN - are taken into consideration to reach a holistic view of SAN activity.

Coupling and heterogeneity modulate pacemaking capability in healthy and diseased two-dimensional sinoatrial node tissue models

Both experimental and modeling studies have attempted to determine mechanisms by which a small anatomical region, such as the sinoatrial node (SAN), can robustly drive electrical activity in the human heart. However, despite many advances from prior research, important questions remain unanswered. This study aimed to investigate, through mathematical modeling, the roles of intercellular coupling and cellular heterogeneity in synchronization and pacemaking within the healthy and diseased SAN. In a multicellular computational model of a monolayer of either human or rabbit SAN cells, simulations revealed that heterogenous cells synchronize their discharge frequency into a unique beating rhythm across a wide range of heterogeneity and intercellular coupling values. However, an unanticipated behavior appeared under pathological conditions where perturbation of ionic currents led to reduced excitability. Under these conditions, an intermediate range of intercellular coupling (900-4000 MΩ) was beneficial to SAN automaticity, enabling a very small portion of tissue (3.4%) to drive propagation, with propagation failure occurring at both lower and higher resistances. This protective effect of intercellular coupling and heterogeneity, seen in both human and rabbit tissues, highlights the remarkable resilience of the SAN. Overall, the model presented in this work allowed insight into how spontaneous beating of the SAN tissue may be preserved in the face of perturbations that can cause individual cells to lose automaticity. The simulations suggest that certain degrees of gap junctional coupling protect the SAN from ionic perturbations that can be caused by drugs or mutations.

Initiation and entrainment of multicellular automaticity via diffusion limited extracellular domains

Electrically excitable cells often spontaneously and synchronously depolarize in vitro and in vivo preparations. It remains unclear how cells entrain and autorhythmically activate above the intrinsic mean activation frequency of isolated cells with or without pacemaking mechanisms. Recent studies suggest that cyclic ion accumulation and depletion in diffusion-limited extracellular volumes modulate electrophysiology by ephaptic mechanisms (nongap junction or synaptic coupling). This report explores how potassium accumulation and depletion in a restricted extracellular domain induces spontaneous action potentials in two different computational models of excitable cells without gap junctional coupling: Hodgkin-Huxley and Luo-Rudy. Importantly, neither model will spontaneously activate on its own without external stimuli. Simulations demonstrate that cells sharing a diffusion-limited extracellular compartment can become autorhythmic and entrained despite intercellular electrical heterogeneity. Autorhythmic frequency is modulated by the cleft volume and potassium fluxes through the cleft. Additionally, inexcitable cells can suppress or induce autorhythmic activity in an excitable cell via a shared cleft. Diffusion-limited shared clefts can also entrain repolarization. Critically, this model predicts a mechanism by which diffusion-limited shared clefts can initiate, entrain, and modulate multicellular automaticity in the absence of gap junctions.

Evolution of mathematical models of cardiomyocyte electrophysiology

Advanced computational techniques and mathematical modeling have become more and more important to the study of cardiac electrophysiology. In this review, we provide a brief history of the evolution of cardiomyocyte electrophysiology models and highlight some of the most important ones that had a major impact on our understanding of the electrical activity of the myocardium and associated transmembrane ion fluxes in normal and pathological states. We also present the use of these models in the study of various arrhythmogenesis mechanisms, particularly the integration of experimental pharmacology data into advanced humanized models for in silico proarrhythmogenic risk prediction as an essential component of the Comprehensive in vitro Proarrhythmia Assay (CiPA) drug safety paradigm.

Mechanistic Insights Into the Reduced Pacemaking Rate of the Rabbit Sinoatrial Node During Postnatal Development: A Simulation Study

Marked age- and development- related differences have been observed in morphology and characteristics of action potentials (AP) of neonatal and adult sinoatrial node (SAN) cells. These may be attributable to a different set of ion channel interactions between the different ages. However, the underlying mechanism(s) have yet to be elucidated. The objective of this study was to determine the mechanisms underlying different spontaneous APs and heart rate between neonatal and adult SAN cells of the rabbit heart by biophysical modeling approaches. A mathematical model of neonatal rabbit SAN cells was developed by modifying the current densities and/or kinetics of ion channels and transporters in an adult cell model based on available experimental data obtained from neonatal SAN cells. The single cell models were then incorporated into a multi-cellular, two-dimensional model of the intact SAN-atrium to investigate the functional impact of altered ion channels during maturation on pacemaking electrical activities and their conduction at the tissue level. Effects of the neurotransmitter acetylcholine on the pacemaking activities in neonatal cells were also investigated and compared to those in the adult. Our results showed: (1) the differences in ion channel properties between neonatal and adult SAN cells are able to account for differences in their APs and the heart rate, providing mechanistic insight into understanding the reduced pacemaking rate of the rabbit sinoatrial node during postnatal development; (2) in the 2D model of the intact SAN-atria, it was shown that cellular changes during postnatal development impaired pacemaking activity through increasing the activation time and reducing the conduction velocity across the SAN; (3) the neonatal SAN model, with its faster beating rates, showed a greater sensitivity to parasympathetic modulation in response to acetylcholine than did the adult model. These results provide novel insights into the understanding of the cellular mechanisms underlying the differences in the cardiac pacemaking activities of the neonatal and adult SAN.

The Cardiac Pacemaker Story-Fundamental Role of the Na+/Ca2+ Exchanger in Spontaneous Automaticity

The electrophysiological mechanism of the sinus node automaticity was previously considered exclusively regulated by the so-called "funny current". However, parallel investigations increasingly emphasized the importance of the Ca²⁺-homeostasis and Na⁺/Ca²⁺ exchanger (NCX). Recently, increasing experimental evidence, as well as insight through mechanistic in silico modeling demonstrates the crucial role of the exchanger in sinus node pacemaking. NCX had a key role in the exciting story of discovery of sinus node pacemaking mechanisms, which recently settled with a consensus on the coupled-clock mechanism after decades of debate. This review focuses on the role of the Na⁺/Ca²⁺ exchanger from the early results and concepts to recent advances and attempts to give a balanced summary of the characteristics of the local, spontaneous, and rhythmic Ca²⁺ releases, the molecular control of the NCX and its role in the fight-or-flight response. Transgenic animal models and pharmacological manipulation of intracellular Ca²⁺ concentration and/or NCX demonstrate the pivotal function of the exchanger in sinus node automaticity. We also highlight where specific hypotheses regarding NCX function have been derived from computational modeling and require experimental validation. Nonselectivity of NCX inhibitors and the complex interplay of processes involved in Ca²⁺ handling render the design and interpretation of these experiments challenging.

