• Open Access

Correlated Fluctuations in Strongly Coupled Binary Networks Beyond Equilibrium

David Dahmen, Hannah Bos, and Moritz Helias
Phys. Rev. X 6, 031024 – Published 19 August 2016

Abstract

Randomly coupled Ising spins constitute the classical model of collective phenomena in disordered systems, with applications covering glassy magnetism and frustration, combinatorial optimization, protein folding, stock market dynamics, and social dynamics. The phase diagram of these systems is obtained in the thermodynamic limit by averaging over the quenched randomness of the couplings. However, many applications require the statistics of activity for a single realization of the possibly asymmetric couplings in finite-sized networks. Examples include reconstruction of couplings from the observed dynamics, representation of probability distributions for sampling-based inference, and learning in the central nervous system based on the dynamic and correlation-dependent modification of synaptic connections. The systematic cumulant expansion for kinetic binary (Ising) threshold units with strong, random, and asymmetric couplings presented here goes beyond mean-field theory and is applicable outside thermodynamic equilibrium; a system of approximate nonlinear equations predicts average activities and pairwise covariances in quantitative agreement with full simulations down to hundreds of units. The linearized theory yields an expansion of the correlation and response functions in collective eigenmodes, leads to an efficient algorithm solving the inverse problem, and shows that correlations are invariant under scaling of the interaction strengths.

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  • Received 9 December 2015

DOI:https://doi.org/10.1103/PhysRevX.6.031024

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Physics of Living SystemsNetworksStatistical Physics & Thermodynamics

Authors & Affiliations

David Dahmen1, Hannah Bos1, and Moritz Helias1,2

  • 1Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, 52425 Jülich, Germany
  • 2Department of Physics, Faculty 1, RWTH Aachen University, 52074 Aachen, Germany

Popular Summary

Networks are composed of interacting units whose dynamics influence each other relative to their pairwise couplings. While network activity on the level of populations has been studied for decades, many applications require the statistics of the activity of individual units. However, one major challenge is that the couplings between units are, in general, nonsymmetric, which results in nonequilibrium dynamics that go beyond the common and well-understood setting of statistical mechanics. Here, we derive the statistics for an ensemble of units that forms such a network.

We theoretically examine a model of stochastic binary units whose activity can be characterized by either 0 or 1. In most cases, the diversity of couplings in complex networks introduces large variability in pairwise unit correlations. A theoretical assessment of this diversity is inaccessible with mean-field theory that replaces the input to each unit by a single effective field determined by the statistics of the couplings. Instead, we opt to derive a closed set of equations for a single realization of the couplings that describes local mean activities, pairwise correlations, and response functions for networks of binary units, the prototypical model of collective network dynamics. Unlike prior work, our approach captures effects of finite network size and holds true for arbitrary couplings. We successfully derive the statistics associated with both individual neurons and pairs of neurons.

We expect that our findings will pave the way for studies of correlation-sensitive plasticity in contexts as diverse as learning and neural network representations of probability distributions. The generic nature of binary interacting units as a model for collective dynamics renders our results applicable in a variety of disciplines.

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Vol. 6, Iss. 3 — July - September 2016

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