Publications (25)100.05 Total impact
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Article: Intracellular Ca2+ waves, afterdepolarizations, and triggered arrhythmias.
Cardiovascular research 04/2012; 95(3):265-8. · 5.80 Impact Factor -
Article: The statistics of calcium-mediated focal excitations on a one-dimensional cable.
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ABSTRACT: It is well known that various cardiac arrhythmias are initiated by an ill-timed excitation that originates from a focal region of the heart. However, up to now, it is not known what governs the timing, location, and morphology of these focal excitations. Recent studies have shown that these excitations can be caused by abnormalities in the calcium (Ca) cycling system. However, the cause-and-effect relationships linking subcellular Ca dynamics and focal activity in cardiac tissue is not completely understood. In this article, we present a minimal model of Ca-mediated focal excitations in cardiac tissue. This model accounts for the stochastic nature of spontaneous Ca release on a one-dimensional cable of cardiac cells. Using this model, we show that the timing of focal excitations is equivalent to a first passage time problem in a spatially extended system. In particular, we find that for a short cable the mean first passage time increases exponentially with the number of cells in tissue, and is critically dependent on the ratio of inward to outward currents near the threshold for an action potential. For long cables excitations occurs due to ectopic foci that occur on a length scale determined by the minimum length of tissue that can induce an action potential. Furthermore, we find that for long cables the mean first passage time decreases as a power law in the number cells. These results provide precise criteria for the occurrence of focal excitations in cardiac tissue, and will serve as a guide to determine the propensity of Ca-mediated triggered arrhythmias in the heart.Biophysical Journal 02/2012; 102(3):461-71. · 3.65 Impact Factor -
Article: A mathematical model of spontaneous calcium release in cardiac myocytes.
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ABSTRACT: In cardiac myocytes, calcium (Ca) can be released from the sarcoplasmic reticulum independently of Ca influx from voltage-dependent membrane channels. This efflux of Ca, referred to as spontaneous Ca release (SCR), is due to Ryanodine receptor fluctuations, which can induce spontaneous Ca sparks, which propagate to form Ca waves. This release of Ca can then induce delayed after-depolarizations (DADs), which can lead to arrhythmogenic-triggered activity in the heart. However, despite its importance, to date there is no mathematical model of SCR that accounts for experimentally observed features of subcellular Ca. In this article, we present an experimentally based model of SCR that reproduces the timing distribution of spontaneous Ca sparks and key features of the propagation of Ca waves emanating from these spontaneous sparks. We have coupled this model to an ionic model for the rabbit ventricular action potential to simulate SCR within several thousand cells in cardiac tissue. We implement this model to study the formation of an ectopic beat on a cable of cells that exhibit SCR-induced DADs.AJP Heart and Circulatory Physiology 02/2011; 300(5):H1794-805. · 3.71 Impact Factor -
Article: Variability in timing of spontaneous calcium release in the intact rat heart is determined by the time course of sarcoplasmic reticulum calcium load.
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ABSTRACT: Abnormalities in intracellular calcium (Ca) cycling during Ca overload can cause triggered activity because spontaneous calcium release (SCR) activates sufficient Ca-sensitive inward currents to induce delayed afterdepolarizations (DADs). However, little is known about the mechanisms relating SCR and triggered activity on the tissue scale. Laser scanning confocal microscopy was used to measure the spatiotemporal properties of SCR within large myocyte populations in intact rat heart. Computer simulations were used to predict how these properties of SCR determine DAD magnitude. We measured the average and standard deviation of the latency distribution of SCR within a large population of myocytes in intact tissue. We found that as external [Ca] is increased, and with faster pacing rates, the average and SD of the latency distribution decreases substantially. This result demonstrates that the timing of SCR occurs with less variability as the sarcoplasmic reticulum (SR) Ca load is increased, causing more sites to release Ca within each cell. We then applied a mathematical model of subcellular Ca cycling to show that a decrease in SCR variability leads to a higher DAD amplitude and is dictated by the rate of SR Ca refilling following an action potential. Our results demonstrate that the variability of the timing of SCR in a population of cells in tissue decreases with SR load and is dictated by the time course of the SR Ca content.Circulation Research 10/2010; 107(9):1117-26. · 9.49 Impact Factor -
Article: Role of coupled gating between cardiac ryanodine receptors in the genesis of triggered arrhythmias.
