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Lilienkamp T, Parlitz U. Terminating transient chaos in spatially extended systems. CHAOS (WOODBURY, N.Y.) 2020; 30:051108. [PMID: 32491910 DOI: 10.1063/5.0011506] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2020] [Accepted: 05/07/2020] [Indexed: 06/11/2023]
Abstract
In many real-life systems, transient chaotic dynamics plays a major role. For instance, the chaotic spiral or scroll wave dynamics of electrical excitation waves during life-threatening cardiac arrhythmias can terminate by itself. Epileptic seizures have recently been related to the collapse of transient chimera states. Controlling chaotic transients, either by maintaining the chaotic dynamics or by terminating it as quickly as possible, is often desired and sometimes even vital (as in the case of cardiac arrhythmias). We discuss in this study that the difference of the underlying structures in state space between a chaotic attractor (persistent chaos) and a chaotic saddle (transient chaos) may have significant implications for efficient control strategies in real life systems. In particular, we demonstrate that in the latter case, chaotic dynamics in spatially extended systems can be terminated via a relatively low number of (spatially and temporally) localized perturbations. We demonstrate as a proof of principle that control and targeting of high-dimensional systems exhibiting transient chaos can be achieved with exceptionally small interactions with the system. This insight may impact future control strategies in real-life systems like cardiac arrhythmias.
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Affiliation(s)
- Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
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Aron M, Herzog S, Parlitz U, Luther S, Lilienkamp T. Spontaneous termination of chaotic spiral wave dynamics in human cardiac ion channel models. PLoS One 2019; 14:e0221401. [PMID: 31461472 PMCID: PMC6713330 DOI: 10.1371/journal.pone.0221401] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2019] [Accepted: 08/06/2019] [Indexed: 11/18/2022] Open
Abstract
Chaotic spiral or scroll wave dynamics can be found in diverse systems. In cardiac dynamics, spiral or scroll waves of electrical excitation determine the dynamics during life-threatening arrhythmias like ventricular fibrillation. In numerical studies it was found that chaotic episodes of spiral and scroll waves can be transient, thus they terminate spontaneously. We show in this study that this behavior can also be observed using models which describe the ion channel dynamics of human cardiomyocytes (Bueno-Orovio-Cherry-Fenton model and the Ten Tusscher-Noble-Noble-Panfilov model). For both models we find that the average lifetime of the chaotic transients grows exponentially with the system size. With this behavior, we classify the systems into the group of type-II supertransients. We observe a significant difference of the breakup behavior between the models, which results in a distinct dynamics during the final phase just before the termination. The observation of a (temporally) stable single-spiral state affects the prevailing description of the dynamics of type-II supertransients as being “quasi-stationary” and also the feasibility of predicting the spontaneous termination of the spiral wave dynamics. In the long term, the relation between the breakup behavior of spiral waves and properties of chaotic transients like predictability or average transient lifetime may contribute to an improved understanding and classification of cardiac arrhythmias.
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Affiliation(s)
- Marcel Aron
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Göttingen, Germany
| | - Sebastian Herzog
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Göttingen, Germany
- Third Institute of Physics, Universität Göttingen, Friedrich-Hund Platz 1, Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Friedrich-Hund Platz 1, Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Göttingen, Germany
- DZHK (German Centre for Cardiovascular Research), Partner Site Göttingen, Göttingen, Germany
| | - Stefan Luther
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Göttingen, Germany
- DZHK (German Centre for Cardiovascular Research), Partner Site Göttingen, Göttingen, Germany
- Department of Pharmacology and Toxicology, University Medical Center Göttingen, Göttingen, Germany
| | - Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- DZHK (German Centre for Cardiovascular Research), Partner Site Göttingen, Göttingen, Germany
- Department of Cardiology and Pneumology, University Medical Center Göttingen, Göttingen, Germany
- * E-mail:
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Gray RA, Pathmanathan P. A Parsimonious Model of the Rabbit Action Potential Elucidates the Minimal Physiological Requirements for Alternans and Spiral Wave Breakup. PLoS Comput Biol 2016; 12:e1005087. [PMID: 27749895 PMCID: PMC5066986 DOI: 10.1371/journal.pcbi.1005087] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2016] [Accepted: 07/21/2016] [Indexed: 11/19/2022] Open
Abstract
