# Connectivity measures

## ROI-to-ROI measures

ROI-to-ROI measures are defined in the same way as the Seed-to-Voxel measures above, simply by substituting the target voxel BOLD timeseres S(x,t) by a target ROI timeseres Rj(t). In particular:

• ROI-to-ROI Correlation (RRC): RRC matrices represent the level of functional connectivity between each pair of ROIs. RRC is defined as the Fisher-transformed bivariate correlation coefficients between two ROIs (Regions Of Interest) BOLD timeseries (ROI BOLD timeseries are computed by averaging voxel timeseries across all voxels within each ROI). Alternatively, bivariate regression coefficient (raw) between the same timeseries

with Ri(t) = BOLD timeseries within i-th ROI, centered to zero mean

r(i,j) = correlation coefficients between i-th and j-th ROIs

Z(i,j) = Fisher-transformed correlation coefficient

Implementation notes: BOLD timeseries are preprocessed and denoised (e.g. aCompCor , detrended, and band-pass filtered) separately for each run/session, then concatenated and normalized to build the S(x,t) and R(t) timeseries above. ROI-to-ROI correlation analyses are defined in the first-level analyses tab, selecting 'functional connectivity (weighted GLM)' and 'ROI-to-ROI' in the analysis type section, and 'bivariate correlation' and 'no weighting' in the analysis options section

• Multivariate ROI-to-ROI Connectivity (mRRC): Semipartial or multivariate mRRC matrices represent the level of effective connectivity (or unique functional connectivity) between two ROIs (i.e. strength of associations discounting those that may be mediated or accounted for by any of the other ROIs). mRRC is defined as the correlation coefficients between two ROI BOLD timeseries, after controlling for other (one or several) additional ROI BOLD timeseries. Alternatively, multivariate regression coefficients (raw) between the same timeseries. To compute these measures a separate multiple regression model between each individual ROI BOLD timeseries (outcome) and all of the other included ROI BOLD timeseries (predictors) is estimated

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

beta(i,j) = multivariate regression coefficient between i-th and j-th ROIs, estimated using least squares (OLS)

Implementation notes: correlation measures are computed by first fitting the appropriate linear regression model, and then re-scaling the resulting regression coefficients. Multivariate ROI-to-ROI connectivity analyses are defined in the first-level analyses tab, selecting 'functional connectivity (weighted GLM)' and 'ROI-to-ROI' in the analysis type section, and 'semipartial correlation'/'multivariate regression' and 'no weighting' in the analysis options section

• Weighted ROI-to-ROI Connectivity (wRRC): Weighted RRC measures represent task- or condition- specific functional connectivity (i.e. functional connectivity during each task/condition). wRRC is defined as the same bivariate, multivariate, and semipartial correlation and regression measures as above but now computed using weighted Least Squares (WLS) with user-defined weights. In task designs, weights are defined as condition timeseries convolved with a canonical hemodynamic response function in order to estimate task-specific functional connectivity measures

with R_k(t) = BOLD timeseries within k-th ROI, orthogonal to task/condition effects and centered to zero mean

w_n(t) = n-th weighting function

h_n(t) = n-th raw task/condition effect

f(t) = canonical hemodynamic response function (spm_hrf); note: * represents a linear convolution operation

beta_n(i,j) = bivariate regression coefficient during n-th condition between i-th and j-th ROI, estimated using weighted least squares (WLS)

Z_n(i,j) = Fisher-transformed correlation coefficient during n-th condition between i-th and j-th ROI

Implementation notes: weighted ROI-to-ROI connectivity analyses are defined in the first-level analyses tab, selecting 'functional connectivity (weighted GLM)' and 'ROI-to-ROI' in the analysis type section, and 'hrf weighting' in the analysis options section. ROI orthogonalization to task effects is defined in the Denoising tab, selecting 'effect of task' in the confounding effects list

• Generalized Psycho-Physiological Interaction (gPPI): gPPI measures represent the level of task-modulated effective connectivity between two ROIs (i.e. changes in functional association strength covarying with the external or experimental factor). gPPI is computed using a separate multiple regression model for each target ROI timeseres (outcome). Each model includes as predictors: a) all of the selected task effects convolved with a canonical hemodynamic response function (main psychological factor in PPI nomenclature); b) each seed ROI BOLD timeseries (main physiological factor in PPI nomenclature); and c) the interaction term specified as the product of (a) and (b) (PPI term). gPPI output is defined as the regression coefficients associated with the interaction term in these models

with R_i(t) = BOLD timeseries within i-th ROI, orthogonal to task effects and centered to zero mean

h_k(t) = k-th raw task/condition effect, centered to zero mean

f(t) = canonical hemodynamic response function (spm_hrf); note: * represents a linear convolution operation

gamma_k(i,j) = interaction term (regression coefficient) between the i-th and j-th ROI BOLD timeseries and the k-th task factor, estimated, together with alpha and beta above, using least squares (OLS)

Implementation notes: this implementation of gPPI in CONN is similar to that in FSL, and differs from the one in SPM, by modeling the interaction in terms of the raw BOLD signal and convolved psychological factors, rather than in terms of the deconvolved BOLD signals and raw psychological factors. ROI-to-ROI gPPI analyses are defined in the first-level analyses tab, selecting 'task modulation (gPPI)' and 'ROI-to-ROI' in the analysis type section, and 'bivariate regression' in the analysis options section