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Phan NA, Wang Y. Mixed-mode oscillations in a three-timescale coupled Morris-Lecar system. CHAOS (WOODBURY, N.Y.) 2024; 34:053119. [PMID: 38717416 PMCID: PMC11087137 DOI: 10.1063/5.0181308] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Accepted: 04/19/2024] [Indexed: 05/12/2024]
Abstract
Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.
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Affiliation(s)
- Ngoc Anh Phan
- Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA
| | - Yangyang Wang
- Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453, USA
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2
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Barrio R, Jover-Galtier JA, Martínez MA, Pérez L, Serrano S. Mathematical birth of Early Afterdepolarizations in a cardiomyocyte model. Math Biosci 2023; 366:109088. [PMID: 37863283 DOI: 10.1016/j.mbs.2023.109088] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Revised: 09/20/2023] [Accepted: 10/17/2023] [Indexed: 10/22/2023]
Abstract
Early Afterdepolarizations (EADs) are abnormal behaviors that can lead to cardiac failure and even cardiac death. In this paper we investigate the occurrence and development of these phenomena in a reduced Luo-Rudy cardiac model. Through a comprehensive dynamical analysis, we map out the distinct patterns observed in the parametric plane, differentiating between normal beats without EADs and pathological beats with EADs. By examining the bifurcation structure of the model, we elucidate the dynamical elements associated with these patterns and their transitions. Using a fast-slow analysis, we explore the emergence and evolution of EADs in the model. Notably, our approach combines the two commonly used fast-slow approaches (1-slow-2-fast and 2-slow-1-fast), and we show how both approaches together provide a more complete understanding of this phenomenon.
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Affiliation(s)
- R Barrio
- IUMA, CoDy and Dpto. Matemática Aplicada, Universidad de Zaragoza, E-50009 Zaragoza, Spain.
| | - J A Jover-Galtier
- IUMA, CoDy and Dpto. Matemática Aplicada, Universidad de Zaragoza, E-50009 Zaragoza, Spain.
| | - M A Martínez
- IUMA, CoDy and Dpto. Matemática Aplicada, Universidad de Zaragoza, E-50009 Zaragoza, Spain.
| | - L Pérez
- Dpto. Matemáticas, University of Oviedo, Oviedo E-33007, Spain.
| | - S Serrano
- IUMA, CoDy and Dpto. Matemática Aplicada, Universidad de Zaragoza, E-50009 Zaragoza, Spain.
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Barrio R, Martínez MÁ, Serrano S, Pueyo E. Dynamical mechanism for generation of arrhythmogenic early afterdepolarizations in cardiac myocytes: Insights from in silico electrophysiological models. Phys Rev E 2022; 106:024402. [PMID: 36109976 DOI: 10.1103/physreve.106.024402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2021] [Accepted: 07/10/2022] [Indexed: 06/15/2023]
Abstract
We analyze the dynamical mechanisms underlying the formation of arrhythmogenic early afterdepolarizations (EADs) in two mathematical models of cardiac cellular electrophysiology: the Sato et al. biophysically detailed model of a rabbit ventricular myocyte of dimension 27 and a reduced version of the Luo-Rudy mammalian myocyte model of dimension 3. Based on a comparison of the two models, with detailed bifurcation analysis using spike-counting techniques and continuation methods in the simple model and numerical explorations in the complex model, we locate the point where the first EAD originates in an unstable branch of periodic orbits. These results serve as a basis to propose a conjectured scheme involving a hysteresis mechanism with the creation of alternans and EADs in the unstable branch. This theoretical scheme fits well with electrophysiological experimental data on EAD generation and hysteresis phenomena. Our findings open the door to the development of novel methods for pro-arrhythmia risk prediction related to EAD generation without actual induction of EADs.
