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Conn: A Functional Connectivity Toolbox for Correlated

and Anticorrelated Brain Networks

Susan Whitﬁeld-Gabrieli and Alfonso Nieto-Castanon

Abstract

Resting state functional connectivity reveals intrinsic, spontaneous networks that elucidate the functional archi-

tecture of the human brain. However, valid statistical analysis used to identify such networks must address sour-

ces of noise in order to avoid possible confounds such as spurious correlations based on non-neuronal sources.

We have developed a functional connectivity toolbox Conn (www.nitrc.org/projects/conn) that implements

the component-based noise correction method (CompCor) strategy for physiological and other noise source re-

duction, additional removal of movement, and temporal covariates, temporal ﬁltering and windowing of the re-

sidual blood oxygen level-dependent (BOLD) contrast signal, ﬁrst-level estimation of multiple standard

functional connectivity magnetic resonance imaging (fcMRI) measures, and second-level random-effect analysis

for resting state as well as task-related data. Compared to methods that rely on global signal regression, the

CompCor noise reduction method allows for interpretation of anticorrelations as there is no regression of the

global signal. The toolbox implements fcMRI measures, such as estimation of seed-to-voxel and region of interest

(ROI)-to-ROI functional correlations, as well as semipartial correlation and bivariate/multivariate regression

analysis for multiple ROI sources, graph theoretical analysis, and novel voxel-to-voxel analysis of functional con-

nectivity. We describe the methods implemented in the Conn toolbox for the analysis of fcMRI data, together with

examples of use and interscan reliability estimates of all the implemented fcMRI measures. The results indicate

that the CompCor method increases the sensitivity and selectivity of fcMRI analysis, and show a high degree

of interscan reliability for many fcMRI measures.

Key words: brain connectivity; CompCor functional connectivity; intrinsic connectivity; noise; resting state

Introduction

Functional connectivity has been broadly deﬁned to

be the statistical association (or temporal correlation)

among two or more anatomically distinct regions (or remote

neurophysiological events) (Friston et al., 1994; Horwitz,

2003; Salvador et al., 2005). Numerous methods have been

used to investigate these temporal correlations, including in-

dependent component analysis (ICA) (e.g., Beckmann et al.,

2005; Calhoun et al., 2001, 2004), seed-driven functional con-

nectivity magnetic resonance imaging (fcMRI) (e.g., Biswal

et al., 1995; Fox et al., 2005; Greicius et al., 2003), and psycho-

physiological interactions used to characterize activation in a

particular brain region in terms of the interaction between

the inﬂuence of another area and an experimental parameter

(Friston et al., 1997; Gitelman et al., 2003). Functional connec-

tivity has been investigated in block (e.g., Hampson et al.,

2002; Koshino et al., 2005) and event-related (Aizenstein

et al., 2009; Fox et al., 2006; Rissman et al., 2004; Siegle et al.,

2007) fMRI activation designs. Further, functional connectivity

is evident during rest in the absence of task-related activation

(Biswal et al., 1995).

Low-frequency resting state networks ( <0.1 Hz) reveal co-

herent, spontaneous ﬂuctuations that delineate the functional

architecture of the human brain (Biswal et al., 1995, 2010;

Buckner et al., 2008; Fox et al., 2005; Fox and Raichle, 2007).

Such networks were initially discovered for the motor system

(Biswal et al., 1995), but have also been discovered for both

task-positive and task-negative (i.e., default, Raichle et al.,

2001) neural systems (Fox et al., 2005; Fransson, 2005; Kelly

et al., 2008; Uddin et al., 2009). Resting state networks have

been shown to be robust and reliable (Chen et al., 2008; Dam-

oiseaux et al., 2006; Shehzad et al., 2009; Zuo et al., 2010a,

2010b), and to exist in infants (Fransson et al., 2007), during

sleep (Fukunaga et al., 2006; Horovitz et al., 2008), under

light sedation (Greicius et al., 2008) and under anesthesia in

primates (Vincent et al., 2007). Such networks have been asso-

ciated with individual differences in healthy people (Mennes

Department of Brain and Cognitive Sciences, Martinos Imaging Center at McGovern Institute for Brain Research, and Poitras Center for

Affective Disorders Research, Massachusetts Institute of Technology, Cambridge, Massachusetts.

BRAIN CONNECTIVITY

Volume 2, Number 3, 2012

ªMary Ann Liebert, Inc.

DOI: 10.1089/brain.2012.0073

125

et al., 2010). Because rest has no behavioral demands, resting

state connectivity is particularly useful for characterizing

functional brain network differences in pediatric and clinical

populations, such as schizophrenia (Whitﬁeld-Gabrieli et al.,

2009; Whitﬁeld-Gabrieli and Ford 2012), ADHD (Castellanos

et al., 2008), autism (Weng et al., 2010), depression (Greicius

et al., 2007; Hamilton et al., 2011), bipolar disorder (Chai

et al., 2012), and Alzheimer’s disease (Buckner et al., 2009;

Greicius et al., 2004; Wang et al., 2007).

The two most common analytical approaches toward ana-

lyzing resting state functional connectivity (RSFC) data are

ICA (e.g., Beckmann et al., 2005, 2009; Greicius et al., 2007;

Stevens et al., 2009) and seed-driven RSFC (e.g., Biswal

et al., 1995; Castellanos et al., 2008; Greicius et al., 2003; Fox

et al., 2005). In seed-driven RSFC analysis, Pearson’s correla-

tion coefﬁcients are calculated between the seed time course

and the time course of all other voxels, after which correlation

coefﬁcients are typically converted to normally distributed

scores using Fisher’s transform to allow for second-level Gen-

eral Linear Model analysis. Correlation maps often depend

on the speciﬁc location of the seed, so that seed-driven

RSFC has been used to dissociate functionally and anatomi-

cally heterogeneous regions of interest (Di Martino et al.,

2008; Margulies et al., 2007; Roy et al., 2009; Uddin et al.,

2010), and to delineate functional topography in the brain

by sharp transitions in correlation patterns that signal func-

tional boundaries across cortex (Cohen et al., 2008).

In functional connectivity analysis, it is critical to appropri-

ately address noise in order to avoid possible confounding ef-

fects (spurious correlations based on non-neuronal sources).

