EiDA: A Lossless Approach for the Dynamic Analysis of Connectivity Patterns in Signals; Application to Resting State fMRI of a Model of Ageing

Complexity science offers a framework for analysing high-dimensional, non-linear interacting systems such as financial markets or activity in the brain, to extract meaningful dynamic information for decision-making or scientific enquiry. By virtue of the data involved, various analytical methods are required for dimensionality reduction, clustering, discrete analysis, continuous flow analysis, and for estimations of complexity. We introduce EiDA (Eigenvector Dynamic Analysis), a closed form analytical methodology to losslessly extract dynamical functional connectivity (dFC) information from instantaneous phase-locking matrices (iPL). EiDA builds on the existing LEiDA approach (Leading Eigenvector Dynamic Analysis), by showing that the iPL matrix is of rank 2, and can thus be completely characterised by two eigenvectors. We give a full analytical derivation of the eigenvectors and their associated eigenvalues. As a second step we propose two alternatives to analyze the time evolution of the iPL matrix or equivalently of instantaneous connectivity patterns: i) Discrete EiDA, which identifies a discrete set of phase locking states using k-means clustering on the decomposed iPL matrices, and ii) Continuous EiDA, which introduces a 2-dimensional position and reconfiguration speed representation of the eigenvectors. In Continuous EiDA, dynamic Functional Connectivity is conceived as a continuous exploration of this 2-D space. Finally, we show that the two non-trivial eigenvalues are interdependent as their sum is equal to the number of signal channels, and define spectral metastability as the standard deviation of the the spectral radius, the first eigenvalue. Finally, we compute informational complexity using the Lempel-Ziv-Welch algorithm. We apply EiDA to a dataset comprising a cohort of M=48 rats among N=44 brain regions, scanned with functional magnetic resonance imaging (fMRI) at T=4 stages during a study of ageing. We previously found that static functional connectivity declined with age. In dFC, we found that using only the leading eigenvector resulted in the loss of dFC information, and that this was exacerbated with ageing. Additionally, we found that while k-means clustering did not yield satisfactory partitioning, continuous EiDA provided a marker for ageing. Specifically we found that reconfiguration speed of the first eigenvector increased significantly over the life-span concurrent with a reduction in spectral metastability. In addition, we found an increase in informational complexity with age, and that this complexity was highly, significantly and inversely correlated (R = 0.95, p < 0.001) with the magnitude of the first eigenvalue of the iPL matrix. Finally, the computation time for EiDA outperforms numerical spectral decomposition algorithms: 2 orders of magnitude faster for 100×100 matrices, and 3 orders of magnitude faster for 10,000×10,000 matrices. EiDA provides an analytically principled method to extract connectivity (relationship) information without loss from high-dimensional time-series, establishes a link between dynamical systems and information complexity in resting-state neuroimaging, and significantly reduces computational time for high-dimensional data.