Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for travelling waves in one-dimensional excitable media. The normal form which has been recently proposed on phenomenological grounds is given in form of a differential delay equation. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both, the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.Comment: to be published in Chao

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that are triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media and their application to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact in cardiac arrhythmias.Peer ReviewedPreprin

Complex patterns of spontaneous initiations and terminations of reentrant circulation in a loop of cardiac tissue

A two-component model is developed that consists of a discrete loop of cardiac cells that circulates action potentials together with a cardiac pacing mechanism. Physiological properties of cells such as restitutions of refractoriness and of conduction velocity are given via experimentally measured functions. The dynamics of circulating pulses and their interactions with the pacer are regulated by two threshold relations. Patterns of spontaneous initiations and terminations of reentry (SITR) generated by this system are studied through numerical simulations and analytical observations. These patterns can be regular or irregular; causes of irregularities are identified as the threshold bistability of reentrant circulation (T-bistability) and in some cases, also phase-resetting interactions with the pacer.Comment: 27 pages, 10 figures, 61 references; A version of this paper (same results) is to appear in the Journal of Theoretical Biology; arXiv V2 adds helpful commments to facilitate reading and corrects minor errors in presentatio

A comparative study fourth order runge kutta-tvd Scheme and fluent software case of inlet flow problems

Inlet as part of aircraft engine plays important role in controlling the rate of airflow entering to the engine. The shape of inlet has to be designed in such away to make the rate of airflow does not change too much with angle of attack and also not much pressure losses at the time, the airflow entering to the compressor section. It is therefore understanding on the flow pattern inside the inlet is important. The present work presents on the use of the Fourth Order Runge Kutta – Harten Yee TVD scheme for the flow analysis inside inlet. The flow is assumed as an inviscid quasi two dimensional compressible flow. As an initial stage of computer code development, here uses three generic inlet models. The first inlet model to allow the problem in hand solved as the case of inlet with expansion wave case. The second inlet model will relate to the case of expansion compression wave. The last inlet model concerns with the inlet which produce series of weak shock wave and end up with a normal shock wave. The comparison result for the same test case with Fluent Software [1, 2] indicates that the developed computer code based on the Fourth Order Runge Kutta – Harten – Yee TVD scheme are very close to each other. However for complex inlet geometry, the problem is in the way how to provide an appropriate mesh model

Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity

It has become widely accepted that the most dangerous cardiac arrhythmias are due to re- entrant waves, i.e., electrical wave(s) that re-circulate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, has made it extremely difficult to pinpoint the detailed mechanisms of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols.Comment: 128 pages, 42 figures (29 color, 13 b&w

Modeling Excitable Systems: Reentrant Tachycardia

Excitable membranes are an important type of nonlinear dynamical system and their study can be used to provide a connection between physical and biological circuits. We discuss two models of excitable membranes important in cardiac and neural tissues. One model is based on the Fitzhugh-Nagumo equations and the other is based on a three-transistor excitable circuit. We construct a circuit that simulates reentrant tachycardia and its treatment by surgical ablation. This project is appropriate for advanced undergraduates as a laboratory capstone project, or as a senior thesis or honors project, and can also be a collaborative project, with one student responsible for the computational predictions and another for the circuit construction and measurements.Comment: 9 pages, twocolumn, revised and published in American Journal of Physic

Using Delay-Differential Equations for Modeling Calcium Cycling in Cardiac Myocytes

The cycling of calcium at the intracellular level of cardiac cells plays a key role in the excitation-contraction process. The interplay between ionic currents, buffering agents, and calcium release from the sarcoplasmic reticulum (SR) is a complex system that has been shown experimentally to exhibit complex dynamics including period-2 states (alternans) and higher-order rhythms. Many of the calcium cycling activities involve the sensing, binding, or diffusion of calcium between intracellular compartments; these are physical processes that take time and typically are modeled by “relaxation” equations where the steady-state value and time course of a particular variable are specified through an ordinary differential equation (ODE) with a time constant. An alternative approach is to use delay-differential equations (DDEs), where the delays in the system correspond to non-instantaneous events. In this thesis, we present a thorough overview of results from calcium cycling experiments and proposed intracellular calcium cycling models, as well as the context of alternans and delay-differential equations in cardiac modeling. We utilize a DDE to model the diffusion of calcium through the SR by replacing the relaxation ODE typically used for this process. The relaxation time constant τa is replaced by a delay δj, which could also be interpreted as the refractoriness of ryanodine receptor channels after releasing calcium from the sarcoplasmic reticulum. This is the first application of delay-differential equations to modeling calcium cycling dynamics, and to modeling cardiac systems at the cellular level. We analyzed the dynamical behaviors of the system and focus on the factors that have been shown to produce alternans and irregular dynamics in experiments and models with cardiac myocytes. We found that chaotic calcium dynamics could occur even for a more physiologically revelant SR calcium release slope than comparable ODE models. Increasing the SR release slope did not affect the calcium dynamics, but only shifted behavior down to lower values of the delay, allowing alternans, higher-order behavior, and chaos to occur for smaller delays than in simulations with a normal SR release slope. For moderate values of the delay, solely alternans and 1:1 steady-state behavior were observed. Above a particular threshold value for the delay, chaos appeared in the dynamics and further increasing the delay caused the system to destabilize under broader ranges of periods. We also compare our results with other models of intracellular calcium cycling and suggest promising avenues for further development of our preliminary work