Standard methods dealing with blood oxygen level-depend-

ent (BOLD) contrast signal noise sources that may be appro-

priate in the context of the estimation of task- or condition-

dependent BOLD signal responses (e.g., regression of subject

movement parameters in standard functional analysis) may

not sufﬁce in the context of the estimation of functional con-

nectivity measures. For activation studies, the risk of only

partially removing BOLD signal noise sources is typically a

potential decrease of sensitivity (increasing type II errors),

whereas for resting connectivity studies, the risk is a potential

decrease of validity (increasing type I errors). Therefore, a

more conservative approach to controlling the effects of

BOLD signal noise sources is warranted in the context of

functional connectivity analysis compared with that of stan-

dard functional analysis. In Chai et al. (2012) we showed

how a method for reducing spurious sources of variance in

BOLD and perfusion-based fMRI, the anatomical compo-

nent-based noise correction method (aCompCor) (Behzadi

et al., 2007), can be particularly useful in the context of

fcMRI analysis, increasing not only the validity, but also the

sensitivity and speciﬁcity of these analyses. Compared to

methods that subtract global signals from noise regions of

interest (ROIs), the CompCor method is more ﬂexible in its

characterization of noise. It models the inﬂuence of noise as

a voxel-speciﬁc linear combination of multiple empirically es-

timated noise sources, which are estimated from the variabil-

ity in BOLD responses within noise ROIs. This is particularly

appropriate for fMRI noise sources as cardiac and respiratory

effects do not have a common spatial distribution in their ef-

fects (e.g., cardiac effects are particularly visible near vessels

and respiratory effects appear more globally and stronger

near edges in the image). Removal of this richer characteriza-

tion of the range of voxel-speciﬁc noise effects and additional

movement and possible task-related covariates, together with

temporal ﬁltering and windowing of the resulting BOLD sig-

nal at each voxel, provides increased protection against pos-

sible confounding effects in RSFC without introducing

artifactual biases in the estimated connectivity measures.

In addition to physiological artifacts, head motion artifacts

have been shown to signiﬁcantly inﬂuence intrinsic func-

tional connectivity measurements (Satterthwaite et al., 2012;

Van Dijk et al., 2012). Moreover, it has been recently demon-

strated that artifacts in the functional time series may result in

substantial changes in RSFC data despite standard compen-

satory regression of motion estimates from the data (Power

et al., 2012). These ﬁndings suggest that rigorous artifact re-

jection in addition to motion regression is especially prudent

for valid interpretation of RSFC. Conn is seamlessly interoper-

able with quality assurance and artifact rejection software, art

(www.nitrc.org/projects/artifact_detect/), such that a ma-

trix of outlier, artifactual time points saved by art may be eas-

ily entered as ﬁrst-level covariates in Conn. The combination

of the Conn’s implementation of the CompCor method of

noise reduction along with the efﬁcient rejection of motion

and artifactual time points allows for better interpretation

of functional connectivity results for both correlated and anti-

correlated networks.

With the increase in popularity of linear functional connec-

tivity analysis, there is still a large degree of variability in the

exact methods used for the analysis of fcMRI data, with dif-

ferences in noise-preprocessing steps as well as differences

in the characterization of fcMRI measures across labs,

which can complicate the interpretation and comparison of

fcMRI results across different studies. To provide a common

framework for the analysis of fcMRI data, we have developed

and made publicly available the Conn toolbox (www.nitrc

.org/projects/conn). The toolbox is compatible with most

data formats—including nifty (nii) and analyze images

(img)—and implements all the processing steps necessary

to perform fcMRI analysis—including spatial preprocessing

of BOLD signal and anatomical volumes, CompCor removal

of noise sources, ﬁrst-level estimation of fcMRI measures, and

second-level random effect analysis—while maintaining the

ﬂexibility to deﬁne and estimate different forms of fcMRI

analysis. The implementation includes standard fcMRI analy-

sis, such as estimation of seed-to-voxel and ROI-to-ROI func-

tional correlations, as well as other forms of fcMRI analysis,

such as bivariate regression analysis, semipartial correlation

and multivariate regression analysis of multiple ROI sources,

graph theoretical analysis of brain networks, and novel voxel-

to-voxel analysis of functional connectivity. In the following

sections we ﬁrst describe in detail the methods used by the

toolbox to compute different functional connectivity mea-

sures (Functional Connectivity Metrics in Conn section), and

then we illustrate examples of use and reliability estimates

of all of these fcMRI measures indicating their validity as po-

tentially useful neuromarkers (Illustration of Functional Con-

nectivity Analysis in Conn section).

Functional Connectivity Metrics Implemented in Conn

The analysis steps involved in the computation of fcMRI

measures, as implemented in the Conn toolbox, are illustrated

in Figure 1. The following sections explain these steps in

detail.

126 WHITFIELD-GABRIELI AND NIETO-CASTANON

Spatial preprocessing

Spatial preprocessing steps for functional connectivity

analysis do not typically differ from those used in the context

of functional activation analysis. Most fcMRI studies include

slice-timing correction, realignment, coregistration and/or

normalization, and spatial smoothing. In addition to these

steps, the toolbox employs segmentation of gray matter,

white matter, and cerebrospinal ﬂuid (CSF) areas for optional

use during removal of temporal confounding factors. All spa-

tial preprocessing steps are implemented using SPM8 (Well-

come Department of Imaging Neuroscience, London, UK; www

.ﬁl.ion.ucl.ac.uk/spm); however, users can choose to omit

this step and use their own spatial preprocessing pipeline.

Temporal processing (treatment of temporal

confounding factors)

Several studies have emphasized the importance of additional

preprocessing steps in fcMRI studies (e.g., Birn et al., 2006; Fox

et al., 2005, 2009; Power et al., 2012; Van Dijk et al., 2010; Weis-

senbacher et al., 2009), including—but not limited to—band-

pass ﬁltering and the inclusion of estimated subject motion

parameters, artifacts, respiratory and cardiac signals, global

BOLD signal, and BOLD signals in white matter and CSF

areas as additional covariates. The main concern is that move-

ment and physiological noise sources can potentially induce spu-

rious correlations among distant voxels, increasing the chance of

false positives and confounding the interpretation of fcMRI

results. These additional preprocessing steps are designed to

help mitigate the impact of motion and physiological noise fac-

tors, increasing the validity and the robustness of fcMRI analysis.

