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: 7.73). 05/2006; 96(14):144101. DOI: 10.1103/PhysRevLett.96.144101 Source: PubMed

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**ABSTRACT:**The formation of oscillating phase clusters in a network of identical Hodgkin–Huxley neurons is studied, along with their dynamic behavior. The neu-rons are synaptically coupled in an all-to-all manner, yet the synaptic coupling char-acteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscil-latory single-cluster state can transition—through N − 1 (possibly perturbed) period-doubling and subsequent bifurcations—to a variety of multiple-cluster states. The clustering dynamic behavior is computationally studied both at the detailed and the coarse-grained levels, and a numerical approach that can enable studying the coarse-grained dynamics in a network of arbitrarily large size is suggested. Among a number of cluster states formed, double clusters, composed of nearly equal sub-network sizes are seen to be stable; interestingly, the heterogeneity parameter in each of the double-cluster components tends to be consistent with the random variable over the entire network: Given a double-cluster state, permuting the dynamical variables of the neu-rons can lead to a combinatorially large number of different, yet similar " fine " states that appear practically identical at the coarse-grained level. For weak heterogeneity we find that correlations rapidly develop, within each cluster, between the neuron's " identity " (its own value of the heterogeneity parameter) and its dynamical state. For single-and double-cluster states we demonstrate an effective coarse-graining ap-S.J. Moon · K.A. Cook · K. Rajendran · I.G. Kevrekidis proach that uses the Polynomial Chaos expansion to succinctly describe the dynam-ics by these quickly established " identity-state " correlations. This coarse-graining approach is utilized, within the equation-free framework, to perform efficient com-putations of the neuron ensemble dynamics.Journal of Mathematical Neuroscience. 01/2015; 5(2). - Ultrasound in Medicine & Biology 08/2011; 37(8). · 2.10 Impact Factor
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**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. · 2.01 Impact Factor

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