Spatiotemporal intracellular calcium dynamics during cardiac alternans

Beyond Alternans: Detection of Higher-Order Periodicity in Ex-Vivo Human Ventricles Before Induction of Ventricular Fibrillation

Background

Repolarization alternans, defined as period-2 oscillation in the repolarization phase of the action potentials, is one of the cornerstones of cardiac electrophysiology as it provides a mechanistic link between cellular dynamics and ventricular fibrillation (VF). Theoretically, higher-order periodicities (e.g., period-4, period-8,...) are expected but have very limited experimental evidence.

Methods

We studied explanted human hearts, obtained from the recipients of heart transplantation at the time of surgery, using optical mapping technique with transmembrane voltage-sensitive fluorescent dyes. The hearts were stimulated at an increasing rate until VF was induced. The signals recorded from the right ventricle endocardial surface just before the induction of VF and in the presence of 1:1 conduction were processed using the Principal Component Analysis and a combinatorial algorithm to detect and quantify higher-order dynamics.

Results

A prominent and statistically significant 1:4 peak (corresponding to period-4 dynamics) was seen in three of the six studied hearts. Local analysis revealed the spatiotemporal distribution of higher-order periods. Period-4 was localized to temporally stable islands. Higher-order oscillations (period-5, 6, and 8) were transient and primarily occurred in arcs parallel to the activation isochrones.

Discussion

We present evidence of higher-order periodicities and the co-existence of such regions with stable non-chaotic areas in ex-vivo human hearts before VF induction. This result is consistent with the period-doubling route to chaos as a possible mechanism of VF initiation, which complements the concordant to discordant alternans mechanism. The presence of higher-order regions may act as niduses of instability that can degenerate into chaotic fibrillation.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Dissecting cell fate dynamics in pediatric glioblastoma through the lens of complex systems and cellular cybernetics

Cancers are complex dynamic ecosystems. Reductionist approaches to science are inadequate in characterizing their self-organized patterns and collective emergent behaviors. Since current approaches to single-cell analysis in cancer systems rely primarily on single time-point multiomics, many of the temporal features and causal adaptive behaviors in cancer dynamics are vastly ignored. As such, tools and concepts from the interdisciplinary paradigm of complex systems theory are introduced herein to decode the cellular cybernetics of cancer differentiation dynamics and behavioral patterns. An intuition for the attractors and complex networks underlying cancer processes such as cell fate decision-making, multiscale pattern formation systems, and epigenetic state-transitions is developed. The applications of complex systems physics in paving targeted therapies and causal pattern discovery in precision oncology are discussed. Pediatric high-grade gliomas are discussed as a model-system to demonstrate that cancers are complex adaptive systems, in which the emergence and selection of heterogeneous cellular states and phenotypic plasticity are driven by complex multiscale network dynamics. In specific, pediatric glioblastoma (GBM) is used as a proof-of-concept model to illustrate the applications of the complex systems framework in understanding GBM cell fate decisions and decoding their adaptive cellular dynamics. The scope of these tools in forecasting cancer cell fate dynamics in the emerging field of computational oncology and patient-centered systems medicine is highlighted.

Voltage-mediated mechanism for calcium wave synchronization and arrhythmogenesis in atrial tissue

A wide range of atrial arrythmias are caused by molecular defects in proteins that regulate calcium (Ca) cycling. In many cases, these defects promote the propagation of subcellular Ca waves in the cell, which can perturb the voltage time course and induce dangerous perturbations of the action potential (AP). However, subcellular Ca waves occur randomly in cells and, therefore, electrical coupling between cells substantially decreases their effect on the AP. In this study, we present evidence that Ca waves in atrial tissue can synchronize in-phase owing to an order-disorder phase transition. In particular, we show that, below a critical pacing rate, Ca waves are desynchronized and therefore do not induce substantial AP fluctuations in tissue. However, above this critical pacing rate, Ca waves gradually synchronize over millions of cells, which leads to a dramatic amplification of AP fluctuations. We exploit an underlying Ising symmetry of paced cardiac tissue to show that this transition exhibits universal properties common to a wide range of physical systems in nature. Finally, we show that in the heart, phase synchronization induces spatially out-of-phase AP duration alternans which drives wave break and reentry. These results suggest that cardiac tissue exhibits a phase transition that is required for subcellular Ca cycling defects to induce a life-threatening arrhythmia.

Cardiac transmembrane ion channels and action potentials: cellular physiology and arrhythmogenic behavior

Cardiac arrhythmias are among the leading causes of mortality. They often arise from alterations in the electrophysiological properties of cardiac cells and their underlying ionic mechanisms. It is therefore critical to further unravel the pathophysiology of the ionic basis of human cardiac electrophysiology in health and disease. In the first part of this review, current knowledge on the differences in ion channel expression and properties of the ionic processes that determine the morphology and properties of cardiac action potentials and calcium dynamics from cardiomyocytes in different regions of the heart are described. Then the cellular mechanisms promoting arrhythmias in congenital or acquired conditions of ion channel function (electrical remodeling) are discussed. The focus is on human-relevant findings obtained with clinical, experimental, and computational studies, given that interspecies differences make the extrapolation from animal experiments to human clinical settings difficult. Deepening the understanding of the diverse pathophysiology of human cellular electrophysiology will help in developing novel and effective antiarrhythmic strategies for specific subpopulations and disease conditions.

Delayed global feedback in the genesis and stability of spatiotemporal excitation patterns in paced biological excitable media

Biological excitable media, such as cardiac or neural cells and tissue, exhibit memory in which a change in the present excitation may affect the behaviors in the next excitation. For example, a change in calcium (Ca²⁺) concentration in a cell in the present excitation may affect the Ca²⁺ dynamics in the next excitation via bi-directional coupling between voltage and Ca²⁺, forming a delayed feedback loop. Since the Ca²⁺ dynamics inside the excitable cells are spatiotemporal while the membrane voltage is a global signal, the feedback loop is then a delayed global feedback (DGF) loop. In this study, we investigate the roles of DGF in the genesis and stability of spatiotemporal excitation patterns in periodically-paced excitable media using mathematical models with different levels of complexity: a model composed of coupled FitzHugh-Nagumo units, a 3-dimensional physiologically-detailed ventricular myocyte model, and a coupled map lattice model. We investigate the dynamics of excitation patterns that are temporal period-2 (P2) and spatially concordant or discordant, such as subcellular concordant or discordant Ca²⁺alternans in cardiac myocytes or spatially concordant or discordant Ca²⁺ and repolarization alternans in cardiac tissue. Our modeling approach allows both computer simulations and rigorous analytical treatments, which lead to the following results and conclusions. When DGF is absent, concordant and discordant P2 patterns occur depending on initial conditions with the discordant P2 patterns being spatially random. When the DGF is negative, only concordant P2 patterns exist. When the DGF is positive, both concordant and discordant P2 patterns can occur. The discordant P2 patterns are still spatially random, but they satisfy that the global signal exhibits a temporal period-1 behavior. The theoretical analyses of the coupled map lattice model reveal the underlying instabilities and bifurcations for the genesis, selection, and stability of spatiotemporal excitation patterns.

