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Bolotov MI, Munyayev VO, Smirnov LA, Osipov GV, Belykh I. Breathing and switching cyclops states in Kuramoto networks with higher-mode coupling. Phys Rev E 2024; 109:054202. [PMID: 38907462 DOI: 10.1103/physreve.109.054202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2024] [Accepted: 04/10/2024] [Indexed: 06/24/2024]
Abstract
Cyclops states are intriguing cluster patterns observed in oscillator networks, including neuronal ensembles. The concept of cyclops states formed by two distinct, coherent clusters and a solitary oscillator was introduced by Munyaev et al. [Phys. Rev. Lett. 130, 107201 (2023)0031-900710.1103/PhysRevLett.130.107201], where we explored the surprising prevalence of such states in repulsive Kuramoto networks of rotators with higher-mode harmonics in the coupling. This paper extends our analysis to understand the mechanisms responsible for destroying the cyclops' states and inducing dynamical patterns called breathing and switching cyclops states. We first analytically study the existence and stability of cyclops states in the Kuramoto-Sakaguchi networks of two-dimensional oscillators with inertia as a function of the second coupling harmonic. We then describe two bifurcation scenarios that give birth to breathing and switching cyclops states. We demonstrate that these states and their hybrids are prevalent across a wide coupling range and are robust against a relatively large intrinsic frequency detuning. Beyond the Kuramoto networks, breathing and switching cyclops states promise to strongly manifest in other physical and biological networks, including coupled theta neurons.
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Affiliation(s)
- Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
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Brice Azangue A, Megam Ngouonkadi EB, Kabong Nono M, Fotsin HB, Sone Ekonde M, Yemele D. Stability and synchronization in neural network with delayed synaptic connections. CHAOS (WOODBURY, N.Y.) 2024; 34:013117. [PMID: 38215223 DOI: 10.1063/5.0175408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2023] [Accepted: 12/04/2023] [Indexed: 01/14/2024]
Abstract
In this paper, we investigate the stability of the synchronous state in a complex network using the master stability function technique. We use the extended Hindmarsh-Rose neuronal model including time delayed electrical, chemical, and hybrid couplings. We find the corresponding master stability equation that describes the whole dynamics for each coupling mode. From the maximum Lyapunov exponent, we deduce the stability state for each coupling mode. We observe that for electrical coupling, there exists a mixing between stable and unstable states. For a good setting of some system parameters, the position and the size of unstable areas can be modified. For chemical coupling, we observe difficulties in having a stable area in the complex plane. For hybrid coupling, we observe a stable behavior in the whole system compared to the case where these couplings are considered separately. The obtained results for each coupling mode help to analyze the stability state of some network topologies by using the corresponding eigenvalues. We observe that using electrical coupling can involve a full or partial stability of the system. In the case of chemical coupling, unstable states are observed whereas in the case of hybrid interactions a full stability of the network is obtained. Temporal analysis of the global synchronization is also done for each coupling mode, and the results show that when the network is stable, the synchronization is globally observed, while in the case when it is unstable, its nodes are not globally synchronized.
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Affiliation(s)
- A Brice Azangue
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
| | - E B Megam Ngouonkadi
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
- Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63 Buea, Cameroon
| | - M Kabong Nono
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
| | - H B Fotsin
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
| | - M Sone Ekonde
- Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63 Buea, Cameroon
| | - D Yemele
- Research Unit of Mechanics and Modeling of Physical Systems, Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 067 Dschang, Cameroon
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Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh I. Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling. PHYSICAL REVIEW LETTERS 2023; 130:107201. [PMID: 36962033 DOI: 10.1103/physrevlett.130.107201] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2022] [Accepted: 02/09/2023] [Indexed: 06/18/2023]
Abstract
Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops' eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.
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Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
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Belykh I, Kuske R, Porfiri M, Simpson DJW. Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications. CHAOS (WOODBURY, N.Y.) 2023; 33:010402. [PMID: 36725634 DOI: 10.1063/5.0138169] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2022] [Accepted: 12/16/2022] [Indexed: 06/18/2023]
Abstract
Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.
