Roles of L-type Ca2+ and delayed-rectifier K+ currents in sinoatrial node pacemaking: insights from stability and bifurcation analyses of a mathematical model

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.

Bifurcations and Proarrhythmic Behaviors in Cardiac Electrical Excitations

The heart is a hierarchical dynamic system consisting of molecules, cells, and tissues, and acts as a pump for blood circulation. The pumping function depends critically on the preceding electrical activity, and disturbances in the pattern of excitation propagation lead to cardiac arrhythmia and pump failure. Excitation phenomena in cardiomyocytes have been modeled as a nonlinear dynamical system. Because of the nonlinearity of excitation phenomena, the system dynamics could be complex, and various analyses have been performed to understand the complex dynamics. Understanding the mechanisms underlying proarrhythmic responses in the heart is crucial for developing new ways to prevent and control cardiac arrhythmias and resulting contractile dysfunction. When the heart changes to a pathological state over time, the action potential (AP) in cardiomyocytes may also change to a different state in shape and duration, often undergoing a qualitative change in behavior. Such a dynamic change is called bifurcation. In this review, we first summarize the contribution of ion channels and transporters to AP formation and our knowledge of ion-transport molecules, then briefly describe bifurcation theory for nonlinear dynamical systems, and finally detail its recent progress, focusing on the research that attempts to understand the developing mechanisms of abnormal excitations in cardiomyocytes from the perspective of bifurcation phenomena.

Quantitative roles of ion channel dynamics on ventricular action potential

Mathematical models for the action potential (AP) generation of the electrically excitable cells including the heart are involved different mechanisms including the voltage-dependent currents with nonlinear time- and voltage-gating properties. From the shape of the AP waveforms to the duration of the refractory periods or heart rhythms are greatly affected by the functions describing the features or the quantities of these ion channels. In this work, a mathematical measure to analyze the regional contributions of voltage-gated channels is defined by dividing the AP into phases, epochs, and intervals of interest. The contribution of each time-dependent current for the newly defined cardiomyocyte model is successfully calculated and it is found that the contribution of dominant ion channels changes substantially not only for each phase but also for different regions of the cardiac AP. Besides, the defined method can also be applied in all Hodgkin-Huxley types of electrically excitable cell models to be able to understand the underlying dynamics better.

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.

Dynamical mechanisms of phase-2 early afterdepolarizations in human ventricular myocytes: insights from bifurcation analyses of two mathematical models

Early afterdepolarization (EAD) is known as a cause of ventricular arrhythmias in long QT syndromes. We theoretically investigated how the rapid (I_Kr) and slow (I_Ks) components of delayed-rectifier K⁺ channel currents, L-type Ca²⁺ channel current (I_CaL), Na⁺/Ca²⁺ exchanger current (I_NCX), Na⁺-K⁺ pump current (I_NaK), intracellular Ca²⁺ (Ca_i) handling via sarcoplasmic reticulum (SR), and intracellular Na⁺ concentration (Na_i) contribute to initiation, termination, and modulation of phase-2 EADs, using two human ventricular myocyte models. Bifurcation structures of dynamical behaviors in model cells were explored by calculating equilibrium points, limit cycles (LCs), and bifurcation points as functions of parameters. EADs were reproduced by numerical simulations. The results are summarized as follows: 1) decreasing I_Ks and/or I_Kr or increasing I_CaL led to EAD generation, to which mid-myocardial cell models were especially susceptible; the parameter regions of EADs overlapped the regions of stable LCs. 2) Two types of EADs (termination mechanisms), I_Ks activation-dependent and I_CaL inactivation-dependent EADs, were detected; I_Ks was not necessarily required for EAD formation. 3) Inhibiting I_NCX suppressed EADs via facilitating Ca²⁺-dependent I_CaL inactivation. 4) Ca_i dynamics (SR Ca²⁺ handling) and Na_i strongly affected bifurcations and EAD generation in model cells via modulating I_CaL, I_NCX, and I_NaK Parameter regions of EADs, often overlapping those of stable LCs, shifted depending on Ca_i and Na_i in stationary and dynamic states. 5) Bradycardia-related induction of EADs was mainly due to decreases in Na_i at lower pacing rates. This study demonstrates that bifurcation analysis allows us to understand the dynamical mechanisms of EAD formation more profoundly.

