Ising model for neural data: Model quality and approximate methods for extracting functional connectivity

    Phys. Rev. E 79, 051915 – Published 19 May 2009
    Yasser Roudi, Joanna Tyrcha, and John Hertz

    Abstract

    We study pairwise Ising models for describing the statistics of multineuron spike trains, using data from a simulated cortical network. We explore efficient ways of finding the optimal couplings in these models and examine their statistical properties. To do this, we extract the optimal couplings for subsets of size up to 200 neurons, essentially exactly, using Boltzmann learning. We then study the quality of several approximate methods for finding the couplings by comparing their results with those found from Boltzmann learning. Two of these methods—inversion of the Thouless-Anderson-Palmer equations and an approximation proposed by Sessak and Monasson—are remarkably accurate. Using these approximations for larger subsets of neurons, we find that extracting couplings using data from a subset smaller than the full network tends systematically to overestimate their magnitude. This effect is described qualitatively by infinite-range spin-glass theory for the normal phase. We also show that a globally correlated input to the neurons in the network leads to a small increase in the average coupling. However, the pair-to-pair variation in the couplings is much larger than this and reflects intrinsic properties of the network. Finally, we study the quality of these models by comparing their entropies with that of the data. We find that they perform well for small subsets of the neurons in the network, but the fit quality starts to deteriorate as the subset size grows, signaling the need to include higher-order correlations to describe the statistics of large networks.

    DOI: http://dx.doi.org/10.1103/PhysRevE.79.051915

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    • Received 18 February 2009
    • Revised 3 April 2009
    • Published 19 May 2009

    © 2009 The American Physical Society

    Authors & Affiliations

    Yasser Roudi1, Joanna Tyrcha2, and John Hertz1,3

    • 1NORDITA, Roslagstullsbacken 23, 10691 Stockholm, Sweden
    • 2Department of Mathematical Statistics, Stockholm University, 10691 Stockholm, Sweden
    • 3The Niels Bohr Institute, Copenhagen University, 2100 Copenhagen Ø, Denmark

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