Period-doubling bifurcation in an array of coupled stochastically excitable elements subjected to global periodic forcing

Calcium- and voltage-driven atrial alternans: Insight from [Ca]i and Vm asynchrony

Cardiac alternans is defined as beat-to-beat alternations in contraction strength, action potential duration (APD), and Ca transient (CaT) amplitude. Cardiac excitation-contraction coupling relies on the activity of two bidirectionally coupled excitable systems, membrane voltage (Vm ) and Ca release. Alternans has been classified as Vm - or Ca-driven, depending whether a disturbance of Vm or [Ca]i regulation drives the alternans. We determined the primary driver of pacing induced alternans in rabbit atrial myocytes, using combined patch clamp and fluorescence [Ca]i and Vm measurements. APD and CaT alternans are typically synchronized; however, uncoupling between APD and CaT regulation can lead to CaT alternans in the absence of APD alternans, and APD alternans can fail to precipitate CaT alternans, suggesting a considerable degree of independence of CaT and APD alternans. Using alternans AP voltage clamp protocols with extra APs showed that most frequently the pre-existing CaT alternans pattern prevailed after the extra-beat, indicating that alternans is Ca-driven. In electrically coupled cell pairs, dyssynchrony of APD and CaT alternans points to autonomous regulation of CaT alternans. Thus, with three novel experimental protocols, we collected evidence for Ca-driven alternans; however, the intimately intertwined regulation of Vm and [Ca]i precludes entirely independent development of CaT and APD alternans.

The 'Reverse FDUF' Mechanism of Atrial Excitation-Contraction Coupling Sustains Calcium Alternans-A Hypothesis

Cardiac calcium alternans is defined as beat-to-beat alternations of Ca transient (CaT) amplitude and has been linked to cardiac arrhythmia, including atrial fibrillation. We investigated the mechanism of atrial alternans in isolated rabbit atrial myocytes using high-resolution line scan confocal Ca imaging. Alternans was induced by increasing the pacing frequency until stable alternans was observed (1.6-2.5 Hz at room temperature). In atrial myocytes, action potential-induced Ca release is initiated in the cell periphery and subsequently propagates towards the cell center by Ca-induced Ca release (CICR) in a Ca wave-like fashion, driven by the newly identified 'fire-diffuse-uptake-fire' (FDUF) mechanism. The development of CaT alternans was accompanied by characteristic changes of the spatio-temporal organization of the CaT. During the later phase of the CaT, central [Ca]_i exceeded peripheral [Ca]_i that was indicative of a reversal of the subcellular [Ca]_i gradient from centripetal to centrifugal. This gradient reversal resulted in a reversal of CICR propagation, causing a secondary Ca release during the large-amplitude alternans CaT, thereby prolonging the CaT, enhancing Ca-release refractoriness and reducing Ca release on the subsequent beat, thus enhancing the degree of CaT alternans. Here, we propose the 'reverse FDUF' mechanism as a novel cellular mechanism of atrial CaT alternans, which explains how the uncoupling of central from peripheral Ca release leads to the reversal of propagating CICR and to alternans.

Mechanism of carvedilol induced action potential and calcium alternans

Carvedilol is a nonspecific β-blocker clinically used for the treatment of cardiovascular diseases but has also been shown to have profound effects on excitation-contraction coupling and Ca signaling at the cellular level. We investigate the mechanism by which carvedilol facilitates Ca transient (CaT) and action potential duration (APD) alternans in rabbit atrial myocytes. Carvedilol lowered the frequency threshold for pacing-induced CaT alternans and facilitated alternans in a concentration-dependent manner. Carvedilol prolonged the sarcoplasmic reticulum (SR) Ca release refractoriness by significantly increasing the time constant τ of recovery of SR Ca release; however, no changes in L-type calcium current recovery from inactivation or SR Ca load were found after carvedilol treatment. Carvedilol enhanced the degree of APD alternans nearly two-fold. Carvedilol slowed the APD restitution kinetics and steepened the APD restitution curve at the pacing frequency (2 Hz) where alternans were elicited. No effect on the CaT or APD alternans ratios was observed in experiments with a different β-blocker (metoprolol), excluding the possibility that the carvedilol effect on CaT and APD alternans was determined by its β-blocking properties. These data suggest that carvedilol contributes to the generation of CaT and APD alternans in atrial myocytes by modulating the restitution of CaT and APD.

