1
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Thamizharasan S, Chandrasekar VK, Senthilvelan M, Senthilkumar DV. Hebbian and anti-Hebbian adaptation-induced dynamical states in adaptive networks. Phys Rev E 2024; 109:014221. [PMID: 38366486 DOI: 10.1103/physreve.109.014221] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2023] [Accepted: 12/13/2023] [Indexed: 02/18/2024]
Abstract
We investigate the interplay of an external forcing and an adaptive network, whose connection weights coevolve with the dynamical states of the phase oscillators. In particular, we consider the Hebbian and anti-Hebbian adaptation mechanisms for the evolution of the connection weights. The Hebbian adaptation manifests several interesting partially synchronized states, such as phase and frequency clusters, bump state, bump frequency phase clusters, and forced entrained clusters, in addition to the completely synchronized and forced entrained states. Anti-Hebbian adaptation facilitates the manifestation of the itinerant chimera characterized by randomly evolving coherent and incoherent domains along with some of the aforementioned dynamical states induced by the Hebbian adaptation. We introduce three distinct measures for the strength of incoherence based on the local standard deviations of the time-averaged frequency and the instantaneous phase of each oscillator, and the time-averaged mean frequency for each bin to corroborate the distinct dynamical states and to demarcate the two parameter phase diagrams. We also arrive at the existence and stability conditions for the forced entrained state using the linear stability analysis, which is found to be consistent with the simulation results.
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Affiliation(s)
- S Thamizharasan
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, Department of Physics, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - M Senthilvelan
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695 551, Kerala, India
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2
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Gil L. Self-adaptation of networks of nonidentical pulse-coupled excitatory and inhibitory oscillators in the presence of distance-related delays to achieve frequency synchronization. Phys Rev E 2023; 108:034211. [PMID: 37849137 DOI: 10.1103/physreve.108.034211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2023] [Accepted: 09/01/2023] [Indexed: 10/19/2023]
Abstract
We show that a network of nonidentical nodes, with excitable dynamics, pulse-coupled, with coupling delays depending on the Euclidean distance between nodes, is able to adapt the topology of its connections to obtain spike frequency synchronization. The adapted network exhibits remarkable properties: sparse, anticluster, necessary presence of a minimum of inhibitory nodes, predominance of connections from inhibitory nodes over those from excitatory nodes, and finally spontaneous spatial structuring of the inhibitory projections: the furthest are the most intense. In a second step, we discuss the possible implications of our findings to neural systems.
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Affiliation(s)
- L Gil
- Université Côte d'Azur, Institut de Physique de Nice, 17 rue Julien Lauprêtre, 06200 Nice, France
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3
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Botha AE, Ansariara M, Emadi S, Kolahchi MR. Chimera Patterns of Synchrony in a Frustrated Array of Hebb Synapses. Front Comput Neurosci 2022; 16:888019. [PMID: 35814347 PMCID: PMC9260432 DOI: 10.3389/fncom.2022.888019] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2022] [Accepted: 05/30/2022] [Indexed: 11/13/2022] Open
Abstract
The union of the Kuramoto–Sakaguchi model and the Hebb dynamics reproduces the Lisman switch through a bistability in synchronized states. Here, we show that, within certain ranges of the frustration parameter, the chimera pattern can emerge, causing a different, time-evolving, distribution in the Hebbian synaptic strengths. We study the stability range of the chimera as a function of the frustration (phase-lag) parameter. Depending on the range of the frustration, two different types of chimeras can appear spontaneously, i.e., from randomized initial conditions. In the first type, the oscillators in the coherent region rotate, on average, slower than those in the incoherent region; while in the second type, the average rotational frequencies of the two regions are reversed, i.e., the coherent region runs, on average, faster than the incoherent region. We also show that non-stationary behavior at finite N can be controlled by adjusting the natural frequency of a single pacemaker oscillator. By slowly cycling the frequency of the pacemaker, we observe hysteresis in the system. Finally, we discuss how we can have a model for learning and memory.
