EiDA: A lossless approach for the dynamic analysis of connectivity patterns in signals; application to resting state fMRI of a model of ageing

Giuseppe De Alteriis, Eilidh MacNicol, Fran Hancock, Diana Cash, Paul Expert, Federico Turkheimer

Abstract:

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 100x100 matrices, and 3 orders of magnitude faster for 10,000x10,000 matrices.