Article

# Coarse graining the dynamics of coupled oscillators.

Department of Chemical Engineering & Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, NJ 08544, USA.

Physical Review Letters (Impact Factor: 7.94). 05/2006; 96(14):144101. DOI: 10.1103/PhysRevLett.96.144101 Source: PubMed

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**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.Physical Review E 09/2011; 84(3 Pt 2):036708. · 2.31 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider a coupled, heterogeneous population of relaxation oscillators used to model rhythmic oscillations in the pre-Bötzinger complex. By choosing specific values of the parameter used to describe the heterogeneity, sampled from the probability distribution of the values of that parameter, we show how the effects of heterogeneity can be studied in a computationally efficient manner. When more than one parameter is heterogeneous, full or sparse tensor product grids are used to select appropriate parameter values. The method allows us to effectively reduce the dimensionality of the model, and it provides a means for systematically investigating the effects of heterogeneity in coupled systems, linking ideas from uncertainty quantification to those for the study of network dynamics.Journal of mathematical neuroscience. 03/2012; 2(1):5. - [Show abstract] [Hide abstract]

**ABSTRACT:**We show how the Equation Free approach for multi-scale computations can be exploited to extract, in a computational rigorous and systematic way the emergent dynamical attributes, from detailed large-scale microscopic stochastic models of neurons that interact on complex networks. In particular we show how bifurcation, stability analysis and estimation of mean appearance times of rare events can be derived bypassing the need for obtaining analytical approximations, providing an “on-demand” model reduction with respect to the underlying degree distribution.Neurocomputing. 10/2011; 74(17):3576–3589.

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