Elucidating the underlying mechanisms of fatal cardiac arrhythmias requires a tight integration of electrophysiological experiments, models, and theory. Existing models of transmembrane action potential (AP) are complex (resulting in over parameterization) and varied (leading to dissimilar predictions). Thus, simpler models are needed to elucidate the "minimal physiological requirements" to reproduce significant observable phenomena using as few parameters as possible. Moreover, models have been derived from experimental studies from a variety of species under a range of environmental conditions (for example, all existing rabbit AP models incorporate a formulation of the rapid sodium current, INa, based on 30 year old data from chick embryo cell aggregates). Here we develop a simple "parsimonious" rabbit AP model that is mathematically identifiable (i.e., not over parameterized) by combining a novel Hodgkin-Huxley formulation of INa with a phenomenological model of repolarization similar to the voltage dependent, time-independent rectifying outward potassium current (IK). The model was calibrated using the following experimental data sets measured from the same species (rabbit) under physiological conditions: dynamic current-voltage (I-V) relationships during the AP upstroke; rapid recovery of AP excitability during the relative refractory period; and steady-state INa inactivation via voltage clamp. Simulations reproduced several important "emergent" phenomena including cellular alternans at rates > 250 bpm as observed in rabbit myocytes, reentrant spiral waves as observed on the surface of the rabbit heart, and spiral wave breakup. Model variants were studied which elucidated the minimal requirements for alternans and spiral wave break up, namely the kinetics of INa inactivation and the non-linear rectification of IK.The simplicity of the model, and the fact that its parameters have physiological meaning, make it ideal for engendering generalizable mechanistic insight and should provide a solid "building-block" to generate more detailed ionic models to represent complex rabbit electrophysiology.
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Affiliation(s)
- Richard A. Gray
- Division of Biomedical Physics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, Maryland, United States of America
- * E-mail:
| | - Pras Pathmanathan
- Division of Biomedical Physics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, Maryland, United States of America
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Aslanidi OV, Colman MA, Zhao J, Smaill BH, Gilbert SH, Hancox JC, Boyett MR, Zhang H. Arrhythmogenic substrate for atrial fibrillation: insights from an integrative computational model of pulmonary veins. ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. ANNUAL INTERNATIONAL CONFERENCE 2013; 2012:203-6. [PMID: 23365867 DOI: 10.1109/embc.2012.6345906] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
Mechanisms underlying the genesis of re-entrant substrate for atrial fibrillation (AF) in the pulmonary veins (PVs) and left atrium (LA) are not well understood. We develop a biophysically detailed computational model for the PVs and surrounding LA tissue. The model integrates canine PV and LA single cell electrophysiology with the respective 3D tissue geometry and fiber orientation reconstructed from micro-CT data. The model simulations demonstrate that a combination of tissue anisotropy and electrical heterogeneity between the PVs and LA causes a break-down of normal electrical excitation wave-fronts. This leads to the generation of a high-frequency re-entrant source near the PV sleeves. Evidence of such sources have been seen clinically in AF patients. In summary, our modeling results provide new insights into the arrhythmogenic mechanisms of re-entrant excitation waves underlying AF.
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Affiliation(s)
- Oleg V Aslanidi
- Division of Imaging Sciences & Biomedical Engineering, King's College London, London SE1 7EH, United Kingdom.
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Virtual tissue engineering of the human atrium: Modelling pharmacological actions on atrial arrhythmogenesis. Eur J Pharm Sci 2012; 46:209-21. [DOI: 10.1016/j.ejps.2011.08.014] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2011] [Revised: 07/28/2011] [Accepted: 08/09/2011] [Indexed: 11/23/2022]
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Qiao C, Wu Y, Lu X, Wang C, Ouyang Q, Wang H. Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient. CHAOS (WOODBURY, N.Y.) 2008; 18:026109. [PMID: 18601511 DOI: 10.1063/1.2927532] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.
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Affiliation(s)
- Chun Qiao
- State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China
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Clayton RH, Panfilov AV. A guide to modelling cardiac electrical activity in anatomically detailed ventricles. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2007; 96:19-43. [PMID: 17825362 DOI: 10.1016/j.pbiomolbio.2007.07.004] [Citation(s) in RCA: 88] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Abstract
One of the most recent trends in cardiac electrophysiology is the development of integrative anatomically accurate models of the heart, which include description of cardiac activity from sub-cellular and cellular level to the level of the whole organ. In order to construct this type of model, a researcher needs to collect a wide range of information from books and journal articles on various aspects of biology, physiology, electrophysiology, numerical mathematics and computer programming. The aim of this methodological article is to survey recent developments in integrative modelling of electrical activity in the ventricles of the heart, and to provide a practical guide to the resources and tools that are available for work in this exciting and challenging area.
