Fig 1.
Presentation of the DySCo framework.
A: What is dynamic Functional Connectivity: i) We can start from any set of brain recordings, where each signal is referred to a brain location (e.g. fMRI, EEG, intracranial recordings in rodents, and more). ii) “Static” Functional Connectivity (FC) is a matrix where each entry is a time aggregated functional measure of interaction between two regions, for example, the Pearson Correlation Coefficient. iii) Dynamic Functional Connectivity (dFC) is a FC matrix (that can be calculated in different ways, see below) that changes with time, under the assumption that patterns of brain interactions are non-stationary. B: Why dFC is important: i) In this toy example, 3 brain signals are recorded, referred to 3 anatomical locations (). In the first half of the recording (blue half)
and
are highly correlated (high FC), while in the second half
and
are highly correlated. Thus, the brain switches between two different spatio-temporal patterns of interaction (pattern 1, blue, and pattern 2, green). ii) Pattern 1 can be seen as a matrix, as a graph, and as a set of main axes of variation in a 3D space (blue), and the same for pattern 2 (green). In this toy example, the switch from a high 1–2 correlation to a high 1–3 correlation can be seen as a change in the connectivity matrix or a rotation of the main axes of variation of the signals in the 3D space. However, by using a “static” approach, this switch would not be captured, and a spurious spatio-temporal pattern (the black one), associated with a spurious set of axes of variation, would appear, which does not reflect any actual brain configuration. This is why dFC is a tool to investigate brain dynamics, by looking at how spatio-temporal patterns of interaction (the shape of the cloud of points) change in time. C: The Dynamic Symmetric Connectivity (DySCo) Matrix analysis framework for dFC: i) DySCo is a comprehensive framework that puts together different dFC approaches. Interestingly, these include the 3 most employed methods for dFC, i.e., sliding window correlation/covariance, co-fluctuation, approaches based on instantaneous phase. They all involve symmetric matrices. ii) DySCo proposes a unified mathematical formalism and a set of measures and algorithms to compute and analyse dFC matrices. In a nutshell, this entails: 1. The selection of a dFC matrix 2. A unique algorithm (the Temporal Covariance EVD) to compute and store the dFC matrices with their eigenvectors and eigenvalues, which is orders of magnitude faster and more memory efficient than naïve approaches 3. A common set of measures to quantify the evolution of dFC in time. These measures allow to perform the analyses that are typically performed in dFC studies. They can be classified in three categories: measures based on the total amount of dynamic interactions (matrix norm); measures based on distance/similarity of dFC patterns (e.g. to perform clustering); measures based on the entropy of the dFC patterns.
Fig 2.
Summary of the DySCo Pipeline.
This schema illustrates the main steps involved in the DySCo framework as well as important methodological decisions that must be made when using the framework. After input of raw data and appropriate pre-processing there are multiple dFC matrices as described in Theory (The DySCo theory). Based upon the choice of dFC matrix, which we define as C ( t ) , subsequent processing steps are employed (such as window size adjustment or extraction of phase) to express these dFC matrices into a single equation (). We next calculate the eigenvalues and eigenvectors associated with the dFC matrices using the Temporal Covariance EVD. The eigenvalue-eigenvector representation contains all the information needed to perform the dFC analyses, and to compute the DySCo measures described in Theory. The three main measures are Norms, Distances, and Entropy (see DySCo measures). From them we can obtain derived measures: from the norm it is possible to compute metastability (see Norm metastability), from the distance it is possible to compute the FCD matrix and the reconfiguration speed (see Distances between dFC operators).
Fig 3.
Computational efficiency of the DySCo framework.
i) Comparison of computational speed of the TCEVD algorithm compared to naïve numerical methods (the MATLAB eigs function, see Investigation of computational efficiency of the TCEVD in the DySCo framework), using randomly generated covariance matrices in a window of size 10. We repeated the experiment 20 times. Thick lines represent the mean computation time, thin lines the ± variance. ii) Comparison of the memory requirements (in bytes) for the storage of the matrices using their upper triangular form (N ( N − 1 ) ∕ 2), versus using the eigenvector decomposition (NT). iii) Comparison of the time required to compute the Euclidean distance between two vectorized matrices versus using the DySCo EVD approach.
Fig 4.
Application of the DySCo measures to a simulated dataset.
i) simulated signals and the five underlying covariance patterns, corresponding to brain states. ii) The sliding window covariance matrix computed using the DySCo formula. iii) Reconfiguration speed with a lag of 100 frames shows peak corresponding to the switches between brain states (the three colors are the three options to compute distance as defined in the theory, see Distances between dFC operators). iv) Functional Connectivity Dynamics matrices.
is the distance between the matrix at time
and the matrix at time tj using the three possible distances proposed in the DySCo framework.
Fig 5.
Illustration of different dynamic matrices.
A: i) We selected two random signals from an example participant and plotted the timecourse of all the measures of the DySCo framework: ii) sliding window correlation, with window sizes ranging from 5 to 50; iii) sliding window covariance, with window sizes ranging from 5 to 50; iv) co-fluctuation; v) instantaneous Phase Alignment. B: i) The average (across different couples of signals, across subjects) correlation in time between instantaneous Phase Alignment and sliding window correlation as a function of the window size. ii) The average (across different couples of signals, across subjects) correlation in time between co-fluctuation and sliding window covariance as a function of the window size.
Fig 6.
Application of DySCo to HCP dataset (all subjects).
A: i) The task structure (gray line), and the HRF convolved task timecourse, in orange (see HCP task fMRI data). ii) Shows the mean reconfiguration speed (green) standard error (shaded) calculated from the obtained eigenvalues across 100 subjects with a window size of 21. The dashed line again shows the task timecourse of the HCP n-back task (r = –0.46, p < 0.001). iii) Shows the mean von Neumann Entropy (blue) standard error (shaded) calculated from the obtained eigenvalues across 100 subjects. The dashed line shows the Task timecourse of the HCP n-back task (r = 0.76 , p < 0.001). iv) Shows the FCD matrix averaged across all subjects. The entry ij of the FCD matrix (see Distances between dFC operators) represents the distance 2 between the dFC matrix at time and the dFC matrix at time
. B: To give an example of evolution in time of the sliding window correlation matrices, we show them by using their first 3 eigenvectors (averaged across all subjects). We display the first half of the recording to maximise space for brain rendering.
Fig 7.
Application of DySCo to HCP dataset (One Example Subject).
A: i) Shows the reconfiguration speed (green) for the single example subject. The dashed line shows the Task timecourse of the HCP n-back task (r = –0.66, p < 0.001). ii) Shows the von Neumann Entropy (blue). The dashed line shows the Task timecourse of the HCP n-back task (r = 0.89, p < 0.001). iii) Shows the FCD matrix for the single example subject. B: To give an example of evolution in time of the sliding window correlation matrices, for the single example subject, we show them by using their first 3 eigenvectors. We display the first half of the recording to maximise space for brain rendering.
Table 1.
Description of the dFC matrices presented above and their main properties.
Fig 8.
Illustration of Von Neumann Entropy.
Two example cases to show the Von Neumann Entropy, in an Example 3-dimensional random signal. In the first case, there is no main axis of variation, thus the eigenvalues are all similar. This corresponds to a high Von-Neumann entropy. In the second case, there is a main axis of variation, which corresponds to a less dispersed eigenvalue spectrum. This corresponds to a low Von-Neumann entropy.