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We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence operator. As a practical application, we show how the operators can be used to symbolically compute local conservation laws(More)
The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. The KdV equation is known to have large families of periodic solutions that are parameterized by hyperelliptic Riemann surfaces. They are generalizations of the famous multi-soliton solutions. We show that all such periodic(More)
Intracellular calcium (Ca) cycling dynamics in cardiac myocytes is regulated by a complex network of spatially distributed organelles, such as sarcoplasmic reticulum (SR), mitochondria, and myofibrils. In this study, we present a mathematical model of intracellular Ca cycling and numerical and computational methods for computer simulations. The model(More)
In heart failure (HF), T-tubule (TT) disruption contributes to dyssynchronous calcium (Ca) release and impaired contraction, but its role in arrhythmogenesis remains unclear. In this study, we investigate the effects of TT disruption and other HF remodeling factors on Ca alternans in ventricular myocytes using computer modeling. A ventricular myocyte model(More)
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(More)
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(More)
Early afterdepolarizations (EADs) and delayed afterdepolarizations (DADs) are voltage oscillations known to cause cardiac arrhythmias. EADs are mainly driven by voltage oscillations in the repolarizing phase of the action potential (AP), while DADs are driven by spontaneous calcium (Ca) release during diastole. Because voltage and Ca are bidirectionally(More)
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(More)
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(More)
The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. In this paper, we derive all periodic traveling wave solutions of the focusing and defocusing mKdV equations. We show that in the defocusing case all such solutions are orbitally stable with respect to subharmonic perturbations:(More)