NEW & NOTEWORTHY

We investigated mechanisms of phase-2 early afterdepolarization (EAD) by bifurcation analyses of human ventricular myocyte (HVM) models. EAD formation in paced HVMs basically depended on bifurcation phenomena in non-paced HVMs, but was strongly affected by intracellular ion concentrations in stationary and dynamic states. EAD generation did not necessarily require I_Ks.

Quantitative Decomposition of Dynamics of Mathematical Cell Models: Method and Application to Ventricular Myocyte Models

Mathematical cell models are effective tools to understand cellular physiological functions precisely. For detailed analysis of model dynamics in order to investigate how much each component affects cellular behaviour, mathematical approaches are essential. This article presents a numerical analysis technique, which is applicable to any complicated cell model formulated as a system of ordinary differential equations, to quantitatively evaluate contributions of respective model components to the model dynamics in the intact situation. The present technique employs a novel mathematical index for decomposed dynamics with respect to each differential variable, along with a concept named instantaneous equilibrium point, which represents the trend of a model variable at some instant. This article also illustrates applications of the method to comprehensive myocardial cell models for analysing insights into the mechanisms of action potential generation and calcium transient. The analysis results exhibit quantitative contributions of individual channel gating mechanisms and ion exchanger activities to membrane repolarization and of calcium fluxes and buffers to raising and descending of the cytosolic calcium level. These analyses quantitatively explicate principle of the model, which leads to a better understanding of cellular dynamics.

Regionally diverse mitochondrial calcium signaling regulates spontaneous pacing in developing cardiomyocytes

The quintessential property of developing cardiomyocytes is their ability to beat spontaneously. The mechanisms underlying spontaneous beating in developing cardiomyocytes are thought to resemble those of adult heart, but have not been directly tested. Contributions of sarcoplasmic and mitochondrial Ca(2+)-signaling vs. If-channel in initiating spontaneous beating were tested in human induced Pluripotent Stem cell-derived cardiomyocytes (hiPS-CM) and rat Neonatal cardiomyocytes (rN-CM). Whole-cell and perforated-patch voltage-clamping and 2-D confocal imaging showed: (1) both cell types beat spontaneously (60-140/min, at 24°C); (2) holding potentials between -70 and 0mV had no significant effects on spontaneous pacing, but suppressed action potential formation; (3) spontaneous pacing at -50mV activated cytosolic Ca(2+)-transients, accompanied by in-phase inward current oscillations that were suppressed by Na(+)-Ca(2+)-exchanger (NCX)- and ryanodine receptor (RyR2)-blockers, but not by Ca(2+)- and If-channels blockers; (4) spreading fluorescence images of cytosolic Ca(2+)-transients emanated repeatedly from preferred central cellular locations during spontaneous beating; (5) mitochondrial un-coupler, FCCP at non-depolarizing concentrations (∼50nM), reversibly suppressed spontaneous pacing; (6) genetically encoded mitochondrial Ca(2+)-biosensor (mitycam-E31Q) detected regionally diverse, and FCCP-sensitive mitochondrial Ca(2+)-uptake and release signals activating during INCX oscillations; (7) If-channel was absent in rN-CM, but activated only negative to -80mV in hiPS-CM; nevertheless blockers of If-channel failed to alter spontaneous pacing.

