Alternans promotion in cardiac electrophysiology models by delay differential equations

Generic generation of noise-driven chaos in stochastic time delay systems: Bridging the gap with high-end simulations

Nonlinear time delay systems produce inherently delay-induced periodic oscillations, which are, however, too idealistic compared to observations. We exhibit a unified stochastic framework to systematically rectify such oscillations into oscillatory patterns with enriched temporal variabilities through generic, nonlinear responses to stochastic perturbations. Two paradigms of noise-driven chaos in high dimension are identified, fundamentally different from chaos triggered by parameter-space noise. Noteworthy is a low-dimensional stretch-and-fold mechanism, leading to stochastic strange attractors exhibiting horseshoe-like structures mirroring turbulent transport of passive tracers. The other is high-dimensional , with noise acting along the critical eigendirection and transmitted to "deeper" stable modes through nonlinearity, leading to stochastic attractors exhibiting swarm-like behaviors with power-law and scale break properties. The theory is applied to cloud delay models to parameterize missing physics such as intermittent rain and Lagrangian turbulent effects. The stochastically rectified model reproduces with fidelity complex temporal variabilities of open-cell oscillations exhibited by high-end cloud simulations.

Contribution of Cardiorespiratory Coupling to the Irregular Dynamics of the Human Cardiovascular System

Irregularity is an important aspect of the cardiovascular system dynamics. Numerical indices of irregularity, such as the largest Lyapunov exponent and the correlation dimension estimated from interbeat interval time series, are early markers of cardiovascular diseases. However, there is no consensus on the origin of irregularity in the cardiovascular system. A common hypothesis suggests the importance of nonlinear bidirectional coupling between the cardiovascular system and the respiratory system for irregularity. Experimental investigations of this theory are severely limited by the capabilities of modern medical equipment and the nonstationarity of real biological systems. Therefore, we studied this problem using a mathematical model of the coupled cardiovascular system and respiratory system. We estimated and compared the numerical indices of complexity for a model simulating the cardiovascular dynamics in healthy subjects and a model with blocked regulation of the respiratory frequency and amplitude, which disturbs the coupling between the studied systems.

Delay differential equation-based models of cardiac tissue: Efficient implementation and effects on spiral-wave dynamics

Delay differential equations (DDEs) recently have been used in models of cardiac electrophysiology, particularly in studies focusing on electrical alternans, instabilities, and chaos. A number of processes within cardiac cells involve delays, and DDEs can potentially represent mechanisms that result in complex dynamics both at the cellular level and at the tissue level, including cardiac arrhythmias. However, DDE-based formulations introduce new computational challenges due to the need for storing and retrieving past values of variables at each spatial location. Cardiac tissue simulations that use DDEs may require over 28 GB of memory if the history of variables is not managed carefully. This paper addresses both computational and dynamical issues. First, we present new methods for the numerical solution of DDEs in tissue to mitigate the memory requirements associated with the history of variables. The new methods exploit the different time scales of an action potential to dynamically optimize history size. We find that the proposed methods decrease memory usage by up to 95% in cardiac tissue simulations compared to straightforward history-management algorithms. Second, we use the optimized methods to analyze for the first time the dynamics of wave propagation in two-dimensional cardiac tissue for models that include DDEs. In particular, we study the effects of DDEs on spiral-wave dynamics, including wave breakup and chaos, using a canine myocyte model. We find that by introducing delays to the gating variables governing the calcium current, DDEs can induce spiral-wave breakup in 2D cardiac tissue domains.

Autonomic control is a source of dynamical chaos in the cardiovascular system

The origin of complex irregular dynamics in a cardiovascular system is still being actively debated. Some hypotheses suggest the crucial role of stochastic modulation of cardiovascular parameters, while others argue for the importance of cardiac pacemakers' chaotic deterministic dynamics. In the present study, we estimate the largest Lyapunov exponent and the correlation dimension for the 4-h experimental interbeat intervals and the chaotic signals generated by the mathematical model of the cardiovascular system. We study the complexity of the mathematical model for such cases as the autonomic blockade, the exclusion of all the stochastic components, and the absence of variability of respiration. The obtained results suggest that the complexity of the heart rate variability is largely due to the chaotic dynamics in the loops of autonomic control of circulation.

Multi-fidelity modeling in sequential design for stability identification in dynamic time-delay systems

Time delay arises in a variety of real-world complex systems. A high-fidelity simulation generally renders high accuracy to simulate the dynamic evolution of such complex systems and appraise quantity of interest for process design and response optimization. Identification of limit states exemplifies such applications, which outlines the boundary that separates distinct regions (e.g., stability region) in parameter space. While design of experiments is the common procedure to evaluate decision functions to sketch the boundary, it crucially relies on the quantity and quality of sampling points. This has made it infeasible to explore a large parameter design space with expensive-to-evaluate high-fidelity simulations. Furthermore, the complex contour of stability region in time-delay systems nullifies most existing sequential design paradigms, including adaptive classification approaches. On the other hand, low-fidelity surrogate modeling efficiently emulates a high-fidelity simulation, albeit at the expense of accuracy, not ideal to inspect the system behavior near the critical boundary. In this study, we investigate a multifidelity approach to delineate the stability region in a sequential fashion: sampling points are first evaluated by the low-fidelity surrogate modeling, and only those selected according to the exploration-exploitation trade-off principle are then assessed by a high-fidelity simulation to approximate the stability boundary. The application in a numerical case study of the delayed Mathieu equation as well as a real-world machining process corroborates the proposed approach.