Michael Nivala

University of California, Los Angeles | UCLA

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 perio...

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 mod...

Key points Calcium (Ca ²⁺ ) is fundamental to biological cell function, and Ca ²⁺ waves generating oscillatory Ca ²⁺ signals are widely observed in many cell types. Some experimental studies have shown that Ca ²⁺ waves initiate from random locations within the cell, while other studies have shown that waves occur repetitively from preferred locatio...

A brief introductory article on the role of chaotic synchronization in the context of complex economic systems. The basic framework developed by the late Richard Goodwin in his book, Chaotic Economic Dynamics, of 1990 has been extended to massively complex dynamical systems of chaotic elements. Recent experimental results and speculative applicatio...

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...

-Hypokalemia is known to promote ventricular arrhythmias, especially in combination with Class III antiarrhythmic drugs like dofetilide. Here we evaluated the underlying molecular mechanisms. -Arrhythmias were recorded in isolated rabbit and rat hearts or patch-clamped ventricular myocytes exposed to hypokalemia (1.0-3.5 mmol/l) in the absence or p...

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 bidirectio...

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 myocy...

Spontaneous calcium (Ca) waves must emerge near-synchronously in thousands of contiguous myocytes to produce delayed afterdepolarizations (DADs) in cardiac tissue. Previous studies have shown as Ca load increases, the time to onset of a spontaneous Ca wave after pacing (latency), as well as its variability, decreases. Two proposed mechanisms includ...

Subcellular Calcium (Ca) cycling plays fundamental roles in normal heart dynamics. In cardiac myocytes, the elementary Ca cycling events are Ca sparks: random discretized Ca release events due to random and collective openings of the ryanodine receptor (RyR) channels clustered in Ca release units (CRUs). A typical cardiac myocyte includes about 10,...

Biological systems, such as the heart, are typically regulated by nonlinear dynamics occurring on multiple scales, ranging from random molecular motions to more regular cellular and tissue-level behaviors. In this chapter, we review experimental observations and mechanistic insights gained from mathematical modeling of biological functions across s...

Calcium (Ca) is a ubiquitous second messenger regulating 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 including intracellular and intercellular Ca waves and whole-cell Ca oscillations. In a recent study using a computati...

Intracellular calcium (Ca) waves in cardiac myocytes can cause delayed afterdepolarizations (DADs), which are known triggers of cardiac arrhythmias. How these Ca waves are modulated by diffusive Ca-mediated coupling among Ca release units (CRUs) and promote DADs is not fully understood. Here, we hypothesized that myocytes are most susceptible to DA...

In cardiac myocytes, the elementary Ca cycling events are Ca sparks, spatially discrete Ca release events due to random and collective openings of ryanodine receptor (RyR) channels clustered in close proximity to L-type Ca channels (LCCs), forming what are known as Ca release units (CRUs). A typical cardiac myocyte includes about 10,000 to 20,000 C...

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 subcel...

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 character...

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;...

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 characteri...

It has been shown that transient single mitochondrial depolarizations, known as flickers, tend to occur randomly in space and time. On the other hand, many studies have shown that mitochondrial depolarization waves and whole-cell oscillations occur under oxidative stress. How single mitochondrial flickering events and whole-cell oscillations are me...

The stability of the stationary periodic solutions of the integrable (one-dimensional, cubic) defocusing nonlinear Schrödinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions...

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 hear...

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 pertur...

The homotopy algorithm is a powerful method for indefinite integration of total derivatives. By combining these ideas with straightforward Gaussian elimination, we construct an algorithm for the optimal symbolic integration that contain terms that are not total derivatives. The optimization consists of minimizing the number of terms that remain uni...

The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. In this thesis, a new method for investigating the (nonlinear) orbital stability of periodic solutions of integrable Hamiltonian systems is presented. The method is demonstrated on the KdV equation, proving that all peri...

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 conservat...

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 conservat...