Action potential characterization of human induced pluripotent stem cell-derived cardiomyocytes using automated patch-clamp technology

Recent progress in embryonic stem cell (ESC) and induced pluripotent stem cell (iPSC) research led to high-purity preparations of human cardiomyocytes (CMs) differentiated from these two sources-suitable for tissue regeneration, in vitro models of disease, and cardiac safety pharmacology screening. We performed a detailed characterization of the effects of nifedipine, cisapride, and tetrodotoxin (TTX) on Cor.4U(®) human iPSC-CM, using automated whole-cell patch-clamp recordings with the CytoPatch™ 2 equipment, within a complex assay combining multiple voltage-clamp and current-clamp protocols in a well-defined sequence, and quantitative analysis of several action potential (AP) parameters. We retrieved three electrical phenotypes based on AP shape: ventricular, atrial/nodal, and S-type (with ventricular-like depolarization and lack of plateau). To suppress spontaneous firing, present in many cells, we injected continuously faint hyperpolarizing currents of -10 or -20 pA. We defined quality criteria (both seal and membrane resistance over 1 GΩ), and focused our study on cells with ventricular-like AP. Nifedipine induced marked decreases in AP duration (APD): APD90 (49.8% and 40.8% of control values at 1 and 10 μM, respectively), APD50 (16.1% and 12%); cisapride 0.1 μM increased APD90 to 176.2%; and tetrodotoxin 10 μM decreased maximum slope of phase to 33.3% of control, peak depolarization potential to 76.3% of control, and shortened APD90 on average to 80.4%. These results prove feasibility of automated voltage- and current-clamp recordings on human iPSC-CM and their potential use for in-depth drug evaluation and proarrhythmic liability assessment, as well as for diagnosis and pharmacology tests for cardiac channelopathy patients.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

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.

Cycle length restitution in sinoatrial node cells: a theory for understanding spontaneous action potential dynamics

Normal heart rhythm (sinus rhythm) is governed by the sinoatrial node, a specialized and highly heterogeneous collection of spontaneously active myocytes in the right atrium. Sinoatrial node dysfunction, characterized by slow and/or asynchronous pacemaker activity and even failure, is associated with cardiovascular disease (e.g. heart failure, atrial fibrillation). While tremendous progress has been made in understanding the molecular and ionic basis of automaticity in sinoatrial node cells, the dynamics governing sinoatrial nodel cell synchrony and overall pacemaker function remain unclear. Here, a well-validated computational model of the mouse sinoatrial node cell is used to test the hypothesis that sinoatrial node cell dynamics reflect an inherent restitution property (cycle length restitution) that may give rise to a wide range of behavior from regular periodicity to highly complex, irregular activation. Computer simulations are performed to determine the cycle length restitution curve in the computational model using a newly defined voltage pulse protocol. The ability of the restitution curve to predict sinoatrial node cell dynamics (e.g., the emergence of irregular spontaneous activity) and susceptibility to termination is evaluated. Finally, ionic and tissue level factors (e.g. ion channel conductances, ion concentrations, cell-to-cell coupling) that influence restitution and sinoatrial node cell dynamics are explored. Together, these findings suggest that cycle length restitution may be a useful tool for analyzing cell dynamics and dysfunction in the sinoatrial node.

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.

Effect of hyperpolarization-activated current I(f) on robustness of sinoatrial node pacemaking: theoretical study on influence of intracellular Na(+) concentration

To elucidate the effects of hyperpolarization-activated current I(f) on robustness of sinoatrial node (SAN) pacemaking in connection with intracellular Na(+) concentration (Na(i)) changes, we theoretically investigated 1) the impacts of I(f) on dynamical properties of SAN model cells during inhibition of L-type Ca(2+) channel currents (I(CaL)) or hyperpolarizing loads and 2) I(f)-dependent changes in Na(i) and their effects on dynamical properties of model cells. Bifurcation analyses were performed for Na(i)-variable and Na(i)-fixed versions of mathematical models for rabbit SAN cells; equilibrium points (EPs), limit cycles (LCs), and their stability were determined as functions of model parameters. Increasing I(f) conductance (g(f)) shrank I(CaL) conductance (g(CaL)) regions of unstable EPs and stable LCs (rhythmic firings) in the Na(i)-variable system but slightly broadened that of rhythmic firings at lower g(f) in the Na(i)-fixed system. In the Na(i)-variable system, increased g(f) yielded elevations in Na(i) at EPs and during spontaneous oscillations, which caused EP stabilization and shrinkage in the parameter regions of unstable EPs and rhythmic firings. As g(f) increased, parameter regions of unstable EPs and stable LCs determined for hyperpolarizing loads shrank in the Na(i)-variable system but were enlarged in the Na(i)-fixed system. These findings suggest that 1) I(f) does not enhance but rather attenuates robustness of rabbit SAN cells via facilitating EP stabilization and LC destabilization even in physiological g(f) ranges; and 2) the enhancing effect of I(f) on robustness of pacemaker activity, which could be observed at lower g(f) when Na(i) was fixed, is actually reversed by I(f)-dependent changes in Na(i).

