Eigenvector Dynamics Analysis (EiDA): a mathematical framework for dynamic Functional Connectivity
Poster No:
1545
Submission Type:
Abstract Submission
Authors:
Giuseppe de Alteriis1, Oliver Sherwood2, Alessandro Ciaramella3, Robert Leech4, Federico Turkheimer5, Paul Expert6
Institutions:
1King's College London, London, Greater London, 2Kings College London, London, London, 3Scuola Superiore Sant'Anna, Pisa, Toscana, 4King's College London, London, United Kingdom, 5King's College London, London, London, 6University College London, London, Greater London
First Author:
Co-Author(s):
Introduction:
A key property of brain-wide networks is the dynamic nature of the interactions between their nodes. This is what the field of dynamic Functional Connectivity (dFC) investigates (Hutchison et al., 2013).
dFC is, in general terms, the analysis of a matrix that evolves with time dFC(t). Commonly used dFC(t) matrices are the sliding window correlation/covariances (Allen et al., 2014), coactivations (Esfahlani et al. 2020), or instantaneous Phase Locking (Cabral et al., 2017). It is also common to then extract the eigenvectors of dFC(t), perform clustering on dFC(t) and/or calculate parameters such as average connectivity or entropy.
However, different dFC approaches lack a unified mathematical framework, in which to perform the operations mentioned above. We introduce here Eigenvector Dynamics Analysis (EiDA) which is a unified mathematical framework that allows theoretically sound and fast analysis of dFC(t) data and exact calculations of derived dynamic parameters.
dFC is, in general terms, the analysis of a matrix that evolves with time dFC(t). Commonly used dFC(t) matrices are the sliding window correlation/covariances (Allen et al., 2014), coactivations (Esfahlani et al. 2020), or instantaneous Phase Locking (Cabral et al., 2017). It is also common to then extract the eigenvectors of dFC(t), perform clustering on dFC(t) and/or calculate parameters such as average connectivity or entropy.
However, different dFC approaches lack a unified mathematical framework, in which to perform the operations mentioned above. We introduce here Eigenvector Dynamics Analysis (EiDA) which is a unified mathematical framework that allows theoretically sound and fast analysis of dFC(t) data and exact calculations of derived dynamic parameters.
Methods:
All the dFC(t) matrices are ultra low-ranked, positive semidefinite and symmetric. We propose a new formula for computing the EVD (eigenvector decomposition) of dFC(t), which does not require building the dFC(t) matrices but takes advantage of the ultra-low-rankedness to operate in a low-dimensional space without loss of information.
We also introduce a common set of basic operations for dFC which are general enough to be applied to correlation, covariance, phase locking, and coactivation matrix, and rely on their EVD. We then show that a) the norm of dFC(t) is a proxy for the overall amount of connectivity b) the distance between two matrices is the norm of their difference and c) the reconfiguration speed is the distance between dFC(t) and the lagged dFC(t-τ). As a measure of entropy, we propose the Von Neumann Entropy, which quantifies the diversity of the eigenvalue spectrum.
We showcase the method using a simulated dataset. We generated an i.i.d. Gaussian multivariate signal and multiplied it by the Cholesky decomposition of a known covariance matrix. We used this approach to generate a multivariate time series with 5 time-varying underlying covariance patterns. (simulated brain states), see Fig 2.i.
See https://github.com/Mimbero/MEIDAS_MAIN for more details on the formulas and the software. (both MATLAB and Python available).
We also introduce a common set of basic operations for dFC which are general enough to be applied to correlation, covariance, phase locking, and coactivation matrix, and rely on their EVD. We then show that a) the norm of dFC(t) is a proxy for the overall amount of connectivity b) the distance between two matrices is the norm of their difference and c) the reconfiguration speed is the distance between dFC(t) and the lagged dFC(t-τ). As a measure of entropy, we propose the Von Neumann Entropy, which quantifies the diversity of the eigenvalue spectrum.
We showcase the method using a simulated dataset. We generated an i.i.d. Gaussian multivariate signal and multiplied it by the Cholesky decomposition of a known covariance matrix. We used this approach to generate a multivariate time series with 5 time-varying underlying covariance patterns. (simulated brain states), see Fig 2.i.
See https://github.com/Mimbero/MEIDAS_MAIN for more details on the formulas and the software. (both MATLAB and Python available).