Understanding the mechanisms of pattern formation in biological systems is of great importance. Here we investigate the dynamical mechanisms by which delayed global feedback affects excitation pattern formation and stability in periodically-paced biological excitable media, such as cardiac or neural cells and tissue. We focus on the formation and stability of the temporal period-2 and spatially in-phase and out-of-phase excitation patterns. Using models of different levels of complexity, we show that when the delayed global feedback is negative, only the spatially in-phase patterns are stable. When the feedback is positive, both spatially in-phase and out-of-phase patterns are stable, and the out-of-phase patterns are spatially random but satisfy that the global signals are temporal period-1 solutions.

Dual regulation by subcellular calcium heterogeneity and heart rate variability on cardiac electromechanical dynamics

Heart rate constantly varies under physiological conditions, termed heart rate variability (HRV), and in clinical studies, low HRV is associated with a greater risk of cardiac arrhythmias. Prior work has shown that HRV influences the temporal patterns of electrical activity, specifically the formation of pro-arrhythmic alternans, a beat-to-beat alternation in the action potential duration (APD), or intracellular calcium (Ca) levels. We previously showed that HRV may be anti-arrhythmic by disrupting APD and Ca alternations in a homogeneous cardiac myocyte. Here, we expand on our previous work, incorporating variation in subcellular Ca handling (also known to influence alternans) into a nonlinear map model of a cardiac myocyte composed of diffusively coupled Ca release units (CRUs). Ca-related parameters and initial conditions of each CRU are varied to mimic subcellular Ca heterogeneity, and a stochastic pacing sequence reproduces HRV. We find that subcellular Ca heterogeneity promotes the formation of spatially discordant subcellular alternans patterns, which decreases whole cell Ca and APD alternation for low and moderate HRV, while high subcellular Ca heterogeneity and HRV both promote electromechanical desynchronization. Finally, we find that for low and moderate HRV, both the specific subcellular Ca-related parameters and the pacing sequences influence measures of electromechanical dynamics, while for high HRV, these measures depend predominantly on the pacing sequence. Our results suggest that pro-arrhythmic subcellular discordant alternans tend to form for low levels of HRV, while high HRV may be anti-arrhythmic due to mitigated influence from subcellular Ca heterogeneity and desynchronization of APD from Ca instabilities.

Sensitivity of a data-assimilation system for reconstructing three-dimensional cardiac electrical dynamics

Modelling of cardiac electrical behaviour has led to important mechanistic insights, but important challenges, including uncertainty in model formulations and parameter values, make it difficult to obtain quantitatively accurate results. An alternative approach is combining models with observations from experiments to produce a data-informed reconstruction of system states over time. Here, we extend our earlier data-assimilation studies using an ensemble Kalman filter to reconstruct a three-dimensional time series of states with complex spatio-temporal dynamics using only surface observations of voltage. We consider the effects of several algorithmic and model parameters on the accuracy of reconstructions of known scroll-wave truth states using synthetic observations. In particular, we study the algorithm's sensitivity to parameters governing different parts of the process and its robustness to several model-error conditions. We find that the algorithm can achieve an acceptable level of error in many cases, with the weakest performance occurring for model-error cases and more extreme parameter regimes with more complex dynamics. Analysis of the poorest-performing cases indicates an initial decrease in error followed by an increase when the ensemble spread is reduced. Our results suggest avenues for further improvement through increasing ensemble spread by incorporating additive inflation or using a parameter or multi-model ensemble. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Remodeling Promotes Proarrhythmic Disruption of Calcium Homeostasis in Failing Atrial Myocytes

It is well known that heart failure (HF) typically coexists with atrial fibrillation (AF). However, until now, no clear mechanism has been established that relates HF to AF. In this study, we apply a multiscale computational framework to establish a mechanistic link between atrial myocyte structural remodeling in HF and AF. Using a spatially distributed model of calcium (Ca) signaling, we show that disruption of the spatial relationship between L-type Ca channels (LCCs) and ryanodine receptors results in markedly increased Ca content of the sarcoplasmic reticulum (SR). This increase in SR load is due to changes in the balance between Ca entry via LCCs and Ca extrusion due to the sodium-calcium exchanger after an altered spatial relationship between these signaling proteins. Next, we show that the increased SR load in atrial myocytes predisposes these cells to subcellular Ca waves that occur during the action potential (AP) and are triggered by LCC openings. These waves are common in atrial cells because of the absence of a well-developed t-tubule system in most of these cells. This distinct spatial architecture allows for the presence of a large pool of orphaned ryanodine receptors, which can fire and sustain Ca waves during the AP. Finally, we incorporate our atrial cell model in two-dimensional tissue simulations and demonstrate that triggered wave generation in cells leads to electrical waves in tissue that tend to fractionate to form wavelets of excitation. This fractionation is driven by the underlying stochasticity of subcellular Ca waves, which perturbs AP repolarization and consequently induces localized conduction block in tissue. We outline the mechanism for this effect and argue that it may explain the propensity for atrial arrhythmias in HF.

Region-specific parasympathetic nerve remodeling in the left atrium contributes to creation of a vulnerable substrate for atrial fibrillation

Atrial fibrillation (AF) is the most common heart rhythm disorder and a major cause of stroke. Unfortunately, current therapies for AF are suboptimal, largely because the molecular mechanisms underlying AF are poorly understood. Since the autonomic nervous system is thought to increase vulnerability to AF, we used a rapid atrial pacing (RAP) canine model to investigate the anatomic and electrophysiological characteristics of autonomic remodeling in different regions of the left atrium. RAP led to marked hypertrophy of parent nerve bundles in the posterior left atrium (PLA), resulting in a global increase in parasympathetic and sympathetic innervation throughout the left atrium. Parasympathetic fibers were more heterogeneously distributed in the PLA when compared with other left atrial regions; this led to greater fractionation and disorganization of AF electrograms in the PLA. Computational modeling revealed that heterogeneously distributed parasympathetic activity exacerbates sympathetic substrate for wave break and reentry. We further discovered that levels of nerve growth factor (NGF) were greatest in the left atrial appendage (LAA), where AF was most organized. Preferential NGF release by the LAA - likely a direct function of frequency and regularity of atrial stimulation - may have important implications for creation of a vulnerable AF substrate.