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ABSTRACT: Mutations in the ryanodine receptor (RyR) have been linked to exercise-induced sudden cardiac death. However, the precise sequence of events linking RyR channel mutations to a whole heart arrhythmia is not completely understood. In this paper, we apply a detailed, mathematical model of subcellular calcium (Ca) release, coupled to membrane voltage, to study how defective RyR channels can induce arrhythmogenic-triggered activity. In particular, we show that subcellular Ca activity, such as spontaneous Ca sparks and Ca waves, is highly sensitive to coupled gating between RyR channels in clusters. We show that small changes in coupled gating can induce aberrant Ca release activity, which, under Ca overload conditions, can induce delayed afterdepolarization (DAD). We systematically investigate the properties of subcellular Ca during DAD induction and show that the voltage time course during a DAD is dependent on the timing and number of spontaneous Ca sparks that transition to Ca waves. These results provide a detailed mechanism for the role of coupled gating in the genesis of triggered arrhythmias.AJP Heart and Circulatory Physiology 06/2009; 297(1):H171-80. · 3.71 Impact Factor -
Article: Mechanisms underlying the formation and dynamics of subcellular calcium alternans in the intact rat heart.
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ABSTRACT: Optical mapping of intact cardiac tissue reveals that, in some cases, intracellular calcium (Ca) release can alternate from one beat to the next in a large-small-large sequence, also referred to as Ca transient (CaT) alternans. CaT alternans can also become spatially phase-mismatched within a single cell, when one part of the cell alternates in a large-small-large sequence, whereas a different part alternates in a small-large-small sequence, a phenomenon known as subcellular discordant alternans. The mechanisms for the formation and spatiotemporal evolution of these phase-mismatched patterns are not known. We used confocal Ca imaging to measure CaT alternans at the sarcomeric level within individual myocytes in the intact rat heart. After a sudden change in cycle length (CL), 2 distinct spatial patterns of CaT alternans emerge. CaTs can form spatially phase-mismatched alternans patterns after the first few beats following the change in CL. The phase mismatch persists for many beats, after which it gradually becomes phase matched via the movement of nodes, which are junctures between phase-mismatched cell regions. In other examples, phase-matched alternans gradually become phase-mismatched, via the formation and movement of nodes. In these examples, we observed large beat-to-beat variations in the cell activation times, despite constant CL pacing. Using computer simulations, we explored the underlying mechanisms for these dynamical phenomena. Our results show how heterogeneity at the sarcomeric level, in conjunction with the dynamics of Ca cycling and membrane voltage, can lead to complex spatiotemporal phenomena within myocytes of the intact heart.Circulation Research 02/2009; 104(5):639-49. · 9.49 Impact Factor -
Article: A rabbit ventricular action potential model replicating cardiac dynamics at rapid heart rates.
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ABSTRACT: Mathematical modeling of the cardiac action potential has proven to be a powerful tool for illuminating various aspects of cardiac function, including cardiac arrhythmias. However, no currently available detailed action potential model accurately reproduces the dynamics of the cardiac action potential and intracellular calcium (Ca(i)) cycling at rapid heart rates relevant to ventricular tachycardia and fibrillation. The aim of this study was to develop such a model. Using an existing rabbit ventricular action potential model, we modified the L-type calcium (Ca) current (I(Ca,L)) and Ca(i) cycling formulations based on new experimental patch-clamp data obtained in isolated rabbit ventricular myocytes, using the perforated patch configuration at 35-37 degrees C. Incorporating a minimal seven-state Markovian model of I(Ca,L) that reproduced Ca- and voltage-dependent kinetics in combination with our previously published dynamic Ca(i) cycling model, the new model replicates experimentally observed action potential duration and Ca(i) transient alternans at rapid heart rates, and accurately reproduces experimental action potential duration restitution curves obtained by either dynamic or S1S2 pacing.Biophysical Journal 02/2008; 94(2):392-410. · 3.65 Impact Factor -
Article: Modifying L-type calcium current kinetics: consequences for cardiac excitation and arrhythmia dynamics.
