# Connectivity measures

## Dynamic connectivity measures

**note: these measures are under development, implementation details may change in future revisions**

**Sliding window**: 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

with S(x,t) = BOLD timeseries at voxel x, centered to zero mean

R(t) = BOLD timeseries within seed/ROI, centered to zero mean

w(t) = Hann sliding window

beta(x,t) = bivariate regression coefficient between seed ROI and target voxel x within time-window centered at time t, estimated using weighted least squares (WLS)

** DV(x) = dynamic variability** in connectivity between seed ROI and target voxel x

**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

with R_k(t) = BOLD timeseries within k-th seed/ROI, centered to zero mean

w(t) = Hann sliding window

beta_ij(t) = bivariate regression coefficient between i-th and j-th ROI within time-window centered at time t, estimated using weighted least squares (WLS)

** DV_ij = dynamic variability** in connectivity between i-th and j-th ROI

*Implementation notes: Sliding window analyses are defined first in the Setup.Conditions tab by selecting 'temporal decomposition (sliding-window)' in the 'time-frequency decomposition' field. This will define a number of new conditions each covering an individual temporal window. Selecting these conditions when running any first-level analysis will compute the individual time-centered connectivity measures as well as the summary dynamic variability measures*

**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:

with R_nk(t) = BOLD timeseries within k-th ROI for the n-th subject, centered to zero mean

gamma_l(i,j) = group-level changes in connectivity between i-th and j-th ROIs associated with l-th group-level modulatory component, estimated, together with beta_n(i,j) and h_nl(t) above, using iterative dual regression

The group-level modulatory components Gamma_l(i,j) 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 modulatory components into a series of subject-level components gamma_nk(i,j) is performed using a series of standard first-level gPPI models with the estimated dynamic independent component/circuit timeseries h(t) as gPPI psychological factors:

with R_nk(t) = BOLD timeseries within k-th ROI for the n-th subject, centered to zero mean

** gamma_nk(i,j) = changes in connectivity associated with k-th dynamic independent component/circuit **for the n-th subject between i-th and j-th ROIs, estimated, together with alpha and beta above, using least squares (OLS)

*Implementation notes: Dynamic ICA analyses are defined in the first-level dyn-ICA tab. These analyses produce multiple outputs, including the individual subject-level matrices gamma(i,j) (in the second-level dynICA-circuits 'Spatial Properties' tab) and the variability and frequency of the dynamic component/circuit timeseries h(t) (in the second-level dynICA-circuits 'Temporal Properties' tab)*