# Dynamic Connectivity

## Dynamic connectivity measures

Dynamic connectivity measures are aimed at characterizing and studying sources of temporal variability in functional connectivity patterns. Some of the most common dynamic connectivity techniques are based on sliding-window approaches, where functional connectivity measures of interest are estimated over relatively small time windows in order to analyze potential changes in these measures over time. In addition, the section below also describes dyn-ICA, a technique currently being actively developed and aimed at characterizing clusters of connections that show similar temporal variations in functional connectivity

### Sliding window analyses

Every connectivity measure in CONN, including seed-based, ROI-to-ROI, network and graph measures, can also be estimated from windowed BOLD timeseries using a series of sequential sliding windows. Each individual window is treated as a separate condition, and weighted GLM is used to compute the corresponding condition-/time- specific measures. Variability of these measures across time is then computed as the main measure of interest characterizing dynamic connectivity properties. One example of such sliding-window measures of connectivity is dynamic variability in seed-based or ROI-to-ROI connectivity measures:

Dynamic variability in seed-based connectivity (dvSBC): dvSBC maps represent the degree of temporal variability in functional connectivity between a seed/ROI and every location in the brain. They are defined as the standard deviation in bivariate, multivariate, or semipartial correlation or regression measures between seed ROI and each target voxel, computed using weighted Least Squares (WLS) within a discrete set of temporal sliding windows

e.g. dynamic variability in bivariate regression SBC

where S is the BOLD timeseries (for simplicity all timeseries are considered centered to zero mean), R is the BOLD timeseries within a seed/ROI, w is a Hann sliding window of length 2L, beta is the bivariate regression coefficient map within each time window, estimated using weighted least squares (WLS), and DV is the dynamic variability in SBC connectivity map

Dynamic variability in ROI-to-ROI connectivity (dvRRC): dvRRC matrices represent the degree of temporal variability in functional connectivity between pairs of ROIs. They are defined as the standard deviation in bivariate, multivariate, or semipartial correlation or regression measures between two ROIs, computed using weighted Least Squares (WLS) within a discrete set of temporal sliding windows

e.g. dynamic variability in bivariate regression RRC

where R is the BOLD timeseries within each seed/ROI, w is a Hann sliding window of length 2L, beta is the bivariate regression coefficient matrix, estimated using weighted least squares (WLS), and DV is the dynamic variability in RRC connectivity matrix

### Dynamic Independent Component Analyses (dyn-ICA)

Dynamic ICA matrices represent a measure of different modulatory circuits expression and rate of connectivity change between each pair of ROIs, characterized by the strength and sign of connectivity changes covarying with a given component/circuit timeseries. Dyn-ICA matrices are defined as the gPPI interaction terms between each component/circuit timeseries (data-driven gPPI psychologial factors) and a series of ROI BOLD timeseries (user-defined gPPI physiological factors).

Group-level dynamic ICA is implemented using iterative dual regression on group-level data obtained by concatenation across-subjects, followed by Independent Component Analyses and gPPI back-projection. Specifically, group-level modulatory components Gamma_l(i,j) are first estimated following a simplified gPPI model of the form:

where R are the BOLD timeseries within each ROI and for each subject n (as before, for simplicity all timeseries are considered centered to zero mean), and beta, the subject-independent stationary connectivity matrix of regression coefficients between each pair of ROIs, gamma, the subject-independent matrix of connectivity changes associated with the l-th modulatory component, and h, the subject-specific timecourse of each of the estimated modulatory components, are all estimated using iterative dual regression

The Gamma matrices are then rotated using fastICA with a hyperbolic tangent contrast function, and the ICA mixing matrix W is inverted to compute the dynamic independent component/circuit timeseries:

Last, back-projection of the group-level gamma matrices into a series of subject-specific gamma components is performed using standard first-level gPPI models with the estimated dynamic independent component/circuit timecourses h as gPPI psychological factors:

where R are the BOLD timeseries within each ROI and for each subject n, and gamma is the gPPI matrix of regression coefficients for each modulatory component/circuit, estimated, together with alpha and beta above, using least squares (OLS)