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Affiliation(s)
- A. E. Botha
- Department of Physics, Science Campus, University of South Africa, Private Bag X6, Johannesburg, South Africa
| | - M. Ansariara
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - S. Emadi
- Department of Biological Sciences, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - M. R. Kolahchi
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
- *Correspondence: M. R. Kolahchi
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4
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Thamizharasan S, Chandrasekar VK, Senthilvelan M, Berner R, Schöll E, Senthilkumar DV. Exotic states induced by coevolving connection weights and phases in complex networks. Phys Rev E 2022; 105:034312. [PMID: 35428128 DOI: 10.1103/physreve.105.034312] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2021] [Accepted: 03/07/2022] [Indexed: 06/14/2023]
Abstract
We consider an adaptive network, whose connection weights coevolve in congruence with the dynamical states of the local nodes that are under the influence of an external stimulus. The adaptive dynamical system mimics the adaptive synaptic connections common in neuronal networks. The adaptive network under external forcing displays exotic dynamical states such as itinerant chimeras whose population density of coherent and incoherent domains coevolves with the synaptic connection, bump states, and bump frequency cluster states, which do not exist in adaptive networks without forcing. In addition, the adaptive network also exhibits partial synchronization patterns such as phase and frequency clusters, forced entrained, and incoherent states. We introduce two measures for the strength of incoherence based on the standard deviation of the temporally averaged (mean) frequency and on the mean frequency to classify the emergent dynamical states as well as their transitions. We provide a two-parameter phase diagram showing the wealth of dynamical states. We additionally deduce the stability condition for the frequency-entrained state. We use the paradigmatic Kuramoto model of phase oscillators, which is a simple generic model that has been widely employed in unraveling a plethora of cooperative phenomena in natural and man-made systems.
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Affiliation(s)
- S Thamizharasan
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, Department of Physics, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - M Senthilvelan
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India
| | - Rico Berner
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
- Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473 Potsdam, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, Philippstraße 13, 10115 Berlin, Germany
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695 551, Kerala, India
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5
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Qian W, Papadopoulos L, Lu Z, Kroma-Wiley KA, Pasqualetti F, Bassett DS. Path-dependent dynamics induced by rewiring networks of inertial oscillators. Phys Rev E 2022; 105:024304. [PMID: 35291167 DOI: 10.1103/physreve.105.024304] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2020] [Accepted: 10/14/2021] [Indexed: 06/14/2023]
Abstract
In networks of coupled oscillators, it is of interest to understand how interaction topology affects synchronization. Many studies have gained key insights into this question by studying the classic Kuramoto oscillator model on static networks. However, new questions arise when the network structure is time varying or when the oscillator system is multistable, the latter of which can occur when an inertial term is added to the Kuramoto model. While the consequences of evolving topology and multistability on collective behavior have been examined separately, real-world systems such as gene regulatory networks and the brain may exhibit these properties simultaneously. It is thus relevant to ask how time-varying network connectivity impacts synchronization in systems that can exhibit multistability. To address this question, we study how the dynamics of coupled Kuramoto oscillators with inertia are affected when the topology of the underlying network changes in time. We show that hysteretic synchronization behavior in networks of coupled inertial oscillators can be driven by changes in connection topology alone. Moreover, we find that certain fixed-density rewiring schemes induce significant changes to the level of global synchrony that remain even after the network returns to its initial configuration, and we show that these changes are robust to a wide range of network perturbations. Our findings highlight that the specific progression of network topology over time, in addition to its initial or final static structure, can play a considerable role in modulating the collective behavior of systems evolving on complex networks.
