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Shen C, Chen H, Hou Z, Kurths J. Hybrid multiscale coarse-graining for dynamics on complex networks. Chaos 2018; 28:123122. [PMID: 30599515 DOI: 10.1063/1.5048962] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2018] [Accepted: 12/03/2018] [Indexed: 06/09/2023]
Abstract
We propose a hybrid multiscale coarse-grained (HMCG) method which combines a fine Monte Carlo (MC) simulation on the part of nodes of interest with a more coarse Langevin dynamics on the rest part. We demonstrate the validity of our method by analyzing the equilibrium Ising model and the nonequilibrium susceptible-infected-susceptible model. It is found that HMCG not only works very well in reproducing the phase transitions and critical phenomena of the microscopic models, but also accelerates the evaluation of dynamics with significant computational savings compared to microscopic MC simulations directly for the whole networks. The proposed method is general and can be applied to a wide variety of networked systems just adopting appropriate microscopic simulation methods and coarse graining approaches.
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Affiliation(s)
- Chuansheng Shen
- School of Mathematics and Computational Science, Anqing Normal University, Anqing 246011, China
| | - Hanshuang Chen
- School of Physics and Materials Science, Anhui University, Hefei 230039, China
| | - Zhonghuai Hou
- Hefei National Laboratory for Physical Sciences at Microscales and Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany
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Bertalan T, Wu Y, Laing C, Gear CW, Kevrekidis IG. Coarse-Grained Descriptions of Dynamics for Networks with Both Intrinsic and Structural Heterogeneities. Front Comput Neurosci 2017; 11:43. [PMID: 28659781 PMCID: PMC5467358 DOI: 10.3389/fncom.2017.00043] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2017] [Accepted: 05/12/2017] [Indexed: 11/23/2022] Open
Abstract
Finding accurate reduced descriptions for large, complex, dynamically evolving networks is a crucial enabler to their simulation, analysis, and ultimately design. Here, we propose and illustrate a systematic and powerful approach to obtaining good collective coarse-grained observables—variables successfully summarizing the detailed state of such networks. Finding such variables can naturally lead to successful reduced dynamic models for the networks. The main premise enabling our approach is the assumption that the behavior of a node in the network depends (after a short initial transient) on the node identity: a set of descriptors that quantify the node properties, whether intrinsic (e.g., parameters in the node evolution equations) or structural (imparted to the node by its connectivity in the particular network structure). The approach creates a natural link with modeling and “computational enabling technology” developed in the context of Uncertainty Quantification. In our case, however, we will not focus on ensembles of different realizations of a problem, each with parameters randomly selected from a distribution. We will instead study many coupled heterogeneous units, each characterized by randomly assigned (heterogeneous) parameter value(s). One could then coin the term Heterogeneity Quantification for this approach, which we illustrate through a model dynamic network consisting of coupled oscillators with one intrinsic heterogeneity (oscillator individual frequency) and one structural heterogeneity (oscillator degree in the undirected network). The computational implementation of the approach, its shortcomings and possible extensions are also discussed.
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Affiliation(s)
- Tom Bertalan
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States
| | - Yan Wu
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States.,Department of Bioengineering, University of California, La JollaLa Jolla, CA, United States
| | - Carlo Laing
- Institute of Natural and Mathematical Sciences, Massey UniversityAuckland, New Zealand
| | - C William Gear
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States
| | - Ioannis G Kevrekidis
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States.,Program in Applied and Computational Mathematics, Princeton UniversityPrinceton, NJ, United States
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Ashwin P, Coombes S, Nicks R. Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience. J Math Neurosci 2016; 6:2. [PMID: 26739133 PMCID: PMC4703605 DOI: 10.1186/s13408-015-0033-6] [Citation(s) in RCA: 103] [Impact Index Per Article: 12.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Accepted: 10/30/2015] [Indexed: 05/20/2023]
Abstract
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.