The toolbox allows the speciﬁcation of an arbitrary set of

possible temporal confounding factors, which can be deﬁned

from indirect sources, as subject- and session-speciﬁc time se-

ries (e.g., estimated subject movement parameters and arti-

facts, cardiac or respiratory rates, and possible task effects;

these are indicated in Figure 1 as Design matrix), as well as

BOLD signals obtained from subject-speciﬁc noise ROIs

(white matter and CSF masks, as well as optionally additional

user-deﬁned ROIs). The toolbox implements an anatomical

aCompCor strategy (Behzadi et al., 2007) in which a user-

deﬁned number of orthogonal time series are estimated

using principal component analysis (PCA) of the multivariate

BOLD signal within each of these noise ROIs. This strategy

generalizes the common practice of extracting the average

BOLD time series from one or several seeds located within

the white matter and/or CSF areas. In addition, and for

each original temporal confounding factor, ﬁrst- and higher-

order derivatives of the associated time series can also be de-

ﬁned by the user as additional confounding factors (e.g., Fox

et al., 2005). Each of the deﬁned temporal confounding factors

is then regressed from the BOLD time series at each voxel (sep-

arately for each session), and the resulting residual time series

are band-pass ﬁltered. In particular, the removal of temporal

confounding factors, from an observed signal BOLD*(v,t)at

voxel vand time t, takes the following form:

BOLD(,t)=BOLD(,t)X

N

n=1

^

an()cn(t)

X

K

k=1X

Mk

n=1

^

bkn()dkn (t)

(1)

FIG. 1. Schematic representation of fcMRI analysis steps. BOLD, blood oxygen level-dependent; CSF, cerebrospinal ﬂuid;

fcMRI, functional connectivity magnetic resonance imaging; ROI, region of interest.

CONN: A RS-FCMRI TOOLBOX 127

where c

n

(t) represents Ntemporal confounds deﬁned

explicitly through subject- and session-speciﬁc time series or

implicitly as temporal derivatives of these signals (e.g., sub-

ject motion parameters); d

kn

(t) represents those confounds de-

ﬁned implicitly from Knoise ROIs, each characterized by M

k

component time series from each noise ROI (e.g., average sig-

nal, principal components, or temporal derivatives of these

signals within white matter and CSF areas); and a

n

(v) and

b

kn

(v) represent voxel-speciﬁc weights for each of these fac-

tors estimated using linear regression (see Appendix A.1 for

further details of these preprocessing steps).

The toolbox graphical user interface (GUI) encourages

users to explore the effect of these additional preprocessing

steps by displaying the histogram of voxel-to-voxel func-

tional connectivity values (correlation coefﬁcients between

the BOLD time series of a random subset of voxels) before

and after regression of the selected temporal confounding fac-

tors. This display typically shows a heavily skewed distribu-

tion of connectivity values in the presence of motion and/or

physiological noise sources, which is approximately centered

and normalized by the regression process (Fig. 2). This explo-

ration can help users optimize the choice of preprocessing

steps as well as help detect anomalous subjects/sessions.

ROI time series

Functional connectivity measures are typically computed

either between every pair of voxels (voxel-to-voxel analysis),

between a seed voxel or area and every other voxel (seed-to-

voxel analysis), or between each pair of seed areas (ROI-to-

ROI analysis). The toolbox allows the deﬁnition of seed

areas using standard practices, including individual mask

image volumes [where an ROI is deﬁned by all voxels with

values above zero, e.g., WFU pickatlas ﬁles (Maldjian et al.,

2003) or functional mask ﬁles deﬁned using SPM save func-

tionality], text ﬁles (listing Montreal Neurological Institute

(MNI) coordinates of ROI voxels), and atlas image volumes

where multiple ROIs can be jointly deﬁned using a single

image volume. Each ROI is characterized by voxels sharing

the same identiﬁer number, for example, talairach atlas (Lan-

caster et al., 2000). ROIs can be deﬁned separately for each

subject (subject-speciﬁc ROIs) or jointly across all subjects

(e.g., MNI space).

The average BOLD time series is computed across all the

voxels within each ROI. In addition, the toolbox allows the

extraction of additional temporal components from each

area resulting from a principal component decomposition of

the temporal covariance matrix (as for the noise ROIs

above), as well as the estimation of higher-order temporal de-

rivatives of these original BOLD signals. In general the fol-

lowing ROI time series can be computed from each seed area:

xn,m(t)=X

2Ox

wm()qn

qtnBOLD(,t) (2)

BOLD(m,t): BOLD timeseries at voxel mand time t

O

x

: voxels in seed area

m: order of PCA component (0 for straight average)

n: order of temporal derivative (0 for original signal)

In combination with the average BOLD signal within an

ROI, PCA component signals allow multivariate analysis of

functional connectivity patterns. In addition, temporal deriva-

tive BOLD signals when used in combination with multivariate

measures of connectivity (e.g., multivariate regression or semi-

partial correlation measures) allow the exploration of temporal

lags or more complex linear dynamics between two areas.

Linear fcMRI measures

The toolbox focuses on linear measures of functional

connectivity between two sources: zero-lagged bivariate-

correlation and bivariate-regression coefﬁcients, and their

associated multivariate measures, semipartial-correlation and

multivariate-regression coefﬁcients (Table 1). Bivariate correla-

tion and regression coefﬁcients measure the level of linear as-

sociation of the BOLD time series between each pair of

sources when considered in isolation. In contrast, semipartial

correlation and multivariate regression coefﬁcients consider

multiple sources simultaneously and estimate the unique

contribution of each source using a general linear model. In

bivariate and semipartial correlation analyses, effect sizes

represent correlation coefﬁcients (their values squared can

be interpreted as the percentage of the target BOLD signal

variance explained by each source BOLD signal). In bivariate

and multivariate regression analyses, effect sizes represent %

FIG. 2. Effect of temporal preprocessing steps on the distri-

bution of voxel-to-voxel BOLD signal correlation values. The

average distribution (across subjects and sessions) is shown

as thick lines, and its 5% and 95% percentiles are shown as

ﬁlled areas. After temporal preprocessing, voxel-to-voxel

functional connectivity estimates show a reduction in bias

and an associated increase in reliability across subjects and

sessions (see text for details).