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Affiliation(s)
- William Qian
- Department of Physics & Astronomy, College of Arts & Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Lia Papadopoulos
- Department of Physics & Astronomy, College of Arts & Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Zhixin Lu
- Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Keith A Kroma-Wiley
- Department of Physics & Astronomy, College of Arts & Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Fabio Pasqualetti
- Department of Mechanical Engineering, University of California, Riverside, California 92521, USA
| | - Dani S Bassett
- Department of Physics & Astronomy, College of Arts & Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Mechanical Engineering, University of California, Riverside, California 92521, USA
- Department of Neurology, Perelman School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
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6
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Calabrese C, Bardy BG, De Lellis P, di Bernardo M. Modeling Frequency Reduction in Human Groups Performing a Joint Oscillatory Task. Front Psychol 2022; 12:753758. [PMID: 35058838 PMCID: PMC8765722 DOI: 10.3389/fpsyg.2021.753758] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2021] [Accepted: 12/07/2021] [Indexed: 11/13/2022] Open
Abstract
In human groups performing oscillatory tasks, it has been observed that the frequency of participants' oscillations reduces when compared to that acquired in solo. This experimental observation is not captured by the standard Kuramoto oscillators, often employed to model human synchronization. In this work, we aim at capturing this observed phenomenon by proposing three alternative modifications of the standard Kuramoto model that are based on three different biologically-relevant hypotheses underlying group synchronization. The three models are tuned, validated and compared against experiments on a group synchronization task, which is a multi-agent extension of the so-called mirror game.
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Affiliation(s)
- Carmela Calabrese
- Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy.,EuroMov Digital Health in Motion, University of Montpellier IMT Mines Ales, Montpellier, France
| | - Benoît G Bardy
- EuroMov Digital Health in Motion, University of Montpellier IMT Mines Ales, Montpellier, France
| | - Pietro De Lellis
- Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
| | - Mario di Bernardo
- Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
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7
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Sawicki J, Berner R, Löser T, Schöll E. Modeling Tumor Disease and Sepsis by Networks of Adaptively Coupled Phase Oscillators. FRONTIERS IN NETWORK PHYSIOLOGY 2022; 1:730385. [PMID: 36925568 PMCID: PMC10013027 DOI: 10.3389/fnetp.2021.730385] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2021] [Accepted: 11/19/2021] [Indexed: 06/18/2023]
Abstract
In this study, we provide a dynamical systems perspective to the modelling of pathological states induced by tumors or infection. A unified disease model is established using the innate immune system as the reference point. We propose a two-layer network model for carcinogenesis and sepsis based upon the interaction of parenchymal cells and immune cells via cytokines, and the co-evolutionary dynamics of parenchymal, immune cells, and cytokines. Our aim is to show that the complex cellular cooperation between parenchyma and stroma (immune layer) in the physiological and pathological case can be qualitatively and functionally described by a simple paradigmatic model of phase oscillators. By this, we explain carcinogenesis, tumor progression, and sepsis by destabilization of the healthy homeostatic state (frequency synchronized), and emergence of a pathological state (desynchronized or multifrequency cluster). The coupled dynamics of parenchymal cells (metabolism) and nonspecific immune cells (reaction of innate immune system) are represented by nodes of a duplex layer. The cytokine interaction is modeled by adaptive coupling weights between the nodes representing the immune cells (with fast adaptation time scale) and the parenchymal cells (slow adaptation time scale) and between the pairs of parenchymal and immune cells in the duplex network (fixed bidirectional coupling). Thereby, carcinogenesis, organ dysfunction in sepsis, and recurrence risk can be described in a correct functional context.
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Affiliation(s)
- Jakub Sawicki
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
| | - Rico Berner
- Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany
| | | | - Eckehard Schöll
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, Berlin, Germany
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8
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Berner R, Sawicki J, Schöll E. Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks. PHYSICAL REVIEW LETTERS 2020; 124:088301. [PMID: 32167358 DOI: 10.1103/physrevlett.124.088301] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2019] [Revised: 12/05/2019] [Accepted: 01/16/2020] [Indexed: 06/10/2023]
Abstract
We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuroscience and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multilayer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.