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Affiliation(s)
- Peter Ashwin
- Centre for Systems Dynamics and Control, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, Exeter, EX4 4QF, UK.
| | - Stephen Coombes
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
| | - Rachel Nicks
- School of Mathematics, University of Birmingham, Watson Building, Birmingham, B15 2TT, UK.
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Siettos C, Starke J. Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools. Wiley Interdiscip Rev Syst Biol Med 2016; 8:438-58. [PMID: 27340949 DOI: 10.1002/wsbm.1348] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2016] [Revised: 05/01/2016] [Accepted: 05/14/2016] [Indexed: 11/09/2022]
Abstract
The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.
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Affiliation(s)
- Constantinos Siettos
- School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Athens, Greece
| | - Jens Starke
- School of Mathematical Sciences, Queen Mary University of London, London, UK
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Marschler C, Starke J, Liu P, Kevrekidis IG. Coarse-grained particle model for pedestrian flow using diffusion maps. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 89:013304. [PMID: 24580357 DOI: 10.1103/physreve.89.013304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2013] [Indexed: 06/03/2023]
Abstract
Interacting particle systems constitute the dynamic model of choice in a variety of application areas. A prominent example is pedestrian dynamics, where good design of escape routes for large buildings and public areas can improve evacuation in emergency situations, avoiding exit blocking and the ensuing panic. Here we employ diffusion maps to study the coarse-grained dynamics of two pedestrian crowds trying to pass through a door from opposite sides. These macroscopic variables and the associated smooth embeddings lead to a better description and a clearer understanding of the nature of the transition to oscillatory dynamics. We also compare the results to those obtained through intuitively chosen macroscopic variables.
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Affiliation(s)
- Christian Marschler
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, DK-2800 Kongens Lyngby, Denmark
| | - Jens Starke
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, DK-2800 Kongens Lyngby, Denmark
| | - Ping Liu
- Department for Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Ioannis G Kevrekidis
- Department for Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Spiliotis KG, Siettos CI. A timestepper-based approach for the coarse-grained analysis of microscopic neuronal simulators on networks: Bifurcation and rare-events micro- to macro-computations. Neurocomputing 2011; 74:3576-89. [DOI: 10.1016/j.neucom.2011.06.018] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Rajendran K, Kevrekidis IG. Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 84:036708. [PMID: 22060530 DOI: 10.1103/physreve.84.036708] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2011] [Revised: 07/07/2011] [Indexed: 05/31/2023]
Abstract
We present a computer-assisted approach to coarse graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low dimensional. Our first choice of coarse variables consists of the components of the oscillator states--their (complex) phase angles--along the leading eigenvectors of this Laplacian. We then use the equation-free framework, circumventing the derivation of explicit coarse-grained equations, to perform computational tasks such as coarse projective integration, coarse fixed-point, and coarse limit-cycle computations. In a second step, we explore an approach to incorporating oscillator heterogeneity in the coarse-graining process. The approach is based on the observation of fast-developing correlations between oscillator state and oscillator intrinsic properties and establishes a connection with tools developed in the context of uncertainty quantification.
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Affiliation(s)
- Karthikeyan Rajendran
- Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA.
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Shen C, Chen H, Hou Z, Xin H. Coarse-grained Monte Carlo simulations of the phase transition of the Potts model on weighted networks. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 83:066109. [PMID: 21797443 DOI: 10.1103/physreve.83.066109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2011] [Revised: 05/11/2011] [Indexed: 05/31/2023]
Abstract
Developing an effective coarse-grained (CG) approach is a promising way for studying dynamics on large size networks. In the present work, we have proposed a strength-based CG (s-CG) method to study critical phenomena of the Potts model on weighted complex networks. By merging nodes with close strengths together, the original network is reduced to a CG network with much smaller size, on which the CG Hamiltonian can be well defined. In particular, we make an error analysis and show that our s-CG approach satisfies the condition of statistical consistency, which demands that the equilibrium probability distribution of the CG model matches that of the microscopic counterpart. Extensive numerical simulations are performed on scale-free networks and random networks, without or with strength correlation, showing that this s-CG approach works very well in reproducing the phase diagrams, fluctuations, and finite-size effects of the microscopic model, while the d-CG approach proposed in our recent work [Phys. Rev. E 82, 011107 (2010)] does not.