Complex patterns of subcellular cardiac alternans

Cardiac alternans, in which the membrane potential and the intracellular calcium concentration exhibit alternating durations and peak amplitudes at consecutive beats, constitute a precursor to fatal cardiac arrhythmia such as sudden cardiac death. A crucial question therefore concerns the onset of cardiac alternans. Typically, alternans are only reported when they are fully developed. Here, we present a modelling approach to explore recently discovered microscopic alternans, which represent one of the earliest manifestations of cardiac alternans. In this case, the regular periodic dynamics of the local intracellular calcium concentration is already unstable, while the whole-cell behaviour suggests a healthy cell state. In particular, we use our model to investigate the impact of calcium diffusion in both the cytosol and the sarcoplasmic reticulum on the formation of microscopic calcium alternans. We find that for dominant cytosolic coupling, calcium alternans emerge via the traditional period doubling bifurcation. In contrast, dominant luminal coupling leads to a novel route to calcium alternans through a saddle-node bifurcation at the network level. Combining semi-analytical and computational approaches, we compute areas of stability in parameter space and find that as we cross from stable to unstable regions, the emergent patterns of the intracellular calcium concentration change abruptly in a fashion that is highly dependent upon position along the stability boundary. Our results demonstrate that microscopic calcium alternans may possess a much richer dynamical repertoire than previously thought and further strengthen the role of luminal calcium in shaping cardiac calcium dynamics.

Multiscale Modeling of Dyadic Structure-Function Relation in Ventricular Cardiac Myocytes

Cardiovascular disease is often related to defects of subcellular components in cardiac myocytes, specifically in the dyadic cleft, which include changes in cleft geometry and channel placement. Modeling of these pathological changes requires both spatially resolved cleft as well as whole cell level descriptions. We use a multiscale model to create dyadic structure-function relationships to explore the impact of molecular changes on whole cell electrophysiology and calcium cycling. This multiscale model incorporates stochastic simulation of individual L-type calcium channels and ryanodine receptor channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca²⁺ and Ca²⁺-binding molecules in the bulk of the cell. We found action potential duration, systolic, and diastolic [Ca²⁺] to respond most sensitively to changes in L-type calcium channel current. The ryanodine receptor channel cluster structure inside dyadic clefts was found to affect all biomarkers investigated. The shape of clusters observed in experiments by Jayasinghe et al. and channel density within the cluster (characterized by mean occupancy) showed the strongest correlation to the effects on biomarkers.

Stochastic coupled map model of subcellular calcium cycling in cardiac cells

In this study, we analyze a nonlinear map model of intracellular calcium (Ca) and voltage in cardiac cells. In this model, Ca release from the sarcoplasmic reticulum (SR) occurs at spatially distributed dyadic junctions that are diffusively coupled. At these junctions, release occurs with a probability that depends on key variables such as the SR load and the diastolic interval. Using this model, we explore how nonlinearity and stochasticity determine the spatial distribution of Ca release events within a cardiac cell. In particular, we identify a novel synchronization transition, which occurs at rapid pacing rates, in which the global Ca transient transitions from a period 2 response to a period 1 response. In the global period 2 response dyadic junctions fire in unison, on average, on alternate beats, while in the period 1 regime, Ca release at individual dyads is highly irregular. A close examination of the spatial distribution of Ca reveals that in the period 1 regime, the system coarsens into spatially out-of-phase regions with a length scale much smaller than the system size, but larger than the spacing between dyads. We have also explored in detail the coupling to membrane voltage. We study first the case of positive coupling, where a large Ca transient promotes a long action potential duration (APD). Here, the coupling to voltage synchronizes Ca release so that the system exhibits a robust period 2 response that is independent of initial conditions. On the other hand, in the case of negative coupling, where a large Ca transient tends to shorten the APD, we find a multitude of metastable states which consist of complex spatially discordant alternans patterns. Using an analogy to equilibrium statistical mechanics, we show that the spatial patterns observed can be explained by a mapping to the Potts model, with an additional term that accounts for a global coupling of spin states. Using this analogy, we argue that Ca cycling in cardiac cells exhibits complex spatiotemporal patterns that emerge via first or second order phase transitions. These results show that voltage and Ca can interact in order to induce complex subcellular responses, which can potentially lead to heart rhythm disorders.

Oxidative stress creates a unique, CaMKII-mediated substrate for atrial fibrillation in heart failure

The precise mechanisms by which oxidative stress (OS) causes atrial fibrillation (AF) are not known. Since AF frequently originates in the posterior left atrium (PLA), we hypothesized that OS, via calmodulin-dependent protein kinase II (CaMKII) signaling, creates a fertile substrate in the PLA for triggered activity and reentry. In a canine heart failure (HF) model, OS generation and oxidized-CaMKII-induced (Ox-CaMKII-induced) RyR2 and Nav1.5 signaling were increased preferentially in the PLA (compared with left atrial appendage). Triggered Ca2+ waves (TCWs) in HF PLA myocytes were particularly sensitive to acute ROS inhibition. Computational modeling confirmed a direct relationship between OS/CaMKII signaling and TCW generation. CaMKII phosphorylated Nav1.5 (CaMKII-p-Nav1.5 [S571]) was located preferentially at the intercalated disc (ID), being nearly absent at the lateral membrane. Furthermore, a decrease in ankyrin-G (AnkG) in HF led to patchy dropout of CaMKII-p-Nav1.5 at the ID, causing its distribution to become spatially heterogeneous; this corresponded to preferential slowing and inhomogeneity of conduction noted in the HF PLA. Computational modeling illustrated how conduction slowing (e.g., due to increase in CaMKII-p-Nav1.5) interacts with fibrosis to cause reentry in the PLA. We conclude that OS via CaMKII leads to substrate for triggered activity and reentry in HF PLA by mechanisms independent of but complementary to fibrosis.