Improved Computational Identification of Drug Response Using Optical Measurements of Human Stem Cell Derived Cardiomyocytes in Microphysiological Systems

Cardiomyocytes derived from human induced pluripotent stem cells (hiPSC-CMs) hold great potential for drug screening applications. However, their usefulness is limited by the relative immaturity of the cells' electrophysiological properties as compared to native cardiomyocytes in the adult human heart. In this work, we extend and improve on methodology to address this limitation, building on previously introduced computational procedures which predict drug effects for adult cells based on changes in optical measurements of action potentials and Ca2+ transients made in stem cell derived cardiac microtissues. This methodology quantifies ion channel changes through the inversion of data into a mathematical model, and maps this response to an adult phenotype through the assumption of functional invariance of fundamental intracellular and membrane channels during maturation. Here, we utilize an updated action potential model to represent both hiPSC-CMs and adult cardiomyocytes, apply an IC50-based model of dose-dependent drug effects, and introduce a continuation-based optimization algorithm for analysis of dose escalation measurements using five drugs with known effects. The improved methodology can identify drug induced changes more efficiently, and quantitate important metrics such as IC50 in line with published values. Consequently, the updated methodology is a step towards employing computational procedures to elucidate drug effects in adult cardiomyocytes for new drugs using stem cell-derived experimental tissues.

Analysis of heterogeneous cardiac pacemaker tissue models and traveling wave dynamics

The sinoatrial-node (SAN) is a complex heterogeneous tissue that generates a stable rhythm in healthy hearts, yet a general mechanistic explanation for when and how this tissue remains stable is lacking. Although computational and theoretical analyses could elucidate these phenomena, such methods have rarely been used in realistic (large-dimensional) gap-junction coupled heterogeneous pacemaker tissue models. In this study, we adapt a recent model of pacemaker cells (Severi et al., 2012), incorporating biophysical representations of ion channel and intracellular calcium dynamics, to capture physiological features of a heterogeneous population of pacemaker cells, in particular "center" and "peripheral" cells with distinct intrinsic frequencies and action potential morphology. Large-scale simulations of the SAN tissue, represented by a heterogeneous tissue structure of pacemaker cells, exhibit a rich repertoire of behaviors, including complete synchrony, traveling waves of activity originating from periphery to center, and transient traveling waves originating from the center. We use phase reduction methods that do not require fully simulating the large-scale model to capture these observations. Moreover, the phase reduced models accurately predict key properties of the tissue electrical dynamics, including wave frequencies when synchronization occurs, and wave propagation direction in a variety of tissue models. With the reduced phase models, we analyze the relationship between cell distributions and coupling strengths and the resulting transient dynamics. Further, the reduced phase model predicts parameter regimes of irregular electrical dynamics. Thus, we demonstrate that phase reduced oscillator models applied to realistic pacemaker tissue is a useful tool for investigating the spatial-temporal dynamics of cardiac pacemaker activity.

Cardiac automaticity: basic concepts and clinical observations

PURPOSE

The purpose of this report was to review the basic mechanisms underlying cardiac automaticity. Second, we describe our clinical observations related to the anatomical and functional characteristics of sinus automaticity.

METHODS

We first reviewed the main discoveries regarding the mechanisms responsible for cardiac automaticity. We then analyzed our clinical experience regarding the location of sinus automaticity in two unique populations: those with inappropriate sinus tachycardia and those with a dominant pacemaker located outside the crista terminalis region.

RESULTS

We studied 26 patients with inappropriate sinus tachycardia (age 34 ± 8 years; 21 females). Non-contact endocardial mapping (Ensite 3000, Endocardial Solutions) was performed in 19 patients and high-density contact mapping (Carto-3, Biosense Webster with PentaRay catheter) in 7 patients. The site of earliest atrial activation shifted after each RF application within and outside the crista terminalis region, indicating a wide distribution of atrial pacemaker sites. We also analyzed 11 patients with dominant pacemakers located outside the crista terminalis (age 27 ± 7 years; five females). In all patients, the rhythm was the dominant pacemaker both at rest and during exercise and located in the right atrial appendage in 6 patients, in the left atrial appendage in 4 patients, and in the mitral annulus in 1 patient. Following ablation, earliest atrial activation shifted to the region of the crista terminalis at a slower rate.

CONCLUSIONS

Membrane and sub-membrane mechanisms interact to generate cardiac automaticity. The present observations in patients with inappropriate sinus tachycardia and dominant pacemakers are consistent with a wide distribution of pacemaker sites within and outside the boundaries of the crista terminalis.

A Singular Role of IK1 Promoting the Development of Cardiac Automaticity during Cardiomyocyte Differentiation by IK1 -Induced Activation of Pacemaker Current

The inward rectifier potassium current (I_K1) is generally thought to suppress cardiac automaticity by hyperpolarizing membrane potential (MP). We recently observed that I_K1 could promote the spontaneously-firing automaticity induced by upregulation of pacemaker funny current (I_f) in adult ventricular cardiomyocytes (CMs). However, the intriguing ability of I_K1 to activate I_f and thereby promote automaticity has not been explored. In this study, we combined mathematical and experimental assays and found that only I_K1 and I_f, at a proper-ratio of densities, were sufficient to generate rhythmic MP-oscillations even in unexcitable cells (i.e. HEK293T cells and undifferentiated mouse embryonic stem cells [ESCs]). We termed this effect I_K1-induced I_f activation. Consistent with previous findings, our electrophysiological recordings observed that around 50% of mouse (m) and human (h) ESC-differentiated CMs could spontaneously fire action potentials (APs). We found that spontaneously-firing ESC-CMs displayed more hyperpolarized maximum diastolic potential and more outward I_K1 current than quiescent-yet-excitable m/hESC-CMs. Rather than classical depolarization pacing, quiescent mESC-CMs were able to fire APs spontaneously with an electrode-injected small outward-current that hyperpolarizes MP. The automaticity to spontaneously fire APs was also promoted in quiescent hESC-CMs by an I_K1-specific agonist zacopride. In addition, we found that the number of spontaneously-firing m/hESC-CMs was significantly decreased when I_f was acutely upregulated by Ad-CGI-HCN infection. Our study reveals a novel role of I_K1 promoting the development of cardiac automaticity in m/hESC-CMs through a mechanism of I_K1-induced I_f activation and demonstrates a synergistic interaction between I_K1 and I_f that regulates cardiac automaticity.