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Affiliation(s)
- Roberto Barrio
- Department of Applied Mathematics and IUMA, Computational Dynamics group, University of Zaragoza, E-50009 Zaragoza, Spain
| | - M Ángeles Martínez
- Department of Applied Mathematics and IUMA, Computational Dynamics group, University of Zaragoza, E-50009 Zaragoza, Spain
| | - Sergio Serrano
- Department of Applied Mathematics and IUMA, Computational Dynamics group, University of Zaragoza, E-50009 Zaragoza, Spain
| | - Esther Pueyo
- I3A, University of Zaragoza, IIS Aragón and CIBER-BBN, E-50018 Zaragoza, Spain
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Tsumoto K, Kurata Y. Bifurcations and Proarrhythmic Behaviors in Cardiac Electrical Excitations. Biomolecules 2022; 12:459. [PMID: 35327651 PMCID: PMC8946197 DOI: 10.3390/biom12030459] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2022] [Revised: 03/09/2022] [Accepted: 03/14/2022] [Indexed: 12/23/2022] Open
Abstract
The heart is a hierarchical dynamic system consisting of molecules, cells, and tissues, and acts as a pump for blood circulation. The pumping function depends critically on the preceding electrical activity, and disturbances in the pattern of excitation propagation lead to cardiac arrhythmia and pump failure. Excitation phenomena in cardiomyocytes have been modeled as a nonlinear dynamical system. Because of the nonlinearity of excitation phenomena, the system dynamics could be complex, and various analyses have been performed to understand the complex dynamics. Understanding the mechanisms underlying proarrhythmic responses in the heart is crucial for developing new ways to prevent and control cardiac arrhythmias and resulting contractile dysfunction. When the heart changes to a pathological state over time, the action potential (AP) in cardiomyocytes may also change to a different state in shape and duration, often undergoing a qualitative change in behavior. Such a dynamic change is called bifurcation. In this review, we first summarize the contribution of ion channels and transporters to AP formation and our knowledge of ion-transport molecules, then briefly describe bifurcation theory for nonlinear dynamical systems, and finally detail its recent progress, focusing on the research that attempts to understand the developing mechanisms of abnormal excitations in cardiomyocytes from the perspective of bifurcation phenomena.
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Affiliation(s)
| | - Yasutaka Kurata
- Department of Physiology II, Kanazawa Medical University, Uchinada 920-0293, Japan;
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Barrio R, Martínez MÁ, Pueyo E, Serrano S. Dynamical analysis of early afterdepolarization patterns in a biophysically detailed cardiac model. CHAOS (WOODBURY, N.Y.) 2021; 31:073137. [PMID: 34340346 DOI: 10.1063/5.0055965] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2021] [Accepted: 06/28/2021] [Indexed: 06/13/2023]
Abstract
Arrhythmogenic early afterdepolarizations (EADs) are investigated in a biophysically detailed mathematical model of a rabbit ventricular myocyte, providing their location in the parameter phase space and describing their dynamical mechanisms. Simulations using the Sato model, defined by 27 state variables and 177 parameters, are conducted to generate electrical action potentials (APs) for different values of the pacing cycle length and other parameters related to sodium and calcium concentrations. A detailed study of the different AP patterns with or without EADs is carried out, showing the presence of a high variety of temporal AP configurations with chaotic and quasiperiodic behaviors. Regions of bistability are identified and, importantly, linked to transitions between different behaviors. Using sweeping techniques, one-, two-, and three-parameter phase spaces are provided, allowing ascertainment of the role of the selected parameters as well as location of the transition regions. A Devil's staircase, with symbolic sequence analysis, is proposed to describe transitions in the ratio between the number of voltage (EAD and AP) peaks and the number of APs. To conclude, the obtained results are linked to recent studies for low-dimensional models and a conjecture is made for the internal dynamical structure of the transition region from non-EAD to EAD behavior using fold and cusp bifurcations and maximal canards.