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Affiliation(s)
- Rico Berner
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
- Institut für Mathematik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
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9
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Chowdhury SN, Ghosh D, Hens C. Effect of repulsive links on frustration in attractively coupled networks. Phys Rev E 2020; 101:022310. [PMID: 32168719 DOI: 10.1103/physreve.101.022310] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Accepted: 01/31/2020] [Indexed: 06/10/2023]
Abstract
We investigate the impact of attractive-repulsive interaction in networks of limit cycle oscillators. Mainly we focus on the design principle for generating an antiphase state between adjacent nodes in a complex network. We establish that a partial negative control throughout the branches of a spanning tree inside the positively coupled limit cycle oscillators works efficiently well in comparison with randomly chosen negative links to establish zero frustration (antiphase synchronization) in bipartite graphs. Based on the emergence of zero frustration, we develop a universal 0-π rule to understand the antiphase synchronization in a bipartite graph. Further, this rule is used to construct a nonbipartite graph for a given nonzero frustrated value. We finally show the generality of 0-π rule by implementing it in arbitrary undirected nonbipartite graphs of attractive-repulsively coupled limit cycle oscillators and successfully calculate the nonzero frustration value, which matches with numerical data. The validation of the rule is checked through the bifurcation analysis of small networks. Our work may unveil the underlying mechanism of several synchronization phenomena that exist in a network of oscillators having a mixed type of coupling.
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Affiliation(s)
- Sayantan Nag Chowdhury
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India
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10
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Berner R, Fialkowski J, Kasatkin D, Nekorkin V, Yanchuk S, Schöll E. Hierarchical frequency clusters in adaptive networks of phase oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:103134. [PMID: 31675820 DOI: 10.1063/1.5097835] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Accepted: 10/03/2019] [Indexed: 06/10/2023]
Abstract
Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency multiclusters by investigating the effect of the time scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multiclusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.
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Affiliation(s)
- Rico Berner
- Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
| | - Jan Fialkowski
- Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
| | - Dmitry Kasatkin
- Institute of Applied Physics of RAS, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Vladimir Nekorkin
- Institute of Applied Physics of RAS, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Serhiy Yanchuk
- Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany
| | - Eckehard Schöll
- Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
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11
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Karimian M, Dibenedetto D, Moerel M, Burwick T, Westra RL, De Weerd P, Senden M. Effects of synaptic and myelin plasticity on learning in a network of Kuramoto phase oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:083122. [PMID: 31472483 DOI: 10.1063/1.5092786] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2019] [Accepted: 08/01/2019] [Indexed: 06/10/2023]
Abstract
Models of learning typically focus on synaptic plasticity. However, learning is the result of both synaptic and myelin plasticity. Specifically, synaptic changes often co-occur and interact with myelin changes, leading to complex dynamic interactions between these processes. Here, we investigate the implications of these interactions for the coupling behavior of a system of Kuramoto oscillators. To that end, we construct a fully connected, one-dimensional ring network of phase oscillators whose coupling strength (reflecting synaptic strength) as well as conduction velocity (reflecting myelination) are each regulated by a Hebbian learning rule. We evaluate the behavior of the system in terms of structural (pairwise connection strength and conduction velocity) and functional connectivity (local and global synchronization behavior). We find that adaptive myelination is able to both functionally decouple structurally connected oscillators as well as to functionally couple structurally disconnected oscillators. With regard to the latter, we find that for conditions in which a system limited to synaptic plasticity develops two distinct clusters both structurally and functionally, additional adaptive myelination allows for functional communication across these structural clusters. These results confirm that network states following learning may be different when myelin plasticity is considered in addition to synaptic plasticity, pointing toward the relevance of integrating both factors in computational models of learning.