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Affiliation(s)
- R H Clayton
- Department of Computer Science, University of Sheffield, Regent Court, 211 Portobello Street, Sheffield, S1 4DP, UK.
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Wessel N, Kurths J, Ditto W, Bauernschmitt R. Introduction: Cardiovascular physics. CHAOS (WOODBURY, N.Y.) 2007; 17:015101. [PMID: 17411258 DOI: 10.1063/1.2718395] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/14/2023]
Abstract
The number of patients suffering from cardiovascular diseases increases unproportionally high with the increase of the human population and aging, leading to very high expenses in the public health system. Therefore, the challenge of cardiovascular physics is to develop high-sophisticated methods which are able to, on the one hand, supplement and replace expensive medical devices and, on the other hand, improve the medical diagnostics with decreasing the patient's risk. Cardiovascular physics-which interconnects medicine, physics, biology, engineering, and mathematics-is based on interdisciplinary collaboration of specialists from the above scientific fields and attempts to gain deeper insights into pathophysiology and treatment options. This paper summarizes advances in cardiovascular physics with emphasis on a workshop held in Bad Honnef, Germany, in May 2005. The meeting attracted an interdisciplinary audience and led to a number of papers covering the main research fields of cardiovascular physics, including data analysis, modeling, and medical application. The variety of problems addressed by this issue underlines the complexity of the cardiovascular system. It could be demonstrated in this Focus Issue, that data analyses and modeling methods from cardiovascular physics have the ability to lead to significant improvements in different medical fields. Consequently, this Focus Issue of Chaos is a status report that may invite all interested readers to join the community and find competent discussion and cooperation partners.
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Affiliation(s)
- Niels Wessel
- Department of Physics, University of Potsdam, Am Neuen Palais 10, Potsdam, 14415, Germany
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Holden AV, Aslanidi OV, Benson AP, Clayton RH, Halley G, Li P, Tong WC. The virtual ventricular wall: a tool for exploring cardiac propagation and arrhythmogenesis. J Biol Phys 2006; 32:355-68. [PMID: 19669471 PMCID: PMC2651522 DOI: 10.1007/s10867-006-9020-1] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2006] [Accepted: 04/28/2006] [Indexed: 11/25/2022] Open
Abstract
Methods for the experimental and clinical investigation of cardiac arrhythmias are limited to inferring propagation within the myocardium, from surface measurements, or from electrodes at a few sites within the cardiac wall. Biophysically and anatomically detailed computational models of cardiac tissues offer a powerful way for studying the electrical propagation processes and arrhythmias within the virtual heart. We use virtual tissues to study and visualise the effects of patho- and physiological conditions, and pharmacological interventions on transmural propagation in the virtual ventricular walls. Class III drug actions are quantitatively explained by changes induced in the transmural dispersion of action potential duration. We illustrate the automated construction of a virtual anisotropic ventricle from Diffusion Tensor MRI for individual hearts, and use it to explore mechanisms leading to ventricular fibrillation. The virtual ventricular wall provides an effective tool for exploring, evaluating and visualising processes during the initiation and maintenance of ventricular arrhythmias.
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Affiliation(s)
- Arun V Holden
- Computational Biology Laboratory, Institute of Membrane and Systems Biology, University of Leeds, Leeds LS2 9JT, UK.
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Aslanidi OV, Clayton RH, Lambert JL, Holden AV. Dynamical and cellular electrophysiological mechanisms of ECG changes during ischaemia. J Theor Biol 2005; 237:369-81. [PMID: 15979649 DOI: 10.1016/j.jtbi.2005.04.022] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2005] [Revised: 04/22/2005] [Accepted: 04/25/2005] [Indexed: 11/17/2022]
Abstract
The interpretation of normal and pathological electrocardiographic (ECG) patterns in terms of the underlying cellular and tissue electrophysiology is rudimentary, as the existing theories rely on geometrical aspects. We relate effects of sub-endocardial ischaemia on the ST-segment depression in ECG to patterns of transmural action potential propagation in a one-dimensional virtual ventricular wall. Our computational study exposes two electrophysiological mechanisms of ST depression: dynamic-predominantly positive spatial gradients in the membrane potential during abnormal repolarization of the wall, produced by action potential duration changes in the ischaemic region; and static-a negative spatial gradient of the resting membrane potential between the normal and ischaemic regions. Hyperkalaemia is the major contributor to both these mechanisms at the cellular level. These results complement simulations of the effects of cardiac geometry on the ECG, and dissect spatio-temporal and cellular electrophysiological mechanisms of ST depression seen in sub-endocardial ischaemia.