Table 1. Deﬁnition of Linear Measures

of Functional Connectivity

Bivariate regression b=(xtx)1(xty)

Bivariate correlation r=(xtx)1=2b(yty)1=2

Multivariate regression B =(XtX)1(XtY)

Semipartial correlation R =(XtX)1

1=2BYtY

1=2

x and y represent two BOLD time series vectors (centered), X and Y

represent matrices created by concatenating horizontally one or sev-

eral x and y vectors, and the brackets [] represent the operation of

zeroing all the nondiagonal elements in a matrix.

128 WHITFIELD-GABRIELI AND NIETO-CASTANON

changes in BOLD activity at each target associated with a 1%

change of BOLD activity at each source ROI. Before being en-

tered into second-level between-subjects analysis, a Fisher

transformation (inverse hyperbolic tangent function) is ap-

plied to all bivariate and semipartial correlation measures

in order to improve the normality assumptions of standard

second-level general linear models.

Voxel-to-voxel measures

The toolbox also computes a complete voxel-to-voxel func-

tional correlation matrix for each subject. From the residual

BOLD time series at every voxel within an a priori gray mat-

ter mask (isotropic 2-mm voxels), the matrix of voxel-to-voxel

bivariate correlation coefﬁcients is computed. To minimize

storage and computation requirements (explicit storage of

this matrix could occupy above 300 Gb for each subject),

this matrix is instead characterized without loss of precision

by its eigenvectors and associated eigenvalues (see Appendix

A.2). In addition to downsampling the voxel-to-voxel correla-

tion matrices to any desired target resolution, the toolbox also

computes several voxel-level measures of functional connec-

tivity directly from the original voxel-to-voxel correlation

matrix (see Table 2). Integrated local correlation (ILC) (Desh-

pande et al. 2007) characterizes the average local connectivity

between each voxel and its neighbors. Radial correlation con-

trast (RCC) (Goelman, 2004) characterizes the spatial asym-

metry of the local connectivity pattern between each voxel

and its neighbors. Intrinsic connectivity contrast (ICC) (Mar-

tuzzi et al. 2011) and radial similarity contrast (RSC) (Kim

et al., 2010) are novel measures similar to ILC and RCC

measures, but characterizing the global connectivity pattern

between each voxel and the rest of the brain (instead of the

local connectivity pattern around each voxel). In particular,

ICC characterizes the strength of the global connectivity pat-

tern between each voxel and the rest of the brain, while RSC

characterizes the global similarity between the connectivity

patterns of neighboring voxels. In addition to these measures

the toolbox allows simple and fast implementation of other

user-deﬁned voxel-level fcMRI measures, as long as these

measures can also be characterized as a function of the eigen-

vectors/eigenvalues of the voxel-to-voxel correlation matrix.

The Illustration of Functional Connectivity Analysis in Conn

section [Illustration of voxel-to-voxel analysis (optimal placement

of fcMRI seeds) subsection] illustrates the application of one

such user-deﬁned measure to investigate between-session

similarity of functional connectivity patterns, and Appendix

A.2 describes the characterization of this measure as a func-

tion of the eigenvectors/eigenvalues of the voxel-to-voxel

connectivity matrix.

Task-related and resting state fcMRI

The previous sections characterize the steps necessary to

perform ﬁrst-level (within-subjects) connectivity analysis of

resting state BOLD time series, as well as time series derived

from the residuals of BOLD time series in block- and event-

related designs after removing modeled task or condition

effects (Fair et al., 2007) (e.g., simply by including these mod-

eled condition effects as additional temporal confounding

factors). The toolbox also allows condition-dependent func-

tional connectivity analysis of block design studies, such as

fcMRI analysis of interleaved resting periods or analysis of

functional connectivity within task blocks. In these cases

and after the session-speciﬁc treatment of temporal con-

founds, the BOLD time series is divided into scans associated

with each blocked presentation. To take into account the he-

modynamic delay, block regressors for each condition are

convolved with a canonical hemodynamic response function,

a combination of two gamma functions, and rectiﬁed {ﬁltered

to keep the positive part of the original time series; y[n]=

max(0,x[n])}. All of the scans with nonzero effects in the

resulting time series are concatenated for each condition

and across all sessions, weighting each scan by the value of

these time series. Alternatively it is also possible to use a

Hann function (e.g., a ‘‘Hann function’’ window, shaped as

a half cycle of a sine-squared function) instead of the rectiﬁed

hrf function that more heavily de-weights the scans at the be-

ginning and end of each block, as well as to omit any form of

within-block weighting. In the case of block design studies, it

is also recommended to include standard task regressors

(block regressors convolved with a canonical hemodynamic

response function) and their ﬁrst-derivative terms as addi-

tional covariates in the temporal preprocessing step. This

step helps avoid possible between-condition main effects

from affecting within-condition connectivity estimates in

the presence of possible voxel-speciﬁc differences in hemody-

namic delay. Resting state analysis is treated like a special

case of task-speciﬁc analysis where only one condition span-

ning the entire scanner acquisition length is considered.

Second-level analysis

Following the computation for each subject of seed-to-

voxel connectivity maps, ROI-to-ROI connectivity matrices,

and voxel-level fcMRI measures from voxel-to-voxel analysis,

each one of these measures can then be entered into a second-

level general linear model to obtain population-level estimates

and inferences. Speciﬁc hypotheses can then be tested using

between-subjects contrasts (e.g., comparing functional con-

nectivity patterns between two groups of subjects), between-

condition contrasts (e.g., comparing task- or condition-speciﬁc

connectivity patterns between two conditions), between-

source contrasts (e.g., comparing functional connectivity

Table 2. Voxel-Level Functional Connectivity

MRI Measures Derived from the Voxel-to-Voxel

Connectivity Matrix r(x,y)

Integrated local correlation P

y2O

hr(x y) r(x, y)

Radial correlation contrast P

y2O

hr(x y) q

qxk

r(x, y)

Global correlation strength 1

jOjX

y2O

jr(x, y)j2

Radial similarity contrast 1

jOjX

y2O

q

qx

r(x, y)

2

Integrated local correlation and radial correlation contrast charac-

terize properties of the local pattern of connectivity (between each

voxel and its neighbors). Global correlation strength and radial sim-

ilarity contrast characterize properties of the global pattern of connec-

tivity (between each voxel and the entire brain).

x and y represent the spatial locations of two arbitrary voxels, h

r

represents a Gaussian convolution kernel of width r, and Orepre-

sents the set of all brain voxels.

