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Ishii H, Kori H. Diffusive coupling facilitates and impedes noise-induced escape in interacting bistable elements. Sci Rep 2024; 14:11019. [PMID: 38744886 PMCID: PMC11094171 DOI: 10.1038/s41598-024-61058-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2024] [Accepted: 04/30/2024] [Indexed: 05/16/2024] Open
Abstract
Diverse complex systems often undergo sudden changes in their states, such as epileptic seizures, climate changes, and social uprisings. Such behavior has been modeled by noise-induced escape of bistable elements, which is the escape from an attracting state driven by a fluctuation in the system's state. We consider a system of interacting bistable elements and investigate the effect of diffusive coupling among elements on the process of noise-induced escape. We focus on the influence of the coupling strength over the escape time, which is the time it takes for noise-induced escape to occur. We performed numerical simulations and observed that weak coupling reduced the mean escape time, whereas strong coupling impeded escape. We argue that, although diffusive coupling both facilitates and impedes escape, the facilitating effect is dominant when coupling is weak. For weak coupling cases, we develop an approximate theory that can predict the mean and variance of escape times. In contrast, strong coupling reduces the effective noise intensity to impede escape. Our results suggest that diffusive coupling among multistable elements contributes to regulating the rate of transitions among attracting states.
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Affiliation(s)
- Hidemasa Ishii
- Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Chiba, 277-8561, Japan.
| | - Hiroshi Kori
- Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Chiba, 277-8561, Japan
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Ashwin P, Creaser J, Tsaneva-Atanasova K. Quasipotentials for coupled escape problems and the gate-height bifurcation. Phys Rev E 2023; 107:014213. [PMID: 36797857 DOI: 10.1103/physreve.107.014213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Accepted: 12/21/2022] [Indexed: 06/18/2023]
Abstract
The escape statistics of a gradient dynamical system perturbed by noise can be estimated using properties of the associated potential landscape. More generally, the Freidlin and Wentzell quasipotential (QP) can be used for similar purposes, but computing this is nontrivial and it is only defined relative to some starting point. In this paper we focus on computing quasipotentials for coupled bistable units, numerically solving a Hamilton- Jacobi-Bellman type problem. We analyze noise induced transitions using the QP in cases where there is no potential for the coupled system. Gates (points on the boundary of basin of attraction that have minimal QP relative to that attractor) are used to understand the escape rates from the basin, but these gates can undergo a global change as coupling strength is changed. Such a global gate-height bifurcation is a generic qualitative transition in the escape properties of parametrized nongradient dynamical systems for small noise.
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Affiliation(s)
- Peter Ashwin
- Department of Mathematics and Statistics, University of Exeter, Exeter EX4 4QF, United Kingdom
| | - Jennifer Creaser
- Department of Mathematics and Statistics, University of Exeter, Exeter EX4 4QF, United Kingdom
| | - Krasimira Tsaneva-Atanasova
- Department of Mathematics and Statistics, and EPSRC Hub for Quantitative Modelling in Healthcare, University of Exeter, Exeter EX4 4QJ, United Kingdom and Institute for Advanced Study, Technical University of Munich, Lichtenbergstrasse 2 a, D-85748 Garching, Germany
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Salman M, Bick C, Krischer K. Bifurcations of clusters and collective oscillations in networks of bistable units. CHAOS (WOODBURY, N.Y.) 2021; 31:113140. [PMID: 34881589 DOI: 10.1063/5.0067989] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2021] [Accepted: 10/31/2021] [Indexed: 06/13/2023]
Abstract
We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a cluster subspace, where the electrodes take at most three distinct states. We systematically analyze the dynamics and bifurcations of the model equations: We consider the dynamics on cluster subspaces of successively increasing dimension and analyze the bifurcations occurring therein. Most importantly, the system exhibits an equivariant transcritical bifurcation of limit cycles. From this bifurcation, several limit cycles branch, one of which is stable for arbitrarily many bistable units.
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Affiliation(s)
- Munir Salman
- Physics Department, Technical University of Munich, James-Franck-Str. 1, 85748 Garching, Germany
| | - Christian Bick
- Institute for Advanced Study, Technical University of Munich, Lichtenbergstr. 2, 85748 Garching, Germany
| | - Katharina Krischer
- Physics Department, Technical University of Munich, James-Franck-Str. 1, 85748 Garching, Germany
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Junges L, Woldman W, Benjamin OJ, Terry JR. Epilepsy surgery: Evaluating robustness using dynamic network models. CHAOS (WOODBURY, N.Y.) 2020; 30:113106. [PMID: 33261362 DOI: 10.1063/5.0022171] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2020] [Accepted: 10/08/2020] [Indexed: 06/12/2023]
Abstract
Epilepsy is one of the most common neurological conditions affecting over 65 million people worldwide. Over one third of people with epilepsy are considered refractory: they do not respond to drug treatments. For this significant cohort of people, surgery is a potentially transformative treatment. However, only a small minority of people with refractory epilepsy are considered suitable for surgery, and long-term seizure freedom is only achieved in half the cases. Recently, several computational approaches have been proposed to support presurgical planning. Typically, these approaches use a dynamic network model to explore the potential impact of surgical resection in silico. The network component of the model is informed by clinical imaging data and is considered static thereafter. This assumption critically overlooks the plasticity of the brain and, therefore, how continued evolution of the brain network post-surgery may impact upon the success of a resection in the longer term. In this work, we use a simplified dynamic network model, which describes transitions to seizures, to systematically explore how the network structure influences seizure propensity, both before and after virtual resections. We illustrate key results in small networks, before extending our findings to larger networks. We demonstrate how the evolution of brain networks post resection can result in a return to increased seizure propensity. Our results effectively determine the robustness of a given resection to possible network reconfigurations and so provide a potential strategy for optimizing long-term seizure freedom.
