1
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Papo D, Buldú JM. Does the brain behave like a (complex) network? I. Dynamics. Phys Life Rev 2024; 48:47-98. [PMID: 38145591 DOI: 10.1016/j.plrev.2023.12.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2023] [Accepted: 12/10/2023] [Indexed: 12/27/2023]
Abstract
Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.
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Affiliation(s)
- D Papo
- Department of Neuroscience and Rehabilitation, Section of Physiology, University of Ferrara, Ferrara, Italy; Center for Translational Neurophysiology, Fondazione Istituto Italiano di Tecnologia, Ferrara, Italy.
| | - J M Buldú
- Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, Madrid, Spain
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2
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Piuvezam HC, Marin B, Copelli M, Muñoz MA. Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics. Phys Rev E 2023; 108:034110. [PMID: 37849106 DOI: 10.1103/physreve.108.034110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2023] [Accepted: 08/04/2023] [Indexed: 10/19/2023]
Abstract
The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of such transitions, including continuous ones with scaling behavior belonging to known universality classes-such as directed percolation and tricritical directed percolation-as well as six distinct ones. In particular, we show that under some special circumstances, at a so-called "Hopf tricritical directed percolation" transition, rather unconventional behavior is observed, including the emergence of scaling breakdown. Other transitions are discontinuous and show different types of anomalies in scaling and/or exhibit mixed features of continuous and discontinuous transitions. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are, thus, of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of nonequilibrium phase transitions into quiescent or absorbing states.
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Affiliation(s)
| | - Bóris Marin
- Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, São Bernardo do Campo, Brazil
| | - Mauro Copelli
- Departamento de Física, Universidade Federal de Pernambuco, Recife PE 50670-901, Brazil
| | - Miguel A Muñoz
- Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada, Spain
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3
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Rocha RP, Koçillari L, Suweis S, De Filippo De Grazia M, de Schotten MT, Zorzi M, Corbetta M. Recovery of neural dynamics criticality in personalized whole-brain models of stroke. Nat Commun 2022; 13:3683. [PMID: 35760787 PMCID: PMC9237050 DOI: 10.1038/s41467-022-30892-6] [Citation(s) in RCA: 12] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2020] [Accepted: 05/16/2022] [Indexed: 01/13/2023] Open
Abstract
The critical brain hypothesis states that biological neuronal networks, because of their structural and functional architecture, work near phase transitions for optimal response to internal and external inputs. Criticality thus provides optimal function and behavioral capabilities. We test this hypothesis by examining the influence of brain injury (strokes) on the criticality of neural dynamics estimated at the level of single participants using directly measured individual structural connectomes and whole-brain models. Lesions engender a sub-critical state that recovers over time in parallel with behavior. The improvement of criticality is associated with the re-modeling of specific white-matter connections. We show that personalized whole-brain dynamical models poised at criticality track neural dynamics, alteration post-stroke, and behavior at the level of single participants. The authors investigate the influence of brain injury (strokes) on the criticality of neural dynamics using directly measured connectomes and whole-brain models. They show that lesions engender a sub-critical state that recovers over time in parallel with behavior.
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Affiliation(s)
- Rodrigo P Rocha
- Departamento de Física, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, SC, Brazil. .,Department of Physics, School of Philosophy, Sciences and Letters of Ribeirão Preto, University of São Paulo, Ribeirão Preto, SP, Brazil. .,Padova Neuroscience Center, Università di Padova, Padova, Italy.