CONN: A RS-FCMRI TOOLBOX 129

patterns between two seeds), and combinations of these con-

trasts (e.g., testing group by condition interactions). False pos-

itive control in ROI-to-ROI analysis is implemented using

uncorrected or false discovery rate (FDR)-corrected p-values.

Uncorrected p-values are appropriate when the researcher’s

original hypotheses involve only the connectivity between

two a priori ROIs and FDR-corrected p-values are appropriate

when the researcher’s original hypotheses involve the connec-

tivity between larger sets of ROIs and do not specify a priori

which ROIs are expected to show an effect. False positive

control in voxel-level analysis is implemented through a com-

bination of a voxel-level height threshold (deﬁned by uncor-

rected or FDR-corrected voxel-level p-values) and a cluster-

level extent threshold (deﬁned by uncorrected, family wise

error [FWE]-corrected, or FDR-corrected cluster-level p-values).

Graph theoretical analysis

The toolbox also computes several graph theoretical mea-

sures (Achard and Bullmore, 2007; Bullmore and Sporns,

2009; Latora and Marchiori, 2001; Watts and Strogatz, 1998)

characterizing structural properties of the estimated ROI-to-

ROI functional connectivity networks, and allows users

to perform group-level analysis of these measures. Each

subject-speciﬁc ROI-to-ROI connectivity matrix is thresh-

olded at a ﬁxed level. This threshold can be based on raw con-

nectivity values, normalized z-scores, or percentile scores

(resulting in graphs with ﬁxed network-level cost). Supra-

threshold connectivity values deﬁne an adjacency matrix

characterizing a graph with nodes associated with ROIs,

and edges associated with the strength of functional connec-

tivity among these ROIs. For each node nin a graph G,cost

is deﬁned as the proportion of connected neighbors, global

efﬁciency is deﬁned as the average inverse shortest path dis-

tance from node nto all other nodes in the graph, and local ef-

ﬁciency is deﬁned as the average global efﬁciency across all

nodes in the local subgraph of node n(the subgraph consist-

ing only of nodes neighboring node n). In addition, equiva-

lent network-level summary measures can be deﬁned by

averaging across all nodes of the network (Table 3). Popula-

tion-level inferences on these graph theoretical measures are

obtained using a second-level general linear model as in the

fcMRI analysis above.

Illustration of Functional Connectivity Analysis in Conn

In this section several examples of fcMRI analysis per-

formed with Conn are illustrated. These examples are chosen

to illustrate some of the standard approaches available for

RSFC analysis, as well as to demonstrate the reliability of

the functional connectivity measures computed by the Conn

toolbox. The analyses were based on a publically available

resting state dataset (NYU CSC TestRetest dataset, www

.nitrc.org/projects/nyu_trt), which has been previously ana-

lyzed in detail demonstrating the reliability of functional con-

nectivity measures (Shehzad et al., 2009). This dataset consists

of echo planar imaging (EPI) images of 25 participants col-

lected on three occasions: (1) the ﬁrst resting state scan in a

scan session, (2) 5–11 months after the ﬁrst resting state

scan, and (3) about 30 ( <45) min after the second resting

state scan. Resting state scans consist of 197 continuous EPI

functional volumes (TR =2000 ms; TE =25 ms; ﬂip angle =90;

39 slices, matrix =64 ·64, FOV =192 mm; isotropic 3-mm

acquisition voxel size).

Preprocessing of BOLD time courses

Spatial preprocessing of functional volumes included re-

alignment, normalization, and smoothing (8-mm FWHM

Gaussian ﬁlter), using SPM8 default parameter choices. Ana-

tomical volumes were segmented into gray matter, white

matter, and CSF areas, and the resulting masks were eroded

(one voxel erosion, isotropic 2-mm voxel size) to minimize

partial volume effects. The temporal time series characterizing

the estimated subject motion (three-rotation and three-

translation parameters, plus another six parameters repre-

senting their ﬁrst-order temporal derivatives), as well as

the BOLD time series within the subject-speciﬁc white matter

mask (three PCA parameters) and CSF mask (three PCA

parameters), were used as temporal covariates and removed

from the BOLD functional data using linear regression, and

the resulting residual BOLD time series were band-pass ﬁl-

tered (0.01 Hz <f<0.10 Hz). Figure 2 illustrates the effect of

removing temporal covariates on the distribution of voxel-

to-voxel BOLD signal correlation values. A random subset

of 256 voxels (the same voxels across subjects and sessions)

was used to compute the sample distribution of voxel-to-

voxel BOLD signal correlation values separately for each

subject and session, before and after removal of the deﬁned

temporal covariates.

Estimated voxel-to-voxel correlations using the raw BOLD

signals typically show distributions with some degree of pos-

itive bias, and with large differences between sessions and

subjects. In contrast, after temporal preprocessing, the esti-

mated voxel-to-voxel correlations appear more centered and

with very similar distributions across sessions and subjects.

To quantify this observation, we computed measures of inter-

session reliability of the voxel-to-voxel connectivity measures

from the raw BOLD signal, and compared them with the

Table 3. Deﬁnition of Graph Theoretical Measures Characterizing Structural Properties

of Functional Connectivity Networks

ROI-level measures Network-level measures

Cost Cn(G)=1

jGj1jGnjC(G)=1

jGjP

neG

Cn(G)

Global efficiency Eglobal

n(G)=1

jGj1P

m6¼n2G

d1

nm (G)Eglobal(G)=1

jGjP

n2G

Eglobal

n(G)

Local efficiency Elocal

n(G)=Eglobal(Gn)Elocal (G)=1

jGjP

n2G

Elocal

n(G)

d

nm

(G) represents the shortest path distance between nodes nand min graph G, and jGjrepresents the number of nodes in graph G.