Synchronization of Triggered Waves in Atrial Tissue

When an atrial cell is paced rapidly, calcium (Ca) waves can form on the cell boundary and propagate to the cell interior. These waves are referred to as "triggered waves" because they are initiated by Ca influx from the L-type Ca channel and occur during the action potential. However, the consequences of triggered waves in atrial tissue are not known. Here, we develop a phenomenological model of Ca cycling in atrial myocytes that accounts for the formation of triggered waves. Using this model, we show that a fundamental requirement for triggered waves to induce abnormal electrical activity in tissue is that these waves must be synchronized over large populations of cells. This is partly because triggered waves induce a long action potential duration (APD) followed by a short APD. Thus, if these events are not synchronized between cells, then they will on average cancel and have minimal effects on the APD in tissue. Using our computational model, we identify two distinct mechanisms for triggered wave synchronization. The first relies on cycle length (CL) variability, which can prolong the CL at a given beat. In cardiac tissue, we show that CL prolongation leads to a substantial amplification of APD because of the synchronization of triggered waves. A second synchronization mechanism applies in a parameter regime in which the cell exhibits stochastic alternans in which a triggered wave fires, on average, only every other beat. In this scenario, we identify a slow synchronization mechanism that relies on the bidirectional feedback between the APD in tissue and triggered wave initiation. On large cables, this synchronization mechanism leads to spatially discordant APD alternans with spatial variations on a scale of hundreds of cells. We argue that these spatial patterns can potentially serve as an arrhythmogenic substrate for the initiation of atrial fibrillation.

NCX-Mediated Subcellular Ca2+ Dynamics Underlying Early Afterdepolarizations in LQT2 Cardiomyocytes

Long QT syndrome type 2 (LQT2) is a congenital disease characterized by loss of function mutations in hERG potassium channels (I_Kr). LQT2 is associated with fatal ventricular arrhythmias promoted by triggered activity in the form of early afterdepolarizations (EADs). We previously demonstrated that intracellular Ca²⁺ handling is remodeled in LQT2 myocytes. Remodeling leads to aberrant late RyR-mediated Ca²⁺ releases that drive forward-mode Na⁺-Ca²⁺ exchanger (NCX) current and slow repolarization to promote reopening of L-type calcium channels and EADs. Forward-mode NCX was found to be enhanced despite the fact that these late releases do not significantly alter the whole-cell cytosolic calcium concentration during a vulnerable period of phase 2 of the action potential corresponding to the onset of EADs. Here, we use a multiscale ventricular myocyte model to explain this finding. We show that because the local NCX current is a saturating nonlinear function of the local submembrane calcium concentration, a larger number of smaller-amplitude discrete Ca²⁺ release events can produce a large increase in whole-cell forward-mode NCX current without increasing significantly the whole-cell cytosolic calcium concentration. Furthermore, we develop novel insights, to our knowledge, into how alterations of stochastic RyR activity at the single-channel level cause late aberrant Ca²⁺ release events. Experimental measurements in transgenic LTQ2 rabbits confirm the critical arrhythmogenic role of NCX and identify this current as a potential target for antiarrhythmic therapies in LQT2.

Locating Order-Disorder Phase Transition in a Cardiac System

To prevent sudden cardiac death, predicting where in the cardiac system an order-disorder phase transition into ventricular fibrillation begins is as important as when it begins. We present a computationally efficient, information-theoretic approach to predicting the locations of the wavebreaks. Such wavebreaks initiate fibrillation in a cardiac system where the order-disorder behavior is controlled by a single driving component, mimicking electrical misfiring from the pulmonary veins or from the Purkinje fibers. Communication analysis between the driving component and each component of the system reveals that channel capacity, mutual information and transfer entropy can locate the wavebreaks. This approach is applicable to interventional therapies to prevent sudden death, and to a wide range of systems to mitigate or prevent imminent phase transitions.

Stochastic Pacing Inhibits Spatially Discordant Cardiac Alternans

Depressed heart rate variability is a well-established risk factor for sudden cardiac death in survivors of acute myocardial infarction and for those with congestive heart failure. Although measurements of heart rate variability provide a valuable prognostic tool, it is unclear whether reduced heart rate variability itself is proarrhythmic or if it simply correlates with the severity of autonomic nervous system dysfunction. In this work, we investigate a possible mechanism by which heart rate variability could protect against cardiac arrhythmia. Specifically, in numerical simulations, we observe an inverse relationship between the variance of stochastic pacing and the occurrence of spatially discordant alternans, an arrhythmia that is widely believed to facilitate the development of cardiac fibrillation. By analyzing the effects of conduction velocity restitution, cellular dynamics, electrotonic coupling, and stochastic pacing on the nodal dynamics of spatially discordant alternans, we provide intuition for this observed behavior and propose control strategies to inhibit discordant alternans.

Alternans promotion in cardiac electrophysiology models by delay differential equations

Cardiac electrical alternans is a state of alternation between long and short action potentials and is frequently associated with harmful cardiac conditions. Different dynamic mechanisms can give rise to alternans; however, many cardiac models based on ordinary differential equations are not able to reproduce this phenomenon. A previous study showed that alternans can be induced by the introduction of delay differential equations (DDEs) in the formulations of the ion channel gating variables of a canine myocyte model. The present work demonstrates that this technique is not model-specific by successfully promoting alternans using DDEs for five cardiac electrophysiology models that describe different types of myocytes, with varying degrees of complexity. By analyzing results across the different models, we observe two potential requirements for alternans promotion via DDEs for ionic gates: (i) the gate must have a significant influence on the action potential duration and (ii) a delay must significantly impair the gate's recovery between consecutive action potentials.

Stochastic spontaneous calcium release events and sodium channelopathies promote ventricular arrhythmias

Premature ventricular complexes (PVCs), the first initiating beats of a variety of cardiac arrhythmias, have been associated with spontaneous calcium release (SCR) events at the cell level. However, the mechanisms underlying the degeneration of such PVCs into arrhythmias are not fully understood. The objective of this study was to investigate the conditions under which SCR-mediated PVCs can lead to ventricular arrhythmias. In particular, we sought to determine whether sodium (Na⁺) current loss-of-function in the structurally normal ventricles provides a substrate for unidirectional conduction block and reentry initiated by SCR-mediated PVCs. To achieve this goal, a stochastic model of SCR was incorporated into an anatomically accurate compute model of the rabbit ventricles with the His-Purkinje system (HPS). Simulations with reduced Na⁺ current due to a negative-shift in the steady-state channel inactivation showed that SCR-mediated delayed afterdepolarizations led to PVC formation in the HPS, where the electrotonic load was lower, conduction block, and reentry in the 3D myocardium. Moreover, arrhythmia initiation was only possible when intrinsic electrophysiological heterogeneity in action potential within the ventricles was present. In conclusion, while benign in healthy individuals SCR-mediated PVCs can lead to life-threatening ventricular arrhythmias when combined with Na⁺ channelopathies.