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ABSTRACT: The L-type Ca current (I(Ca,L)), essential for normal cardiac function, also regulates dynamic action potential (AP) properties that promote ventricular fibrillation. Blocking I(Ca,L) can prevent ventricular fibrillation, but only at levels suppressing contractility. We speculated that, instead of blocking I(Ca,L), modifying its shape by altering kinetic features could produce equivalent anti-fibrillatory effects without depressing contractility. To test this concept experimentally, we overexpressed a mutant Ca-insensitive calmodulin (CaM(1234)) in rabbit ventricular myocytes to inhibit Ca-dependent I(Ca,L) inactivation, combined with the ATP-sensitive K current agonist pinacidil or I(Ca,L) blocker verapamil to maintain AP duration (APD) near control levels. Cell shortening was enhanced in pinacidil-treated myocytes, but depressed in verapamil-treated myocytes. Both combinations flattened APD restitution slope and prevented APD alternans, similar to I(Ca,L) blockade. To predict the arrhythmogenic consequences, we simulated the cellular effects using a new AP model, which reproduced flattening of APD restitution slope and prevention of APD/Ca(i) transient alternans but maintained a normal Ca(i) transient. In simulated two-dimensional cardiac tissue, these changes prevented the arrhythmogenic spatially discordant APD/Ca(i) transient alternans and spiral wave breakup. These findings provide a proof-of-concept test that I(Ca,L) can be targeted to increase dynamic wave stability without depressing contractility, which may have promise as an antifibrillatory strategy.Biophysical Journal 02/2008; 94(2):411-23. · 3.65 Impact Factor -
Article: Macroscopic consequences of calcium signaling in microdomains: a first-passage-time approach.
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ABSTRACT: Calcium (Ca) plays an important role in regulating various cellular processes. In a variety of cell types, Ca signaling occurs within microdomains where channels deliver localized pulses of Ca activating a nearby collection of Ca-sensitive receptors. The small number of channels involved ensures that the signaling process is stochastic. The aggregate response of several thousand of these microdomains yields a whole-cell response which dictates the cell behavior. Here, we study the statistical properties of a population of these microdomains in response to a trigger signal. We use a first-passage-time approach to show analytically how Ca release in the whole cell depends on properties of Ca channels within microdomains. Using these results we explain for the first time the underlying mechanism for the graded relationship between Ca influx and Ca release in cardiac cells.Physical Review E 12/2007; 76(5 Pt 1):051920. · 2.26 Impact Factor -
Article: MODIFYING L-TYPE CALCIUM CURRENT KINETICS: CONSEQUENCES FOR CARDIAC EXCITATION AND ARRHYTHMIA DYNAMICS.
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ABSTRACT: The L-type Ca current (ICa,L), essential for normal cardiac function, also regulates dynamic action potential (AP) properties that promote ventricular fibrillation (VF). Blocking ICa,L can prevent VF, but only at levels suppressing contractility. We speculated that, instead of blocking ICa,L, modifying its shape by altering kinetic features could produce equivalent anti-fibrillatory effects without depressing contractility. To test this concept experimentally, we over-expressed a mutant Ca-insensitive calmodulin (CaM1234) in rabbit ventricular myocytes to inhibit Ca-dependent ICa,L inactivation, combined with the ATP-sensitive K current agonist pinacidil or ICa,L blocker verapamil to maintain AP duration (APD) near control levels. Cell shortening was enhanced in pinacidil-treated myocytes, but depressed in verapamil-treated myocytes. Both combinations flattened APD restitution slope and prevented APD alternans, similar to ICa,L blockade. To predict the arrhythmogenic consequences, we simulated the cellular effects using a new AP model, which reproduced flattening of APD restitution slope and prevention of APD/Cai transient alternans but maintained a normal Cai transient. In simulated 2D cardiac tissue, these changes prevented the arrhythmogenic spatially discordant APD/Cai transient alternans and spiral wave break up. These findings provide a proof-of-concept test that ICa,L can be targeted to increase dynamic wave stability without depressing contractility, which may have promise as an anti-fibrillatory strategy.Biophysical Journal 11/2007; · 3.65 Impact Factor -
Article: Influence of channel subunit composition on L-type Ca2+ current kinetics and cardiac wave stability.