ROI, region of interest.

130 WHITFIELD-GABRIELI AND NIETO-CASTANON

same measures after temporal preprocessing. The reliability

(average intersession correlation) of the resulting group-

level voxel-to-voxel connectivity estimates between ran-

domly selected voxels was r=0.52 from the raw BOLD signal,

r=0.62 when using subject motion covariates, and r=0.70

when additionally using CompCor method of white matter

and CSF noise covariates, and their corresponding intraclass

correlation coefﬁcients (one-way random effects, Shehzad

et al. 2009) were 0.22, 0.55, and 0.71, respectively. These re-

sults highlight that reliability of group-level voxel-to-voxel

connectivity measures increases dramatically with the addi-

tional methods of noise reduction implemented in the tempo-

ral preprocessing steps of the Conn toolbox, potentially due to

their effect reducing physiological and other noise-dependent

biases on functional connectivity estimates.

Illustration of seed-to-voxel bivariate correlation analysis

A posterior cingulate cortex (PCC) region [a spherical ROI

with MNI coordinates (6,52,40) and radius 10 mm (Fox

et al., 2005)] was used as the seed. The PCC seed shows pos-

itive functional connectivity with a network of default areas

(shown in red in Fig. 3 top) and negative functional connec-

tivity (shown in blue) with task-related regions. In addition,

three separate within-session estimates of the connectivity

strength between the PCC seed and each voxel were com-

puted. These session-speciﬁc estimates represent the Fisher-

transformed correlation coefﬁcients for each voxel averaged

across all subjects and converted back to raw correlation coef-

ﬁcient values. Within-session estimates (Fig. 3) show a high

degree of reliability (intersession correlation r=0.95, mean ab-

solute error 0.03) when comparing group-level estimates

of functional connectivity strength across repeated runs or

sessions. Similarly, high interscan reliability (r=0.97) was

found when repeating these analyses using bivariate regres-

sion measures instead of bivariate correlation measures.

Illustration of seed-to-voxel semipartial

correlation analysis

Multivariate seed-to-voxel analysis was also performed to

explore the unique connectivity with the PCC area that is not

mediated by other default network areas. The average BOLD

time series within the PCC area were used as sources of the

FIG. 3. Seed-to-voxel functional connectivity with PCC seed area. Top: Spatial patterns of group-level seed-to-voxel connectivity

measures (bivariate correlation) collapsed across the three sessions available from each subject. Red: positive connectivity, blue:

negative connectivity. Results are thresholded at FWE-corrected cluster-level p<0.05 (with FDR-corrected two-sided p<0.05

height threshold). Bottom: Intersession reliability. Correlations between session-speciﬁc estimates ofgroup-level seed-to-voxel con-

nectivity measures, between session 2 and session 1 (5–11-month difference between the sessions), and between session 2 and ses-

sion 3 (30-min difference between the sessions). FWE, family wise error; FDR, falsediscoveryrate; PCC, posterior cingulate cortex.

CONN: A RS-FCMRI TOOLBOX 131

seed-to-voxel analysis. A multivariate representation of

the activation within three control ROIs—medial prefrontal

cortex (MPFC), left lateral parietal, and right lateral parie-

tal—characterizing, for each ROI, the average BOLD activa-

tion plus four orthogonal components derived from a

principal component decomposition of the within-ROI

BOLD time series were used as control variables. Semipartial

correlation values with the PCC seed were estimated for each

voxel. PCC shows unique positive and negative functional

connectivity with a large network of areas that are not medi-

ated by other default network regions (Fig. 4). In addition,

three separate within-session estimates of the semipartial cor-

relation coefﬁcients between the PCC seed and each voxel

were computed. These session-speciﬁc estimates represent

the Fisher-transformed semipartial correlation coefﬁcients

for each voxel averaged across all subjects and converted

back to raw correlation coefﬁcient values. Within-session es-

timates show a high degree of reliability (intersession correla-

tion r=0.82, mean absolute error 0.03) when comparing

group-level estimates of unique functional connectivity

strength across repeated runs or sessions (Fig. 4). Similar

interscan reliability (r=0.88) was found when repeating

these analyses using multivariate regression measures in-

stead of semipartial correlation measures.

Illustration of ROI-to-ROI analysis

This analysis uses the same PCC seed area as the previous

seed-to-voxel analysis, and estimates the ROI-to-ROI func-

tional connectivity (bivariate correlation measure) between

this seed and a set of 84 ROIs deﬁning the Brodmann areas

(talairach atlas; Lancaster et al., 2000). Group-level estimates

of ROI-to-ROI connectivity show a high degree of reliability

(Fig. 5; intersession correlation r=0.99, mean absolute error

0.01). Similar interscan reliability (r=0.98) was found when

repeating these analyses using bivariate regression measures

instead of bivariate correlation measures.

Illustration of graph metrics analysis

The entire matrix of ROI-to-ROI functional connectivity

values (bivariate correlation measure) was computed for

each subject using the Brodmann area ROIs, and thresholded

at a ﬁxed network-level cost value to deﬁne an undirected

graph characterizing the entire network of functional connec-

tions between these ROIs. Negative functional connectivity

values were disregarded in these analyses. The network

global and local efﬁciency was computed for a range of pos-

sible cost value (K) thresholds and compared to a random

graph and to a lattice graph with the same network size

FIG. 4. Seed-to-voxel analysis of unique connectivity with PCC seed area (controlled by MPFC, left and right LP). Top: Spatial

pattern of group-level effects of the semipartial correlation coefﬁcients collapsed across the three sessions available from each sub-

ject. Results are thresholded at FWE-corrected cluster-level p<0.05 (with FDR-corrected two-sided p<0.05 height threshold). Bot-

tom: Intersession reliability of semipartial correlation measures with PCC seed. MPFC, medial prefrontal cortex; LP, lateral parietal.