Computational analysis of the human sinus node action potential: model development and effects of mutations

KEY POINTS

We constructed a comprehensive mathematical model of the spontaneous electrical activity of a human sinoatrial node (SAN) pacemaker cell, starting from the recent Severi-DiFrancesco model of rabbit SAN cells. Our model is based on electrophysiological data from isolated human SAN pacemaker cells and closely matches the action potentials and calcium transient that were recorded experimentally. Simulated ion channelopathies explain the clinically observed changes in heart rate in corresponding mutation carriers, providing an independent qualitative validation of the model. The model shows that the modulatory role of the 'funny current' (If ) in the pacing rate of human SAN pacemaker cells is highly similar to that of rabbit SAN cells, despite its considerably lower amplitude. The model may prove useful in the design of experiments and the development of heart-rate modulating drugs.

ABSTRACT

The sinoatrial node (SAN) is the normal pacemaker of the mammalian heart.  Over several decades, a large amount of data on the ionic mechanisms underlying the spontaneous electrical activity of SAN pacemaker cells has been obtained, mostly in experiments on single cells isolated from rabbit SAN. This wealth of data has allowed the development of mathematical models of the electrical activity of rabbit SAN pacemaker cells. The present study aimed to construct a comprehensive model of the electrical activity of a human SAN pacemaker cell using recently obtained electrophysiological data from human SAN pacemaker cells.  We based our model on the recent Severi-DiFrancesco model of a rabbit SAN pacemaker cell. The action potential and calcium transient of the resulting model are close to the experimentally recorded values. The model has a much smaller 'funny current' (If ) than do rabbit cells, although its modulatory role is highly similar. Changes in pacing rate upon the implementation of mutations associated with sinus node dysfunction agree with the clinical observations. This agreement holds for both loss-of-function and gain-of-function mutations in the HCN4, SCN5A and KCNQ1 genes, underlying ion channelopathies in If , fast sodium current and slow delayed rectifier potassium current, respectively. We conclude that our human SAN cell model can be a useful tool in the design of experiments and the development of drugs that aim to modulate heart rate.

Model of electrical activity in cardiac tissue under electromagnetic induction

Complex electrical activities in cardiac tissue can set up time-varying electromagnetic field. Magnetic flux is introduced into the Fitzhugh-Nagumo model to describe the effect of electromagnetic induction, and then memristor is used to realize the feedback of magnetic flux on the membrane potential in cardiac tissue. It is found that a spiral wave can be triggered and developed by setting specific initials in the media, that is to say, the media still support the survival of standing spiral waves under electromagnetic induction. Furthermore, electromagnetic radiation is considered on this model as external stimuli, it is found that spiral waves encounter breakup and turbulent electrical activities are observed, and it can give guidance to understand the occurrence of sudden heart disorder subjected to heavily electromagnetic radiation.

Advances in the management of heart failure: the role of ivabradine

A high resting heart rate (≥70–75 b.p.m.) is a risk factor for patients with heart failure (HF) with reduced ejection fraction (EF), probably in the sense of accelerated atherosclerosis, with an increased morbidity and mortality. Beta-blockers not only reduce heart rate but also have negative inotropic and blood pressure-lowering effects, and therefore, in many patients, they cannot be given in the recommended dose. Ivabradine specifically inhibits the pacemaker current (funny current, I_f) of the sinoatrial node cells, resulting in therapeutic heart rate lowering without any negative inotropic and blood pressure-lowering effect. According to the European Society of Cardiology guidelines, ivabradine should be considered to reduce the risk of HF hospitalization and cardiovascular death in symptomatic patients with a reduced left ventricular EF ≤35% and sinus rhythm ≥70 b.p.m. despite treatment with an evidence-based dose of beta-blocker or a dose below the recommended dose (recommendation class “IIa” = weight of evidence/opinion is in favor of usefulness/efficacy: “should be considered”; level of evidence “B” = data derived from a single randomized clinical trial or large nonrandomized studies). Using a heart rate cutoff of ≥ 75 b.p.m., as licensed by the European Medicines Agency, treatment with ivabradine 5–7.5 mg b.i.d. reduces cardiovascular mortality by 17%, HF mortality by 39% and HF hospitalization rate by 30%. A high resting heart rate is not only a risk factor in HF with reduced EF but also at least a risk marker in HF with preserved EF, in acute HF and also in special forms of HF. In this review, we discuss the proven role of ivabradine in the validated indication “HF with reduced EF” together with interesting preliminary findings, and the potential role of ivabradine in further, specific forms of HF.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Rabbit-specific computational modelling of ventricular cell electrophysiology: Using populations of models to explore variability in the response to ischemia

Computational modelling, combined with experimental investigations, is a powerful method for investigating complex cardiac electrophysiological behaviour. The use of rabbit-specific models, due to the similarities of cardiac electrophysiology in this species with human, is especially prevalent. In this paper, we first briefly review rabbit-specific computational modelling of ventricular cell electrophysiology, multi-cellular simulations including cellular heterogeneity, and acute ischemia. This mini-review is followed by an original computational investigation of variability in the electrophysiological response of two experimentally-calibrated populations of rabbit-specific ventricular myocyte action potential models to acute ischemia. We performed a systematic exploration of the response of the model populations to varying degrees of ischemia and individual ischemic parameters, to investigate their individual and combined effects on action potential duration and refractoriness. This revealed complex interactions between model population variability and ischemic factors, which combined to enhance variability during ischemia. This represents an important step towards an improved understanding of the role that physiological variability may play in electrophysiological alterations during acute ischemia.

A computational model of a human single sinoatrial node cell

For the investigation of the spontaneous rhythmical activity response in the application of cardiac neuromodulation, we formulated a human sinoatrial node (SAN) cell model. With the aim of decreasing elevated heart rate (HR), we want to establish a hardware-in-the-loop system including this model for the analysis of optimal stimulation patterns of the neurostimulation system. Base model structures are adopted from rabbit SAN cell models available in literature and conveyed with Hodgkin-Huxley-type model equations describing the complex time and voltage dependent activation and deactivation processes of individual ion channels. The resulting model consists of 15 currents which are currently known to be responsible for the generation of the membrane action potential (AP). The model reproduces AP frequencies equivalent to those measured in isolated human SAN cells with a resulting HR of 71.8 bpm. Model validation via simulation of the inhibitory effect of ivabradine showed accordance with experimental results obtained in human studies. Furthermore, we could validate the model in regard to its HR effects upon parasympathetic stimulation with results obtained in a human trial study.

