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Tyree TJ, Murphy P, Rappel WJ. Annihilation dynamics during spiral defect chaos revealed by particle models. CHAOS (WOODBURY, N.Y.) 2024; 34:053131. [PMID: 38787314 PMCID: PMC11141445 DOI: 10.1063/5.0203319] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2024] [Accepted: 04/30/2024] [Indexed: 05/25/2024]
Abstract
Pair-annihilation events are ubiquitous in a variety of spatially extended systems and are often studied using computationally expensive simulations. Here, we develop an approach in which we simulate the pair-annihilation of spiral wave tips in cardiac models using a computationally efficient particle model. Spiral wave tips are represented as particles with dynamics governed by diffusive behavior and short-ranged attraction. The parameters for diffusion and attraction are obtained by comparing particle motion to the trajectories of spiral wave tips in cardiac models during spiral defect chaos. The particle model reproduces the annihilation rates of the cardiac models and can determine the statistics of spiral wave dynamics, including its mean termination time. We show that increasing the attraction coefficient sharply decreases the mean termination time, making it a possible target for pharmaceutical intervention.
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Affiliation(s)
- Timothy J. Tyree
- Department of Physics, University of California San Diego, San Diego, California 92093, USA
| | - Patrick Murphy
- Department of Mathematics and Statistics, San Jose State University, San Jose, California 95192, USA
| | - Wouter-Jan Rappel
- Department of Physics, University of California San Diego, San Diego, California 92093, USA
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2
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Hussaini S, Lädke SL, Schröder-Schetelig J, Venkatesan V, Quiñonez Uribe RA, Richter C, Majumder R, Luther S. Dissolution of spiral wave's core using cardiac optogenetics. PLoS Comput Biol 2023; 19:e1011660. [PMID: 38060618 PMCID: PMC10729946 DOI: 10.1371/journal.pcbi.1011660] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2023] [Revised: 12/19/2023] [Accepted: 11/04/2023] [Indexed: 12/20/2023] Open
Abstract
Rotating spiral waves in the heart are associated with life-threatening cardiac arrhythmias such as ventricular tachycardia and fibrillation. These arrhythmias are treated by a process called defibrillation, which forces electrical resynchronization of the heart tissue by delivering a single global high-voltage shock directly to the heart. This method leads to immediate termination of spiral waves. However, this may not be the only mechanism underlying successful defibrillation, as certain scenarios have also been reported, where the arrhythmia terminated slowly, over a finite period of time. Here, we investigate the slow termination dynamics of an arrhythmia in optogenetically modified murine cardiac tissue both in silico and ex vivo during global illumination at low light intensities. Optical imaging of an intact mouse heart during a ventricular arrhythmia shows slow termination of the arrhythmia, which is due to action potential prolongation observed during the last rotation of the wave. Our numerical studies show that when the core of a spiral is illuminated, it begins to expand, pushing the spiral arm towards the inexcitable boundary of the domain, leading to termination of the spiral wave. We believe that these fundamental findings lead to a better understanding of arrhythmia dynamics during slow termination, which in turn has implications for the improvement and development of new cardiac defibrillation techniques.
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Affiliation(s)
- Sayedeh Hussaini
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Germany
| | - Sarah L. Lädke
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
| | - Johannes Schröder-Schetelig
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Germany
| | - Vishalini Venkatesan
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
| | - Raúl A. Quiñonez Uribe
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Germany
| | - Claudia Richter
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Germany
- WG Cardiovascular Optogenetics, Lab Animal Science Unit, Leibniz Institute for Primate research, Göttingen, Germany
| | - Rupamanjari Majumder
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Germany
| | - Stefan Luther
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Germany
- Institute for the Dynamics of Complex Systems, Göttingen University, Germany
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3
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Aron M, Lilienkamp T, Luther S, Parlitz U. Optimising low-energy defibrillation in 2D cardiac tissue with a genetic algorithm. FRONTIERS IN NETWORK PHYSIOLOGY 2023; 3:1172454. [PMID: 37555132 PMCID: PMC10406519 DOI: 10.3389/fnetp.2023.1172454] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/23/2023] [Accepted: 06/28/2023] [Indexed: 08/10/2023]
Abstract
Sequences of low-energy electrical pulses can effectively terminate ventricular fibrillation (VF) and avoid the side effects of conventional high-energy electrical defibrillation shocks, including tissue damage, traumatic pain, and worsening of prognosis. However, the systematic optimisation of sequences of low-energy pulses remains a major challenge. Using 2D simulations of homogeneous cardiac tissue and a genetic algorithm, we demonstrate the optimisation of sequences with non-uniform pulse energies and time intervals between consecutive pulses for efficient VF termination. We further identify model-dependent reductions of total pacing energy ranging from ∼4% to ∼80% compared to reference adaptive-deceleration pacing (ADP) protocols of equal success rate (100%).