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Affiliation(s)
- Roberto Barrio
- Department of Applied Mathematics and IUMA, Computational Dynamics Group, University of Zaragoza, E-50009 Zaragoza, Spain
| | - M Ángeles Martínez
- Department of Applied Mathematics and IUMA, Computational Dynamics Group, University of Zaragoza, E-50009 Zaragoza, Spain
| | - Esther Pueyo
- I3A, University of Zaragoza, IIS Aragón and CIBER-BBN, E-50018 Zaragoza, Spain
| | - Sergio Serrano
- Department of Applied Mathematics and IUMA, Computational Dynamics Group, University of Zaragoza, E-50009 Zaragoza, Spain
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Diekman CO, Wei N. Circadian Rhythms of Early Afterdepolarizations and Ventricular Arrhythmias in a Cardiomyocyte Model. Biophys J 2020; 120:319-333. [PMID: 33285114 DOI: 10.1016/j.bpj.2020.11.2264] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2019] [Revised: 02/25/2020] [Accepted: 11/10/2020] [Indexed: 11/30/2022] Open
Abstract
Sudden cardiac arrest is a malfunction of the heart's electrical system, typically caused by ventricular arrhythmias, that can lead to sudden cardiac death (SCD) within minutes. Epidemiological studies have shown that SCD and ventricular arrhythmias are more likely to occur in the morning than in the evening, and laboratory studies indicate that these daily rhythms in adverse cardiovascular events are at least partially under the control of the endogenous circadian timekeeping system. However, the biophysical mechanisms linking molecular circadian clocks to cardiac arrhythmogenesis are not fully understood. Recent experiments have shown that L-type calcium channels exhibit circadian rhythms in both expression and function in guinea pig ventricular cardiomyocytes. We developed an electrophysiological model of these cells to simulate the effect of circadian variation in L-type calcium conductance. In our simulations, we found that there is a circadian pattern in the occurrence of early afterdepolarizations (EADs), which are abnormal depolarizations during the repolarization phase of a cardiac action potential that can trigger fatal ventricular arrhythmias. Specifically, the model produces EADs in the morning, but not at other times of day. We show that the model exhibits a codimension-2 Takens-Bogdanov bifurcation that serves as an organizing center for different types of EAD dynamics. We also simulated a two-dimensional spatial version of this model across a circadian cycle. We found that there is a circadian pattern in the breakup of spiral waves, which represents ventricular fibrillation in cardiac tissue. Specifically, the model produces spiral wave breakup in the morning, but not in the evening. Our computational study is the first, to our knowledge, to propose a link between circadian rhythms and EAD formation and suggests that the efficacy of drugs targeting EAD-mediated arrhythmias may depend on the time of day that they are administered.
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Affiliation(s)
- Casey O Diekman
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey; EPSRC Centre for Predictive Modelling in Healthcare, Living Systems Institute, University of Exeter, Exeter, United Kingdom.
| | - Ning Wei
- Department of Mathematics, Purdue University, West Lafayette, Indiana
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Kimrey J, Vo T, Bertram R. Canard analysis reveals why a large Ca2+ window current promotes early afterdepolarizations in cardiac myocytes. PLoS Comput Biol 2020; 16:e1008341. [PMID: 33147207 PMCID: PMC7641359 DOI: 10.1371/journal.pcbi.1008341] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2020] [Accepted: 09/14/2020] [Indexed: 12/19/2022] Open
Abstract
The pumping of blood through the heart is due to a wave of muscle contractions that are in turn due to a wave of electrical activity initiated at the sinoatrial node. At the cellular level, this wave of electrical activity corresponds to the sequential excitation of electrically coupled cardiac cells. Under some conditions, the normally-long action potentials of cardiac cells are extended even further by small oscillations called early afterdepolarizations (EADs) that can occur either during the plateau phase or repolarizing phase of the action potential. Hence, cellular EADs have been implicated as a driver of potentially lethal cardiac arrhythmias. One of the major determinants of cellular EAD production and repolarization failure is the size of the overlap region between Ca2+ channel activation and inactivation, called the window region. In this article, we interpret the role of the window region in terms of the fast-slow structure of a low-dimensional model for ventricular action potential generation. We demonstrate that the effects of manipulation of the size of the window region can be understood from the point of view of canard theory. We use canard theory to explain why enlarging the size of the window region elicits EADs and why shrinking the window region can eliminate them. We also use the canard mechanism to explain why some manipulations in the size of the window region have a stronger influence on cellular electrical behavior than others. This dynamical viewpoint gives predictive power that is beyond that of the biophysical explanation alone while also uncovering a common mechanism for phenomena observed in experiments on both atrial and ventricular cardiac cells.