Cell-specific Dynamic Clamp analysis of the role of funny If current in cardiac pacemaking

We used the Dynamic Clamp technique for i) comparative validation of conflicting computational models of the hyperpolarization-activated funny current, If, and ii) quantification of the role of If in mediating autonomic modulation of heart rate. Experimental protocols based on the injection of a real-time recalculated synthetic If current in sinoatrial rabbit cells were developed. Preliminary results of experiments mimicking the autonomic modulation of If demonstrated the need for a customization procedure to compensate for cellular heterogeneity. For this reason, we used a cell-specific approach, scaling the maximal conductance of the injected current based on the cell's spontaneous firing rate. The pacemaking rate, which was significantly reduced after application of Ivabradine, was restored by the injection of synthetic current based on the Severi-DiFrancesco formulation, while the injection of synthetic current based on the Maltsev-Lakatta formulation did not produce any significant variation. A positive virtual shift of the If activation curve, mimicking the Isoprenaline effects, led to a significant increase in pacemaking rate (+17.3 ± 6.7%, p < 0.01), although of lower magnitude than that induced by real Isoprenaline (+45.0 ± 26.1%). Similarly, a negative virtual shift of the activation curve significantly lowered the pacemaking rate (-11.8 ± 1.9%, p < 0.001), as did the application of real Acetylcholine (-20.5 ± 5.1%). The Dynamic Clamp approach, applied to the If study in cardiomyocytes for the first time and rate-adapted to manage intercellular variability, indicated that: i) the quantitative description of the If current in the Severi-DiFrancesco model accurately reproduces the effects of the real current on rabbit sinoatrial cell pacemaking rate and ii) a significant portion (50-60%) of the physiological autonomic rate modulation is due to the shift of the If activation curve.

From two competing oscillators to one coupled-clock pacemaker cell system

At the beginning of this century, debates regarding “what are the main control mechanisms that ignite the action potential (AP) in heart pacemaker cells” dominated the electrophysiology field. The original theory which prevailed for over 50 years had advocated that the ensemble of surface membrane ion channels (i.e., “M-clock”) is sufficient to ignite rhythmic APs. However, more recent experimental evidence in a variety of mammals has shown that the sarcoplasmic reticulum (SR) acts as a “Ca²⁺-clock” rhythmically discharges diastolic local Ca²⁺ releases (LCRs) beneath the cell surface membrane. LCRs activate an inward current (likely that of the Na⁺/Ca²⁺ exchanger) that prompts the surface membrane “M-clock” to ignite an AP. Theoretical and experimental evidence has mounted to indicate that this clock “crosstalk” operates on a beat-to-beat basis and determines both the AP firing rate and rhythm. Our review is focused on the evolution of experimental definition and numerical modeling of the coupled-clock concept, on how mechanisms intrinsic to pacemaker cell determine both the heart rate and rhythm, and on future directions to develop further the coupled-clock pacemaker cell concept.

Pacemaker activity of the human sinoatrial node: an update on the effects of mutations in HCN4 on the hyperpolarization-activated current

Since 2003, several loss-of-function mutations in the HCN4 gene, which encodes the HCN4 protein, have been associated with sinus node dysfunction. In human sinoatrial node (SAN), HCN4 is the most abundant of the four isoforms of the HCN family. Tetramers of HCN subunits constitute the ion channels that conduct the hyperpolarization-activated "funny" current (If), which plays an important modulating role in SAN pacemaker activity. Voltage-clamp experiments on HCN4 channels expressed in COS-7, CHO and HEK-293 cells, as well as in Xenopus oocytes have revealed changes in the expression and kinetics of mutant channels, but the extent to which especially the kinetic changes would affect If flowing during a human SAN action potential often remains unresolved. In our contribution to the Topical Collection on Human Single Nucleotide Polymorphisms and Disease Diagnostics, we provide an updated review of the mutation-induced changes in the expression and kinetics of HCN4 channels and provide an overview of their effects on If during the time course of a human SAN action potential, as assessed in simulated action potential clamp experiments. Future research may solve apparent inconsistencies between data from clinical studies and data from in vitro and in silico experiments.

Pacemaker activity of the human sinoatrial node: effects of HCN4 mutations on the hyperpolarization-activated current

The hyperpolarization-activated 'funny' current, If, plays an important modulating role in the pacemaker activity of the human sinoatrial node (SAN). If is carried by hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which are tetramers built of four HCN subunits. In human SAN, HCN4 is the most abundant of the four isoforms of the HCN family. Since 2003, several loss-of-function mutations in the HCN4 gene, which encodes the HCN4 protein, or in the KCNE2 gene, which encodes the MiRP1 accessory β-subunit, have been associated with sinus node dysfunction. Voltage-clamp experiments on HCN4 channels expressed in COS-7 cells, Xenopus oocytes, or HEK-293 cells have revealed changes in the expression and kinetics of mutant channels, but the extent to which these changes would affect If flowing during a human SAN action potential is unresolved. Here, we review the changes in expression and kinetics of HCN4 mutant channels and provide an overview of their effects on If during the time course of a human SAN action potential, both under resting conditions and upon adrenergic stimulation. These effects are assessed in simulated action potential clamp experiments, with action potentials recorded from isolated human SAN pacemaker cells as command potential and kinetics of If based on voltage-clamp data from these cells. Results from in vitro and in silico experiments show several inconsistencies with clinical observations, pointing to challenges for future research.

Stem cell-based biological pacemakers from proof of principle to therapy: a review

Electronic pacemakers are the standard therapy for bradycardia-related symptoms but have shortcomings. Over the past 15 years, experimental evidence has demonstrated that gene and cell-based therapies can create a biological pacemaker. Recently, physiologically acceptable rates have been reported with an adenovirus-based approach. However, adenovirus-based protein expression does not last more than 4 weeks, which limits its clinical applicability. Cell-based platforms are potential candidates for longer expression. Currently there are two cell-based approaches being tested: (i) mesenchymal stem cells used as a suitcase for delivering pacemaker genes and (ii) pluripotent stem cells differentiated down a cardiac lineage with endogenous pacemaker activity. This review examines the current achievements in engineering a biological pacemaker, defines the patient population for whom this device would be useful and identifies the challenges still ahead before cell therapy can replace current electronic devices.