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.

The development of L-type Ca2+ current mediated alternans does not depend on the restitution slope in canine ventricular myocardium

Cardiac alternans have crucial importance in the onset of ventricular fibrillation. The early explanation for alternans development was the voltage-driven mechanism, where the action potential (AP) restitution steepness was considered as crucial determining factor. Recent results suggest that restitution slope is an inadequate predictor for alternans development, but several studies still claim the role of membrane potential as underlying mechanism of alternans. These controversial data indicate that the relationship of restitution and alternans development is not completely understood. APs were measured by conventional microelectrode technique from canine right ventricular papillary muscles. Ionic currents combined with fluorescent measurements were recorded by patch-clamp technique. APs combined with fluorescent measurements were monitored by sharp microelectrodes. Rapid pacing evoked restitution-independent AP duration (APD) alternans. When non-alternating AP voltage command was used, Ca²⁺_i-transient (CaT) alternans were not observed. When alternating rectangular voltage pulses were applied, CaT alternans were proportional to I_CaL amplitude alternans. Selective I_CaL inhibition did not influence the fast phase of APD restitution. In this study we found that I_CaL has minor contribution in shaping the fast phase of restitution curve suggesting that I_CaL—if it plays important role in the alternans mechanism—could be an additional factor that attenuates the reliability of APD restitution slope to predict alternans.

Electrophysiological Mechanisms Underlying T-Wave Alternans and Their Role in Arrhythmogenesis

T-wave alternans (TWA) reflects every-other-beat alterations in the morphology of the electrocardiogram ST segment or T wave in the setting of a constant heart rate, hence, in the absence of heart rate variability. It is believed to be associated with the dispersion of repolarization and has been used as a non-invasive marker for predicting the risk of malignant cardiac arrhythmias and sudden cardiac death as numerous studies have shown. This review aims to provide up-to-date review on both experimental and simulation studies in elucidating possible mechanisms underlying the genesis of TWA at the cellular level, as well as the genesis of spatially concordant/discordant alternans at the tissue level, and their transition to cardiac arrhythmia. Recent progress and future perspectives in antiarrhythmic therapies associated with TWA are also discussed.

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.

Minimal model for calcium alternans due to SR release refractoriness

In the heart, rapid pacing rates may induce alternations in the strength of cardiac contraction, termed pulsus alternans. Often, this is due to an instability in the dynamics of the intracellular calcium concentration, whose transients become larger and smaller at consecutive beats. This alternation has been linked experimentally and theoretically to two different mechanisms: an instability due to (1) a strong dependence of calcium release on sarcoplasmic reticulum (SR) load, together with a slow calcium reuptake into the SR or (2) to SR release refractoriness, due to a slow recovery of the ryanodine receptors (RyR2) from inactivation. The relationship between calcium alternans and refractoriness of the RyR2 has been more elusive than the corresponding SR Ca load mechanism. To study the former, we reduce a general calcium model, which mimics the deterministic evolution of a calcium release unit, to its most basic elements. We show that calcium alternans can be understood using a simple nonlinear equation for calcium concentration at the dyadic space, coupled to a relaxation equation for the number of recovered RyR2s. Depending on the number of RyR2s that are recovered at the beginning of a stimulation, the increase in calcium concentration may pass, or not, over an excitability threshold that limits the occurrence of a large calcium transient. When the recovery of the RyR2 is slow, this produces naturally a period doubling bifurcation, resulting in calcium alternans. We then study the effects of inactivation, calcium diffusion, and release conductance for the onset of alternans. We find that the development of alternans requires a well-defined value of diffusion while it is less sensitive to the values of inactivation or release conductance.

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.