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Affiliation(s)
- M Karimian
- Maastricht Centre for Systems Biology (MaCSBio), Maastricht University, 6229 ER Maastricht, The Netherlands
| | - D Dibenedetto
- Maastricht Centre for Systems Biology (MaCSBio), Maastricht University, 6229 ER Maastricht, The Netherlands
| | - M Moerel
- Maastricht Centre for Systems Biology (MaCSBio), Maastricht University, 6229 ER Maastricht, The Netherlands
| | - T Burwick
- Maastricht Centre for Systems Biology (MaCSBio), Maastricht University, 6229 ER Maastricht, The Netherlands
| | - R L Westra
- Maastricht Centre for Systems Biology (MaCSBio), Maastricht University, 6229 ER Maastricht, The Netherlands
| | - P De Weerd
- Maastricht Centre for Systems Biology (MaCSBio), Maastricht University, 6229 ER Maastricht, The Netherlands
| | - M Senden
- Department of Cognitive Neuroscience, Faculty of Psychology and Neuroscience, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands
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12
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Khoshkhou M, Montakhab A. Beta-Rhythm Oscillations and Synchronization Transition in Network Models of Izhikevich Neurons: Effect of Topology and Synaptic Type. Front Comput Neurosci 2018; 12:59. [PMID: 30154708 PMCID: PMC6103382 DOI: 10.3389/fncom.2018.00059] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2018] [Accepted: 07/09/2018] [Indexed: 11/13/2022] Open
Abstract
Despite their significant functional roles, beta-band oscillations are least understood. Synchronization in neuronal networks have attracted much attention in recent years with the main focus on transition type. Whether one obtains explosive transition or a continuous transition is an important feature of the neuronal network which can depend on network structure as well as synaptic types. In this study we consider the effect of synaptic interaction (electrical and chemical) as well as structural connectivity on synchronization transition in network models of Izhikevich neurons which spike regularly with beta rhythms. We find a wide range of behavior including continuous transition, explosive transition, as well as lack of global order. The stronger electrical synapses are more conducive to synchronization and can even lead to explosive synchronization. The key network element which determines the order of transition is found to be the clustering coefficient and not the small world effect, or the existence of hubs in a network. These results are in contrast to previous results which use phase oscillator models such as the Kuramoto model. Furthermore, we show that the patterns of synchronization changes when one goes to the gamma band. We attribute such a change to the change in the refractory period of Izhikevich neurons which changes significantly with frequency.
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Affiliation(s)
- Mahsa Khoshkhou
- Department of Physics, College of Sciences, Shiraz University, Shiraz, Iran
| | - Afshin Montakhab
- Department of Physics, College of Sciences, Shiraz University, Shiraz, Iran
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13
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Papadopoulos L, Kim JZ, Kurths J, Bassett DS. Development of structural correlations and synchronization from adaptive rewiring in networks of Kuramoto oscillators. CHAOS (WOODBURY, N.Y.) 2017; 27:073115. [PMID: 28764402 PMCID: PMC5552408 DOI: 10.1063/1.4994819] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2017] [Accepted: 07/07/2017] [Indexed: 05/06/2023]
Abstract
Synchronization of non-identical oscillators coupled through complex networks is an important example of collective behavior, and it is interesting to ask how the structural organization of network interactions influences this process. Several studies have explored and uncovered optimal topologies for synchronization by making purposeful alterations to a network. On the other hand, the connectivity patterns of many natural systems are often not static, but are rather modulated over time according to their dynamics. However, this co-evolution and the extent to which the dynamics of the individual units can shape the organization of the network itself are less well understood. Here, we study initially randomly connected but locally adaptive networks of Kuramoto oscillators. In particular, the system employs a co-evolutionary rewiring strategy that depends only on the instantaneous, pairwise phase differences of neighboring oscillators, and that conserves the total number of edges, allowing the effects of local reorganization to be isolated. We find that a simple rule-which preserves connections between more out-of-phase oscillators while rewiring connections between more in-phase oscillators-can cause initially disordered networks to organize into more structured topologies that support enhanced synchronization dynamics. We examine how this process unfolds over time, finding a dependence on the intrinsic frequencies of the oscillators, the global coupling, and the network density, in terms of how the adaptive mechanism reorganizes the network and influences the dynamics. Importantly, for large enough coupling and after sufficient adaptation, the resulting networks exhibit interesting characteristics, including degree-frequency and frequency-neighbor frequency correlations. These properties have previously been associated with optimal synchronization or explosive transitions in which the networks were constructed using global information. On the contrary, by considering a time-dependent interplay between structure and dynamics, this work offers a mechanism through which emergent phenomena and organization can arise in complex systems utilizing local rules.