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Affiliation(s)
- Igor Belykh
- Department of Mathematics and Statistics & Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-4110, USA
| | - Rachel Kuske
- School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30313, USA
| | - Maurizio Porfiri
- Center for Urban Science and Progress, Department of Mechanical and Aerospace Engineering, and Department of Biomedical Engineering, Tandon School of Engineering, New York University, Brooklyn, New York 11201, USA
| | - David J W Simpson
- School of Mathematical and Computational Sciences, Massey University, Palmerston North 4410, New Zealand
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Tang L, Smith K, Daley K, Belykh I. When multilayer links exchange their roles in synchronization. Phys Rev E 2022; 106:024214. [PMID: 36109922 DOI: 10.1103/physreve.106.024214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2022] [Accepted: 08/09/2022] [Indexed: 06/15/2023]
Abstract
Real world networks contain multiple layers of links whose interactions can lead to extraordinary collective dynamics, including synchronization. The fundamental problem of assessing how network topology controls synchronization in multilayer networks remains open due to serious limitations of the existing stability methods. Towards removing this obstacle, we propose an approximation method which significantly enhances the predictive power of the master stability function for stable synchronization in multilayer networks. For a class of saddle-focus oscillators, including Rössler and piecewise linear systems, our method reduces the complex stability analysis to simply solving a set of linear algebraic equations. Using the method, we analytically predict surprising effects due to multilayer coupling. In particular, we prove that two coupling layers-one of which would alone hamper synchronization and the other would foster it-reverse their roles when used in a multilayer network. We also analytically demonstrate that increasing the size of a globally coupled layer, that in isolation would induce stable synchronization, makes the multilayer network unsynchronizable.
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Affiliation(s)
- Longkun Tang
- School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
| | - Kelley Smith
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
| | - Kevin Daley
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
| | - Igor Belykh
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
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Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh IV. Stability of rotatory solitary states in Kuramoto networks with inertia. Phys Rev E 2022; 105:024203. [PMID: 35291064 DOI: 10.1103/physreve.105.024203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (1) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (2) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras.
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Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Institute of Applied Physics, Russian Academy of Sciences, Ul'yanova Str. 46, Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor V Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302, USA
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Franović I, Eydam S, Semenova N, Zakharova A. Unbalanced clustering and solitary states in coupled excitable systems. CHAOS (WOODBURY, N.Y.) 2022; 32:011104. [PMID: 35105111 DOI: 10.1063/5.0077022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 12/28/2021] [Indexed: 06/14/2023]
Abstract
We discover the mechanisms of emergence and the link between two types of symmetry-broken states, the unbalanced periodic two-cluster states and solitary states, in coupled excitable systems with attractive and repulsive interactions. The prevalent solitary states in non-locally coupled arrays, whose self-organization is based on successive (order preserving) spiking of units, derive their dynamical features from the corresponding unbalanced cluster states in globally coupled networks. Apart from the states with successive spiking, we also find cluster and solitary states where the interplay of excitability and local multiscale dynamics gives rise to so-called leap-frog activity patterns with an alternating order of spiking between the units. We show that the noise affects the system dynamics by suppressing the multistability of cluster states and by inducing pattern homogenization, transforming solitary states into patterns of patched synchrony.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Sebastian Eydam
- Neural Circuits and Computations Unit, RIKEN Center for Brain Science, 2-1 Hirosawa, 351-0106 Wako, Japan
| | - Nadezhda Semenova
- Institute of Physics and Department of Fundamental Medicine and Medical Technology, Saratov State University, Astrakhanskaya str. 83, Saratov 410012, Russia
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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Bahramian A, Parastesh F, Pham VT, Kapitaniak T, Jafari S, Perc M. Collective behavior in a two-layer neuronal network with time-varying chemical connections that are controlled by a Petri net. CHAOS (WOODBURY, N.Y.) 2021; 31:033138. [PMID: 33810759 DOI: 10.1063/5.0045840] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2021] [Accepted: 03/02/2021] [Indexed: 06/12/2023]
Abstract
In this paper, we propose and study a two-layer network composed of a Petri net in the first layer and a ring of coupled Hindmarsh-Rose neurons in the second layer. Petri nets are appropriate platforms not only for describing sequential processes but also for modeling information circulation in complex systems. Networks of neurons, on the other hand, are commonly used to study synchronization and other forms of collective behavior. Thus, merging both frameworks into a single model promises fascinating new insights into neuronal collective behavior that is subject to changes in network connectivity. In our case, the Petri net in the first layer manages the existence of excitatory and inhibitory links among the neurons in the second layer, thereby making the chemical connections time-varying. We focus on the emergence of different types of collective behavior in the model, such as synchronization, chimeras, and solitary states, by considering different inhibitory and excitatory tokens in the Petri net. We find that the existence of only inhibitory or excitatory tokens disturbs the synchronization of electrically coupled neurons and leads toward chimera and solitary states.