132 WHITFIELD-GABRIELI AND NIETO-CASTANON

and cost (Fig. 6). Small world properties were observed in the

range of costs 0.05 <K<0.25, where global efﬁciency is greater

than that of a lattice graph and local efﬁciency is greater than

that of a random graph (Achard and Bullmore, 2007). Using

an intermediate K=0.15 cost threshold level, the global efﬁ-

ciency of each ROI, a measure of the centrality of each ROI

within the network, was computed and averaged across all

subjects. This measure showed a high degree of reliability

when comparing session-speciﬁc estimates of global efﬁ-

ciency across repeated runs or sessions (Fig. 6; intersession

correlation r=0.95, mean absolute error 0.01). Similar inter-

scan reliability was found for other graph theoretical mea-

sures (local efﬁciency r=0.90; cost r=0.95).

Illustration of voxel-to-voxel analysis (RSC)

This analysis investigates the similarity, at each voxel, be-

tween the global functional connectivity patterns of this

voxel and those of its neighbors. The voxel-to-voxel func-

tional connectivity matrix was computed separately for

each session using an isotropic 2-mm voxels within an a priori

gray matter mask (SPM apriori/grey.nii mask thresholded at

FIG. 5. ROI-to-ROI functional

connectivity with PCC seed area. Top:

ROIs deﬁned from talairach atlas

Brodmann areas that show positive

(red) and negative (blue) functional

connectivity with PCC are shown (for

display clarity each ROI is identiﬁed

by its centroid positions). Results are

thresholded at FDR-corrected p<0.05.

Bottom: Intersession reliability of

ROI-to-ROI group-level functional

connectivity measures.

CONN: A RS-FCMRI TOOLBOX 133

FIG. 6. ROI-level analysis of global efﬁciency. Top: Global efﬁciency of each ROI (a measure of ROI centrality, and shown

proportional to circle sizes in the left display) in the network deﬁned by positively associated ROIs (ROIs deﬁned from talair-

ach atlas Brodmann areas). Small world properties, where global efﬁciency is greater than that of a lattice graph and local ef-

ﬁciency is greater than that of a random graph, are observed at the chosen cost threshold level (K=0.15). Bottom: Intersession

reliability of the estimated group-level measures of global efﬁciency for each ROI.

134

p>0.25; N=212,792 voxels). The RSC measure computes the

norm of the difference between the functional connectivity

patterns (rows of the voxel-to-voxel matrix) of neighboring

voxels. Average group-level RSC across all sessions is

shown in Figure 7 (top). Within-session estimates (Fig. 7 bot-

tom) show a high degree of reliability (intersession correla-

tion r=0.98, mean absolute error 0.002) when comparing

group-level estimates of RSC across repeated runs or ses-

sions. Similar interscan reliability was found for other

voxel-level fcMRI measures derived from voxel-to-voxel

analysis (ILC r=0.98; RCC r=0.99; GCS r=0.97).

Illustration of voxel-to-voxel analysis (optimal placement

of fcMRI seeds)

This analysis investigates the reliability of seed-to-voxel

functional connectivity estimates across all possible seed loca-

tions. They were implemented as voxel-to-voxel analysis

using a user-deﬁned measure characterizing the between-

session similarity of functional connectivity patterns at each

voxel. Isotropic 2-mm voxels within an a priori gray matter

mask (SPM apriori/grey.nii mask thresholded at p>0.25)

were used for this analysis (N=212,792 voxels). The matrix

of voxel-to-voxel functional connectivity values (bivariate

correlation matrix, with size N·N) was parameterized sepa-

rately for each subject and for each session (Appendix A.2).

Separately for each row of this matrix (subject-speciﬁc func-

tional connectivity estimates between a given seed voxel

and all of the gray matter voxels) the intersession correlation

was computed and averaged across each pair of sessions and

across all subjects. The resulting measures (for each voxel)

characterize the average subject-level intersession reliability

of seed-to-voxel analysis when using each voxel as a possible

seed location (c.f. group-level reliability measure used in the

previous sections) (Fig. 8). Intersession reliability values of

subject-level connectivity estimates ranged between r=0.01

and r=0.62 (average r=0.29) across all possible seed loca-

tions. Local peaks in this map characterize optimal seed loca-

tions (they result in seed-to-voxel functional connectivity

patterns that are more robust across sessions than those pat-

terns resulting when using neighboring seed locations). Peak

values with intersession reliability above r=0.50 are show in

Figure 8 (bottom). Robust seed locations were identiﬁed in

default network areas—PCC, MPFC, and lateral parietal—

in close agreement with standard seed locations for these

areas (Fox et al., 2005). In addition, other robust locations in-

cluded superior temporal gyrus (one anterior temporal

source, and a different posterior source close to supramargi-

nal gyrus), superior frontal gyrus, and cingulate gyrus. The

results showed high degree of hemispheric symmetry, with

all of the peaks (except medial peaks: MPFC, PCC, and cingu-

late gyrus) having a corresponding peak with similar location

and reliability in the opposite hemisphere. Since the seed with

highest reliability (0,56,28) was close but slightly inferior to

the a priori PCC seed location used in the previous sections

(6,52,40), we deﬁned for comparison a new seed location

using a spherical ROI of 10 mm centered at the new coordina-

tes (0,56,28). The group-level and subject-level intersession

reliability of the seed-to-voxel functional connectivity estima-

tes when using this new seed deﬁnition was r=0.97 and

FIG. 7. Voxel-to-voxel analysis of radial similarity contrast measure. Top: Average group-level radial similarity contrast at

each voxel. Darker shades for a voxel indicate higher similarity between the global functional connectivity patterns of this

voxel and those of its neighbors. Bottom: Intersession reliability of the group-level radial similarity contrast measure.

CONN: A RS-FCMRI TOOLBOX 135

r=0.64, respectively (compared with r=0.95 and r=0.55, re-

spectively, when using the original PCC deﬁnition).

Discussion

RSFC analysis offers an important characterization of func-

tional brain connectivity for both normal and patient popula-

tions. This article describes the methods used to compute a

variety of functional connectivity measures in the Conn tool-

box and illustrates the interscan reliability of these measures.

The Conn toolbox offers a large suite of connectivity analyses

packaged in a user-friendly GUI. The toolbox can be best used

in conjunction with SPM but it is compatible with other anal-

ysis packages. The output of most processing and analysis

procedures are stored as NIFTI volumes (e.g., the time series

post noise reduction and correlation and Z-maps from seed

voxel analysis) that may be used for further interrogation. For

example, researchers may enter the subject-level Z-maps

(Fisher-transformed subject-level correlation coefﬁcients

when performing bivariate-correlation analysis) in their anal-

ysis package of choice for additional second-level analysis.