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.

Modern concepts concerning the origin of the heartbeat

Physiological processes governing the heart beat have been under investigation for several hundred years. Major advances have been made in the recent past. A review of the present paradigm is presented here, including a look back at important steps that led us to where we are today, alongside a glimpse into the exciting future of pacemaker research.

A quantitative comparison of the behavior of human ventricular cardiac electrophysiology models in tissue

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

SEM, TEM, and IHC Analysis of the Sinus Node and Its Implications for the Cardiac Conduction System

More than 100 years after the discovery of the sinus node (SN) by Keith and Flack, the function and structure of the SN have not been completely established yet. The anatomic architecture of the SN has often been described as devoid of an organized structure; the origin of the sinus impulse is still a matter of debate, and a definite description of the long postulated internodal specialized tract conducting the impulse from the SN to the atrioventricular node (AVN) is still missing. In our previously published study, we proposed a morphologically ordered structure for the SN. As a confirmation of what was presented then, we have added the results of additional observations regarding the structural particularities of the SN. We investigated the morphology of the sinus node in the human hearts of healthy individuals using histochemical, immunohistochemical, optical, and electron microscopy (SEM, TEM). Our results confirmed that the SN presents a previously unseen highly organized architecture.

Hyperpolarization-activated current, If, in mathematical models of rabbit sinoatrial node pacemaker cells

A typical feature of sinoatrial (SA) node pacemaker cells is the presence of an ionic current that activates upon hyperpolarization. The role of this hyperpolarization-activated current, I_f, which is also known as the “funny current” or “pacemaker current,” in the spontaneous pacemaker activity of SA nodal cells remains a matter of intense debate. Whereas some conclude that I_f plays a fundamental role in the generation of pacemaker activity and its rate control, others conclude that the role of I_f is limited to a modest contribution to rate control. The ongoing debate is often accompanied with arguments from computer simulations, either to support one's personal view or to invalidate that of the antagonist. In the present paper, we review the various mathematical descriptions of I_f that have been used in computer simulations and compare their strikingly different characteristics with our experimental data. We identify caveats and propose a novel model for I_f based on our experimental data.

Beat-to-Beat Variation in Periodicity of Local Calcium Releases Contributes to Intrinsic Variations of Spontaneous Cycle Length in Isolated Single Sinoatrial Node Cells

Spontaneous, submembrane local Ca²⁺ releases (LCRs) generated by the sarcoplasmic reticulum in sinoatrial nodal cells, the cells of the primary cardiac pacemaker, activate inward Na⁺/Ca²⁺-exchange current to accelerate the diastolic depolarization rate, and therefore to impact on cycle length. Since LCRs are generated by Ca²⁺ release channel (i.e. ryanodine receptor) openings, they exhibit a degree of stochastic behavior, manifested as notable cycle-to-cycle variations in the time of their occurrence.

Aim

The present study tested whether variation in LCR periodicity contributes to intrinsic (beat-to-beat) cycle length variability in single sinoatrial nodal cells.

Methods

We imaged single rabbit sinoatrial nodal cells using a 2D-camera to capture LCRs over the entire cell, and, in selected cells, simultaneously measured action potentials by perforated patch clamp.

Results

LCRs begin to occur on the descending part of the action potential-induced whole-cell Ca²⁺ transient, at about the time of the maximum diastolic potential. Shortly after the maximum diastolic potential (mean 54±7.7 ms, n = 14), the ensemble of waxing LCR activity converts the decay of the global Ca²⁺ transient into a rise, resulting in a late, whole-cell diastolic Ca²⁺ elevation, accompanied by a notable acceleration in diastolic depolarization rate. On average, cells (n = 9) generate 13.2±3.7 LCRs per cycle (mean±SEM), varying in size (7.1±4.2 µm) and duration (44.2±27.1 ms), with both size and duration being greater for later-occurring LCRs. While the timing of each LCR occurrence also varies, the LCR period (i.e. the time from the preceding Ca²⁺ transient peak to an LCR’s subsequent occurrence) averaged for all LCRs in a given cycle closely predicts the time of occurrence of the next action potential, i.e. the cycle length.

Conclusion

Intrinsic cycle length variability in single sinoatrial nodal cells is linked to beat-to-beat variations in the average period of individual LCRs each cycle.

Numerical models based on a minimal set of sarcolemmal electrogenic proteins and an intracellular Ca(2+) clock generate robust, flexible, and energy-efficient cardiac pacemaking

Recent evidence supports the idea that robust and, importantly, FLEXIBLE automaticity of cardiac pacemaker cells is conferred by a coupled system of membrane ion currents (an "M-clock") and a sarcoplasmic reticulum (SR)-based Ca(2+) oscillator ("Ca(2+)clock") that generates spontaneous diastolic Ca(2+) releases. This study identified numerical models of a human biological pacemaker that features robust and flexible automaticity generated by a minimal set of electrogenic proteins and a Ca(2+)clock. Following the Occam's razor principle (principle of parsimony), M-clock components of unknown molecular origin were excluded from Maltsev-Lakatta pacemaker cell model and thirteen different model types of only 4 or 5 components were derived and explored by a parametric sensitivity analysis. The extended ranges of SR Ca(2+) pumping (i.e. Ca(2+)clock performance) and conductance of ion currents were sampled, yielding a large variety of parameter combination, i.e. specific model sets. We tested each set's ability to simulate autonomic modulation of human heart rate (minimum rate of 50 to 70bpm; maximum rate of 140 to 210bpm) in response to stimulation of cholinergic and β-adrenergic receptors. We found that only those models that include a Ca(2+)clock (including the minimal 4-parameter model "ICaL+IKr+INCX+Ca(2+)clock") were able to reproduce the full range of autonomic modulation. Inclusion of If or ICaT decreased the flexibility, but increased the robustness of the models (a relatively larger number of sets did not fail during testing). The new models comprised of components with clear molecular identity (i.e. lacking IbNa & Ist) portray a more realistic pacemaking: A smaller Na(+) influx is expected to demand less energy for Na(+) extrusion. The new large database of the reduced coupled-clock numerical models may serve as a useful tool for the design of biological pacemakers. It will also provide a conceptual basis for a general theory of robust, flexible, and energy-efficient pacemaking based on realistic components.