Mechanism for Triggered Waves in Atrial Myocytes

Excitation-contraction coupling in atrial cells is mediated by calcium (Ca) signaling between L-type Ca channels and Ryanodine receptors that occurs mainly at the cell boundary. This unique architecture dictates essential aspects of Ca signaling under both normal and diseased conditions. In this study we apply laser scanning confocal microscopy, along with an experimentally based computational model, to understand the Ca cycling dynamics of an atrial cell subjected to rapid pacing. Our main finding is that when an atrial cell is paced under Ca overload conditions, Ca waves can then nucleate on the cell boundary and propagate to the cell interior. These propagating Ca waves are referred to as "triggered waves" because they are initiated by L-type Ca channel openings during the action potential. These excitations are distinct from spontaneous Ca waves originating from random fluctuations of Ryanodine receptor channels, and which occur after much longer waiting times. Furthermore, we argue that the onset of these triggered waves is a highly nonlinear function of the sarcoplasmic reticulum Ca load. This strong nonlinearity leads to aperiodic response of Ca at rapid pacing rates that is caused by the complex interplay between paced Ca release and triggered waves. We argue further that this feature of atrial cells leads to dynamic instabilities that may underlie atrial arrhythmias. These studies will serve as a starting point to explore the nonlinear dynamics of atrial cells and will yield insights into the trigger and maintenance of atrial fibrillation.

Studying dyadic structure-function relationships: a review of current modeling approaches and new insights into Ca2+ (mis)handling

Excitation–contraction coupling in cardiac myocytes requires calcium influx through L-type calcium channels in the sarcolemma, which gates calcium release through sarcoplasmic reticulum ryanodine receptors in a process known as calcium-induced calcium release, producing a myoplasmic calcium transient and enabling cardiomyocyte contraction. The spatio-temporal dynamics of calcium release, buffering, and reuptake into the sarcoplasmic reticulum play a central role in excitation–contraction coupling in both normal and diseased cardiac myocytes. However, further quantitative understanding of these cells’ calcium machinery and the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease requires accurate knowledge of cardiac ultrastructure, protein distribution and subcellular function. As current imaging techniques are limited in spatial resolution, limiting insight into changes in calcium handling, computational models of excitation–contraction coupling have been increasingly employed to probe these structure–function relationships. This review will focus on the development of structural models of cardiac calcium dynamics at the subcellular level, orienting the reader broadly towards the development of models of subcellular calcium handling in cardiomyocytes. Specific focus will be given to progress in recent years in terms of multi-scale modeling employing resolved spatial models of subcellular calcium machinery. A review of the state-of-the-art will be followed by a review of emergent insights into calcium-dependent etiologies in heart disease and, finally, we will offer a perspective on future directions for related computational modeling and simulation efforts.

Stochastic initiation and termination of calcium-mediated triggered activity in cardiac myocytes

Cardiac myocytes normally initiate action potentials in response to a current stimulus that depolarizes the membrane above an excitation threshold. Aberrant excitation can also occur due to spontaneous calcium (Ca²⁺) release (SCR) from intracellular stores after the end of a preceding action potential. SCR drives the Na⁺/Ca²⁺ exchange current inducing a "delayed afterdepolarization" that can in turn trigger an action potential if the excitation threshold is reached. This "triggered activity" is known to cause arrhythmias, but how it is initiated and terminated is not understood. Using computer simulations of a ventricular myocyte model, we show that initiation and termination are inherently random events. We determine the probability of those events from statistical measurements of the number of beats before initiation and before termination, respectively, which follow geometric distributions. Moreover, we elucidate the origin of randomness by a statistical analysis of SCR events, which do not follow a Poisson process observed in other eukaryotic cells. Due to synchronization of Ca²⁺ releases during the action potential upstroke, waiting times of SCR events after the upstroke are narrowly distributed, whereas SCR amplitudes follow a broad normal distribution with a width determined by fluctuations in the number of independent Ca²⁺ wave foci. This distribution enables us to compute the probabilities of initiation and termination of bursts of triggered activity that are maintained by a positive feedback between the action potential upstroke and SCR. Our results establish a theoretical framework for interpreting complex and varied manifestations of triggered activity relevant to cardiac arrhythmias.

A unified theory of calcium alternans in ventricular myocytes

Intracellular calcium (Ca²⁺) alternans is a dynamical phenomenon in ventricular myocytes, which is linked to the genesis of lethal arrhythmias. Iterated map models of intracellular Ca²⁺ cycling dynamics in ventricular myocytes under periodic pacing have been developed to study the mechanisms of Ca²⁺ alternans. Two mechanisms of Ca²⁺ alternans have been demonstrated in these models: one relies mainly on fractional sarcoplasmic reticulum Ca²⁺ release and uptake, and the other on refractoriness and other properties of Ca²⁺ sparks. Each of the two mechanisms can partially explain the experimental observations, but both have their inconsistencies with the experimental results. Here we developed an iterated map model that is composed of two coupled iterated maps, which unifies the two mechanisms into a single cohesive mathematical framework. The unified theory can consistently explain the seemingly contradictory experimental observations and shows that the two mechanisms work synergistically to promote Ca²⁺ alternans. Predictions of the theory were examined in a physiologically-detailed spatial Ca²⁺ cycling model of ventricular myocytes.

Nonlinear onset of calcium wave propagation in cardiac cells

Spontaneous calcium (Ca) waves in cardiac myocytes are known to underlie a wide range of cardiac arrhythmias. However, it is not understood which physiological parameters determine the onset of waves. In this study, we explore the relationship between Ca signaling between ion channels and the nucleation of Ca waves. In particular, we apply a master equation approach to analyze the stochastic interaction between neighboring clusters of ryanodine receptor (RyR) channels. Using this analysis, we show that signaling between clusters can be described as a barrier hopping process with exponential sensitivity to system parameters. A consequence of this feature is that the probability that Ca release at a cluster induces release at a neighboring cluster exhibits a sigmoid dependence on the Ca content in the cell. This nonlinearity originates from the regulation of RyR opening due to more than one Ca ion binding site, in conjunction with Ca mediated cooperativity between RyR channels in clusters. We apply a spatially distributed stochastic model of Ca cycling to analyze the physiological consequences of this nonlinearity, and show that it explains the sharp onset of Ca wave nucleation in cardiac cells. Furthermore, we show that this sharp onset can serve as a mechanism for Ca alternans under physiologically relevant conditions. Thus our findings identify the nonlinear features of Ca signaling which potentially underlie the onset of Ca waves and Ca alternans in cardiac cells.