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Affiliation(s)
- O V Aslanidi
- School of Biomedical Sciences, University of Leeds, Leeds LS2 9JT, UK
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Holden AV. The sensitivity of the heart to static magnetic fields. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2005; 87:289-320. [PMID: 15556667 DOI: 10.1016/j.pbiomolbio.2004.08.015] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Abstract
Static magnetic fields induce flow potentials in arterial flows in and around the heart, that have been detected as distortions in the ECG. The resultant currents flowing through the myocardium could alter the rate or rhythm of the heart. No such changes have been seen in animal experiments, or with humans, in static fields up to 8 T. The possible effects of such currents induced by fields larger than 8 T on cardiac pacemaker rate, and arrhythmogenesis are reviewed, using virtual cardiac tissues-computational models of cardiac electrophysiology. Arrhythmogenesis can be by the initiation of ectopic beats, or by re-entry, whose probability of occurrence is increased by any increase in the electrical heterogeneity, in particular, the action potential duration heterogeneity of the ventricle. Focal ectopic activity would be readily detectable, but since re-entrant arrhythmias are very rare events, even a large increase in their probability of occurrence still leaves them unlikely to be observed. Both of these two arrhythmogenic mechanisms would show a steep sigmoidal, or threshold dependence on induced current intensity, with the threshold for increasing the vulnerability to re-entry less than the threshold for initiating activity. Failure to observe them at fields less than 8 T provides only a lower bound for any threshold for arrhythmogenesis.
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Affiliation(s)
- Arun V Holden
- Computational Biology Laboratory, School of Biomedical Sciences, University of Leeds, Leeds LS2 9JT, UK
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Clayton RH, Holden AV. Propagation of normal beats and re-entry in a computational model of ventricular cardiac tissue with regional differences in action potential shape and duration. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2004; 85:473-99. [PMID: 15142758 DOI: 10.1016/j.pbiomolbio.2003.12.002] [Citation(s) in RCA: 44] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
There is substantial experimental evidence from studies using both intact tissue and isolated single cells to support the existence of different cell types within the ventricular wall of the heart, each possessing different electrical properties. However other studies have failed to find these differences, and instead support the idea that electrical coupling in vivo between regions with different cell types smoothes out differences in action potential shape and duration. In this study we have used a computational model of electrical activation in heterogenous 2D and 3D cardiac tissue to investigate the propagation of both normal beats and arrhythmias. We used the Luo-Rudy dynamic model for guinea pig ventricular cells, with simplified Ca2+ handling and transmural heterogeneity in IKs and Ito. With normal cell-to-cell coupling, a layer of M cells was not necessary for the formation of an upright T wave in the simulated electrocardiogram, and the amplitude and configuration of the T wave was not greatly affected by the thickness and configuration of the M cell layer. Transmural gradients in repolarisation pushed re-entrant waves with an intramural filament towards either the base or the apex of the ventricles, and caused transient break up of re-entry with a transmural filament.
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Affiliation(s)
- Richard H Clayton
- Department of Computer Science, University of Sheffield and School of Biomedical Sciences, Regent Court, 211 Portobello Street, Sheffield S1 4DP, UK.
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Christini DJ, Glass L. Introduction: Mapping and control of complex cardiac arrhythmias. CHAOS (WOODBURY, N.Y.) 2002; 12:732-739. [PMID: 12779601 DOI: 10.1063/1.1504061] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
This paper serves as an introduction to the Focus Issue on mapping and control of complex cardiac arrhythmias. We first introduce basic concepts of cardiac electrophysiology and describe the main clinical methods being used to treat arrhythmia. We then provide a brief summary of the main themes contained in the articles in this Focus Issue. In recent years there have been important advances in the ability to map the spread of excitation in intact hearts and in laboratory settings. This work has been combined with simulations that use increasingly realistic geometry and physiology. Waves of excitation and contraction in the heart do not always propagate with constant velocity but are often subject to instabilities that may lead to fluctuations in velocity and cycle time. Such instabilities are often treated best in the context of simple one- or two-dimensional geometries. An understanding of the mechanisms of propagation and wave stability is leading to the implementation of different stimulation protocols in an effort to modify or eliminate abnormal rhythms. (c) 2002 American Institute of Physics.
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Affiliation(s)
- David J. Christini
- Division of Cardiology, Department of Medicine, Weill Medical College of Cornell University, and Department of Physiology and Biophysics, Weill Graduate School of Medical Sciences of Cornell University, New York, New York 10021Centre for Nonlinear Dynamics, Department of Physiology, McGill University, 3655 Montreal, H3G1Y6, Quebec, Canada
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