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

## TL;DR

This paper introduces a novel framework for analyzing dynamic brain functional connectivity by representing it as trajectories on the space of covariance matrices, using advanced metrics and dimensionality reduction, achieving high classification accuracy.

## Contribution

It develops a new metric-based approach for analyzing FC trajectories on the space of covariance matrices and introduces a dimensionality reduction technique for large-scale data.

## Key findings

- Task classification rates match or outperform state-of-the-art methods.
- The framework effectively captures dynamic FC as trajectories on SPDMs.
- Proposed methods are validated on Human Connectome Project data.

## 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.