Long-Lasting Sparks: Multi-Metastability and Release Competition in the Calcium Release Unit Network

Calcium (Ca) sparks are elementary events of biological Ca signaling. A normal Ca spark has a brief duration in the range of 10 to 100 ms, but long-lasting sparks with durations of several hundred milliseconds to seconds are also widely observed. Experiments have shown that the transition from normal to long-lasting sparks can occur when ryanodine receptor (RyR) open probability is either increased or decreased. Here, we demonstrate theoretically and computationally that long-lasting sparks emerge as a collective dynamical behavior of the network of diffusively coupled Ca release units (CRUs). We show that normal sparks occur when the CRU network is monostable and excitable, while long-lasting sparks occur when the network dynamics possesses multiple metastable attractors, each attractor corresponding to a different spatial firing pattern of sparks. We further highlight the mechanisms and conditions that produce long-lasting sparks, demonstrating the existence of an optimal range of RyR open probability favoring long-lasting sparks. We find that when CRU firings are sparse and sarcoplasmic reticulum (SR) Ca load is high, increasing RyR open probability promotes long-lasting sparks by potentiating Ca-induced Ca release (CICR). In contrast, when CICR is already strong enough to produce frequent firings, decreasing RyR open probability counter-intuitively promotes long-lasting sparks by decreasing spark frequency. The decrease in spark frequency promotes intra-SR Ca diffusion from neighboring non-firing CRUs to the firing CRUs, which helps to maintain the local SR Ca concentration of the firing CRUs above a critical level to sustain firing. In this setting, decreasing RyR open probability further suppresses long-lasting sparks by weakening CICR. Since a long-lasting spark terminates via the Kramers’ escape process over a potential barrier, its duration exhibits an exponential distribution determined by the barrier height and noise strength, which is modulated differently by different ways of altering the Ca release flux strength.

Calcium (Ca) sparks, resulting from Ca-induced Ca release, are elementary events of biological Ca signaling. Sparks are normally brief, but long-lasting sparks have been widely observed experimentally under various conditions. The underlying mechanisms of spark duration or termination and the corresponding determinants remain a topic of debate. In this study, we demonstrate theoretically and computationally that normal brief sparks are excitable transients, while long-lasting sparks are multiple metastable states emerging in the diffusively coupled Ca release unit network, as a result of cooperativity and release competition among the Ca release units. Termination of a long-lasting spark is a Kramers’ escape process over a potential barrier, and the spark duration is the first-passage time, exhibiting an exponential distribution.

Slow [Na]i Changes and Positive Feedback Between Membrane Potential and [Ca]i Underlie Intermittent Early Afterdepolarizations and Arrhythmias

BACKGROUND

Most cardiac arrhythmias occur intermittently. As a cellular precursor of lethal cardiac arrhythmias, early afterdepolarizations (EADs) during action potentials(APs) have been extensively investigated, and mechanisms for the occurrence of EADs on a beat-to-beat basis have been proposed. However, no previous study explains slow fluctuations in EADs, which may underlie intermittency of EAD trains and consequent arrhythmias. We hypothesize that the feedback of intracellular calcium and sodium concentrations ([Na](i) and [Ca](i)) that influence membrane voltage (V) can explain EAD intermittency.

METHODS AND RESULTS

AP recordings in rabbit ventricular myocytes revealed intermittent EADs, with slow fluctuations between runs of APs with EADs present or absent. We then used dynamical systems analysis and detailed mathematical models of rabbit ventricular myocytes that replicate the observed behavior and investigated the underlying mechanism. We found that a dominance of inward Na-Ca exchanger current (I(NCX)) over Ca-dependent inactivation of L-type Ca current (I(CaL)) forms a positive feedback between [Ca](i) and V, thus resulting in 2 stable AP states, with and without EADs (ie, bistability). Slow changes in [Na](i) determine the transition between these 2 states, forming a bistable on-off switch of EADs. Tissue simulations showed that this bistable switch of cellular EADs provided both a trigger and a functional substrate for intermittent arrhythmias in homogeneous tissues.

CONCLUSIONS

Our study demonstrates that the interaction among V, [Ca](i), and [Na](i) causes slow on-off switching (or bistability) of AP duration in cardiac myocytes and EAD-mediated arrhythmias and suggests a novel possible mechanism for intermittency of cardiac arrhythmias.

A multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: from detailed cleft dynamics to the whole cell concentration profiles

Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca²⁺ content within tens of seconds. Our concept for a multiscale mathematical model of Ca²⁺ -induced Ca²⁺ release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca²⁺ and Ca²⁺-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca²⁺-concentration profiles using three previously published RyR-channel Markov schemes.

Predicting the onset of period-doubling bifurcations in noisy cardiac systems

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

Stochastic spontaneous calcium release events trigger premature ventricular complexes by overcoming electrotonic load

AIMS

Premature ventricular complexes (PVCs) due to spontaneous calcium (Ca) release (SCR) events at the cell level can precipitate ventricular arrhythmias. However, the mechanistic link between SCRs and PVC formation remains incompletely understood. The aim of this study was to investigate the conditions under which delayed afterdepolarizations resulting from stochastic subcellular SCR events can overcome electrotonic source-sink mismatch, leading to PVC initiation.

METHODS AND RESULTS

A stochastic subcellular-scale mathematical model of SCR was incorporated in a realistic model of the rabbit ventricles and Purkinje system (PS). Elevated levels of diastolic sarcoplasmic reticulum Ca(2+) (CaSR) were imposed until triggered activity was observed, allowing us to compile statistics on probability, timing, and location of PVCs. At CaSR≥ 1500 µmol/L PVCs originated in the PS. When SCR was incapacitated in the PS, PVCs also emerged in the ventricles, but at a higher CaSR (≥1550 µmol/L) and with longer waiting times. For each model configuration tested, the probability of PVC occurrence increased from 0 to 100% within a well-defined critical CaSR range; this transition was much more abrupt in organ-scale models (∼50 µmol/L CaSR range) than in the tissue strand (∼100 µmol/L) or single-cell (∼450 µmol/L) models. Among PVCs originating in the PS, ∼68% were located near Purkinje-ventricular junctions (<1 mm).

CONCLUSION

SCR events overcome source-sink mismatch to trigger PVCs at a critical CaSR threshold. Above this threshold, PVCs emerge due to increased probability and reduced variability in timing of SCR events, leading to significant diastolic depolarization. Sites of lower electronic load, such as the PS, are preferential locations for triggering.