| | - Loren Koçillari
- Padova Neuroscience Center, Università di Padova, Padova, Italy.,Laboratory of Neural Computation, Istituto Italiano di Tecnologia, 38068, Rovereto, Italy.,Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, I-35131, Padova, Italy
| | - Samir Suweis
- Padova Neuroscience Center, Università di Padova, Padova, Italy.,Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, I-35131, Padova, Italy
| | | | - Michel Thiebaut de Schotten
- Brain Connectivity and Behaviour Laboratory, BCBlab, Sorbonne Universities, Paris, France.,Groupe d'Imagerie Neurofonctionnelle, Institut des Maladies Neurodégénératives-UMR 5293, CNRS, CEA University of Bordeaux, Bordeaux, France
| | - Marco Zorzi
- IRCCS San Camillo Hospital, Venice, Italy.,Dipartimento di Psicologia Generale, Università di Padova, Padova, Italy
| | - Maurizio Corbetta
- Padova Neuroscience Center, Università di Padova, Padova, Italy.,Dipartimento di Neuroscienze, Università di Padova, Padova, Italy.,Venetian Institute of Molecular Medicine (VIMM), Fondazione Biomedica, Padova, Italy
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4
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Diaz MMS, Trejo EJA, Martin DA, Cannas SA, Grigera TS, Chialvo DR. Similar local neuronal dynamics may lead to different collective behavior. Phys Rev E 2021; 104:064309. [PMID: 35030861 DOI: 10.1103/physreve.104.064309] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2021] [Accepted: 12/10/2021] [Indexed: 11/07/2022]
Abstract
This report is concerned with the relevance of the microscopic rules that implement individual neuronal activation, in determining the collective dynamics, under variations of the network topology. To fix ideas we study the dynamics of two cellular automaton models, commonly used, rather in-distinctively, as the building blocks of large-scale neuronal networks. One model, due to Greenberg and Hastings (GH), can be described by evolution equations mimicking an integrate-and-fire process, while the other model, due to Kinouchi and Copelli (KC), represents an abstract branching process, where a single active neuron activates a given number of postsynaptic neurons according to a prescribed "activity" branching ratio. Despite the apparent similarity between the local neuronal dynamics of the two models, it is shown that they exhibit very different collective dynamics as a function of the network topology. The GH model shows qualitatively different dynamical regimes as the network topology is varied, including transients to a ground (inactive) state, continuous and discontinuous dynamical phase transitions. In contrast, the KC model only exhibits a continuous phase transition, independently of the network topology. These results highlight the importance of paying attention to the microscopic rules chosen to model the interneuronal interactions in large-scale numerical simulations, in particular when the network topology is far from a mean-field description. One such case is the extensive work being done in the context of the Human Connectome, where a wide variety of types of models are being used to understand the brain collective dynamics.
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Affiliation(s)
- Margarita M Sánchez Diaz
- Center for Complex Systems and Brain Sciences (CEMSC), Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Instituto de Ciencias Físicas (ICIFI), CONICET and Universidad Nacional de San Martín, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina
| | - Eyisto J Aguilar Trejo
- Center for Complex Systems and Brain Sciences (CEMSC), Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Instituto de Ciencias Físicas (ICIFI), CONICET and Universidad Nacional de San Martín, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290, 1425 Buenos Aires, Argentina
| | - Daniel A Martin
- Center for Complex Systems and Brain Sciences (CEMSC), Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Instituto de Ciencias Físicas (ICIFI), CONICET and Universidad Nacional de San Martín, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290, 1425 Buenos Aires, Argentina
| | - Sergio A Cannas
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290, 1425 Buenos Aires, Argentina.,Instituto de Física Enrique Gaviola (IFEG-CONICET), Facultad de Matemática Astronomía Física y Computación, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
| | - Tomás S Grigera
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290, 1425 Buenos Aires, Argentina.,Instituto de Física de Líquidos y Sistemas Biológicos (IFLySiB), CONICET and Universidad Nacional de La Plata, Calle 59 no. 789, B1900BTE La Plata, Buenos Aires, Argentina.,Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 1900 La Plata, Buenos Aires, Argentina
| | - Dante R Chialvo
- Center for Complex Systems and Brain Sciences (CEMSC), Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Instituto de Ciencias Físicas (ICIFI), CONICET and Universidad Nacional de San Martín, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina.,Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290, 1425 Buenos Aires, Argentina
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5
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Safari A, Moretti P, Diez I, Cortes JM, Muñoz MA. Persistence of hierarchical network organization and emergent topologies in models of functional connectivity. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.02.096] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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6
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The effect of noise on the synchronization dynamics of the Kuramoto model on a large human connectome graph. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2020.04.161] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
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7
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Abstract
Concepts shape the interpretation of facts. One of the most popular concepts in systems neuroscience is that of ‘hierarchy’. However, this concept has been interpreted in many different ways, which are not well aligned. This observation suggests that the concept is ill defined. Using the example of the organization of the primate visual cortical system, we explore several contexts in which ‘hierarchy’ is currently used in the description of brain networks. We distinguish at least four different uses, specifically, ‘hierarchy’ as a topological sequence of projections, as a gradient of features, as a progression of scales, or as a sorting of laminar projection patterns. We discuss the interpretation and functional implications of the different notions of ‘hierarchy’ in these contexts and suggest that more specific terms than ‘hierarchy’ should be used for a deeper understanding of the different dimensions of the organization of brain networks. This article is part of the theme issue ‘Unifying the essential concepts of biological networks: biological insights and philosophical foundations’.