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ABSTRACT: Previous studies have demonstrated that the slope of the function relating the action potential duration (APD) and the diastolic interval, known as the APD restitution curve, plays an important role in the initiation and maintenance of ventricular fibrillation. Since the APD restitution slope critically depends on the kinetics of the L-type Ca(2+) current, we hypothesized that manipulation of the subunit composition of these channels may represent a powerful strategy to control cardiac arrhythmias. We studied the kinetic properties of the human L-type Ca(2+) channel (Ca(v)1.2) coexpressed with the alpha(2)delta-subunit alone (alpha(1C) + alpha(2)delta) or in combination with beta(2a), beta(2b), or beta(3) subunits (alpha(1C) + alpha(2)delta + beta), using Ca(2+) as the charge carrier. We then incorporated the kinetic properties observed experimentally into the L-type Ca(2+) current mathematical model of the cardiac action potential to demonstrate that the APD restitution slope can be selectively controlled by altering the subunit composition of the Ca(2+) channel. Assuming that beta(2b) most closely resembles the native cardiac L-type Ca(2+) current, the absence of beta, as well as the coexpression of beta(2a), was found to flatten restitution slope and stabilize spiral waves. These results imply that subunit modification of L-type Ca(2+) channels can potentially be used as an antifibrillatory strategy.AJP Heart and Circulatory Physiology 10/2007; 293(3):H1805-15. · 3.71 Impact Factor -
Article: A RABBIT VENTRICULAR ACTION POTENTIAL MODEL REPLICATING CARDIAC DYNAMICS AT RAPID HEART RATES.
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ABSTRACT: Mathematical modeling of the cardiac action potential has proven to be a powerful tool for illuminating various aspects of cardiac function, including cardiac arrhythmias. However, no currently available detailed action potential model accurately reproduces the dynamics of the cardiac action potential and intracellular calcium (Cai) cycling at rapid heart rates relevant to ventricular tachycardia and fibrillation. The aim of this study was to develop such a model. Using the rabbit ventricular action potential model by Shannon et al. (1), we modified the L-type calcium (Ca) current (ICa,L) and Cai cycling formulations based on new experimental patch clamp data obtained in isolated rabbit ventricular myocytes, using the perforated patch configuration at 35-37 degrees C. Incorporating a minimal seven-state Markov model of ICa,L that reproduced Ca- and voltage-dependent kinetics in combination with our previously published dynamic Cai cycling model, the new model replicates experimentally-observed action potential duration (APD) and Cai transient alternans at rapid heart rates, and accurately reproduces experimental APD restitution curves obtained by either dynamic or S1S2 pacing.Biophysical Journal 06/2007; · 3.65 Impact Factor -
Article: Inferring the cellular origin of voltage and calcium alternans from the spatial scales of phase reversal during discordant alternans.
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ABSTRACT: Beat-to-beat alternation of the action potential duration (APD) in paced cardiac cells has been linked to the onset of lethal arrhythmias. Both experimental and theoretical studies have shown that alternans at the single cell level can be caused by unstable membrane voltage (V(m)) dynamics linked to steep APD-restitution, or unstable intracellular calcium (Ca) cycling linked to high sensitivity of Ca release from the sarcoplasmic reticulum on sarcoplasmic reticulum Ca load. Identifying which of these two mechanisms is the primary cause of cellular alternans, however, has remained difficult since Ca and V(m) are bidirectionally coupled. Here, we use numerical simulations of a physiologically detailed ionic model to show that the origin of alternans can be inferred by measuring the length scales over which APD and Ca(i) alternans reverse phase during spatially discordant alternans. The main conclusion is that these scales are comparable to a few millimeters and equal when alternans is driven by APD restitution, but differ markedly when alternans is driven predominantly by unstable Ca cycling. In the latter case, APD alternans still reverses phase on a millimeter tissue scale due to electrotonic coupling, while Ca alternans reverses phase on a submillimeter cellular scale. These results show that experimentally accessible measurements of Ca(i) and V(m) in cardiac tissue can be used to shed light on the cellular origin of alternans.Biophysical Journal 03/2007; 92(4):L33-5. · 3.65 Impact Factor -
Article: Nonlinear dynamics of cardiac excitation-contraction coupling: an iterated map study.