Exploring the implicit interlayer regulatory mechanism between cells and tissue: stochastic mathematical analyses of the spontaneous ordering in beating synchronization

The present study focused on beating synchronization, and tried to elucidate the interlayer regulatory mechanisms between the cells and clump in beating synchronization with using the stochastic simulations which realize the beating synchronizations in beating cells with low cell-cell conductance. Firstly, the fluctuation in interbeat intervals (IBIs) of beating cells encouraged the process of beating synchronization, which was identified as the stochastic resonance. Secondly, fluctuation in the synchronized IBIs of a clump decreased as the number of beating cells increased. The decrease in IBI fluctuation due to clump formation implied both a decline of the electrophysiological plasticity of each beating cell and an enhancement of the electrophysiological stability of the clump. These findings were identified as the community effects. Because IBI fluctuation and the community effect facilitated the beating stability of the cell and clump, these factors contributed to the spontaneous ordering in beating synchronization. Thirdly, the cellular layouts in clump affected the synchronized beating rhythms. The synchronized beating rhythm in clump was implicitly regulated by a complicated synergistic effect among IBI fluctuation of each beating cell, the community effect and the cellular layout. This finding was indispensable for leading an elucidation of mechanism of emergence. The stochastic simulations showed the necessity of considering the synergistic effect, to elucidate the interlayer regulatory mechanisms in biological system.

An updated computational model of rabbit sinoatrial action potential to investigate the mechanisms of heart rate modulation

The cellular basis of cardiac pacemaking is still debated. Reliable computational models of the sinoatrial node (SAN) action potential (AP) may help gain a deeper understanding of the phenomenon. Recently, novel models incorporating detailed Ca(2+)-handling dynamics have been proposed, but they fail to reproduce a number of experimental data, and more specifically effects of 'funny' (I(f)) current modifications. We therefore developed a SAN AP model, based on available experimental data, in an attempt to reproduce physiological and pharmacological heart rate modulation. Cell compartmentalization and intracellular Ca(2+)-handling mechanisms were formulated as in the Maltsev-Lakatta model, focusing on Ca(2+)-cycling processes. Membrane current equations were revised on the basis of published experimental data. Modifications of the formulation of currents/pumps/exchangers to simulate I(f) blockers, autonomic modulators and Ca(2+)-dependent mechanisms (ivabradine, caesium, acetylcholine, isoprenaline, BAPTA) were derived from experimental data. The model generates AP waveforms typical of rabbit SAN cells, whose parameters fall within the experimental ranges: 352 ms cycle length, 80 mV AP amplitude, -58 mV maximum diastolic potential (MDP), 108 ms APD(50), and 7.1 Vs(-1) maximum upstroke velocity. Rate modulation by I(f) -blocking drugs agrees with experimental findings: 20% and 22% caesium-induced (5mM) and ivabradine-induced (3 μM) rate reductions, respectively, due to changes in diastolic depolarization (DD) slope, with no changes in either MDP or take-off potential (TOP). The model consistently reproduces the effects of autonomic modulation: 20% rate decrease with 10 nM acetylcholine and 28%increase with 1 μM isoprenaline, again entirely due to increase in the DD slope,with no changes in either MDP or TOP. Model testing of BAPTA effects showed slowing of rate, -26%, without cessation of beating. Our up-to-date model describes satisfactorily experimental data concerning autonomic stimulation, funny-channel blockade and inhibition of the Ca(2+)-related system by BAPTA, making it a useful tool for further investigation. Simulation results suggest that a detailed description of the intracellular Ca(2+) fluxes is fully compatible with the observation that I(f) is a major component of pacemaking and rate modulation.

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.

Systematic characterization of the ionic basis of rabbit cellular electrophysiology using two ventricular models

Several mathematical models of rabbit ventricular action potential (AP) have been proposed to investigate mechanisms of arrhythmias and excitation-contraction coupling. Our study aims at systematically characterizing how ionic current properties modulate the main cellular biomarkers of arrhythmic risk using two widely-used rabbit ventricular models, and comparing simulation results using the two models with experimental data available for rabbit. A sensitivity analysis of AP properties, Ca²⁺ and Na⁺ dynamics, and their rate dependence to variations (±15% and ±30%) in the main transmembrane current conductances and kinetics was performed using the Shannon et al. (2004) and the Mahajan et al. (2008a,b) AP rabbit models. The effects of severe transmembrane current blocks (up to 100%) on steady-state AP and calcium transients, and AP duration (APD) restitution curves were also simulated using both models. Our simulations show that, in both virtual rabbit cardiomyocytes, APD is significantly modified by most repolarization currents, AP triangulation is regulated mostly by the inward rectifier K⁺ current (I(K1)) whereas APD rate adaptation as well as [Na⁺](i) rate dependence is influenced by the Na⁺/K⁺ pump current (I(NaK)). In addition, steady-state [Ca²⁺](i) levels, APD restitution properties and [Ca²⁺](i) rate dependence are strongly dependent on I(NaK), the L-Type Ca²⁺ current (I(CaL)) and the Na⁺/Ca²⁺ exchanger current (I(NaCa)), although the relative role of these currents is markedly model dependent. Furthermore, our results show that simulations using both models agree with many experimentally-reported electrophysiological characteristics. However, our study shows that the Shannon et al. model mimics rabbit electrophysiology more accurately at normal pacing rates, whereas Mahajan et al. model behaves more appropriately at faster rates. Our results reinforce the usefulness of sensitivity analysis for further understanding of cellular electrophysiology and validation of cardiac AP models.

3D virtual human atria: A computational platform for studying clinical atrial fibrillation