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Affiliation(s)
- Lia Papadopoulos
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Jason Z Kim
- Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research - Telegraphenberg A 31, 14473 Potsdam, Germany
| | - Danielle S Bassett
- Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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14
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Bronski JC, He Y, Li X, Liu Y, Sponseller DR, Wolbert S. The stability of fixed points for a Kuramoto model with Hebbian interactions. CHAOS (WOODBURY, N.Y.) 2017; 27:053110. [PMID: 28576101 DOI: 10.1063/1.4983524] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We consider a variation of the Kuramoto model with dynamic coupling, where the coupling strengths are allowed to evolve in response to the phase difference between the oscillators, a model first considered by Ha, Noh, and Park. We demonstrate that the fixed points of this model, as well as their stability, can be completely expressed in terms of the fixed points and stability of the analogous classical Kuramoto problem where the coupling strengths are fixed to a constant (the same for all edges). In particular, for the "all-to-all" network, where the underlying graph is the complete graph, the problem reduces to the problem of understanding the fixed points and stability of the all-to-all Kuramoto model with equal edge weights, a problem that is well understood.
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Affiliation(s)
- Jared C Bronski
- Department of Mathematics, University of Illinois, 1409 W Green St., Urbana, Illinois 61801, USA
| | - Yizhang He
- Department of Mathematics, University of Illinois, 1409 W Green St., Urbana, Illinois 61801, USA
| | - Xinye Li
- Department of Mathematics, University of Illinois, 1409 W Green St., Urbana, Illinois 61801, USA
| | - Yue Liu
- Department of Mathematics, University of Illinois, 1409 W Green St., Urbana, Illinois 61801, USA
| | - Danielle Rae Sponseller
- Department of Mathematics, University of Illinois, 1409 W Green St., Urbana, Illinois 61801, USA
| | - Seth Wolbert
- Department of Mathematics, University of Illinois, 1409 W Green St., Urbana, Illinois 61801, USA
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Describing synchronization and topological excitations in arrays of magnetic spin torque oscillators through the Kuramoto model. Sci Rep 2016; 6:32528. [PMID: 27580938 PMCID: PMC5007507 DOI: 10.1038/srep32528] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2016] [Accepted: 08/10/2016] [Indexed: 11/08/2022] Open
Abstract
The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength between STO we show that this imposes an upper limit for the maximum number of oscillators that can be synchronized. Further, we show that the lack of long range order is associated with the formation of topological defects in the phase field similar to the two-dimensional XY model of ferromagnetism. Our results shed new light on the synchronization of STO, where controlling the mutual synchronization of several oscillators is considered crucial for applications.
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Ferrari FAS, Viana RL, Lopes SR, Stoop R. Phase synchronization of coupled bursting neurons and the generalized Kuramoto model. Neural Netw 2015; 66:107-18. [PMID: 25828961 DOI: 10.1016/j.neunet.2015.03.003] [Citation(s) in RCA: 45] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2014] [Revised: 02/24/2015] [Accepted: 03/03/2015] [Indexed: 11/30/2022]
Abstract
Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a partially phase-synchronized state as the coupling strength is varied. The numerically determined critical coupling strength value for this transition to occur is compared with theoretical results valid for the generalized Kuramoto model.
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Affiliation(s)
- F A S Ferrari
- Department of Physics, Federal University of Paraná, 81531-990 Curitiba, Paraná, Brazil
| | - R L Viana
- Department of Physics, Federal University of Paraná, 81531-990 Curitiba, Paraná, Brazil.
| | - S R Lopes
- Department of Physics, Federal University of Paraná, 81531-990 Curitiba, Paraná, Brazil
| | - R Stoop
- Institute of Neuroinformatics, University of Zürich and Eidgenössische Technische Hochschule Zürich, 8057 Zürich, Switzerland
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