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Affiliation(s)
- Claus C Hilgetag
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany.,Department of Health Sciences, Boston University, Boston, MA, USA
| | - Alexandros Goulas
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
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8
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Ódor G, Kelling J. Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs. Sci Rep 2019; 9:19621. [PMID: 31873076 PMCID: PMC6928153 DOI: 10.1038/s41598-019-54769-9] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2019] [Accepted: 11/15/2019] [Indexed: 11/19/2022] Open
Abstract
The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τt ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τt ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.
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Affiliation(s)
- Géza Ódor
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O.Box 49, H-1525, Budapest, Hungary
| | - Jeffrey Kelling
- Department of Information Services and Computing, Helmholtz-Zentrum Dresden - Rossendorf, P.O.Box 51 01 19, 01314, Dresden, Germany.
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9
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Zarepour M, Perotti JI, Billoni OV, Chialvo DR, Cannas SA. Universal and nonuniversal neural dynamics on small world connectomes: A finite-size scaling analysis. Phys Rev E 2019; 100:052138. [PMID: 31870025 DOI: 10.1103/physreve.100.052138] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2019] [Indexed: 06/10/2023]
Abstract
Evidence of critical dynamics has been found recently in both experiments and models of large-scale brain dynamics. The understanding of the nature and features of such a critical regime is hampered by the relatively small size of the available connectome, which prevents, among other things, the determination of its associated universality class. To circumvent that, here we study a neural model defined on a class of small-world networks that share some topological features with the human connectome. We find that varying the topological parameters can give rise to a scale-invariant behavior either belonging to the mean-field percolation universality class or having nonuniversal critical exponents. In addition, we find certain regions of the topological parameter space where the system presents a discontinuous, i.e., noncritical, dynamical phase transition into a percolated state. Overall, these results shed light on the interplay of dynamical and topological roots of the complex brain dynamics.
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Affiliation(s)
- Mahdi Zarepour
- Instituto de Física Enrique Gaviola, CONICET, Ciudad Universitaria, 5000 Córdoba, Córdoba, Argentina
| | - Juan I Perotti
- Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, Instituto de Física Enrique Gaviola, CONICET, Ciudad Universitaria, 5000 Córdoba, Córdoba, Argentina
| | - Orlando V Billoni
- Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, Instituto de Física Enrique Gaviola, CONICET, Ciudad Universitaria, 5000 Córdoba, Córdoba, Argentina
| | - Dante R Chialvo
- Center for Complex Systems and Brain Sciences, Instituto de Ciencias Físicas, Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia, 1650 San Martín, Buenos Aires, Argentina
| | - Sergio A Cannas
- Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, Instituto de Física Enrique Gaviola, CONICET, Ciudad Universitaria, 5000 Córdoba, Córdoba, Argentina
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10
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Ódor G. Robustness of Griffiths effects in homeostatic connectome models. Phys Rev E 2019; 99:012113. [PMID: 30780274 DOI: 10.1103/physreve.99.012113] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2018] [Indexed: 01/08/2023]
Abstract
I provide numerical evidence for the robustness of the Griffiths phase (GP) reported previously in dynamical threshold model simulations on a large human brain network with N=836733 connected nodes. The model, with equalized network sensitivity, is extended in two ways: introduction of refractory states or by randomized time-dependent thresholds. The nonuniversal power-law dynamics in an extended control parameter region survives these modifications for a short refractory state and weak disorder. In case of temporal disorder the GP shrinks and for stronger heterogeneity disappears, leaving behind a mean-field type of critical transition. Activity avalanche size distributions below the critical point decay faster than in the original model, but the addition of inhibitory interactions sets it back to the range of experimental values.