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Affiliation(s)
- Alireza Bahramian
- Department of Biomedical Engineering, Amirkabir University of Technology, No. 350, Hafez Ave., Valiasr Square, Tehran 159163-4311, Iran
| | - Fatemeh Parastesh
- Department of Biomedical Engineering, Amirkabir University of Technology, No. 350, Hafez Ave., Valiasr Square, Tehran 159163-4311, Iran
| | - Viet-Thanh Pham
- Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam
| | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Sajad Jafari
- Center for Computational Biology, Chennai Institute of Technology, Chennai, Tamil Nadu 600069, India
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
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Saggio ML, Crisp D, Scott JM, Karoly P, Kuhlmann L, Nakatani M, Murai T, Dümpelmann M, Schulze-Bonhage A, Ikeda A, Cook M, Gliske SV, Lin J, Bernard C, Jirsa V, Stacey WC. A taxonomy of seizure dynamotypes. eLife 2020; 9:55632. [PMID: 32691734 PMCID: PMC7375810 DOI: 10.7554/elife.55632] [Citation(s) in RCA: 53] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2020] [Accepted: 06/12/2020] [Indexed: 01/02/2023] Open
Abstract
Seizures are a disruption of normal brain activity present across a vast range of species and conditions. We introduce an organizing principle that leads to the first objective Taxonomy of Seizure Dynamics (TSD) based on bifurcation theory. The ‘dynamotype’ of a seizure is the dynamic composition that defines its observable characteristics, including how it starts, evolves and ends. Analyzing over 2000 focal-onset seizures from multiple centers, we find evidence of all 16 dynamotypes predicted in TSD. We demonstrate that patients’ dynamotypes evolve during their lifetime and display complex but systematic variations including hierarchy (certain types are more common), non-bijectivity (a patient may display multiple types) and pairing preference (multiple types may occur during one seizure). TSD provides a way to stratify patients in complement to present clinical classifications, a language to describe the most critical features of seizure dynamics, and a framework to guide future research focused on dynamical properties. Epileptic seizures have been recognized for centuries. But it was only in the 1930s that it was realized that seizures are the result of out-of-control electrical activity in the brain. By placing electrodes on the scalp, doctors can identify when and where in the brain a seizure begins. But they cannot tell much about how the seizure behaves, that is, how it starts, stops or spreads to other areas. This makes it difficult to control and prevent seizures. It also helps explain why almost a third of patients with epilepsy continue to have seizures despite being on medication. Saggio, Crisp et al. have now approached this problem from a new angle using methods adapted from physics and engineering. In these fields, “dynamics research” has been used with great success to predict and control the behavior of complex systems like electrical power grids. Saggio, Crisp et al. reasoned that applying the same approach to the brain would reveal the dynamics of seizures and that such information could then be used to categorize seizures into groups with similar properties. This would in effect create for seizures what the periodic table is for the elements. Applying the dynamics research method to seizure data from more than a hundred patients from across the world revealed 16 types of seizure dynamics. These “dynamotypes” had distinct characteristics. Some were more common than others, and some tended to occur together. Individual patients showed different dynamotypes over time. By constructing a way to classify seizures based on the relationships between the dynamotypes, Saggio, Crisp et al. provide a new tool for clinicians and researchers studying epilepsy. Previous clinical tools have focused on the physical symptoms of a seizure (referred to as the phenotype) or its potential genetic causes (genotype). The current approach complements these tools by adding the dynamotype: how seizures start, spread and stop in the brain. This approach has the potential to lead to new branches of research and better understanding and treatment of seizures.