Despite a vast amount of experimental and clinical data on the underlying ionic, cellular and tissue substrates, the mechanisms of common atrial arrhythmias (such as atrial fibrillation, AF) arising from the functional interactions at the whole atria level remain unclear. Computational modelling provides a quantitative framework for integrating such multi-scale data and understanding the arrhythmogenic behaviour that emerges from the collective spatio-temporal dynamics in all parts of the heart. In this study, we have developed a multi-scale hierarchy of biophysically detailed computational models for the human atria--the 3D virtual human atria. Primarily, diffusion tensor MRI reconstruction of the tissue geometry and fibre orientation in the human sinoatrial node (SAN) and surrounding atrial muscle was integrated into the 3D model of the whole atria dissected from the Visible Human dataset. The anatomical models were combined with the heterogeneous atrial action potential (AP) models, and used to simulate the AP conduction in the human atria under various conditions: SAN pacemaking and atrial activation in the normal rhythm, break-down of regular AP wave-fronts during rapid atrial pacing, and the genesis of multiple re-entrant wavelets characteristic of AF. Contributions of different properties of the tissue to mechanisms of the normal rhythm and arrhythmogenesis were investigated. Primarily, the simulations showed that tissue heterogeneity caused the break-down of the normal AP wave-fronts at rapid pacing rates, which initiated a pair of re-entrant spiral waves; and tissue anisotropy resulted in a further break-down of the spiral waves into multiple meandering wavelets characteristic of AF. The 3D virtual atria model itself was incorporated into the torso model to simulate the body surface ECG patterns in the normal and arrhythmic conditions. Therefore, a state-of-the-art computational platform has been developed, which can be used for studying multi-scale electrical phenomena during atrial conduction and AF arrhythmogenesis. Results of such simulations can be directly compared with electrophysiological and endocardial mapping data, as well as clinical ECG recordings. The virtual human atria can provide in-depth insights into 3D excitation propagation processes within atrial walls of a whole heart in vivo, which is beyond the current technical capabilities of experimental or clinical set-ups.

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 Ca(2+)-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.

Integrative modeling of the cardiac ventricular myocyte

Cardiac electrophysiology is a discipline with a rich 50-year history of experimental research coupled with integrative modeling which has enabled us to achieve a quantitative understanding of the relationships between molecular function and the integrated behavior of the cardiac myocyte in health and disease. In this paper, we review the development of integrative computational models of the cardiac myocyte. We begin with a historical overview of key cardiac cell models that helped shape the field. We then narrow our focus to models of the cardiac ventricular myocyte and describe these models in the context of their subcellular functional systems including dynamic models of voltage-gated ion channels, mitochondrial energy production, ATP-dependent and electrogenic membrane transporters, intracellular Ca dynamics, mechanical contraction, and regulatory signal transduction pathways. We describe key advances and limitations of the models as well as point to new directions for future modeling research. WIREs Syst Biol Med 2011 3 392-413 DOI: 10.1002/wsbm.122

Bifurcation analysis and effects of changing ionic conductances on pacemaker rhythm in a sinoatrial node cell model

The electrical excitation (action potential generation) of sinoatrial node (cardiac pacemaker) cells is directly related to various ion channels (pore-forming proteins) in cell membranes. In order to analyze the relation between action potential generation and ion channels, we use the Yanagihara-Noma-Irisawa (YNI) model of sinoatrial node cells, which is described by the Hodgkin-Huxley-type equations with seven variables. In this paper, we analyze the global bifurcation structure of the YNI model by varying various conductances of ion channels, and examine the effects of these conductance changes on pacemaker rhythm (frequency of action potential generation). The coupling effect on pacemaker rhythm is also examined approximately by applying external current to the YNI model.

Roles of hyperpolarization-activated current If in sinoatrial node pacemaking: insights from bifurcation analysis of mathematical models

To elucidate the roles of hyperpolarization-activated current (I(f)) in sinoatrial node (SAN) pacemaking, we theoretically investigated 1) the effects of I(f) on stability and bifurcation during hyperpolarization of SAN cells; 2) combined effects of I(f) and the sustained inward current (I(st)) or Na(+) channel current (I(Na)) on robustness of pacemaking against hyperpolarization; and 3) whether blocking I(f) abolishes pacemaker activity under certain conditions. Bifurcation analyses were performed for mathematical models of rabbit SAN cells; equilibrium points (EPs), periodic orbits, and their stability were determined as functions of parameters. Unstable steady-state potential region determined with applications of constant bias currents shrunk as I(f) density increased. In the central SAN cell, the critical acetylcholine concentration at which bifurcations, to yield a stable EP and quiescence, occur was increased by smaller I(f), but decreased by larger I(f). In contrast, the critical acetylcholine concentration and conductance of gap junctions between SAN and atrial cells at bifurcations progressively increased with enhancing I(f) in the peripheral SAN cell. These effects of I(f) were significantly attenuated by eliminating I(st) or I(Na), or by accelerating their inactivation. Under hyperpolarized conditions, blocking I(f) abolished SAN pacemaking via bifurcations. These results suggest that 1) I(f) itself cannot destabilize EPs; 2) I(f) improves SAN cell robustness against parasympathetic stimulation via preventing bifurcations in the presence of I(st) or I(Na); 3) I(f) dramatically enhances peripheral cell robustness against electrotonic loads of the atrium in combination with I(Na); and 4) pacemaker activity of hyperpolarized SAN cells could be abolished by blocking I(f).

Cardiac cell modelling: observations from the heart of the cardiac physiome project

In this manuscript we review the state of cardiac cell modelling in the context of international initiatives such as the IUPS Physiome and Virtual Physiological Human Projects, which aim to integrate computational models across scales and physics. In particular we focus on the relationship between experimental data and model parameterisation across a range of model types and cellular physiological systems. Finally, in the context of parameter identification and model reuse within the Cardiac Physiome, we suggest some future priority areas for this field.

A novel quantitative explanation for the autonomic modulation of cardiac pacemaker cell automaticity via a dynamic system of sarcolemmal and intracellular proteins

Classical numerical models have attributed the regulation of normal cardiac automaticity in sinoatrial node cells (SANCs) largely to G protein-coupled receptor (GPCR) modulation of sarcolemmal ion currents. More recent experimental evidence, however, has indicated that GPCR modulation of SANCs automaticity involves spontaneous, rhythmic, local Ca(2+) releases (LCRs) from the sarcoplasmic reticulum (SR). We explored the GPCR rate modulation of SANCs using a unique and novel numerical model of SANCs in which Ca(2+)-release characteristics are graded by variations in the SR Ca(2+) pumping capability, mimicking the modulation by phospholamban regulated by cAMP-mediated, PKA-activated signaling. The model faithfully predicted the entire range of physiological chronotropic modulation of SANCs by the activation of beta-adrenergic receptors or cholinergic receptors only when experimentally documented changes of sarcolemmal ion channels are combined with a simultaneous increase/decrease in SR Ca(2+) pumping capability. The novel numerical mechanism of GPCR rate modulation is based on numerous complex synergistic interactions between sarcolemmal and intracellular processes via membrane voltage and Ca(2+). Major interactions include changes of diastolic Na(+)/Ca(2+) exchanger current that couple earlier/later diastolic Ca(2+) releases (predicting the experimentally defined LCR period shift) of increased/decreased amplitude (predicting changes in LCR signal mass, i.e., the product of LCR spatial size, amplitude, and number per cycle) to the diastolic depolarization and ultimately to the spontaneous action potential firing rate. Concomitantly, larger/smaller and more/less frequent activation of L-type Ca(2+) current shifts the cellular Ca(2+) balance to support the respective Ca(2+) cycling changes. In conclusion, our model simulations corroborate recent experimental results in rabbit SANCs pointing to a new paradigm for GPCR heart rate modulation by a complex system of dynamically coupled sarcolemmal and intracellular proteins.

