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Analyzing Dynamical Brain Functional Connectivity As Trajectories on Space of Covariance Matrices

Mengyu Dai, Zhengwu Zhang, Anuj Srivastava

TL;DR

This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs).

Abstract

Human brain functional connectivity (FC) is often measured as the similarity of functional MRI responses across brain regions when a brain is either resting or performing a task. This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs). We use a recently developed metric on the space of SPDMs for quantifying differences across FC observations, and for clustering and classification of FC trajectories. To facilitate large scale and high-dimensional data analysis, we propose a novel, metric-based dimensionality reduction technique to reduce data from large SPDMs to small SPDMs. We illustrate this comprehensive framework using data from the Human Connectome Project (HCP) database for multiple subjects and tasks, with task classification rates that match or outperform state-of-the-art techniques.

Analyzing Dynamical Brain Functional Connectivity As Trajectories on Space of Covariance Matrices

TL;DR

This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs).

Abstract

Human brain functional connectivity (FC) is often measured as the similarity of functional MRI responses across brain regions when a brain is either resting or performing a task. This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs). We use a recently developed metric on the space of SPDMs for quantifying differences across FC observations, and for clustering and classification of FC trajectories. To facilitate large scale and high-dimensional data analysis, we propose a novel, metric-based dimensionality reduction technique to reduce data from large SPDMs to small SPDMs. We illustrate this comprehensive framework using data from the Human Connectome Project (HCP) database for multiple subjects and tasks, with task classification rates that match or outperform state-of-the-art techniques.

Paper Structure

This paper contains 15 sections, 2 theorems, 17 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Covariance Estimation
  4. Riemannian Structure on Symmetric Positive-Definite Matrices (SPDMs)
  5. Riemannian Metric for FC Trajectories
  6. Dimension Reduction for SPDMs
  7. Experiments
  8. Simulation Studies on Dimension Reduction and Alignment
  9. Real Data Experiments
  10. Dynamic FCs under different sets of ROIs and a fixed task
  11. Dynamic FCs under different tasks but for same ROIs
  12. Discussion
  13. Discussion on pseudoinverse of $\hat{P}$
  14. Proof of Lemma \ref{['lemma:lm1']}
  15. Proof of Lemma \ref{['lemma:lm2']}

Key Result

Lemma 1

Under the conditions specified above, we have, for all $i$, $j$,

Figures (8)

  • Figure 1: Determinants of covariance matrices in 80 trajectories from 4 different task activities in the HCP data ($n=4$).
  • Figure 2: Pairwise distances between 200 covariance matrices before and after dimension reduction. (a) shows the distance matrix before dimension reduction, and (b), (c), (d) show the distance matrices after reducing the dimension to d=20, 10 and 5, respectively. (e) shows $\|D - D_d \|_F$vs dimension.
  • Figure 3: Visualization of connectivities using precision matrices estimated by reconstructed SPDMs from different dimensions.
  • Figure 4: The recovered warping functions before and after different levels of dimension reduction. x-axis and y-axis represent time before and after warping respectively.
  • Figure 5: Time cost for calculating distance between two $333 \times 333 \times 20$ trajectories before and after alignment with different degrees of dimension reduction.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2