Dynamic patterns in a two-dimensional neural field with refractoriness

The formation of dynamic patterns such as localized propagating waves is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended systems including neural systems, in which they might play important functional roles. Here we derive a type of two-dimensional neural-field model with refractoriness to study the formation mechanism of localized waves. After comparing this model with existing neural-field models, we show that it is able to generate a variety of localized patterns, including stationary bumps, localized waves rotating along a circular path, and localized waves with longer-range propagation. We construct explicit bump solutions for the two-dimensional neural field and conduct a linear stability analysis on how a stationary bump transitions to a propagating wave under different spatial eigenmode perturbations. The neural-field model is then partially solved in a comoving frame to obtain localized wave solutions, whose spatial profiles are in good agreement with those obtained from simulations. We demonstrate that when there are multiple such propagating waves, they exhibit rich propagation dynamics, including propagation along periodically oscillating and irregular trajectories; these propagation dynamics are quantitatively characterized. In addition, we show that these waves can have repulsive or merging collisions, depending on their collision angles and the refractoriness parameter. Due to its analytical tractability, the two-dimensional neural-field model provides a modeling framework for studying localized propagating waves and their interactions.

Synchronization and phase redistribution in self-replicating populations of coupled oscillators and excitable elements

We study the dynamics of phase synchronization in growing populations of discrete phase oscillatory systems when the division process is coupled to the distribution of oscillator phases. Using mean-field theory, linear-stability analysis, and numerical simulations, we demonstrate that coupling between population growth and synchrony can lead to a wide range of dynamical behavior, including extinction of synchronized oscillations, the emergence of asynchronous states with unequal state (phase) distributions, bistability between oscillatory and asynchronous states or between two asynchronous states, a switch between continuous (supercritical) and discontinuous (subcritical) transitions, and modulation of the frequency of bulk oscillations.

The mechanisms of calcium cycling and action potential dynamics in cardiac alternans

RATIONALE

Alternans is a risk factor for cardiac arrhythmia, including atrial fibrillation. At the cellular level alternans manifests as beat-to-beat alternations in contraction, action potential duration (APD), and magnitude of the Ca(2+) transient (CaT). Electromechanical and CaT alternans are highly correlated, however, it has remained controversial whether the primary cause of alternans is a disturbance of cellular Ca(2+) signaling or electrical membrane properties.

OBJECTIVE

To determine whether a primary failure of intracellular Ca(2+) regulation or disturbances in membrane potential and AP regulation are responsible for the occurrence of alternans in atrial myocytes.

METHODS AND RESULTS

Pacing-induced APD and CaT alternans were studied in single rabbit atrial and ventricular myocytes using combined [Ca(2+)]i and electrophysiological measurements. In current-clamp experiments, APD and CaT alternans strongly correlated in time and magnitude. CaT alternans was observed without alternation in L-type Ca(2+) current, however, elimination of intracellular Ca(2+) release abolished APD alternans, indicating that [Ca(2+)]i dynamics have a profound effect on the occurrence of CaT alternans. Trains of 2 distinctive voltage commands in form of APs recorded during large and small alternans CaTs were applied to voltage-clamped cells. CaT alternans was observed with and without alternation in the voltage command shape. During alternans AP-clamp large CaTs coincided with both long and short AP waveforms, indicating that CaT alternans develop irrespective of AP dynamics.

CONCLUSIONS

The primary mechanism underlying alternans in atrial cells, similarly to ventricular cells, resides in a disturbance of Ca(2+) signaling, whereas APD alternans are a secondary consequence, mediated by Ca(2+)-dependent AP modulation.

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.

Dynamic patterns and their interactions in networks of excitable elements

Formation of localized propagating patterns is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended, excitable systems in which individual elements have resting, activated, and refractory states. Here we study a type of stochastic three-state excitable network model that has been recently developed; this model is able to generate a rich range of pattern dynamics, including localized wandering patterns and localized propagating patterns with crescent shapes and long-range propagation. The collective dynamics of these localized patterns have anomalous subdiffusive dynamics before symmetry breaking and anomalous superdiffusive dynamics after that, showing long-range spatiotemporal coherence in the system. In this study, the stability of the localized wandering patterns is analyzed by treating an individual localized pattern as a subpopulation to develop its average response function. This stability analysis indicates that when the average refractory period is greater than a certain value, there are too many elements in the refractory state after being activated to allow the subpopulation to support a self-sustained pattern; this is consistent with symmetry breaking identified by using an order parameter. Furthermore, in a broad parameter space, the simple network model is able to generate a range of interactions between different localized propagating patterns including repulsive collisions and partial and full annihilations, and interactions between localized propagating patterns and the refractory wake behind others; in this study, these interaction dynamics are systematically quantified based on their relative propagation directions and the resultant angles between them before and after their collisions. These results suggest that the model potentially provides a modeling framework to understand the formation of localized propagating patterns in a broad class of systems with excitable properties.