A new approach to modelling the dynamics of cardiac action potentials

This thesis is concerned with the development of a new approach to the modelling of cardiac action potentials. Electrophysiological models of the heart have become very accurate in recent years giving rise to extremely complicated systems of differential equations. Although describing the behaviour of cardiac cells well, the models are computationally demanding for numerical simulations and are very difficult to analyse from a mathematical (dynamical-systems) viewpoint. Simplified mathematical models that capture the underlying dynamics to a certain extent are therefore frequently used. However, from a physiological viewpoint these equations are unrealistic and often fail to reproduce important quantitative properties of the tissue. In this thesis we introduce a different approach to the mathematical modelling of cardiac action potentials with the aim of gaining a clearer insight into the origin of the dynamics of electrophysiological models. Chapter 1 contains an introduction to the research and outlines the main aims of the work. In Chapter 2 various background material is introduced. This includes some basic electrophysiology, ideas currently used in mathematical modelling of excitable media, and details of models previously developed for the study of cardiac tissue. In Chapter 3, following a detailed analysis of an early physiological model, we develop a mathematical model based on the currents involved. This model reproduces, to good accuracy, action potentials of heart tissue and we discuss the essential ideas behind the dynamics. In Chapter 4 the mathematical model developed in the previous chapter is analysed in more detail and simpler equations using similar ideas are introduced. Various types of action potentials of varying behaviours are studied. In Chapter 5 we investigate some spatial simulations of the new mathematical models. We principally concentrate on one-dimensional studies but towards the end of the chapter we look at some two-dimensional simulations. Finally, in Chapter 6, we discuss our conclusions and some possible ideas for further related work. Details of our methods of numerical simulation are included in Appendix A

Optogenetic Control of Cardiac Arrhythmias

The regular, coordinated contraction of the heart muscle is orchestrated by periodic waves generated by the heart’s natural pacemaker and transmitted through the heart’s electrical conduction system. Abnormalities occurring anywhere within the cardiac electrical conduction system can disrupt the propagation of these waves. Such dis- ruptions often lead to the development of high frequency spiral waves that override normal pacemaker activity and compromise cardiac function. The occurrence of high frequency spiral waves in the heart is associated with cardiac rhythm disorders such as tachycardia and fibrillation. While tachycardia may be terminated by rapid periodic stimulation known as anti-tachycardia pacing (ATP), life-threatening ventricular fibril- lation requires a single high-voltage electric shock that resets all the activity and restore the normal heart function. However, despite the high success rate of defibrillation, it is associated with significant side effects including tissue damage, intense pain and trauma. Thus, extensive research is conducted for developing low-energy alternatives to conventional defibrillation. An example of such an alternative is the low-energy anti-fibrillation pacing (LEAP). However, the clinical application of this technique, and other evolving techniques requires a detailed understanding of the dynamics of spiral waves that occur during arrhythmias. Optogenetics is a tool, that has recently gained popularity in the cardiac research, which serves as a probe to study biological processes. It involves genetically modifying cardiac muscle cells such that they become light sensitive, and then using light of specific wavelengths to control the electrical activity of these cells. Cardiac optogenetics opens up new ways of investigating the mechanisms underlying the onset, maintenance and control of cardiac arrhythmias. In this thesis, I employ optogenetics as a tool to control the dynamics of a spiral wave, in both computer simulations and in experiments.In the first study, I use optogenetics to investigate the mechanisms underlying de- fibrillation. Analogous to the conventional single electric-shock, I apply a single globally-illuminating light pulse to a two-dimensional cardiac tissue to study how wave termination occurs during defibrillation. My studies show a characteristic transient dynamics leading to the termination of the spiral wave at low light intensities, while at high intensities, the spiral waves terminate immediately. Next, I move on to explore the use of optogenetics to study spiral wave termina- tion via drift, theoretically well-known mechanism of arrhythmia termination in the context of electrical stimulation (e.g. ATP). I show that spiral wave drift can be induced by structured illumination patterns using lights of low intensity, that result in a spatial modulation of cardiac excitability. I observe that drift occurs in the positive direction of light intensity gradient, where the spiral also rotates with a longer period. I further show how modulation of the excitability in space can be used to control the dynamics of a spiral wave, resulting in the termination of the wave by collision with the domain boundary. Based on these observations, I propose a possible mechanism of optogenetic defibrillation. In the next chapter, I use optogenetics to demonstrate control over the dynamics of the spiral waves by periodic stimulation with light of different intensities and pacing frequencies resulting in a temporal modulation of cardiac excitability. I demonstrate how the temporal modulation of excitability leads to efficient termination of arrhythmia. In addition, I use computer simulations to identify mechanisms responsible for arrhyth- mia termination for sub- and supra-threshold light intensities. My numerical results are supported by experimental studies on intact hearts, extracted from transgenic mice. Finally, I demonstrate that cardiac optogenetics not only allows control of excita- tion waves, but also by generating new waves through the induction of wave breaks. We demonstrate the effects of high sub-threshold illumination on the morphology of the propagating wave, leading to the creation of new excitation windows in space that can serve as potential sites for re-entry initiation. In summary, this thesis investigates several approaches to control arrhythmia dy- namics using optogenetics. The experimental and numerical results demonstrate the potential of feedback-induced resonant pacing as a low-energy method to control arrhythmia.2022-01-1