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Affiliation(s)
- Maria Luisa Saggio
- Aix Marseille Univ, Inserm, INS, Institut de Neurosciences des Systèmes, Marseille, France, Marseille, France
| | - Dakota Crisp
- Department of Biomedical Engineering, BioInterfaces Institute, University of Michigan, Ann Arbor, United States
| | - Jared M Scott
- Department of Biomedical Engineering, BioInterfaces Institute, University of Michigan, Ann Arbor, United States
| | - Philippa Karoly
- Graeme Clark Institute, The University of Melbourne, Melbourne, Australia
| | - Levin Kuhlmann
- Department of Medicine, St. Vincent's Hospital, The University of Melbourne, Melbourne, Australia.,Faculty of Information Technology, Monash University, Clayton, Australia
| | - Mitsuyoshi Nakatani
- Aix Marseille Univ, Inserm, INS, Institut de Neurosciences des Systèmes, Marseille, France, Marseille, France
| | - Tomohiko Murai
- Department of Epilepsy, Movement Disorders and Physiology, Kyoto University Graduate School of Medicine, Kyoto, Japan
| | - Matthias Dümpelmann
- Epilepsy Center, Medical Center - University of Freiburg, Freiburg im Breisgau, Germany.,Faculty of Medicine, University of Freiburg, Freiburg im Breisgau, Germany
| | - Andreas Schulze-Bonhage
- Epilepsy Center, Medical Center - University of Freiburg, Freiburg im Breisgau, Germany.,Faculty of Medicine, University of Freiburg, Freiburg im Breisgau, Germany.,Center for Basics in NeuroModulation (NeuroModul Basics), Epilepsy Center, Medical Center - University of Freiburg, Freiburg im Breisgau, Germany
| | - Akio Ikeda
- Department of Epilepsy, Movement Disorders and Physiology, Kyoto University Graduate School of Medicine, Kyoto, Japan
| | - Mark Cook
- Graeme Clark Institute, The University of Melbourne, Melbourne, Australia.,Department of Medicine, St. Vincent's Hospital, The University of Melbourne, Melbourne, Australia
| | - Stephen V Gliske
- Department of Neurology, University of Michigan, Ann Arbor, United States
| | - Jack Lin
- Department of Neurology, University of Michigan, Ann Arbor, United States
| | - Christophe Bernard
- Aix Marseille Univ, Inserm, INS, Institut de Neurosciences des Systèmes, Marseille, France, Marseille, France
| | - Viktor Jirsa
- Aix Marseille Univ, Inserm, INS, Institut de Neurosciences des Systèmes, Marseille, France, Marseille, France
| | - William C Stacey
- Department of Biomedical Engineering, BioInterfaces Institute, University of Michigan, Ann Arbor, United States.,Department of Neurology, University of Michigan, Ann Arbor, United States
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Daley K, Zhao K, Belykh IV. Synchronizability of directed networks: The power of non-existent ties. CHAOS (WOODBURY, N.Y.) 2020; 30:043102. [PMID: 32357666 DOI: 10.1063/1.5134920] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2019] [Accepted: 03/16/2020] [Indexed: 06/11/2023]
Abstract
The understanding of how synchronization in directed networks is influenced by structural changes in network topology is far from complete. While the addition of an edge always promotes synchronization in a wide class of undirected networks, this addition may impede synchronization in directed networks. In this paper, we develop the augmented graph stability method, which allows for explicitly connecting the stability of synchronization to changes in network topology. The transformation of a directed network into a symmetrized-and-augmented undirected network is the central component of this new method. This transformation is executed by symmetrizing and weighting the underlying connection graph and adding new undirected edges with consideration made for the mean degree imbalance of each pair of nodes. These new edges represent "non-existent ties" in the original directed network and often control the location of critical nodes whose directed connections can be altered to manipulate the stability of synchronization in a desired way. In particular, we show that the addition of small-world shortcuts to directed networks, which makes "non-existent ties" disappear, can worsen the synchronizability, thereby revealing a destructive role of small-world connections in directed networks. An extension of our method may open the door to studying synchronization in directed multilayer networks, which cannot be effectively handled by the eigenvalue-based methods.