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.

Intracellular calcium signals display an avalanche-like behavior over multiple lengthscales

Many natural phenomena display “self-organized criticality” (SOC), (Bak et al., 1987). This refers to spatially extended systems for which patterns of activity characterized by different lengthscales can occur with a probability density that follows a power law with pattern size. Differently from power laws at phase transitions, systems displaying SOC do not need the tuning of an external parameter. Here we analyze intracellular calcium (Ca²⁺) signals, a key component of the signaling toolkit of almost any cell type. Ca²⁺ signals can either be spatially restricted (local) or propagate throughout the cell (global). Different models have suggested that the transition from local to global signals is similar to that of directed percolation. Directed percolation has been associated, in turn, to the appearance of SOC. In this paper we discuss these issues within the framework of simple models of Ca²⁺ signal propagation. We also analyze the size distribution of local signals (“puffs”) observed in immature Xenopus Laevis oocytes. The puff amplitude distribution obtained from observed local signals is not Gaussian with a noticeable fraction of large size events. The experimental distribution of puff areas in the spatio-temporal record of the image has a long tail that is approximately log-normal. The distribution can also be fitted with a power law relationship albeit with a smaller goodness of fit. The power law behavior is encountered within a simple model that includes some coupling among individual signals for a wide range of parameter values. An analysis of the model shows that a global elevation of the Ca²⁺ concentration plays a major role in determining whether the puff size distribution is long-tailed or not. This suggests that Ca²⁺-clearing from the cytosol is key to determine whether IP₃-mediated Ca²⁺ signals can display a SOC-like behavior or not.

Criticality in intracellular calcium signaling in cardiac myocytes

Calcium (Ca) is a ubiquitous second messenger that regulates many biological functions. The elementary events of local Ca signaling are Ca sparks, which occur randomly in time and space, and integrate to produce global signaling events such as intra- and intercellular Ca waves and whole-cell Ca oscillations. Despite extensive experimental characterization in many systems, the transition from local random to global synchronous events is still poorly understood. Here we show that criticality, a ubiquitous dynamical phenomenon in nature, is responsible for the transition from local to global Ca signaling. We demonstrate this first in a computational model of Ca signaling in a cardiac myocyte and then experimentally in mouse ventricular myocytes, complemented by a theoretical agent-based model to delineate the underlying dynamics. We show that the interaction between the Ca release units via Ca-induced Ca release causes self-organization of Ca spark clusters. When the coupling between Ca release units is weak, the cluster-size distribution is exponential. As the interactions become strong, the cluster-size distribution changes to a power-law distribution, which is characteristic of criticality in thermodynamic and complex nonlinear systems, and facilitates the formation and propagation of Ca waves and whole-cell Ca oscillations. Our findings illustrate how criticality is harnessed by a biological cell to regulate Ca signaling via self-organization of random subcellular events into cellular-scale oscillations, and provide a general theoretical framework for the transition from local Ca signaling to global Ca signaling in biological cells.

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.

Dynamic pattern formation and collisions in networks of excitable elements

Spatially extended, excitable systems with resting, activated, and refractory states, and emergent localized propagating patterns, are widespread in nature. Here a unique type of three-state excitable network model is shown to generate such dynamic patterns with rich collective dynamics. It is shown that symmetry breaking leads to the formation of dynamical patterns, leading to a change from local subdiffusive wandering to directed superdiffusive propagation. Furthermore, the model yields a rich repertoire of collision dynamics between localized propagating patterns and between propagating patterns and the refractory wakes of others. This work is particularly motivated by recent experimental studies of neural systems that exhibit localized propagating patterns, exemplifying a far wider class of excitable systems.