hERG K+ Channels: Structure, Function, and Clinical Significance

The human ether-a-go-go related gene (hERG) encodes the pore-forming subunit of the rapid component of the delayed rectifier K+ channel, Kv11.1, which are expressed in the heart, various brain regions, smooth muscle cells, endocrine cells, and a wide range of tumor cell lines. However, it is the role that Kv11.1 channels play in the heart that has been best characterized, for two main reasons. First, it is the gene product involved in chromosome 7-associated long QT syndrome (LQTS), an inherited disorder associated with a markedly increased risk of ventricular arrhythmias and sudden cardiac death. Second, blockade of Kv11.1, by a wide range of prescription medications, causes drug-induced QT prolongation with an increase in risk of sudden cardiac arrest. In the first part of this review, the properties of Kv11.1 channels, including biogenesis, trafficking, gating, and pharmacology are discussed, while the second part focuses on the pathophysiology of Kv11.1 channels.

Roles of sarcoplasmic reticulum Ca2+ cycling and Na+/Ca2+ exchanger in sinoatrial node pacemaking: Insights from bifurcation analysis of mathematical models

To elucidate the roles of sarcoplasmic reticulum (SR) Ca2+ cycling and Na+/Ca2+ exchanger (NCX) in sinoatrial node (SAN) pacemaking, we have applied stability and bifurcation analyses to a coupled-clock system model developed by Maltsev and Lakatta ( Am J Physiol Heart Circ Physiol 296: H594-H615, 2009). Equilibrium point (EP) at which the system is stationary (i.e., the oscillatory system fails to function), periodic orbit (limit cycle), and their stability were determined as functions of model parameters. The stability analysis to detect bifurcation points confirmed crucial importance of SR Ca2+ pumping rate constant ( Pup), NCX density ( kNCX), and L-type Ca2+ channel conductance for the system function reported in previous parameter-dependent numerical simulations. We showed, however, that the model cell does not exhibit self-sustained automaticity of SR Ca2+ release at any clamped voltage and therefore needs further tuning to reproduce oscillatory local Ca2+ release and net membrane current reported experimentally at −10 mV. Our further extended bifurcation analyses revealed important novel features of the pacemaker system that go beyond prior numerical simulations in relation to the roles of SR Ca2+ cycling and NCX in SAN pacemaking. Specifically, we found that 1) NCX contributes to EP instability and enhancement of robustness in the full system during normal spontaneous action potential firings, while stabilizing EPs to prevent sustained Ca2+ oscillations under voltage clamping; 2) SR requires relatively large kNCX and subsarcolemmal Ca2+ diffusion barrier (i.e., subspace) to contribute to EP destabilization and enhancement of robustness; and 3) decrementing Pup or kNCX decreased the full system robustness against hyperpolarizing loads because EP stabilization and cessation of pacemaking were observed at the lower critical amplitude of hyperpolarizing bias currents, suggesting that SR Ca2+ cycling contributes to enhancement of the full system robustness by modulating NCX currents and promoting EP destabilization.

Nonlinear dynamics in cardiology

The dynamics of many cardiac arrhythmias, as well as the nature of transitions between different heart rhythms, have long been considered evidence of nonlinear phenomena playing a direct role in cardiac arrhythmogenesis. In most types of cardiac disease, the pathology develops slowly and gradually, often over many years. In contrast, arrhythmias often occur suddenly. In nonlinear systems, sudden changes in qualitative dynamics can, counterintuitively, result from a gradual change in a system parameter-this is known as a bifurcation. Here, we review how nonlinearities in cardiac electrophysiology influence normal and abnormal rhythms and how bifurcations change the dynamics. In particular, we focus on the many recent developments in computational modeling at the cellular level that are focused on intracellular calcium dynamics. We discuss two areas where recent experimental and modeling work has suggested the importance of nonlinearities in calcium dynamics: repolarization alternans and pacemaker cell automaticity.