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Affiliation(s)
- Marcel Aron
- Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Göttingen, Germany
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute for the Dynamics of Complex Systems, Georg-August-Universität Göttingen, Göttingen, Germany
- German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Göttingen, Germany
| | - Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Computational Physics for Life Science, Nuremberg Institute of Technology Georg Simon Ohm, Nuremberg, Germany
| | - Stefan Luther
- Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Göttingen, Germany
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute for the Dynamics of Complex Systems, Georg-August-Universität Göttingen, Göttingen, Germany
- German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Institute for the Dynamics of Complex Systems, Georg-August-Universität Göttingen, Göttingen, Germany
- German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Göttingen, Germany
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4
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Pattanayak D, Mishra A, Bairagi N, Dana SK. Multimodal distribution of transient time of predator extinction in a three-species food chain. CHAOS (WOODBURY, N.Y.) 2023; 33:043122. [PMID: 37097935 DOI: 10.1063/5.0136372] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2022] [Accepted: 03/20/2023] [Indexed: 06/19/2023]
Abstract
The transient dynamics capture the time history in the behavior of a system before reaching an attractor. This paper deals with the statistics of transient dynamics in a classic tri-trophic food chain with bistability. The species of the food chain model either coexist or undergo a partial extinction with predator death after a transient time depending upon the initial population density. The distribution of transient time to predator extinction shows interesting patterns of inhomogeneity and anisotropy in the basin of the predator-free state. More precisely, the distribution shows a multimodal character when the initial points are located near a basin boundary and a unimodal character when chosen from a location far away from the boundary. The distribution is also anisotropic because the number of modes depends on the direction of the local of initial points. We define two new metrics, viz., homogeneity index and local isotropic index, to characterize the distinctive features of the distribution. We explain the origin of such multimodal distributions and try to present their ecological implications.
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Affiliation(s)
- Debarghya Pattanayak
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Arindam Mishra
- Department of Physics, National University of Singapore, Singapore 117551
| | - Nandadulal Bairagi
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
- Department of Mathematics, National Institute of Technology, Durgapur 713209, India
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Abstract
The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.
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Affiliation(s)
- Wouter-Jan Rappel
- Department of Physics, University of California San Diego, La Jolla, CA 92037
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6
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Liu C, Dong JQ, Chen QJ, Huang ZG, Huang L, Zhou HJ, Lai YC. Controlled generation of self-sustained oscillations in complex artificial neural networks. CHAOS (WOODBURY, N.Y.) 2021; 31:113127. [PMID: 34881621 DOI: 10.1063/5.0069333] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2021] [Accepted: 10/20/2021] [Indexed: 06/13/2023]
Abstract
Spatially distinct, self-sustained oscillations in artificial neural networks are fundamental to information encoding, storage, and processing in these systems. Here, we develop a method to induce a large variety of self-sustained oscillatory patterns in artificial neural networks and a controlling strategy to switch between different patterns. The basic principle is that, given a complex network, one can find a set of nodes-the minimum feedback vertex set (mFVS), whose removal or inhibition will result in a tree-like network without any loop structure. Reintroducing a few or even a single mFVS node into the tree-like artificial neural network can recover one or a few of the loops and lead to self-sustained oscillation patterns based on these loops. Reactivating various mFVS nodes or their combinations can then generate a large number of distinct neuronal firing patterns with a broad distribution of the oscillation period. When the system is near a critical state, chaos can arise, providing a natural platform for pattern switching with remarkable flexibility. With mFVS guided control, complex networks of artificial neurons can thus be exploited as potential prototypes for local, analog type of processing paradigms.
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Affiliation(s)
- Chang Liu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Jia-Qi Dong
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Qing-Jian Chen
- Key Laboratory of Biomedical Information Engineering of Ministry of Education, Key Laboratory of Neuro-informatics & Rehabilitation Engineering of Ministry of Civil Affairs, and Institute of Health and Rehabilitation Science, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China
| | - Zi-Gang Huang
- Key Laboratory of Biomedical Information Engineering of Ministry of Education, Key Laboratory of Neuro-informatics & Rehabilitation Engineering of Ministry of Civil Affairs, and Institute of Health and Rehabilitation Science, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China
| | - Liang Huang
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Hai-Jun Zhou
- CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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7
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Ray A, Pal A, Ghosh D, Dana SK, Hens C. Mitigating long transient time in deterministic systems by resetting. CHAOS (WOODBURY, N.Y.) 2021; 31:011103. [PMID: 33754784 DOI: 10.1063/5.0038374] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2020] [Accepted: 12/24/2020] [Indexed: 06/12/2023]
Abstract
How long does a trajectory take to reach a stable equilibrium point in the basin of attraction of a dynamical system? This is a question of quite general interest and has stimulated a lot of activities in dynamical and stochastic systems where the metric of this estimation is often known as the transient or first passage time. In nonlinear systems, one often experiences long transients due to their underlying dynamics. We apply resetting or restart, an emerging concept in statistical physics and stochastic process, to mitigate the detrimental effects of prolonged transients in deterministic dynamical systems. We show that resetting the intrinsic dynamics intermittently to a spatial control line that passes through the equilibrium point can dramatically expedite its completion, resulting in a huge reduction in mean transient time and fluctuations around it. Moreover, our study reveals the emergence of an optimal restart time that globally minimizes the mean transient time. We corroborate the results with detailed numerical studies on two canonical setups in deterministic dynamical systems, namely, the Stuart-Landau oscillator and the Lorenz system. The key features-expedition of transient time-are found to be very generic under different resetting strategies. Our analysis opens up a door to control the mean and fluctuations in transient time by unifying the original dynamics with an external stochastic or periodic timer and poses open questions on the optimal way to harness transients in dynamical systems.