Calcium alternans is due to an order-disorder phase transition in cardiac cells

Electromechanical alternans is a beat-to-beat alternation in the strength of contraction of a cardiac cell, which can be caused by an instability of calcium cycling. Using a distributed model of subcellular calcium we show that alternans occurs via an order-disorder phase transition which exhibits critical slowing down and a diverging correlation length. We apply finite size scaling along with a mapping to a stochastic coupled map model, to show that this transition in two dimensions is characterized by critical exponents consistent with the Ising universality class. These findings highlight the important role of cooperativity in biological cells, and suggest novel approaches to investigate the onset of the alternans instability in the heart.

Hyperphosphorylation of RyRs underlies triggered activity in transgenic rabbit model of LQT2 syndrome

RATIONALE

Loss-of-function mutations in human ether go-go (HERG) potassium channels underlie long QT syndrome type 2 (LQT2) and are associated with fatal ventricular tachyarrhythmia. Previously, most studies focused on plasma membrane-related pathways involved in arrhythmogenesis in long QT syndrome, whereas proarrhythmic changes in intracellular Ca(2+) handling remained unexplored.

OBJECTIVE

We investigated the remodeling of Ca(2+) homeostasis in ventricular cardiomyocytes derived from transgenic rabbit model of LQT2 to determine whether these changes contribute to triggered activity in the form of early after depolarizations (EADs).

METHODS AND RESULTS

Confocal Ca(2+) imaging revealed decrease in amplitude of Ca(2+) transients and sarcoplasmic reticulum Ca(2+) content in LQT2 myocytes. Experiments using sarcoplasmic reticulum-entrapped Ca(2+) indicator demonstrated enhanced ryanodine receptor (RyR)-mediated sarcoplasmic reticulum Ca(2+) leak in LQT2 cells. Western blot analyses showed increased phosphorylation of RyR in LQT2 myocytes versus controls. Coimmunoprecipitation experiments demonstrated loss of protein phosphatases type 1 and type 2 from the RyR complex. Stimulation of LQT2 cells with β-adrenergic agonist isoproterenol resulted in prolongation of the plateau of action potentials accompanied by aberrant Ca(2+) releases and EADs, which were abolished by inhibition of Ca(2+)/calmodulin-dependent protein kinase type 2. Computer simulations showed that late aberrant Ca(2+) releases caused by RyR hyperactivity promote EADs and underlie the enhanced triggered activity through increased forward mode of Na(+)/Ca(2+) exchanger type 1.

CONCLUSIONS

Hyperactive, hyperphosphorylated RyRs because of reduced local phosphatase activity enhance triggered activity in LQT2 syndrome. EADs are promoted by aberrant RyR-mediated Ca(2+) releases that are present despite a reduction of sarcoplasmic reticulum content. Those releases increase forward mode Na(+)/Ca(2+) exchanger type 1, thereby slowing repolarization and enabling L-type Ca(2+) current reactivation.

Spatiotemporal dynamics of calcium-driven cardiac alternans

We investigate the dynamics of spatially discordant alternans (SDA) driven by an instability of intracellular calcium cycling using both amplitude equations [P. S. Skardal, A. Karma, and J. G. Restrepo, Phys. Rev. Lett. 108, 108103 (2012)] and ionic model simulations. We focus on the common case where the bidirectional coupling of intracellular calcium concentration and membrane voltage dynamics produces calcium and voltage alternans that are temporally in phase. We find that, close to the alternans bifurcation, SDA is manifested as a smooth wavy modulation of the amplitudes of both repolarization and calcium transient (CaT) alternans, similarly to the well-studied case of voltage-driven alternans. In contrast, further away from the bifurcation, the amplitude of CaT alternans jumps discontinuously at the nodes separating out-of-phase regions, while the amplitude of repolarization alternans remains smooth. We identify universal dynamical features of SDA pattern formation and evolution in the presence of those jumps. We show that node motion of discontinuous SDA patterns is strongly hysteretic even in homogeneous tissue due to the novel phenomenon of "unidirectional pinning": node movement can only be induced towards, but not away from, the pacing site in response to a change of pacing rate or physiological parameter. In addition, we show that the wavelength of discontinuous SDA patterns scales linearly with the conduction velocity restitution length scale, in contrast to the wavelength of smooth patterns that scales sublinearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations due to the property that unidirectional node motion collapses multiple jumps accumulating in nodal regions into a single jump. Amplitude equation predictions are in good overall agreement with ionic model simulations. Finally, we briefly discuss physiological implications of our findings. In particular, we suggest that due to the tendency of conduction blocks to form near nodes, the presence of unidirectional pinning makes calcium-driven alternans potentially more arrhythmogenic than voltage-driven alternans.

Inhibition of the late sodium current slows t-tubule disruption during the progression of hypertensive heart disease in the rat

The treatment of heart failure (HF) is challenging and morbidity and mortality are high. The goal of this study was to determine if inhibition of the late Na(+) current with ranolazine during early hypertensive heart disease might slow or stop disease progression. Spontaneously hypertensive rats (aged 7 mo) were subjected to echocardiographic study and then fed either control chow (CON) or chow containing 0.5% ranolazine (RAN) for 3 mo. Animals were then restudied, and each heart was removed for measurements of t-tubule organization and Ca(2+) transients using confocal microscopy of the intact heart. RAN halted left ventricular hypertrophy as determined from both echocardiographic and cell dimension (length but not width) measurements. RAN reduced the number of myocytes with t-tubule disruption and the proportion of myocytes with defects in intracellular Ca(2+) cycling. RAN also prevented the slowing of the rate of restitution of Ca(2+) release and the increased vulnerability to rate-induced Ca(2+) alternans. Differences between CON- and RAN-treated animals were not a result of different expression levels of voltage-dependent Ca(2+) channel 1.2, sarco(endo)plasmic reticulum Ca(2+)-ATPase 2a, ryanodine receptor type 2, Na(+)/Ca(2+) exchanger-1, or voltage-gated Na(+) channel 1.5. Furthermore, myocytes with defective Ca(2+) transients in CON rats showed improved Ca(2+) cycling immediately upon acute exposure to RAN. Increased late Na(+) current likely plays a role in the progression of cardiac hypertrophy, a key pathological step in the development of HF. Early, chronic inhibition of this current slows both hypertrophy and development of ultrastructural and physiological defects associated with the progression to HF.

Calcium alternans in cardiac myocytes: order from disorder

Calcium alternans is associated with T-wave alternans and pulsus alternans, harbingers of increased mortality in the setting of heart disease. Recent experimental, computational, and theoretical studies have led to new insights into the mechanisms of Ca alternans, specifically how disordered behaviors dominated by stochastic processes at the subcellular level become organized into ordered periodic behaviors. In this article, we summarize the recent progress in this area, outlining a holistic theoretical framework in which the complex effects of Ca cycling proteins on Ca alternans are linked to three key properties of the cardiac Ca cycling network: randomness, refractoriness, and recruitment. We also illustrate how this '3R theory' can reconcile many seemingly contradictory experimental observations.