Onset of atrial arrhythmias elicited by autonomic modulation of rabbit sinoatrial node activity: a modeling study

Neuronal modulation of the sinoatrial node (SAN) plays a crucial role in the initiation and maintenance of atrial arrhythmias (AF), although the exact mechanisms remain unclear. We used a computer model of a rabbit right atrium (RA) with a heterogeneous SAN and detailed ionic current descriptions for atrial and SAN myocytes to explore reentry initiation associated with autonomic activity. Heterogeneous acetylcholine (ACh)-dependent ionic responses along with L-type Ca current (I(Ca,L)) upregulation were incorporated in the SAN only. During control, activation was typical with the leading pacemaker site located close to the superior vena cava or the intercaval region. With cholinergic stimulation, activation patterns frequently included caudal shifts of the leading pacemaker site and occasional double breakouts. The model became increasingly arrhythmogenic for the ACh concentration >20 nM and for large I(Ca,L) conductance. Reentries obtained included counterclockwise rotors in the free wall, clockwise reentry circulating between the SAN and free wall, and typical flutter. The SAN was the cause of reentry with a common leading sequence of events: a bradycardic beat with shifting in the caudal direction, followed by a premature beat or unidirectional block within the SAN. Electrotonic loading, and not just overdrive pacing, squelches competing pacemaker sites in the SAN. Cholinergic stimulation concomitant with I(Ca,L) upregulation shifts leading pacemaker site and can lead to reentry. A heterogeneous response to autonomic innervation, a large myocardial load, and an extensive SAN in the intercaval region are required for neurally induced SAN-triggered reentry.

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).

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 novel method to quantify contribution of channels and transporters to membrane potential dynamics

The action potential, once triggered in ventricular or atrial myocytes, automatically proceeds on its time course or is generated spontaneously in sinoatrial node pacemaker cells. It is induced by complex interactions among such cellular components as ion channels, transporters, intracellular ion concentrations, and signaling molecules. We have developed what is, to our knowledge, a new method using a mathematical model to quantify the contribution of each cellular component to the automatic time courses of the action potential. In this method, an equilibrium value, which the membrane potential is approaching at a given moment, is calculated along the time course of the membrane potential. The calculation itself is based on the time-varying conductance and the reversal potentials of individual ion channels and electrogenic ion transporters. Since the equilibrium potential moves in advance of the membrane potential change, we refer to it as the lead potential, V(L). The contribution of an individual current was successfully quantified by comparing dV(L)/dt before and after fixing the time-dependent change of a component of interest, such as the variations in the open probability of a channel or the turnover rate of an ion transporter. In addition to the action potential, the lead-potential analysis should also be applicable in all types of membrane excitation in many different kinds of cells.

Stochastic vagal modulation of cardiac pacemaking may lead to erroneous identification of cardiac "chaos"

Fluctuations in the time interval between two consecutive R-waves of electrocardiogram during normal sinus rhythm may result from irregularities in the autonomic drive of the pacemaking sinoatrial node (SAN). We use a biophysically detailed mathematical model of the action potentials of rabbit SAN to quantify the effects of fluctuations in acetylcholine (ACh) on the pacemaker activity of the SAN and its variability. Fluctuations in ACh concentration model the effect of stochastic activity in the vagal parasympathetic fibers that innervate the SAN and produce varying rates of depolarization during the pacemaker potential, leading to fluctuations in cycle length (CL). Both the estimated maximal Lyapunov exponent and the noise limit of the resultant sequence of fluctuating CLs suggest chaotic dynamics. Apparently chaotic heart rate variability (HRV) seen in sinus rhythm can be produced by stochastic modulation of the SAN. The identification of HRV data as chaotic by use of time series measures such as a positive maximal Lyapunov exponent or positive noise limit requires both caution and a quantitative, predictive mechanistic model that is fully deterministic.