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Affiliation(s)
- Leandro Junges
- Centre for Systems Modelling and Quantitative Biomedicine, University of Birmingham, Birmingham B15 2TT, United Kingdom
| | - Wessel Woldman
- Centre for Systems Modelling and Quantitative Biomedicine, University of Birmingham, Birmingham B15 2TT, United Kingdom
| | - Oscar J Benjamin
- Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, United Kingdom
| | - John R Terry
- Centre for Systems Modelling and Quantitative Biomedicine, University of Birmingham, Birmingham B15 2TT, United Kingdom
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Creaser J, Lin C, Ridler T, Brown JT, D’Souza W, Seneviratne U, Cook M, Terry JR, Tsaneva-Atanasova K. Domino-like transient dynamics at seizure onset in epilepsy. PLoS Comput Biol 2020; 16:e1008206. [PMID: 32986695 PMCID: PMC7544071 DOI: 10.1371/journal.pcbi.1008206] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2019] [Revised: 10/08/2020] [Accepted: 07/29/2020] [Indexed: 12/20/2022] Open
Abstract
The International League Against Epilepsy (ILAE) groups seizures into "focal", "generalized" and "unknown" based on whether the seizure onset is confined to a brain region in one hemisphere, arises in several brain region simultaneously, or is not known, respectively. This separation fails to account for the rich diversity of clinically and experimentally observed spatiotemporal patterns of seizure onset and even less so for the properties of the brain networks generating them. We consider three different patterns of domino-like seizure onset in Idiopathic Generalized Epilepsy (IGE) and present a novel approach to classification of seizures. To understand how these patterns are generated on networks requires understanding of the relationship between intrinsic node dynamics and coupling between nodes in the presence of noise, which currently is unknown. We investigate this interplay here in the framework of domino-like recruitment across a network. In particular, we use a phenomenological model of seizure onset with heterogeneous coupling and node properties, and show that in combination they generate a range of domino-like onset patterns observed in the IGE seizures. We further explore the individual contribution of heterogeneous node dynamics and coupling by interpreting in-vitro experimental data in which the speed of onset can be chemically modulated. This work contributes to a better understanding of possible drivers for the spatiotemporal patterns observed at seizure onset and may ultimately contribute to a more personalized approach to classification of seizure types in clinical practice.
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Affiliation(s)
- Jennifer Creaser
- Department of Mathematics, University of Exeter, Exeter, EX4 4QF, UK
- EPSRC Centre for Predictive modeling in Healthcare, University of Exeter, Exeter, EX4 4QJ, UK
| | - Congping Lin
- Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
- Hubei Key Lab of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
| | - Thomas Ridler
- Institute of Biomedical and Clinical Sciences, College of Medicine and Health, University of Exeter, EX4 4PS, UK
| | - Jonathan T. Brown
- Institute of Biomedical and Clinical Sciences, College of Medicine and Health, University of Exeter, EX4 4PS, UK
| | - Wendyl D’Souza
- Department of Medicine, St. Vincent’s Hospital, University of Melbourne, Melbourne, VIC 3065, Australia
| | - Udaya Seneviratne
- Department of Medicine, St. Vincent’s Hospital, University of Melbourne, Melbourne, VIC 3065, Australia
- Department of Neuroscience, Monash Medical Centre, Melbourne, VIC 3168, Australia
| | - Mark Cook
- Department of Medicine, St. Vincent’s Hospital, University of Melbourne, Melbourne, VIC 3065, Australia
- Graeme Clark Institute, University of Melbourne, Parkville, VIC 3010, Australia
| | - John R. Terry
- EPSRC Centre for Predictive modeling in Healthcare, University of Exeter, Exeter, EX4 4QJ, UK
- Centre for Systems Modelling and Quantitative Biomedicine, University of Birmingham, Birmingham, B15 2TT, UK
| | - Krasimira Tsaneva-Atanasova
- Department of Mathematics, University of Exeter, Exeter, EX4 4QF, UK
- EPSRC Centre for Predictive modeling in Healthcare, University of Exeter, Exeter, EX4 4QJ, UK
- Living System Institute, University of Exeter, Exeter, EX4 4QJ, UK
- Institute for Advanced Study, Technical University of Munich, Lichtenbergstrasse 2a, D-85748 Garching, Germany
- Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, 105 Acad. G. Bonchev Str, 1113 Sofia, Bulgaria
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Bonciolini G, Ebi D, Boujo E, Noiray N. Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation. ROYAL SOCIETY OPEN SCIENCE 2018; 5:172078. [PMID: 29657803 PMCID: PMC5882727 DOI: 10.1098/rsos.172078] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/04/2017] [Accepted: 02/19/2018] [Indexed: 06/08/2023]
Abstract
Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.
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Affiliation(s)
- Giacomo Bonciolini
- CAPS Laboratory, MAVT department ETH Zürich, Sonneggstrasse 3, 8092, Zurich, Switzerland
| | - Dominik Ebi
- Laboratory for Thermal Processes and Combustion, Paul Scherrer Institute, 5232 Villigen, Switzerland
| | - Edouard Boujo
- CAPS Laboratory, MAVT department ETH Zürich, Sonneggstrasse 3, 8092, Zurich, Switzerland
| | - Nicolas Noiray
- CAPS Laboratory, MAVT department ETH Zürich, Sonneggstrasse 3, 8092, Zurich, Switzerland
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