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ABSTRACT: Cardiac myocytes are excitable cells in which an external current stimulus depolarizes the membrane potential to elicit an action potential. This action potential then triggers calcium release from intracellular stores, which mediates contraction. Conversely, intracellular calcium also modulates membrane currents, affecting action potential morphology and action potential duration (APD). The interactions between action potential and calcium, termed excitation-contraction coupling, give rise to a rich spectrum of nonlinear dynamics, especially at rapid heart rates, which are important for cardiac contraction and the development of lethal arrhythmias. In this study, we developed a nonlinear iterated map model to investigate the dynamics of cardiac excitation-contraction coupling in a periodically stimulated cell. We first studied the nonlinear dynamics due to APD restitution, a functional relation between APD and its preceding diastolic interval. We then studied the nonlinear dynamics due to intracellular calcium cycling when total cell calcium is constant or varies at a beat-to-beat basis. Finally, we studied the nonlinear dynamics due to the bidirectional coupling of the two dynamical systems. Saddle-node bifurcations leading to bistability, period-doubling bifurcations leading to alternans, and period-doubling routes to chaos can independently occur in both action potential or intracellular calcium cycling subsystems as heart rate increases. A Hopf bifurcation leading to quasiperiodicity occurs when the two dynamical systems are coupled. Although these dynamics are predicted from low-dimensional iterated maps, the approach here provides valuable information which can be used as a basis to explore dynamical features of physiologically detailed ionic models, to illuminate experimental findings, and to design experimentally testable predictions for new biological experiments.Physical Review E 02/2007; 75(1 Pt 1):011927. · 2.26 Impact Factor -
Article: Dynamic origin of spatially discordant alternans in cardiac tissue.
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ABSTRACT: Alternans, a condition in which there is a beat-to-beat alternation in the electromechanical response of a periodically stimulated cardiac cell, has been linked to the genesis of life-threatening ventricular arrhythmias. Optical mapping of membrane voltage (V(m)) and intracellular calcium (Ca(i)) on the surface of animal hearts reveals complex spatial patterns of alternans. In particular, spatially discordant alternans has been observed in which regions with a large-small-large action potential duration (APD) alternate out-of-phase adjacent to regions of small-large-small APD. However, the underlying mechanisms that lead to the initiation of discordant alternans and govern its spatiotemporal properties are not well understood. Using mathematical modeling, we show that dynamic changes in the spatial distribution of discordant alternans can be used to pinpoint the underlying mechanisms. Optical mapping of V(m) and Ca(i) in paced rabbit hearts revealed that spatially discordant alternans induced by rapid pacing exhibits properties consistent with a purely dynamical mechanism as shown in theoretical studies. Our results support the viewpoint that spatially discordant alternans in the heart can be formed via a dynamical pattern formation process which does not require tissue heterogeneity.Biophysical Journal 02/2007; 92(2):448-60. · 3.65 Impact Factor -
Article: Nonlinear dynamics of paced cardiac cells.
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ABSTRACT: When a cardiac cell is rapidly paced it can exhibit a beat-to-beat alternation in the action potential duration (APD) and the intracellular calcium transient. This dynamical instability at the cellular level has been shown to correlate with the genesis of cardiac arrhythmias and has motivated the application of nonlinear dynamics in cardiology. In this article, we review mathematical approaches to describe the underlying mechanisms for alternans using beat-to-beat iterated maps. We explain the development and properties of these maps, and show that they provide a fruitful framework to understand dynamical instabilities of voltage and calcium in paced cardiac cells.Annals of the New York Academy of Sciences 11/2006; 1080:376-94. · 3.15 Impact Factor -
Article: Spatially discordant alternans in cardiac tissue: role of calcium cycling.
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ABSTRACT: Spatially discordant alternans, where the action potential duration (APD) and intracellular calcium transient (Ca(i)) alternate with opposite phase in different regions of tissue, is known to promote wave break and reentry. However, this phenomenon is not completely understood. It is known that alternans at the cellular level can be caused by dynamical instabilities arising from either membrane voltage (V(m)) attributable to steep APD restitution or to calcium (Ca) cycling. Here, we used a mathematical model of intracellular Ca cycling, coupled with membrane ion currents, to investigate the dynamics of V(m) and Ca(i) transient alternans in an isolated cell, in two electrotonically coupled cells, and in 1D spatially homogeneous tissue. Our main finding is a novel instability mechanism in which the bidirectional coupling of V(m) and Ca(i) can drive the Ca(i) transient of two neighboring cells to be out of phase. This instability is manifested in cardiac tissue by the dynamical formation of spatially discordant alternans. In this case, Ca(i) transient alternans can reverse phase over a length scale of one cell, whereas APD alternans reverses phase over a much longer length scale set by the electrotonic coupling. We analyze this mechanism in detail and show that it is a robust consequence of experimentally established properties of the bidirectional coupling between Ca cycling and V(m) dynamics. Finally, we address the experimental relevance of these findings and suggest physiological conditions under which these patterns can be observed.Circulation Research 10/2006; 99(5):520-7. · 9.49 Impact Factor -
Article: From pulsus to pulseless: the saga of cardiac alternans.