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Affiliation(s)
- Arnob Ray
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Arnab Pal
- School of Chemistry, Faculty of Exact Sciences and The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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8
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Lilienkamp T, Parlitz U. Susceptibility of transient chimera states. Phys Rev E 2020; 102:032219. [PMID: 33075925 DOI: 10.1103/physreve.102.032219] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Accepted: 08/26/2020] [Indexed: 11/07/2022]
Abstract
Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a 'terminal transient phase" (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.
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Affiliation(s)
- Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.,German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.,German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany.,Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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9
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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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10
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Medeiros ES, Medrano-T RO, Caldas IL, Tél T, Feudel U. State-dependent vulnerability of synchronization. Phys Rev E 2019; 100:052201. [PMID: 31869887 DOI: 10.1103/physreve.100.052201] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2019] [Indexed: 11/07/2022]
Abstract
A state-dependent vulnerability of synchronization is shown to exist in a complex network composed of numerically simulated electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the current state of its trajectory. We address such state dependence by systematically perturbing the synchronized system at states equally distributed along its trajectory. We find the states at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive states by defining a safety index between them. Finally, we demonstrate that the observed vulnerability is due to the existence of an unstable chaotic set in the system's state space.
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Affiliation(s)
- Everton S Medeiros
- Institute of Physics, University of São Paulo, Rua do Matão, Travessa R 187, 05508-090, São Paulo, Brazil.,Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Straße 9-11, Box 2503, 26111 Oldenburg, Germany
| | - Rene O Medrano-T
- Department of Physics, Federal University of São Paulo, Rua São Nicolau, 210, 09913-030, São Paulo, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, Rua do Matão, Travessa R 187, 05508-090, São Paulo, Brazil
| | - Tamás Tél
- Institute for Theoretical Physics, Eötvös Loránd University, Pázmány Péter Sétány 1/A, H-1117 Budapest, Hungary.,MTA-ELTE Theoretical Physics Research Group, Pázmány Péter Sétány 1/A, H-1117 Budapest, Hungary
| | - Ulrike Feudel
- Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Straße 9-11, Box 2503, 26111 Oldenburg, 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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12
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Lilienkamp T, Parlitz U. Scaling behavior of the terminal transient phase. Phys Rev E 2018; 98:022215. [PMID: 30253596 DOI: 10.1103/physreve.98.022215] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2018] [Indexed: 06/08/2023]
Abstract
Transient chaos can emerge in a variety of diverse systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. The end of the chaotic episode can either be desired or not, depending on the specific system and application. In both cases, however, a prediction of the end of the chaotic dynamics is required. Despite the general challenges of reliably predicting chaotic dynamics for a long time period, the recent observation of a "terminal transient phase" of chaotic transients provides new insights into the transition from chaos to the subsequent (nonchaotic) regime. In spatially extended systems and also low-dimensional maps it was shown that the structure of the state space changes already a significant amount of time before the actual end of the chaotic dynamics. In this way, the terminal transient phase provides the conceptual foundation for a possible prediction of the upcoming end of the chaotic episode a significant amount of time in advance. In this study, we strengthen the general validity of the terminal transient phase by verifying its existence in another spatially extended model (Gray-Scott model) and the Hénon map, where in the latter case the underlying mechanisms can be understood in an intuitive way. Furthermore, we show that the temporal length of the terminal transient phase remains approximately constant, when changing the system size (Gray-Scott) or parameters (Hénon map) of the investigated models, although the average lifetime of the observed chaotic transients sensitively depends on these variations. Since the timescale of the terminal transient phase is in this sense relatively robust, this insight might be essential for possible applications, where the ratio between the length of the terminal transient phase and the relevant timescale of the dynamics may probably be crucial when a reasonable prediction (thus a sufficient time before) the end of the chaotic episode is required.
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Affiliation(s)
- Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
- Institute for Nonlinear Dynamics, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
- Institute for Nonlinear Dynamics, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
- DZHK (German Centre for Cardiovascular Research), partner site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany
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