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Affiliation(s)
- Géza Ódor
- Research Institute for Technical Physics and Materials Science, Centre for Energy Research of the Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
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11
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Rocha RP, Koçillari L, Suweis S, Corbetta M, Maritan A. Homeostatic plasticity and emergence of functional networks in a whole-brain model at criticality. Sci Rep 2018; 8:15682. [PMID: 30356174 PMCID: PMC6200722 DOI: 10.1038/s41598-018-33923-9] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2018] [Accepted: 09/27/2018] [Indexed: 11/09/2022] Open
Abstract
Understanding the relationship between large-scale structural and functional brain networks remains a crucial issue in modern neuroscience. Recently, there has been growing interest in investigating the role of homeostatic plasticity mechanisms, across different spatiotemporal scales, in regulating network activity and brain functioning against a wide range of environmental conditions and brain states (e.g., during learning, development, ageing, neurological diseases). In the present study, we investigate how the inclusion of homeostatic plasticity in a stochastic whole-brain model, implemented as a normalization of the incoming node's excitatory input, affects the macroscopic activity during rest and the formation of functional networks. Importantly, we address the structure-function relationship both at the group and individual-based levels. In this work, we show that normalization of the node's excitatory input improves the correspondence between simulated neural patterns of the model and various brain functional data. Indeed, we find that the best match is achieved when the model control parameter is in its critical value and that normalization minimizes both the variability of the critical points and neuronal activity patterns among subjects. Therefore, our results suggest that the inclusion of homeostatic principles lead to more realistic brain activity consistent with the hallmarks of criticality. Our theoretical framework open new perspectives in personalized brain modeling with potential applications to investigate the deviation from criticality due to structural lesions (e.g. stroke) or brain disorders.
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Affiliation(s)
- Rodrigo P Rocha
- Department of Physics, School of Philosophy, Sciences and Letters of Ribeirão Preto, University of São Paulo, Ribeirão Preto, SP, Brazil. .,Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, I-35131, Padova, Italy. .,Padova Neuroscience Center, Università di Padova, Padova, Italy.
| | - Loren Koçillari
- Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, I-35131, Padova, Italy.,Padova Neuroscience Center, Università di Padova, Padova, Italy
| | - Samir Suweis
- Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, I-35131, Padova, Italy.,Padova Neuroscience Center, Università di Padova, Padova, Italy
| | - Maurizio Corbetta
- Padova Neuroscience Center, Università di Padova, Padova, Italy.,Dipartimento di Neuroscienze, Università di Padova, Padova, Italy.,Departments of Neurology, Radiology, Neuroscience, and Bioengineering, Washington University, School of Medicine, St. Louis, USA
| | - Amos Maritan
- Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, I-35131, Padova, Italy.,Padova Neuroscience Center, Università di Padova, Padova, Italy
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12
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Ódor G, Hartmann B. Heterogeneity effects in power grid network models. Phys Rev E 2018; 98:022305. [PMID: 30253599 DOI: 10.1103/physreve.98.022305] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2018] [Indexed: 06/08/2023]
Abstract
We have compared the phase synchronization transition of the second-order Kuramoto model on two-dimensional (2D) lattices and on large, synthetic power grid networks, generated from real data. The latter are weighted, hierarchical modular networks. Due to the inertia the synchronization transitions are of first-order type, characterized by fast relaxation and hysteresis by varying the global coupling parameter K. Finite-size scaling analysis shows that there is no real phase transition in the thermodynamic limit, unlike in the mean-field model. The order parameter and its fluctuations depend on the network size without any real singular behavior. In case of power grids the phase synchronization breaks down at lower global couplings, than in case of 2D lattices of the same sizes, but the hysteresis is much narrower or negligible due to the low connectivity of the graphs. The temporal behavior of desynchronization avalanches after a sudden quench to low K values has been followed and duration distributions with power-law tails have been detected. This suggests rare region effects, caused by frozen disorder, resulting in heavy-tailed distributions, even without a self-organization mechanism as a consequence of a catastrophic drop event in the couplings.