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Affiliation(s)
- Chuansheng Shen
- Hefei National Laboratory for Physical Sciences at Microscales & Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China
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Chen H, Hou Z, Xin H, Yan Y. Statistically consistent coarse-grained simulations for critical phenomena in complex networks. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 82:011107. [PMID: 20866565 DOI: 10.1103/physreve.82.011107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2010] [Revised: 05/17/2010] [Indexed: 05/29/2023]
Abstract
We propose a degree-based coarse-graining approach that not just accelerates the evaluation of dynamics on complex networks, but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows. For the Ising model and susceptible-infected-susceptible epidemic model, we introduce these required conditions explicitly and further prove that they are satisfied by our coarse-grained network construction within the annealed network approximation. Finally, we numerically show that the phase transitions and fluctuations on the coarse-grained network are all in good agreements with those on the original one.
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Affiliation(s)
- Hanshuang Chen
- Department of Chemical Physics, Hefei National Laboratory for Physical Sciences at Microscales, University of Science and Technology of China, Hefei, Anhui 230026, China
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Bold KA, Zou Y, Kevrekidis IG, Henson MA. An equation-free approach to analyzing heterogeneous cell population dynamics. J Math Biol 2007; 55:331-52. [PMID: 17429634 DOI: 10.1007/s00285-007-0086-6] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2006] [Revised: 03/02/2007] [Indexed: 10/23/2022]
Abstract
We propose a computational approach to modeling the collective dynamics of populations of coupled, heterogeneous biological oscillators. We consider the synchronization of yeast glycolytic oscillators coupled by the membrane exchange of an intracellular metabolite; the heterogeneity consists of a single random parameter, which accounts for glucose influx into each cell. In contrast to Monte Carlo simulations, distributions of intracellular species of these yeast cells are represented by a few leading order generalized Polynomial Chaos (gPC) coefficients, thus reducing the dynamics of an ensemble of oscillators to dynamics of their (typically significantly fewer) representative gPC coefficients. Equation-free (EF) methods are employed to efficiently evolve this coarse description in time and compute the coarse-grained stationary state and/or limit cycle solutions, circumventing the derivation of explicit, closed-form evolution equations. Coarse projective integration and fixed-point algorithms are used to compute collective oscillatory solutions for the cell population and quantify their stability. These techniques are extended to the special case of a "rogue" oscillator; a cell sufficiently different from the rest "escapes" the bulk synchronized behavior and oscillates with a markedly different amplitude. The approach holds promise for accelerating the computer-assisted analysis of detailed models of coupled heterogeneous cell or agent populations.
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Affiliation(s)
- Katherine A Bold
- Department of Chemical Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
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Moon SJ, Nabet B, Leonard NE, Levin SA, Kevrekidis IG. Heterogeneous animal group models and their group-level alignment dynamics: an equation-free approach. J Theor Biol 2006; 246:100-12. [PMID: 17258774 DOI: 10.1016/j.jtbi.2006.12.018] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2006] [Revised: 12/08/2006] [Accepted: 12/11/2006] [Indexed: 11/25/2022]
Abstract
We study coarse-grained (group-level) alignment dynamics of individual-based animal group models for heterogeneous populations consisting of informed (on preferred directions) and uninformed individuals. The orientation of each individual is characterized by an angle, whose dynamics are nonlinearly coupled with those of all the other individuals, with an explicit dependence on the difference between the individual's orientation and the instantaneous average direction. Choosing convenient coarse-grained variables (suggested by uncertainty quantification methods) that account for rapidly developing correlations during initial transients, we perform efficient computations of coarse-grained steady states and their bifurcation analysis. We circumvent the derivation of coarse-grained governing equations, following an equation-free computational approach.
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Affiliation(s)
- Sung Joon Moon
- Department of Chemical Engineering & Program in Applied and Computational Mathematics (PACM), USA
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