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Affiliation(s)
- Kevin Daley
- Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
| | - Kun Zhao
- Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
| | - Igor V Belykh
- Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
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11
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Ge P, Cao H. Synchronization of Rulkov neuron networks coupled by excitatory and inhibitory chemical synapses. CHAOS (WOODBURY, N.Y.) 2019; 29:023129. [PMID: 30823734 DOI: 10.1063/1.5053908] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2018] [Accepted: 02/01/2019] [Indexed: 06/09/2023]
Abstract
This paper takes into account a neuron network model in which the excitatory and the inhibitory Rulkov neurons interact each other through excitatory and inhibitory chemical coupling, respectively. Firstly, for two or more identical or non-identical Rulkov neurons, the existence conditions of the synchronization manifold of the fixed points are investigated, which have received less attention over the past decades. Secondly, the master stability equation of the arbitrarily connected neuron network under the existence conditions of the synchronization manifold is discussed. Thirdly, taking three identical Rulkov neurons as an example, some new results are presented: (1) topological structures that can make the synchronization manifold exist are given, (2) the stability of synchronization when different parameters change is discussed, and (3) the roles of the control parameters, the ratio, as well as the size of the coupling strength and sigmoid function are analyzed. Finally, for the chemical coupling between two non-identical neurons, the transversal system is given and the effect of two coupling strengths on synchronization is analyzed.
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Affiliation(s)
- Penghe Ge
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
| | - Hongjun Cao
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
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12
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Reimbayev R, Daley K, Belykh I. When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2017; 375:rsta.2016.0282. [PMID: 28507227 PMCID: PMC5434073 DOI: 10.1098/rsta.2016.0282] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/08/2017] [Indexed: 05/24/2023]
Abstract
Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. In this paper, we study the emergence of synchronization in networks of bursting neurons as a highly non-trivial, combined effect of electrical and inhibitory connections. We report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism, which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.
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Affiliation(s)
- Reimbay Reimbayev
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, GA 30303, USA
| | - Kevin Daley
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, GA 30303, USA
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, GA 30303, USA
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13
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Sathiyadevi K, Karthiga S, Chandrasekar VK, Senthilkumar DV, Lakshmanan M. Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings. Phys Rev E 2017; 95:042301. [PMID: 28505842 DOI: 10.1103/physreve.95.042301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2016] [Indexed: 06/07/2023]
Abstract
Spontaneous symmetry breaking is an important phenomenon observed in various fields including physics and biology. In this connection, we here show that the trade-off between attractive and repulsive couplings can induce spontaneous symmetry breaking in a homogeneous system of coupled oscillators. With a simple model of a system of two coupled Stuart-Landau oscillators, we demonstrate how the tendency of attractive coupling in inducing in-phase synchronized (IPS) oscillations and the tendency of repulsive coupling in inducing out-of-phase synchronized oscillations compete with each other and give rise to symmetry breaking oscillatory states and interesting multistabilities. Further, we provide explicit expressions for synchronized and antisynchronized oscillatory states as well as the so called oscillation death (OD) state and study their stability. If the Hopf bifurcation parameter (λ) is greater than the natural frequency (ω) of the system, the attractive coupling favors the emergence of an antisymmetric OD state via a Hopf bifurcation whereas the repulsive coupling favors the emergence of a similar state through a saddle-node bifurcation. We show that an increase in the repulsive coupling not only destabilizes the IPS state but also facilitates the reentrance of the IPS state.