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.

Regional difference in dynamical property of sinoatrial node pacemaking: role of na+ channel current

To elucidate the regional differences in sinoatrial node pacemaking mechanisms, we investigated 1), bifurcation structures during current blocks or hyperpolarization of the central and peripheral cells, 2), ionic bases of regional differences in bifurcation structures, and 3), the role of Na(+) channel current (I(Na)) in peripheral cell pacemaking. Bifurcation analyses were performed for mathematical models of the rabbit sinoatrial node central and peripheral cells; equilibrium points, periodic orbits, and their stability were determined as functions of parameters. Structural stability against applications of acetylcholine or electrotonic modulations of the atrium was also evaluated. Blocking L-type Ca(2+) channel current (I(Ca,L)) stabilized equilibrium points and abolished pacemaking in both the center and periphery. Critical acetylcholine concentration and gap junction conductance for pacemaker cessation were higher in the periphery than in the center, being dramatically reduced by blocking I(Na). Under hyperpolarized conditions, blocking I(Na), but not eliminating I(Ca,L), abolished peripheral cell pacemaking. These results suggest that 1), I(Ca,L) is responsible for basal pacemaking in both the central and peripheral cells, 2), the peripheral cell is more robust in withstanding hyperpolarizing loads than the central cell, 3), I(Na) improves the structural stability to hyperpolarizing loads, and 4), I(Na)-dependent pacemaking is possible in hyperpolarized peripheral cells.

Computationally efficient strategy for modeling the effect of ion current modifiers

Electrophysiological studies often seek to relate changes in ion current properties caused by a chemical modifier to changes in cellular properties. Therefore, quantifying concentration-dependent effects of modifiers on ion currents is a topic of importance. In this paper, we sought a mathematical method for using ion current data to predict the effect of several theoretical ion current modifiers on cellular and tissue properties that is computationally efficient without compromising predictive power. We focused on the K+ current I(K,r) as an example case due to its link to long QT syndrome and arrhythmias, but these methods should be generally applicable to other electrophysiological studies. We compared predictions using a Markov model with mass action binding of the modifiers to specific conformational states of the channel to predictions generated by two simplified models. We investigated scaling I(K,r) conductance, and found that although this method produced predictions that agreed qualitatively with the more complicated model, it did not generate quantitatively consistent predictions for all modifiers tested. Our simulations showed that a more computationally efficient Hodgkin-Huxley model that incorporates the effect of modifiers through functional changes in the current produced quantitatively consistent predictions of concentration-dependent changes in cell and tissue properties for all modifiers tested.

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.

Effects of pacemaker currents on creation and modulation of human ventricular pacemaker: theoretical study with application to biological pacemaker engineering

A cardiac biological pacemaker (BP) has been created by suppression of the inward rectifier K+ current ( IK1) or overexpression of the hyperpolarization-activated current ( Ih). We theoretically investigated the effects of incorporating Ih, T-type Ca2+ current ( ICa,T), sustained inward current ( Ist), and/or low-voltage-activated L-type Ca2+ channel current ( ICa,LD) on 1) creation of BP cells, 2) robustness of BP activity to electrotonic loads of nonpacemaking (NP) cells, and 3) BP cell ability to drive NP cells. We used a single-cell model for human ventricular myocytes (HVMs) and also coupled-cell models composed of BP and NP cells. Bifurcation structures of the model cells were explored during changes in conductance of the currents and gap junction. Incorporating the pacemaker currents did not yield BP activity in HVM with normal IK1 but increased the critical IK1 conductance for BP activity to emerge. Expressing Ih appeared to be most helpful in facilitating creation of BP cells via IK1 suppression. In the coupled-cell model, Ist significantly enlarged the gap conductance ( GC) region where stable BP cell pacemaking and NP cell driving occur, reducing the number of BP cells required for robust pacemaking and driving. In contrast, Ih enlarged the GC region of pacemaking and driving only when IK1 of the NP cell was relatively low. ICa,T or ICa,LD exerted effects similar to those of Ist but caused shrinkage or irregularity of BP oscillations. These findings suggest that expressing Ist most effectively improves the structural stability of BPs to electrotonic loads and the BP ability to drive the ventricle.

