# Connectivity measures

## Seed-Based Connectivity measures

**Seed-Based Correlation (SBC)**: SBC maps represent the level of functional connectivity between the seed/ROI and every location in the brain. SBC is defined as the Fisher-transformed bivariate correlation coefficients between an ROI (Region Of Interest) BOLD timeseries (averaged across all voxels within an ROI) and an individual voxel BOLD timeseries. Alternatively, bivariate regression coefficients (raw) between the same timeseries.

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

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

r(x) = correlation coefficients between seed ROI and target voxel

**Z(x) = Fisher-transformed correlation coefficient**

*Implementation notes: BOLD timeseries are preprocessed and denoised (e.g. aCompCor , motion and outlier regressed, 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. Seed-based correlation analyses are defined in the first-level analyses tab, selecting 'functional connectivity (weighted GLM)' and 'Seed-to-Voxel' in the analysis type section, and 'bivariate correlation' and 'no weighting' in the analysis options section*

**Multivariate****Seed-Based Connectivity (mSBC)**: Semipartial or multivariate SBC maps represent the level of effective connectivity (or*unique*functional connectivity) between a seed/ROI and every location in the brain (i.e. strength of associations discounting those that may be mediated or accounted for by other seeds/ROIs). mSBC is defined as the semipartial correlation coefficients between an ROI BOLD timeseries and an individual voxel BOLD timeseries, after controlling for other (one or several) ROI BOLD timeseries. Alternatively, multivariate regression coefficients (raw) between the same timeseries. To compute these measures a separate multiple regression model between all of the selected ROI BOLD timeseries (predictors) and each individual target voxel BOLD timeseries (outcome) is estimated

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

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

**beta_k(x) = multivariate regression coefficient** between k-th ROI and voxel x, 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 seed-based connectivity analyses are defined in the first-level analyses tab, selecting 'functional connectivity (weighted GLM)' and 'Seed-to-Voxel' in the analysis type section, and 'semipartial correlation'/'multivariate regression' and 'no weighting' in the analysis options section*

**Weighted****Seed-Based Connectivity (wSBC)**: Weighted SBC measures represent task- or condition- specific functional connectivity (i.e. functional connectivity during each task/condition). wSBC 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

(e.g. task-based bivariate correlation wSBC)

with S(x,t) = BOLD timeseries at voxel x, orthogonal to task/condition effects and centered to zero mean

R(t) = BOLD timeseries within 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(x) = bivariate regression coefficient during n-th condition between seed/ROI and voxel x, estimated using weighted least squares (WLS)

**Z_n(x) = Fisher-transformed correlation coefficient during n-th task/condition **between seed/ROI and voxel x

*Implementation notes: weighted seed-based connectivity analyses are defined in the first-level analyses tab, selecting 'functional connectivity (weighted GLM)' and 'Seed-to-Voxel' in the analysis type section, and 'hrf weighting' in the analysis options section. BOLD timeseries 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 a seed/ROI and every location in the brain (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 voxel BOLD 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) the 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 S(x,t) = BOLD timeseries at voxel x, orthogonal to task effects and centered to zero mean

R(t) = BOLD timeseries within 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(x) = interaction term (regression coefficient) **at target voxel x between k-th task factor and ROI BOLD timeseries, 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. Seed-based gPPI analyses are defined in the first-level analyses tab, selecting 'task modulation (gPPI)' and 'Seed-to-Voxel' in the analysis type section, and 'bivariate regression' in the analysis options section*