New experimental evidence for mechanism of arrhythmogenic membrane potential alternans based on balance of electrogenic I(NCX)/I(Ca) currents

BACKGROUND

Computer simulations have predicted that the balance of various electrogenic sarcolemmal ion currents may control the amplitude and phase of beat-to-beat alternans of membrane potential (V(m)). However, experimental evidence for the mechanism by which alternans of calcium transients produces alternation of V(m) (V(m)-ALT) is lacking.

OBJECTIVE

To provide experimental evidence that Ca-to-V(m) coupling during alternans is determined by the balanced influence of 2 Ca-sensitive electrogenic sarcolemmal ionic currents: I(NCX) and I(Ca).

METHODS AND RESULTS

V(m)-ALT and Ca-ALT were measured simultaneously from isolated guinea pig myocytes (n = 41) by using perforated patch and Indo-1(AM) fluorescence, respectively. There were 3 study groups: (1) control, (2) I(NCX) predominance created by adenoviral-induced NCX overexpression, and (3) I(Ca) predominance created by I(NCX) inhibition (SEA-0400) or enhanced I(Ca) (As(2)O(3)). During alternans, 14 of 14 control myocytes demonstrated positive Ca-to-V(m) coupling, consistent with I(NCX), but not I(Ca), as the major electrogenic current in modulating action potential duration. Positive Ca-to-V(m) coupling was maintained during I(NCX) predominance in 8 of 8 experiments with concurrent increase in Ca-to-V(m) gain (P <.05), reaffirming the role of increased forward-mode electrogenic I(NCX). Conversely, I(Ca) predominance produced negative Ca-to-V(m) coupling in 14 of 19 myocytes (P < .05) and decreased Ca-to-V(m) gain compared with control (P <.05). Furthermore, computer simulation demonstrated that Ca-to-V(m) coupling changes from negative to positive because of a shift from I(Ca) to I(NCX) predominance with increasing pacing rate.

CONCLUSIONS

These data provide the first direct experimental evidence that coupling in phase and magnitude of Ca-ALT to V(m)-ALT is strongly determined by the relative balance of the prominence of I(NCX) vs I(Ca) currents.

Calcium alternans in a couplon network model of ventricular myocytes: role of sarcoplasmic reticulum load

Intracellular calcium (Ca) alternans in cardiac myocytes have been shown in many experimental studies, and the mechanisms remain incompletely understood. We recently developed a "3R theory" that links Ca sparks to whole cell Ca alternans through three critical properties: randomness of Ca sparks; recruitment of a Ca spark by neighboring Ca sparks; and refractoriness of Ca release units. In this study, we used computer simulation of a physiologically detailed mathematical model of a ventricular myocyte couplon network to study how sarcoplasmic reticulum (SR) Ca load and other physiological parameters, such as ryanodine receptor sensitivity, SR uptake rate, Na-Ca exchange strength, and Ca buffer levels affect Ca alternans in the context of 3R theory. We developed a method to calculate the parameters used in the 3R theory (i.e., the primary spark rate and the recruitment rate) from the physiologically detailed Ca cycling model and paced the model periodically to elicit Ca alternans. We show that alternans only occurs for an intermediate range of the SR Ca load, and the underlying mechanism can be explained via its effects on the 3Rs. Furthermore, we show that altering the physiological parameters not only directly changes the 3Rs but also alters the SR Ca load, having an indirect effect on the 3Rs as well. Therefore, our present study links the SR Ca load and other physiological parameters to whole cell Ca alternans through the framework of the 3R theory, providing a general mechanistic understanding of Ca alternans in ventricular myocytes.

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.

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.

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

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

Understanding cardiac alternans: a piecewise linear modeling framework

Cardiac alternans is a beat-to-beat alternation in action potential duration (APD) and intracellular calcium (Ca(2+)) cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca(2+) and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard [Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 (2006)]. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca(2+) cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca(2+) transport, the model supports spatially varying patterns of alternans. We analyze the onset of this instability with a generalization of the master stability approach to accommodate the nonsmooth nature of our system.

Identification of intracellular calcium dynamics in stimulated cardiomyocytes

We have developed an automatic method for the analysis and identification of dynamical regimes in intracellular calcium patterns from confocal calcium images. The method allows the identification of different dynamical patterns such as spatially concordant and discordant alternans, irregular behavior or phase-locking regimes such as period doubling or halving. The method can be applied to the analysis of different cardiac pathologies related to anomalies at the cellular level such as ventricular reentrant arrhythmias.

Ca2+ alternans in a cardiac myocyte model that uses moment equations to represent heterogeneous junctional SR Ca2+

Multiscale whole-cell models that accurately represent local control of Ca2+-induced Ca2+ release in cardiac myocytes can reproduce high-gain Ca2+ release that is graded with changes in membrane potential. Using a recently introduced formalism that represents heterogeneous local Ca2+ using moment equations, we present a model of cardiac myocyte Ca2+ cycling that exhibits alternating sarcoplasmic reticulum (SR) Ca2+ release when periodically stimulated by depolarizing voltage pulses. The model predicts that the distribution of junctional SR [Ca2+] across a large population of Ca2+ release units is distinct on alternating cycles. Load-release and release-uptake functions computed from this model give insight into how Ca2+ fluxes and stimulation frequency combine to determine the presence or absence of Ca2+ alternans. Our results show that the conditions for the onset of Ca2+ alternans cannot be explained solely by the steepness of the load-release function, but that changes in the release-uptake process also play an important role. We analyze the effect of the junctional SR refilling time constant on Ca2+ alternans and conclude that physiologically realistic models of defective Ca2+ cycling must represent the dynamics of heterogeneous junctional SR [Ca2+] without assuming rapid equilibration of junctional and network SR [Ca2+].

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Introduction to focus issue: intracellular Ca2+ dynamics--a change of modeling paradigm?

Intracellular Ca(2+) concentration dynamics have been perceived as a prototypical deterministic intracellular reaction-diffusion system in biophysics for many years. Recent experimental findings challenge that view and suggest them to be fluctuation driven. That renders this system interesting for nonlinear physics, in general, since we can study the emergence of macroscopic behavior from mesoscopic dynamics. In particular, we can observe the random elemental events, called puffs, and the macroscopic pattern with the same experimental means. Here, we give a short introduction to the current discussion on theoretical and modeling concepts, and this Focus Issue reflecting it.