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.

Multi-scale modeling in biology: how to bridge the gaps between scales?

Human physiological functions are regulated across many orders of magnitude in space and time. Integrating the information and dynamics from one scale to another is critical for the understanding of human physiology and the treatment of diseases. Multi-scale modeling, as a computational approach, has been widely adopted by researchers in computational and systems biology. A key unsolved issue is how to represent appropriately the dynamical behaviors of a high-dimensional model of a lower scale by a low-dimensional model of a higher scale, so that it can be used to investigate complex dynamical behaviors at even higher scales of integration. In the article, we first review the widely-used different modeling methodologies and their applications at different scales. We then discuss the gaps between different modeling methodologies and between scales, and discuss potential methods for bridging the gaps between scales.

Alternans and arrhythmias: from cell to heart

The goal of systems biology is to relate events at the molecular level to more integrated scales from organelle to cell, tissue, and living organism. Here, we review how normal and abnormal excitation-contraction coupling properties emerge from the protein scale, where behaviors are dominated by randomness, to the cell and tissue scales, where heart has to beat with reliable regularity for a lifetime. Beginning with the fundamental unit of excitation-contraction coupling, the couplon where L-type Ca channels in the sarcolemmal membrane adjoin ryanodine receptors in the sarcoplasmic reticulum membrane, we show how a network of couplons with 3 basic properties (random activation, refractoriness, and recruitment) produces the classic physiological properties of excitation-contraction coupling and, under pathophysiological conditions, leads to Ca alternans and Ca waves. Moving to the tissue scale, we discuss how cellular Ca alternans and Ca waves promote both reentrant and focal arrhythmias in the heart. Throughout, we emphasize the qualitatively novel properties that emerge at each new scale of integration.

T-wave alternans and arrhythmogenesis in cardiac diseases

T-wave alternans, a manifestation of repolarization alternans at the cellular level, is associated with lethal cardiac arrhythmias and sudden cardiac death. At the cellular level, several mechanisms can produce repolarization alternans, including: (1) electrical restitution resulting from collective ion channel recovery, which usually occurs at fast heart rates but can also occur at normal heart rates when action potential is prolonged resulting in a short diastolic interval; (2) the transient outward current, which tends to occur at normal or slow heart rates; (3) the dynamics of early after depolarizations, which tends to occur during bradycardia; and (4) intracellular calcium cycling alternans through its interaction with membrane voltage. In this review, we summarize the cellular mechanisms of alternans arising from these different mechanisms, and discuss their roles in arrhythmogenesis in the setting of cardiac disease.

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+].

Spark-induced sparks as a mechanism of intracellular calcium alternans in cardiac myocytes

RATIONALE

Intracellular calcium (Ca) alternans has been widely studied in cardiac myocytes and tissue, yet the underlying mechanism remains controversial.

OBJECTIVE

In this study, we used computational modeling and simulation to study how randomly occurring Ca sparks interact collectively to result in whole-cell Ca alternans.

METHODS AND RESULTS

We developed a spatially distributed intracellular Ca cycling model in which Ca release units (CRUs) are locally coupled by Ca diffusion throughout the myoplasm and sarcoplasmic reticulum (SR) network. Ca sparks occur randomly in the CRU network when periodically paced with a clamped voltage waveform, but Ca alternans develops as the pacing speeds up. Combining computational simulation with theoretical analysis, we show that Ca alternans emerges as a collective behavior of Ca sparks, determined by 3 critical properties of the CRU network from which Ca sparks arise: "randomness" (of Ca spark activation), "refractoriness" (of a CRU after a Ca spark), and "recruitment" (Ca sparks inducing Ca sparks in adjacent CRUs). We also show that the steep nonlinear relationship between fractional SR Ca release and SR Ca load arises naturally as a collective behavior of Ca sparks, and Ca alternans can occur even when SR Ca is held constant.

CONCLUSIONS

We present a general theory for the mechanisms of intracellular Ca alternans, which mechanistically links Ca sparks to whole-cell Ca alternans, and is applicable to Ca alternans in both physiological and pathophysiological conditions.