# Network measures

## Network (voxel-level) measures

Network measures attempt to summarize properties of the entire voxel-to-voxel connectome (all functionals connections between every pair of voxels in the brain) into a series of reduced and interpretable measures at each individual voxel. These include measures that address properties specified a priori, and estimate how those properties are expressed in each individual subject, such as Intrinsic Connectivity (IC), Global Correlation (GCOR), and Local Correlation (LCOR), as well as data-driven measures that are first informed by group-level properties and then attempt to determine how those observed properties are expressed in each individual subject, such as Independent Component Analyses (group-ICA), Principal Component Analyses (group-PCA), and Multivariate Pattern Analyses (group-MVPA).

### Intrinsic Connectivity (IC)

IC maps represent a measure of node 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)

where * r* is the map of voxel-to-voxel correlations between every pair of voxels,

*is a pre-defined mask (by default covering the entire brain), and*

**M**

**IC****is the**

**Intrinsic Connectivity map**

*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 node 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:

where * r* is the map of voxel-to-voxel correlations between every pair of voxels,

*is a pre-defined mask (by default covering the entire brain), and*

**M***is the*

**GCOR****Global Correlation map**. The spatial average of a GCOR map represents the GCOR quality control measure (Saad et al. 2013) which can be used as a subject-level covariate characterizing brain-wide correlation properties

*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 neighboring areas in the brain. LCOR is defined as the average of correlation coefficients between each individual voxel and a region of neighboring voxels (Integrated Local Correlation, Deshpande et al. 2009) :

where * r* is the map of voxel-to-voxel correlations between every pair of voxels,

*is an isotropic Gaussian weighting function with size*

**w***characterizing the size of the local neighborhoods, and*

**sigma***is the*

**LCOR****Local Correlation map**

*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*

### Inter-Hemispheric Correlation (IHC)

IHC maps represent a measure of interhemispheric connectivity at each voxel, characterized by the Fisher-transformed correlation between the BOLD signal at a given voxel and that of another voxel at the same anatomical location at the contralateral hemisphere:

where * r* is the map of voxel-to-voxel correlations between every pair of voxels,

**y(x)***is the mapping between homotopic anatomical locations in controlateral hemispheres defined as the linear combination of one or multiple orthogonal transformations*

*,*

**R***and*

**IHC***is the*

**Inter-Hemispheric Correlation**

**map**

note: for 3D volume-level functional data, the mapping *y**(x) *is defined as voxels with the same spatial coordinates and opposite x-coordinate signs, e.g. a voxel at coordinates (x,y,z) in MNI-space is mapped to a voxel at coordinates (-x,y,z). For 2D surface-level functional data, the mapping *y**(x) *is defined between vertices in left- and right- hemisphere spherical coordinates, and estimated using a 32-basis non-linear mapping matching Freesurfer left- and right- hemisphere surface-curvature templates

*Implementation notes: same normalization and task/condition-specific options available as in*

*other*

*measures above.*

*IHC*

*analyses are defined in the first-level voxel-to-voxel analyses tab, selecting '*

*Inter-Hemispheric Correlation*

*' in the analysis type section*

### Multivariate Correlation (MCOR) (group-MVPA)

MCOR maps represent for each voxel the *m* most salient spatial features of the SBC maps seeded at this same voxel. MCOR maps are defined from a Singular Value Decomposition (SVD, Strang 2007), separately for each seed-voxel, of the patterns of seed-based correlations across all subjects:

where * r* is the map of voxel-to-voxel correlations between every pair of voxels for each subject

*n*,

*is an orthogonal spatial basis characterizing the*

**Q***m*most salient SBC spatial patterns over a pre-defined mask area

**M***(*by default covering the entire brain

*)*for each seed voxel

*x*, and

*are the*

**MCOR***m*-th dimensional

**Multivariate Connectivity maps**for each subject . MCOR maps can be used to perform functional connectivity Multivariate Pattern Analyses (fc-MVPA, Nieto-Castanon 2022) evaluating differences between subjects in the entire patterns of seed-to-voxel connectivity. In the context of fc-MVPA, a set of several MCOR maps (typically the first

*k*maps) forms the eigenpattern scores vector

*(*

**s***x*), which can be used as a low-dimensional representation of the entire pattern of connectivity between the voxel

*x*and the rest of the brain.