·fig1: EiDA, computational speed
Results:
As in Fig 1.ii, the EiDA algorithm outperforms the standard numerical algorithms for eigenvector decomposition by up to 1000x in practical applications. For example, in an HCP voxelwise fMRI timeseries (Van Essen et al. 2013), with a window of 30 frames, the computation of sliding window covariance matrices dFC(t) would be extremely demanding with standard algorithms. EiDA recovered the full information of dFC(t) from 29 eigenvectors (rank=window size-1) and then all the measures of interest, without rebuilding the matrices (in ~3s on an Apple M2 CPU).
In the simulated dataset, EiDA correctly recovered the underlying covariance patterns (Fig. 2.ii). The reconfiguration speed peaks in the time frames where the covariance patterns switch from one state to the other (Fig. 2.iii). K-medoids clustering with the EiDA matrix distance (Distance 2) recovers the 5 underlying covariance patterns, (Fig 2.iv). The Functional Connectivity Dynamics plot (time-to time distance matrix) shows that, with all three possible dFC EiDA distances, it is possible to recover the simulated brain states (Fig. 2.v).
In the simulated dataset, EiDA correctly recovered the underlying covariance patterns (Fig. 2.ii). The reconfiguration speed peaks in the time frames where the covariance patterns switch from one state to the other (Fig. 2.iii). K-medoids clustering with the EiDA matrix distance (Distance 2) recovers the 5 underlying covariance patterns, (Fig 2.iv). The Functional Connectivity Dynamics plot (time-to time distance matrix) shows that, with all three possible dFC EiDA distances, it is possible to recover the simulated brain states (Fig. 2.v).
·fig2: application of EiDA to a synthetic dataset
Conclusions:
We have introduced a generalizable theoretical framework for dFC analyses.
The first benefit is that it unifies most of the main dFC approaches (Fig 1.i).
The second is the gain in computational speed so that otherwise uncomputable matrices can be represented losslessly and all the quantities of interest (norm, distance, entropy) recovered efficiently. This paves the way for parcellation-free analyses and real-time algorithms, where computational efficiency is a stringent requirement.
The first benefit is that it unifies most of the main dFC approaches (Fig 1.i).
The second is the gain in computational speed so that otherwise uncomputable matrices can be represented losslessly and all the quantities of interest (norm, distance, entropy) recovered efficiently. This paves the way for parcellation-free analyses and real-time algorithms, where computational efficiency is a stringent requirement.
Modeling and Analysis Methods:
Connectivity (eg. functional, effective, structural) 1
fMRI Connectivity and Network Modeling 2
Keywords:
Computational Neuroscience
Computing
Data analysis
FUNCTIONAL MRI
Machine Learning
Open-Source Code
Open-Source Software
1|2Indicates the priority used for review
Provide references using author date format
Cabral, J., Vidaurre, D., Marques, P., Magalhães, R., Silva Moreira, P., Miguel Soares, J., Deco, G., Sousa, N., & Kringelbach, M. L. (2017). Cognitive performance in healthy older adults relates to spontaneous switching between states of functional connectivity during rest. Scientific Reports, 7(1), Articolo 1. https://doi.org/10.1038/s41598-017-05425-7
Hutchison, R. M., Womelsdorf, T., Allen, E. A., Bandettini, P. A., Calhoun, V. D., Corbetta, M., Della Penna, S., Duyn, J. H., Glover, G. H., Gonzalez-Castillo, J., Handwerker, D. A., Keilholz, S., Kiviniemi, V., Leopold, D. A., de Pasquale, F., Sporns, O., Walter, M., & Chang, C. (2013). Dynamic functional connectivity: Promise, issues, and interpretations. NeuroImage, 80, 360–378. https://doi.org/10.1016/j.neuroimage.2013.05.079
Zamani Esfahlani, F., Jo, Y., Faskowitz, J., Byrge, L., Kennedy, D. P., Sporns, O., & Betzel, R. F. (2020). High-amplitude cofluctuations in cortical activity drive functional connectivity. Proceedings of the National Academy of Sciences, 117(45), 28393-28401.
Van Essen, D. C., Smith, S. M., Barch, D. M., Behrens, T. E., Yacoub, E., Ugurbil, K., & Wu-Minn HCP Consortium. (2013). The WU-Minn human connectome project: an overview. Neuroimage, 80, 62-79.