# Connectivity measures

## Voxel-level network measures

**Intrinsic Connectivity (IC)**: IC maps represent a measure of network centrality at each voxel, characterized by the strength of connectivity between a given voxel and the rest of the brain. IC is defined as the root mean square of correlation coefficients between each individual voxel and all of the voxels in the brain (Intrinsic Connectivity Contrast, Martuzzi et al. 2011)

with r(x,y) = correlation coefficients between voxels x and y

N = number of voxels in analysis mask M

**IC(x) = Intrinsic Connectivity **at voxel x

*Implementation notes: voxel-to-voxel connectivity matrices r(x,y) are represented in terms of their Singular Value Decomposition (subject-specific SVD) with a maximum of 256 components. Separately for each voxel-to-voxel measure users may further limit the number of components used in the reconstruction of r(x,y) from the singular vectors/values as a form of subject-level dimensionality reduction. Users may also normalize IC measures to have a N(0,1) Gaussian distribution with zero mean and unit variance over all voxels x. Task/condition-specific IC measures are computed starting with the corresponding task/condition-specific correlation r(x,y) computed using weighted GLM. IC analyses are defined in the first-level voxel-to-voxel analyses tab, selecting 'intrinsic connectivity' in the analysis type section*

**Global Correlation (GCOR)**: GCOR maps represents a measure of network centrality at each voxel, characterized by the strength and sign of connectivity between a given voxel and the rest of the brain. GCOR is defined as the average of correlation coefficients between each individual voxel and all of the voxels in the brain.

with r(x,y) = correlation coefficients between voxels x and y

N = number of voxels in analysis mask M

**GCOR(x) = Global Correlation **at voxel x

*Implementation notes: same dimensionality reduction, normalization, and task/condition-specific options available as in IC measure above. GC analyses are defined in the first-level voxel-to-voxel analyses tab, selecting 'global correlation' in the analysis type section. Note: IC and GC measures are inter-related, as IC(x)^2-GCOR(x)^2 equals the variability in seed-based correlations between each voxel x and the rest of the brain*

**Local Correlation (LCOR)**: LCOR maps represent a measure of local coherence at each voxel, characterized by the strength and sign of connectivity between a given voxel and the neighbouring areas in the brain. LCOR is defined as the average of correlation coefficients between each individual voxel and a region of neighbouring voxels (Integrated Local Correlation, Deshpande et al. 2007)

with r(x,y) = correlation coefficients between voxels x and y

w(z) = isotropic Gaussian weighting function

**LCOR(x) = Local Correlation **at voxel x

*Implementation notes: same dimensionality reduction, normalization, and task/condition-specific options available as in IC measure above. Users may choose the size of the Gaussian kernel width sigma characterizing the degree of locality of the analyses. LCOR analyses are defined in the first-level voxel-to-voxel analyses tab, selecting 'local correlation' in the analysis type section. Note: LCOR and GCOR measures are inter-related, as LCOR values converge towards GCOR values for sufficiently large kernel widths sigma *

**Multivariate Connectivity Analyses (group MVPA)**: For each voxel, a low-dimensional representation of the entire pattern of seed-based correlations between this voxel and the rest of the brain. MVPA maps are defined using a Singular Value Decomposition (SVD), separately for each seed-voxel, of the patterns of seed-based correlations across all subjects

with r_n(x,y) = correlation coefficients between voxels x and y for the n-th subject

Qi(x,y) = group-level i-th orthogonal Principal Component spatial map for seed-voxel x, normalized to unit norm

** rho_ni(x) = i-th Multivariate Connectivity spatial map **for seed-voxel x and nth-subject

*Implementation notes: same dimensionality reduction options available as in IC measure above. MVPA analyses are defined in the first-level voxel-to-voxel analyses tab, selecting 'group-MVPA' in the analysis type section. MVPA outputs multiple Multivariate Connectivity spatial maps rho(x), which may be used jointly in multivariate voxel-based second-level analyses as a low-dimensional proxy for the entire pattern of connectivity between each voxel and the rest of the brain*

**Independent Component Analyses (group ICA)**: ICA maps represent a measure of different networks expression and connectivity at each voxel, characterized by the strength and sign of connectivity between a given network timeseries and the rest of the brain. It is defined as the multivariate regression coefficients between each component/network timeseries and an individual voxel BOLD timeseries. CONN's ICA implementation follows Calhoun's group ICA methodology (ICA, Calhoun et al. 2001), with optional subject-level dimensionality reduction, concatenation across subjects, group-level Singular Value Decomposition for dimensionality reduction, a fastICA algorithm for group-level independent component definition, and GICA1 for subject-level back-projection.

with Sn(x,t) = BOLD timeseries at voxel x for the n-th subject, normalized to zero mean and unit norm

Rnk(t) = BOLD timeseries of k-th component for the n-th subject (subject-level network timeseries)

**beta_nk(x) = k-th Independent Component connectivity **at voxel x for the n-th subject, estimated using Least Squares (OLS)

N = number of voxels in analysis mask M

Mk(x) = group-level k-th IC spatial map, estimated using fastICA on concatenated group-level data over analysis mask (using the following group-ICA equations)

with r_n(x,t) = correlation coefficients between voxels x and y for the n-th subject

Mk(x) = group-level k-th Independent Component spatial map

Wki = group-level ICA mixing matrix (estimated using fastICA with a hyperbolic tangent contrast function)

Qi(x) = group-level i-th orthogonal Principal Component spatial map, normalized to unit norm

*Implementation notes: ICA analyses are defined in the first-level voxel-to-voxel analyses tab, selecting 'group-ICA' in the analysis type section. These analyses produce multiple outputs, including the individual subject-level maps S(x,t) (in the second-level ICA-networks 'Spatial Properties' tab) and the variability and frequency of the timeseries Rk(t) (in the second-level ICA-networks 'Temporal Properties' tab)*

**Principal Component Analyses (group PCA)**: CONN's PCA implementation is identical to the above ICA methodology, simply skipping the ICA rotation/weighting step and using directly the Singular Value Decomposition group-level dimensionality reduction spatial maps instead

with Sn(x,t) = BOLD timeseries at voxel x for the n-th subject, normalized to zero mean and unit norm

Rnk(t) = BOLD timeseries of k-th component for the n-th subject (subject-level network timeseries)

**beta_nk(x) = k-th Principal Component connectivity **at voxel x for the n-th subject, estimated using Least Squares (OLS)

N = number of voxels in analysis mask M

Qk(x) = group-level k-th orthogonal Principal Component spatial map, estimated using a Singular Value Decomposition (SVD) of the group-level data over analysis mask (using the following group-PCA equations)

with r_n(x,y) = correlation coefficients between voxels x and y for the n-th subject

Qi(x) = group-level i-th orthogonal Principal Component spatial map, normalized to unit norm

Yni(t) = group-level i-th orthogonal Principal Component timeseries for the n-th concatenated subject

*Implementation notes: PCA analyses are defined in the first-level voxel-to-voxel analyses tab, selecting 'group-PCA' in the analysis type section. These analyses produce multiple outputs, including the individual subject-level maps S(x,t) (in the second-level PCA-networks 'Spatial Properties' tab) and the variability and frequency of the timeseries Rk(t) (in the second-level PCA-networks 'Temporal Properties' tab)*