*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 Correlation spatial maps MCOR(x), which may be used jointly in multivariate 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. CONN's ICA implementation follows Calhoun's group-ICA methodology (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 (Hyvarinen 1999, with G1/tanh, G2/gauss, or G3/pow3 non-linear contrast function), and GICA1 or GICA3 for subject-level back-projection (see Calhoun et al. 2001 for method details)

*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 Calhoun's group-ICA methodology but without the ICA rotation/weighting step, so the spatial maps represent the group-level maximal-variance components instead

*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)*

### Notes on dimensionality reduction

All of the above properties are either directly or indirectly defined from **r***, *the maps of voxel-to-voxel correlations between every pair of voxels for each subject and condition. It is often useful to represent these symmetric voxel-to-voxel correlation maps in terms of their orthogonal Singular Value Decomposition (SVD) components:

where * r* is the map of voxel-to-voxel correlations between every pair of voxels for one individual subject and condition,

*is an orthogonal spatial basis characterizing each of the*

**Q***m*maximal-variance spatial components or eigenvectors of

*, and sigma are the eigenvalues of*

**r***characterizing the variance associated with each of these components.*

**r**When using this representation, it is also possible to remove from consideration those components explaining minimal residual variance in the BOLD signal after all standard preprocessing and denoising steps, simply by using a value of *m *lower than the rank of the matrix * r*. This is often useful for computational simplicity, but also as an additional subject-level denoising strategy, as well as to minimize potential differences in effective degrees of freedom of the residual BOLD signal across subjects. This procedure is referred to as "subject-level dimensionality reduction", and in CONN it can be optionally applied to any of the above network measures. The default recommended value in CONN keeps the first 64 components to characterize the voxel-to-voxel correlation matrix separately for each individual subject and experimental condition (see Whitfield-Gabrieli and Nieto-Castanon 2012 for method details; see Calhoun et al. 2001 for subject-level dimensionality reduction in the context of group-level ICA)

## References

Calhoun, V. D., Adali, T., Pearlson, G. D., & Pekar, J. J. (2001). A method for making group inferences from functional MRI data using independent component analysis. *Human brain mapping*, *14*(3), 140-151

Deshpande, G., LaConte, S., Peltier, S., & Hu, X. (2009). Integrated local correlation: a new measure of local coherence in fMRI data. *Human brain mapping*, *30*(1), 13-23

Hyvarinen, A. (1999). Fast and robust fixed-point algorithms for independent component analysis. *IEEE transactions on Neural Networks*, *10*(3), 626-634

Martuzzi, R., Ramani, R., Qiu, M., Shen, X., Papademetris, X., & Constable, R. T. (2011). A whole-brain voxel based measure of intrinsic connectivity contrast reveals local changes in tissue connectivity with anesthetic without a priori assumptions on thresholds or regions of interest. *Neuroimage*, *58*(4), 1044-1050

Norman, K. A., Polyn, S. M., Detre, G. J., & Haxby, J. V. (2006). Beyond mind-reading: multi-voxel pattern analysis of fMRI data. *Trends in cognitive sciences*, *10*(9), 424-430

Saad, Z. S., Reynolds, R. C., Jo, H. J., Gotts, S. J., Chen, G., Martin, A., & Cox, R. W. (2013). Correcting brain-wide correlation differences in resting-state FMRI. *Brain connectivity*, *3*(4), 339-352

Strang, G. (1993). *Introduction to linear algebra* (Vol. 3). Wellesley, MA: Wellesley-Cambridge Press

Whitfield-Gabrieli, S., & Nieto-Castanon, A. (2012). Conn: a functional connectivity toolbox for correlated and anticorrelated brain networks. *Brain connectivity*, *2*(3), 125-141

## How to compute network (voxel-level) measures in CONN

All CONN's network (voxel-level) measures can be computed using any of the following options:

### Option 1: using CONN's gui

If you have already imported and denoised your data in CONN (either through the GUI or batch commands) go to CONN's *Analyses (1st-level)* tab, and select '*Voxel-to-Voxel*' connectivity measures. Select '*Create new first-level analysis*' and give this analysis a name (e.g. **LCOR**), then in the 'Analysis type' field select the type of network measure you would like to compute (e.g. **LocalCorrelation**). All options there will be set by default to standard values appropriate to the chosen measure. Modify these options if needed and simply click '**Done**' and '**Start**' to compute the corresponding maps for each subject and condition (optionally change the '*local processing*' option available in that window to '*distributed processing*' if you want to parallelize this pipeline across multiple processors or nodes in an HPC cluster)

### Option 2: using CONN's batch commands

Similarly, if you have already imported and denoised your data in CONN (either through the GUI or batch commands) , you may compute any network maps across all subjects and conditions using Matlab command syntax:

conn_batch( 'vvAnalysis.name', 'LCOR', 'vvAnalysis.measures', 'LocalCorrelation', 'vvAnalysis.done', true )

optionally adding to this command any desired alternative field name/value pairs (see *doc conn_batch* for additional details), for example:

conn_batch( 'filename', '/data/Cambridge/conn_Cambridge.mat', ...

'vvAnalysis.name', 'LCOR', ...

'vvAnalysis.measures.names', 'LocalCorrelation', ...

'vvAnalysis.measures.kernelsupport', 25, ...

'vvAnalysis.done', true)