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ABSTRACT: Computer simulations and nonlinear dynamics have provided invaluable tools for illuminating the underlying mechanisms of cardiac arrhythmias. Here, we review how this approach has led to major insights into the mechanisms of spatially discordant alternans, a key arrhythmogenic factor predisposing the heart to re-entry and lethal arrhythmias. During spatially discordant alternans, the action potential duration (APD) alternates out of phase in different regions of the heart, markedly enhancing dispersion of refractoriness so that ectopic beats have a high probability of inducing reentry. We show how, at the cellular level, instabilities in membrane voltage (ie, steep APD restitution slope) and intracellular Ca (Cai) cycling dynamics cause APD and the Cai transient to alternate and how the characteristics of alternans are affected by different "modes" of the bidirectional coupling between voltage and Cai. We illustrate how, at the tissue level, additional factors, such as conduction velocity restitution and ectopic beats, promote spatially discordant alternans. These insights have illuminated the mechanistic basis underlying the clinical association of cardiac alternans (eg, T wave alternans) with arrhythmia risk, which may lead to novel therapeutic approaches to avert sudden cardiac death.Circulation Research 06/2006; 98(10):1244-53. · 9.49 Impact Factor -
Article: Turing instability mediated by voltage and calcium diffusion in paced cardiac cells.
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ABSTRACT: In cardiac cells, the coupling between the voltage across the cell membrane (V(m)) and the release of calcium (Ca) from intracellular stores is a crucial ingredient of heart function. Under abnormal conditions and/or rapid pacing, both the action potential duration and the peak Ca concentration in the cell can exhibit well known period-doubling oscillations referred to as "alternans," which have been linked to sudden cardiac death. Fast diffusion of V(m) keeps action potential duration alternans spatially synchronized over the approximately 150-mum-length scale of a cell, but slow diffusion of Ca ions allows Ca alternans within a cell to become spatially asynchronous, as observed in some experiments. This finding raises the question: When are Ca alternans spatially in-phase or out-of-phase on subcellular length scales? This question is investigated by using a spatially distributed model of Ca cycling coupled to V(m). Our main finding is the existence of a Turing-type symmetry breaking instability mediated by V(m) and Ca diffusion that causes Ca alternans to become spontaneously out-of-phase at opposite ends of a cardiac cell. Pattern formation is governed by the interplay of short-range activation of Ca alternans, because of a dynamical instability of Ca cycling, and long-range inhibition of Ca alternans by V(m) alternans through Ca-sensitive membrane ionic currents. These results provide a striking example of a Turing instability in a biological context where the morphogens can be clearly identified, as well as a potential link between dynamical instability on subcellular scales and life-threatening cardiac disorders.Proceedings of the National Academy of Sciences 05/2006; 103(15):5670-5. · 9.68 Impact Factor -
Article: Coupled dynamics of voltage and calcium in paced cardiac cells.
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ABSTRACT: We investigate numerically and analytically the coupled dynamics of transmembrane voltage and intracellular calcium cycling in paced cardiac cells using a detailed physiological model, and its reduction to a three-dimensional discrete map. The results provide a theoretical framework to interpret various experimentally observed modes of instability ranging from electromechanically concordant and discordant alternans to quasiperiodic oscillations of voltage and calcium.Physical Review E 03/2005; 71(2 Pt 1):021903. · 2.26 Impact Factor
Top co-authors
Top Journals
Institutions
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2007–2012
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California State University, Northridge
- Department of Physics and Astronomy
Los Angeles, CA, USA
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2008
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UCLA Cardiovascular Research Laboratory
Los Angeles, CA, USA
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2006–2007
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University of California, Los Angeles
- Division of Cardiology
Los Angeles, CA, USA -
University of Southern California
Los Angeles, CA, USA
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2005–2006
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Northeastern University
- Center for Interdisciplinary Research on Complex Systems (CIRCS)
Boston, MA, USA
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2000
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University of Pittsburgh
- Physics and Astronomy
Pittsburgh, PA, USA
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