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Affiliation(s)
- Géza Ódor
- Centre for Energy Research of the Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Bálint Hartmann
- Centre for Energy Research of the Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
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13
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Cota W, Ódor G, Ferreira SC. Griffiths phases in infinite-dimensional, non-hierarchical modular networks. Sci Rep 2018; 8:9144. [PMID: 29904065 PMCID: PMC6002411 DOI: 10.1038/s41598-018-27506-x] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2018] [Accepted: 05/31/2018] [Indexed: 11/28/2022] Open
Abstract
Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effective rare-regions, slowing down the extinction of activation. As a consequence, we find extended control parameter regions with continuously changing dynamical exponents for single network realizations, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs.
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Affiliation(s)
- Wesley Cota
- Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, Minas Gerais, Brazil.
| | - Géza Ódor
- MTA-EK-MFA, Centre for Energy Research of the Hungarian Academy of Sciences, H-1121, P.O. Box 49, Budapest, Hungary
| | - Silvio C Ferreira
- Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, Minas Gerais, Brazil.,National Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil
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14
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Messé A, Hütt MT, Hilgetag CC. Toward a theory of coactivation patterns in excitable neural networks. PLoS Comput Biol 2018; 14:e1006084. [PMID: 29630592 PMCID: PMC5908206 DOI: 10.1371/journal.pcbi.1006084] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2017] [Revised: 04/19/2018] [Accepted: 03/11/2018] [Indexed: 11/18/2022] Open
Abstract
The relationship between the structural connectivity (SC) and functional connectivity (FC) of neural systems is of central importance in brain network science. It is an open question, however, how the SC-FC relationship depends on specific topological features of brain networks or the models used for describing neural dynamics. Using a basic but general model of discrete excitable units that follow a susceptible—excited—refractory activity cycle (SER model), we here analyze how the network activity patterns underlying functional connectivity are shaped by the characteristic topological features of the network. We develop an analytical framework for describing the contribution of essential topological elements, such as common inputs and pacemakers, to the coactivation of nodes, and demonstrate the validity of the approach by comparison of the analytical predictions with numerical simulations of various exemplar networks. The present analytic framework may serve as an initial step for the mechanistic understanding of the contributions of brain network topology to brain dynamics. Functional connectivity, as reflected in the statistical dependencies of distributed activity, is widely used to probe the organization of complex systems such as the brain. While this measure has been helpful for characterizing brain states and highlighting alterations of brain dynamics in various diseases, the mechanisms underlying the generation of FC patterns remain poorly understood. One prominent factor shaping FC is the underlying neural network structure. Using a minimalist model of excitation, we investigate how the topology of excitable neural networks contributes to FC. Specifically, we show that FC can be analytically predicted from the way in which the nodes are embedded in the network and how they are related to basic self-organizing units of excitable dynamics, particularly, short pacemaker cycles. These insights are a step towards a mechanistic understanding of the activation patterns of complex neural networks.
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Affiliation(s)
- Arnaud Messé
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
| | - Marc-Thorsten Hütt
- Department of Life Sciences and Chemistry, Jacobs University, Bremen, Germany
| | - Claus C. Hilgetag
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
- Department of Health Sciences, Boston University, Boston, Massachusetts, United States of America
- * E-mail:
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15
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The topology of large Open Connectome networks for the human brain. Sci Rep 2016; 6:27249. [PMID: 27270602 PMCID: PMC4895133 DOI: 10.1038/srep27249] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Accepted: 05/12/2016] [Indexed: 11/18/2022] Open
Abstract
The structural human connectome (i.e. the network of fiber connections in the brain) can be analyzed at ever finer spatial resolution thanks to advances in neuroimaging. Here we analyze several large data sets for the human brain network made available by the Open Connectome Project. We apply statistical model selection to characterize the degree distributions of graphs containing up to nodes and edges. A three-parameter generalized Weibull (also known as a stretched exponential) distribution is a good fit to most of the observed degree distributions. For almost all networks, simple power laws cannot fit the data, but in some cases there is statistical support for power laws with an exponential cutoff. We also calculate the topological (graph) dimension D and the small-world coefficient σ of these networks. While σ suggests a small-world topology, we found that D < 4 showing that long-distance connections provide only a small correction to the topology of the embedding three-dimensional space.
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