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Affiliation(s)
- Joshua Kimrey
- Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America
| | - Theodore Vo
- School of Mathematics, Monash University, Clayton, Victoria, Australia
| | - Richard Bertram
- Department of Mathematics, and Programs in Neuroscience and Biophysics, Florida State University, Tallahassee, Florida, United States of America
- * E-mail:
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Bifurcations and Slow-Fast Analysis in a Cardiac Cell Model for Investigation of Early Afterdepolarizations. MATHEMATICS 2020. [DOI: 10.3390/math8060880] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
In this study, we teased out the dynamical mechanisms underlying the generation of arrhythmogenic early afterdepolarizations (EADs) in a three-variable model of a mammalian ventricular cell. Based on recently published studies, we consider a 1-fast, 2-slow variable decomposition of the system describing the cellular action potential. We use sweeping techniques, such as the spike-counting method, and bifurcation and continuation methods to identify parametric regions with EADs. We show the existence of isolas of periodic orbits organizing the different EAD patterns and we provide a preliminary classification of our fast–slow decomposition according to the involved dynamical phenomena. This investigation represents a basis for further studies into the organization of EAD patterns in the parameter space and the involved bifurcations.
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Jæger KH, Wall S, Tveito A. Detecting undetectables: Can conductances of action potential models be changed without appreciable change in the transmembrane potential? CHAOS (WOODBURY, N.Y.) 2019; 29:073102. [PMID: 31370420 DOI: 10.1063/1.5087629] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2019] [Accepted: 06/12/2019] [Indexed: 05/23/2023]
Abstract
Mathematical models describing the dynamics of the cardiac action potential are of great value for understanding how changes to the system can disrupt the normal electrical activity of cells and tissue in the heart. However, to represent specific data, these models must be parameterized, and adjustment of the maximum conductances of the individual contributing ionic currents is a commonly used method. Here, we present a method for investigating the uniqueness of such resulting parameterizations. Our key question is: Can the maximum conductances of a model be changed without giving any appreciable changes in the action potential? If so, the model parameters are not unique and this poses a major problem in using the models to identify changes in parameters from data, for instance, to evaluate potential drug effects. We propose a method for evaluating this uniqueness, founded on the singular value decomposition of a matrix consisting of the individual ionic currents. Small singular values of this matrix signify lack of parameter uniqueness and we show that the conclusion from linear analysis of the matrix carries over to provide insight into the uniqueness of the parameters in the nonlinear case. Using numerical experiments, we quantify the identifiability of the maximum conductances of well-known models of the cardiac action potential. Furthermore, we show how the identifiability depends on the time step used in the observation of the currents, how the application of drugs may change identifiability, and, finally, how the stimulation protocol can be used to improve the identifiability of a model.
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Affiliation(s)
| | - Samuel Wall
- Simula Research Laboratory, 1325 Lysaker, Norway
| | - Aslak Tveito
- Simula Research Laboratory, 1325 Lysaker, Norway
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Vo T, Bertram R. Why pacing frequency affects the production of early afterdepolarizations in cardiomyocytes: An explanation revealed by slow-fast analysis of a minimal model. Phys Rev E 2019; 99:052205. [PMID: 31212514 DOI: 10.1103/physreve.99.052205] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2019] [Indexed: 12/28/2022]
Abstract
Early afterdepolarizations (EADs) are pathological voltage oscillations in cardiomyocytes that have been observed in response to a number of pharmacological agents and disease conditions. Phase-2 EADs consist of small voltage fluctuations during the plateau of an action potential, typically under conditions in which the action potential is elongated. Although a single-cell behavior, EADs can lead to tissue-level arrhythmias. Much is currently known about the biophysical mechanisms (i.e., the roles of ion channels and intracellular Ca^{2+} stores) for the various forms of EADs, due partially to the development and analysis of mathematical models. This includes the application of slow-fast analysis, which takes advantage of timescale separation inherent in the system to simplify its analysis. We take this further, using a minimal three-dimensional model to demonstrate that phase-2 EADs are canards formed in the neighborhood of a folded node singularity. This allows us to predict the number of EADs that can be produced for a given parameter set, and provides guidance on parameter changes that facilitate or inhibit EAD production. With this approach, we demonstrate why periodic stimulation, as occurs in intact heart, preferentially facilitates EAD production when applied at low frequencies. We also explain the origin of complex alternan dynamics that can occur with intermediate-frequency stimulation, in which varying numbers of EADs are produced with each pulse. These revelations fall out naturally from an understanding of folded node singularities, but are difficult to glean from knowledge of the biophysical mechanism for EADs alone. Therefore, understanding the canard mechanism is a useful complement to understanding of the biophysical mechanism that has been developed over years of experimental and computational investigations.
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Affiliation(s)
- Theodore Vo
- Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
| | - Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, USA
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