Circadian rhythms are endogenous~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models K E Y W O R D S biological oscillators, circadian clock, circadian rhythms, dynamic systems, mathematical modeling, statistical modeling BOX 1 CIRCADIAN MODELINGCircadian clocks are present within individual cells, and communication among multiple cells gives rise to emergent properties at the tissue level. In mammals, both the master circadian clock in the suprachiasmatic nucleus and tissuelevel peripheral clocks have major effects at the organism level on numerous key physiological functions, including sleep/wake cycles, metabolism, cardiovascular function, reproduction, immune function, neurobehavioral performance, and mood. Misalignment between the master clock and peripheral clocks within an organism, or misalignment between an organism's clocks and its external environment, has adverse physiological consequences. When circadian interventions are needed to improve physiological function, a multiscale understanding of circadian rhythmicity therefore is essential to accurately manipulate this complex oscillatory system. Mathematical modeling is an essential tool to study and analyze complex physiological systems. It has been used to provide insight into the circadian system at multiple levels (i.e., organism, multi-cellular, cellular, molecular, genetic), to design new experiments, and to manipulate and control the components of the system in silico with specificity that

Background: The mechanisms underlying dysfunction in the sinoatrial node (SAN), the heart’s primary pacemaker, are incompletely understood. Electrical and Ca2+-handling remodeling have been implicated in SAN dysfunction associated with heart failure, aging, and diabetes. Cardiomyocyte [Na+]i is also elevated in these diseases, where it contributes to arrhythmogenesis. Here, we sought to investigate the largely unexplored role of Na+ homeostasis in SAN pacemaking and test whether [Na+]i dysregulation may contribute to SAN dysfunction. Methods: We developed a dataset-specific computational model of the murine SAN myocyte and simulated alterations in the major processes of Na+ entry (Na+/Ca2+ exchanger, NCX) and removal (Na+/K+ ATPase, NKA). Results: We found that changes in intracellular Na+ homeostatic processes dynamically regulate SAN electrophysiology. Mild reductions in NKA and NCX function increase myocyte firing rate, whereas a stronger reduction causes bursting activity and loss of automaticity. These pathologic phenotypes mimic those observed experimentally in NCX- and ankyrin-B-deficient mice due to altered feedback between the Ca2+ and membrane potential clocks underlying SAN firing. Conclusions: Our study generates new testable predictions and insight linking Na+ homeostasis to Ca2+ handling and membrane potential dynamics in SAN myocytes that may advance our understanding of SAN (dys)function.

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.