Modeling High-Dimensional Multichannel Brain Signals

Abstract

Our goal is to model and measure functional and effective (directional) connectivity in multichannel brain physiological signals (e.g., electroencephalograms, local field potentials). The difficulties from analyzing these data mainly come from two aspects: first, there are major statistical and computational challenges for modeling and analyzing high-dimensional multichannel brain signals; second, there is no set of universally agreed measures for characterizing connectivity. To model multichannel brain signals, our approach is to fit a vector autoregressive (VAR) model with potentially high lag order so that complex lead-lag temporal dynamics between the channels can be captured. Estimates of the VAR model will be obtained by our proposed hybrid LASSLE (LASSO \(+\) LSE) method which combines regularization (to control for sparsity) and least squares estimation (to improve bias and mean-squared error). Then we employ some measures of connectivity but put an emphasis on partial directed coherence (PDC) which can capture the directional connectivity between channels. PDC is a frequency-specific measure that explains the extent to which the present oscillatory activity in a sender channel influences the future oscillatory activity in a specific receiver channel relative to all possible receivers in the network. The proposed modeling approach provided key insights into potential functional relationships among simultaneously recorded sites during performance of a complex memory task. Specifically, this novel method was successful in quantifying patterns of effective connectivity across electrode locations, and in capturing how these patterns varied across trial epochs and trial types.

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Notes

  1. 1.

    Assume \(b_k = (b_k^1,\ldots ,b_k^J,b_k^{J+1},\ldots ,b_k^q)^T\), where \(b_k^j \ne 0\) for \(j = 1,\ldots ,J\) and \(b_k^j = 0\) for \(j = J+1,\ldots ,q\). Let \(b_k^{(1)} = (b_k^1,\ldots ,b_k^J)^T\) and \(b_k^{(2)} = (b_k^{J+1},\ldots ,b_k^q)^T\). Denote Gram matrix \(\Psi = \frac{1}{n}{{\mathbb {X}}}'{{\mathbb {X}}}= \begin{pmatrix} \Psi _{11} &{} \Psi _{12}\\ \Psi _{21} &{} \Psi _{22} \end{pmatrix}\), then Irrepresentable Condition is satisfied if there exists a positive constant vector \(\eta \), such that \(|\Psi _{21}(\Psi _{11})^{-1}{{\mathrm{sgn}}}(b_k^{(1)})|\le 1-\eta \).

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Acknowledgements

N.J. Fortin’s research was supported in part by the National Science Foundation (Awards IOS-1150292 and BCS-1439267), the Whitehall Foundation (Award 2010-05-84), and the University of California, Irvine. H. Ombao’s work was supported in part by grants from the US NSF Division of Mathematical Sciences (DMS 15-09023) and the Division of Social and Economic Sciences (SES 14-61534).

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Correspondence to Hernando Ombao.

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Hu, L., Fortin, N.J. & Ombao, H. Modeling High-Dimensional Multichannel Brain Signals. Stat Biosci 11, 91–126 (2019). https://doi.org/10.1007/s12561-017-9210-3

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Keywords

  • Electroencephalograms
  • Local field potentials
  • Brain effective connectivity
  • Multivariate time series
  • Vector autoregressive model
  • Partial directed coherence