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Affiliation(s)
- K Sathiyadevi
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - S Karthiga
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695 016, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
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14
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Zhao K, Wohlhueter RM, Li Y. Finishing monkeypox genomes from short reads: assembly analysis and a neural network method. BMC Genomics 2016; 17 Suppl 5:497. [PMID: 27585810 PMCID: PMC5009526 DOI: 10.1186/s12864-016-2826-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
BACKGROUND Poxviruses constitute one of the largest and most complex animal virus families known. The notorious smallpox disease has been eradicated and the virus contained, but its simian sister, monkeypox is an emerging, untreatable infectious disease, killing 1 to 10 % of its human victims. In the case of poxviruses, the emergence of monkeypox outbreaks in humans and the need to monitor potential malicious release of smallpox virus requires development of methods for rapid virus identification. Whole-genome sequencing (WGS) is an emergent technology with increasing application to the diagnosis of diseases and the identification of outbreak pathogens. But "finishing" such a genome is a laborious and time-consuming process, not easily automated. To date the large, complete poxvirus genomes have not been studied comprehensively in terms of applying WGS techniques and evaluating genome assembly algorithms. RESULTS To explore the limitations to finishing a poxvirus genome from short reads, we first analyze the repetitive regions in a monkeypox genome and evaluate genome assembly on the simulated reads. We also report on procedures and insights relevant to the assembly (from realistically short reads) of genomes. Finally, we propose a neural network method (namely Neural-KSP) to "finish" the process by closing gaps remaining after conventional assembly, as the final stage in a protocol to elucidate clinical poxvirus genomic sequences. CONCLUSIONS The protocol may prove useful in any clinical viral isolate (regardless if a reference-strain sequence is available) and especially useful in genomes confounded by many global and local repetitive sequences embedded in them. This work highlights the feasibility of finishing real, complex genomes by systematically analyzing genetic characteristics, thus remedying existing assembly shortcomings with a neural network method. Such finished sequences may enable clinicians to track genetic distance between viral isolates that provides a powerful epidemiological tool.
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Affiliation(s)
- Kun Zhao
- Office of Infectious Diseases, Centers for Disease Control and Prevention, Atlanta, 30333, USA.
| | | | - Yu Li
- Poxvirus and Rabies Branch, Division of High Consequence Pathogens and Pathology, National Center for Emerging and Zoonotic Infectious Diseases, Centers for Disease Control and Prevention, Atlanta, 30333, USA
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15
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Bera BK, Ghosh D, Banerjee T. Imperfect traveling chimera states induced by local synaptic gradient coupling. Phys Rev E 2016; 94:012215. [PMID: 27575131 DOI: 10.1103/physreve.94.012215] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2016] [Indexed: 06/06/2023]
Abstract
In this paper, we report the occurrence of chimera patterns in a network of neuronal oscillators, which are coupled through local, synaptic gradient coupling. We discover a new chimera pattern, namely the imperfect traveling chimera state, where the incoherent traveling domain spreads into the coherent domain of the network. Remarkably, we also find that chimera states arise even for one-way local coupling, which is in contrast to the earlier belief that only nonlocal, global, or nearest-neighbor local coupling can give rise to chimera state; this find further relaxes the essential connectivity requirement of getting a chimera state. We choose a network of identical bursting Hindmarsh-Rose neuronal oscillators, and we show that depending upon the relative strength of the synaptic and gradient coupling, several chimera patterns emerge. We map all the spatiotemporal behaviors in parameter space and identify the transitions among several chimera patterns, an in-phase synchronized state, and a global amplitude death state.