A coupled SYSTEM of intracellular Ca2+ clocks and surface membrane voltage clocks controls the timekeeping mechanism of the heart's pacemaker

Ion channels on the surface membrane of sinoatrial nodal pacemaker cells (SANCs) are the proximal cause of an action potential. Each individual channel type has been thoroughly characterized under voltage clamp, and the ensemble of the ion channel currents reconstructed in silico generates rhythmic action potentials. Thus, this ensemble can be envisioned as a surface "membrane clock" (M clock). Localized subsarcolemmal Ca(2+) releases are generated by the sarcoplasmic reticulum via ryanodine receptors during late diastolic depolarization and are referred to as an intracellular "Ca(2+) clock," because their spontaneous occurrence is periodic during voltage clamp or in detergent-permeabilized SANCs, and in silico as well. In spontaneously firing SANCs, the M and Ca(2+) clocks do not operate in isolation but work together via numerous interactions modulated by membrane voltage, subsarcolemmal Ca(2+), and protein kinase A and CaMKII-dependent protein phosphorylation. Through these interactions, the 2 subsystem clocks become mutually entrained to form a robust, stable, coupled-clock system that drives normal cardiac pacemaker cell automaticity. G protein-coupled receptors signaling creates pacemaker flexibility, ie, effects changes in the rhythmic action potential firing rate, by impacting on these very same factors that regulate robust basal coupled-clock system function. This review examines evidence that forms the basis of this coupled-clock system concept in cardiac SANCs.

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.

What keeps us ticking: a funny current, a calcium clock, or both?

The processes underlying the initiation of the heartbeat, whether due to intracellular metabolism or surface membrane events, have always been a major focus of cardiac research.

About 50 years ago, pioneering work initiated by Silvio Weidmann and others applied the Hodgkin-Huxley formalism of membrane excitation to interpret the cardiac electrical activity, including the pacemaker depolarization (see D. Noble, 1979 The initiation of the heartbeat, Clarendon Press). The underlying idea was that voltage- and time-dependent gating of various surface membrane channels not only generated the cardiac pacemaker action potential (AP), but also controlled the spontaneous depolarization between AP’s and thus determined when the next AP would occur. According to this description, the ensemble of surface membrane ion channels works as a clock that regulates the rate and rhythm of spontaneous AP firing, otherwise known as normal automaticity. A formidable research effort then concentrated in attempting to target which of the surface membrane ion channels had an important role in controlling the spontaneous diastolic depolarization (DD).

Originally, a major role was attributed to the “IK-decay theory”. This was strongly influenced by the previous Hodgkin-Huxley model of nerve AP, which described the slow depolarization following a nerve AP as due to the decay of a K+ current. This model of pacemaker depolarization lasted some 20 years, until it was turned upside-down by a full re-interpretation based on the discovery of the If current. Other ionic currents gated by membrane depolarization, i.e. ICaL, ICaT, IST, non-gated and non-specific background leak currents, and also a current generated by the Na-Ca exchange (NCX) carrier, were also proposed to be involved in pacemaking. Based on a wealth of experimental evidence, If is today considered as the most important ion channel involved in the rate regulation of cardiac pacemaker cells, and is sometimes referred to as “the pacemaker channel.”

Several studies, some of which recent, have also shown that in addition to voltage and time, surface membrane electrogenic molecules are strongly modulated by Ca2+ and phosphorylation. The studies of a sub-group of pacemaker cell researchers focusing upon intracellular Ca2+ movements in pacemaker cells spawned the idea that intracellular Ca2+ is an important player in controlling pacemaker cell automaticity. This elevated the status of NCX current as a major Ca2+-activated electrogenic mechanism. But the fine details of intracellular Ca2+ movements, specifically those beneath the cell membrane during DD, were not at hand, and the concept of Ca2+ involvement in pacemaking stalled, whilst the concept of If control continued to soar—expanding to the design of novel drug development and biological pacemakers.

More recent discoveries over the past decade, made possible by simultaneous submembrane Ca2+ imaging and membrane potential or current recordings with cell-attached patch electrodes, have shown that critically timed Ca2+ releases occur during the DD and activate NCX, causing the late DD to exponentially increase, driving the membrane potential to the threshold for the rapid upstroke of the next AP. Such rhythmic, spontaneous intracellular Ca2+ cycling has been referred to as an “intracellular Ca2+ clock”, i.e. a component that interacts with the classic sarcolemmal membrane voltage clock to form the overall pacemaker clock.

Needless to say, there is presently some degree of uncertainty about the relative roles of Ifvs that of intracellular Ca2+ cycling in controlling the normal pacemaker cell automaticity. The dialogue that ensues aims to present and refute both sides of the issue. Sit back and enjoy the show!

Pacemaker activity of the human sinoatrial node: role of the hyperpolarization-activated current, I(f)

The mechanism of primary, spontaneous cardiac pacemaker activity of the sinoatrial node (SAN) has extensively been studied in several animal species, but is virtually unexplored in man. Understanding the mechanisms of human SAN pacemaker activity is important for developing new therapeutic approaches for controlling the heart rate in the sick sinus syndrome and in diseased myocardium. Here we review the functional role of the hyperpolarization-activated 'funny' current, I(f), in human SAN pacemaker activity. Despite the many animal studies performed over the years, the contribution of I(f) to pacemaker activity is still controversial and not fully established. However, recent clinical data on mutations in the I(f) encoding HCN4 gene, which is thought to be the most abundant isoform of the HCN gene family in SAN, suggest a functional role of I(f) in human pacemaker activity. These clinical findings are supported by recent experimental data from single isolated human SAN cells that provide direct evidence that I(f) contributes to human SAN pacemaker activity. Therefore, controlling heart rate in clinical practice via I(f) blockers offers a valuable approach to lowering heart rate and provides an attractive alternative to conventional treatment for a wide range of patients with confirmed stable angina, while upregulation or artificial expression of I(f) may relieve disease-causing bradycardias.

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.