Creating a cardiac pacemaker by gene therapy

While electronic cardiac pacing in its various modalities represents standard of care for treatment of symptomatic bradyarrhythmias and heart failure, it has limitations ranging from absent or rudimentary autonomic modulation to severe complications. This has prompted experimental studies to design and validate a biological pacemaker that could supplement or replace electronic pacemakers. Advances in cardiac gene therapy have resulted in a number of strategies focused on beta-adrenergic receptors as well as specific ion currents that contribute to pacemaker function. This article reviews basic pacemaker physiology, as well as studies in which gene transfer approaches to develop a biological pacemaker have been designed and validated in vivo. Additional requirements and refinements necessary for successful biopacemaker function by gene transfer are discussed.

Dynamical mechanisms of pacemaker generation in IK1-downregulated human ventricular myocytes: insights from bifurcation analyses of a mathematical model

Dynamical mechanisms of the biological pacemaker (BP) generation in human ventricular myocytes were investigated by bifurcation analyses of a mathematical model. Equilibrium points (EPs), periodic orbits, stability of EPs, and bifurcation points were determined as functions of bifurcation parameters, such as the maximum conductance of inward-rectifier K+ current (I(K1)), for constructing bifurcation diagrams. Stable limit cycles (BP activity) abruptly appeared around an unstable EP via a saddle-node bifurcation when I(K1) was suppressed by 84.6%. After the bifurcation at which a stable EP disappears, the I(K1)-reduced system has an unstable EP only, which is essentially important for stable pacemaking. To elucidate how individual sarcolemmal currents contribute to EP instability and BP generation, we further explored the bifurcation structures of the system during changes in L-type Ca2+ channel current (I(Ca,L)), delayed-rectifier K+ currents (I(K)), or Na(+)/Ca2+ exchanger current (I(NaCa)). Our results suggest that 1), I(Ca,L) is, but I(K) or I(NaCa) is not, responsible for EP instability as a requisite to stable BP generation; 2), I(K) is indispensable for robust pacemaking with large amplitude, high upstroke velocity, and stable frequency; and 3), I(NaCa) is the dominant pacemaker current but is not necessarily required for the generation of spontaneous oscillations.

Vasomotion: cellular background for the oscillator and for the synchronization of smooth muscle cells

1. Vasomotion is the oscillation of vascular tone with frequencies in the range from 1 to 20 min(-1) seen in most vascular beds. The oscillation originates in the vessel wall and is seen both in vivo and in vitro. 2. Recently, our ideas on the cellular mechanisms responsible for vasomotion have improved. Three different types of cellular oscillations have been suggested. One model has suggested that oscillatory release of Ca2+ from intracellular stores is important (the oscillation is based on a cytosolic oscillator). A second proposed mechanism is an oscillation originating in the sarcolemma (a membrane oscillator). A third mechanism is based on an oscillation of glycolysis (metabolic oscillator). For the two latter mechanisms, only limited experimental evidence is available. 3. To understand vasomotion, it is important to understand how the cells synchronize. For the cytosolic oscillators synchronization may occur via activation of Ca2+-sensitive ion channels by oscillatory Ca2+ release. The ensuing membrane potential oscillation feeds back on the intracellular Ca2+ stores and causes synchronization of the Ca2+ release. While membrane oscillators in adjacent smooth muscle cells could be synchronized through the same mechanism that sets up the oscillation in the individual cells, a mechanism to synchronize the metabolic-based oscillators has not been suggested. 4. The interpretation of the experimental observations is supported by theoretical modelling of smooth muscle cells behaviour, and the new insight into the mechanisms of vasomotion has the potential to provide tools to investigate the physiological role of vasomotion.

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.