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Affiliation(s)
- Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700 108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700 108, India
| | - Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India
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16
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Horikawa Y. Effects of self-coupling and asymmetric output on metastable dynamical transient firing patterns in arrays of neurons with bidirectional inhibitory coupling. Neural Netw 2016; 76:13-28. [PMID: 26829604 DOI: 10.1016/j.neunet.2015.12.014] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2015] [Revised: 12/16/2015] [Accepted: 12/25/2015] [Indexed: 10/22/2022]
Abstract
Metastable dynamical transient patterns in arrays of bidirectionally coupled neurons with self-coupling and asymmetric output were studied. First, an array of asymmetric sigmoidal neurons with symmetric inhibitory bidirectional coupling and self-coupling was considered and the bifurcations of its steady solutions were shown. Metastable dynamical transient spatially nonuniform states existed in the presence of a pair of spatially symmetric stable solutions as well as unstable spatially nonuniform solutions in a restricted range of the output gain of a neuron. The duration of the transients increased exponentially with the number of neurons up to the maximum number at which the spatially nonuniform steady solutions were stabilized. The range of the output gain for which they existed reduced as asymmetry in a sigmoidal output function of a neuron increased, while the existence range expanded as the strength of inhibitory self-coupling increased. Next, arrays of spiking neuron models with slow synaptic inhibitory bidirectional coupling and self-coupling were considered with computer simulation. In an array of Class 1 Hindmarsh-Rose type models, in which each neuron showed a graded firing rate, metastable dynamical transient firing patterns were observed in the presence of inhibitory self-coupling. This agreed with the condition for the existence of metastable dynamical transients in an array of sigmoidal neurons. In an array of Class 2 Bonhoeffer-van der Pol models, in which each neuron had a clear threshold between firing and resting, long-lasting transient firing patterns with bursting and irregular motion were observed.
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Affiliation(s)
- Yo Horikawa
- Faculty of Engineering, Kagawa University, Takamatsu, 761-0396, Japan.
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17
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Astakhov S, Gulai A, Fujiwara N, Kurths J. The role of asymmetrical and repulsive coupling in the dynamics of two coupled van der Pol oscillators. CHAOS (WOODBURY, N.Y.) 2016; 26:023102. [PMID: 26931583 DOI: 10.1063/1.4940967] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2015] [Accepted: 01/14/2016] [Indexed: 06/05/2023]
Abstract
A system of two asymmetrically coupled van der Pol oscillators has been studied. We show that the introduction of a small asymmetry in coupling leads to the appearance of a "wideband synchronization channel" in the bifurcational structure of the parameter space. An increase of asymmetry and transition to repulsive interaction leads to the formation of multistability. As the result, the tip of the Arnold's tongue widens due to the formation of folds defined by saddle-node bifurcation curves for the limit cycles on the torus.
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Affiliation(s)
- Sergey Astakhov
- Information Security of Automated Systems Department, Yuri Gagarin State Technical University of Saratov, Politekhnitcheskaya st. 77, Saratov 410054, Russia
| | - Artem Gulai
- Radioelectronics and Telecommunications Department, Yuri Gagarin State Technical University of Saratov, Politekhnitcheskaya st. 77, Saratov 410054, Russia
| | - Naoya Fujiwara
- Center for Spatial Information Science, The University of Tokyo, 277-8568 Chiba, Japan
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany
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18
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Abstract
We study the existence of chimera states in pulse-coupled networks of bursting Hindmarsh-Rose neurons with nonlocal, global, and local (nearest neighbor) couplings. Through a linear stability analysis, we discuss the behavior of the stability function in the incoherent (i.e., disorder), coherent, chimera, and multichimera states. Surprisingly, we find that chimera and multichimera states occur even using local nearest neighbor interaction in a network of identical bursting neurons alone. This is in contrast with the existence of chimera states in populations of nonlocally or globally coupled oscillators. A chemical synaptic coupling function is used which plays a key role in the emergence of chimera states in bursting neurons. The existence of chimera, multichimera, coherent, and disordered states is confirmed by means of the recently introduced statistical measures and mean phase velocity.
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Affiliation(s)
- Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli-620024, India
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