The toolbox also offers a complete batch processing environ-

ment facilitating the implementation of scalable and robust

functional connectivity analysis using a simple common

framework. In addition, the toolbox encourages users to

explore their data at intermediate steps of the analyses (e.g.,

distribution of voxel-to-voxel connectivity estimates, spatial

patterns of potential confounder effects, and individual sub-

ject-level connectivity maps), which can aid in detecting

and correcting potential anomalies in the data as well as iden-

tifying sources of variability that might go unnoticed when

focusing on group-level summary results alone.

Anticorrelations

There has been a debate as to whether observed anticorre-

lations are valid neurophysiological ﬁndings or analytic

artifacts introduced by global signal regression, a common

technique in removing confounds due to physiological and

other noise sources in the BOLD time series (Buckner et al.,

FIG. 8. Voxel-to-voxel analysis studying robustness of seed locations. Top: Intersession reliability maps. This display shows

the intersession correlation between functional connectivity patterns for all possible seed locations. Darker shades for a voxel

indicate that this voxel, when used as seed for standard seed-to-voxel functional connectivity analysis, results in connectivity

patterns that are better replicated across sessions (higher intersession correlations). Bottom: Optimal seed locations, as esti-

mated from the local peaks of the reliability maps above. Seed locations that show local maxima in reliability when comparing

subject-level estimates of functional connectivity strength across repeated runs or sessions (rvalue represents the subject-

speciﬁc intersession correlation averaged across all subjects). All seeds with average r>0.50 are shown. Peak locations are

reported as (x,y,z) Montreal Neurological Institute coordinates. LLP, left lateral parietal; RLP, right lateral parietal.

136 WHITFIELD-GABRIELI AND NIETO-CASTANON

2008; Fox et al., 2009; Murphy et al., 2009; Weissenbacher

et al., 2009). There is general agreement, however, as may

be illustrated with mathematical proof, that a seed voxel anal-

ysis using global signal regression will necessarily show anti-

correlations even if none were truly present in the data

because after global regression the distribution of the correla-

tion coefﬁcients between a voxel and every other voxel in the

brain is shifted such that the sum £0 (Fox et al., 2009; Murphy

et al., 2009). Because of this mathematical consequence of the

shift in correlation distribution and because the global signal

may contain important neural signals as well as noise, it has

been recommended to refrain from interpreting anticorrela-

tions when using global signal regression (Chang and Glover,

2009; Murphy et al., 2009).

However, the CompCor method of noise reduction, which

does not rely on global signal regression or physiological

monitoring, also results in anticorrelations, further support-

ing a biological basis for their existence, and the positive cor-

relations have higher sensitivity and speciﬁcity than global

signal regression (Chai et al., 2012). We believe that the

CompCorr method, as implemented in Conn, yields valid

anticorrelations between large-scale brain networks.

Example illustrations

This article also describes the methods used to compute a

variety of functional connectivity measures in the Conn tool-

box and illustrates the interscan reliability of these mea-

sures. Seed-to-voxel and ROI-to-ROI measures of functional

connectivity show high reliability for well-characterized

seed locations in RSFC, both when using correlation- and

regression-based measures to characterize functional connec-

tivity. Similarly, graph theoretical measures characterizing

structural properties of functional connectivity networks, as

well as voxel-level measures characterizing the local connec-

tivity patterns (between each voxel and its neighbors) and

global connectivity patterns (between each voxel and the

rest of the brain) also show high levels of interscan reliability.

Conclusions

The Conn toolbox offers a common framework to deﬁne

and perform a large suite of connectivity analyses, including

bivariate/semipartial correlations, bivariate/multivariate re-

gression, seed-to-voxel connectivity, ROI-ROI connectivity,

novel voxel-to-voxel connectivity, and graph theoretical mea-

sures for both resting state and task fMRI data. The analyses

in this article show high levels of interscan reliability for a

variety of fcMRI measures corroborating their potential appli-

cation as useful neuromarkers. In addition, the Conn imple-

mentation of the anatomical CompCor method of noise

reduction increases sensitivity and speciﬁcity of functional

connectivity and allows for better interpretability of anticor-

relations as it does not rely on global signal regression. We

hope that the imaging community will beneﬁt from the con-

tribution of the tools.

Acknowledgments

The Poitras Center for Affective Disorders Research at the

McGovern Institute for Brain Research at MIT supported this

work. The authors thank Shay Mozes for initial programming

support and John Gabrieli for comments on the article.

Author Disclosure Statement

The authors of the study have no conﬂict of interest to

declare.

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Address correspondence to:

Susan Whitﬁeld-Gabrieli

Department of Brain and Cognitive Sciences

Martinos Imaging Center at McGovern Institute

for Brain Research

Poitras Center for Affective Disorders Research

Massachusetts Institute of Technology

Cambridge, MA 02139

E-mail: swg@mit.edu

Appendix

Appendix A.1: Treatment of Temporal

Confounding Factors

The observed raw blood oxygen level-dependent (BOLD)

contrast signal s(x,t) at voxel xand time tis characterized as

a linear combination of (1) Ntemporal confounds deﬁned ex-

plicitly through subject- and session-speciﬁc time series c

n

(t)

(e.g., representing subject motion effects); (2) those confounds

deﬁned implicitly from Knoise region of interests (ROIs),

each characterized by M

k

principal component time series

d

kn

(t) (e.g., representing physiological effects observable in

white matter and CSF areas); and (3) an underlying BOLD

time series of interest e(x,t):

s(x,t)=X

N

n=1

^

an(x)cn(t)þX

K

k=1X

Mk

n=1

^

bkn(x)dkn (t)þe(x,t)

(A:1:1)

where a

n

(x) and b

kn

(x) represent voxel-speciﬁc weights for

each of the confounding factors. The factors a

n

(x) and b

n

(x)

CONN: A RS-FCMRI TOOLBOX 139

are estimated using ordinary least squares, and the BOLD sig-

nal of interest e(x,t) is approximated as the residuals in the lin-

ear model ﬁt. The noise ROI time series d

kn

(t) for each ROI k

are estimated using principal component analysis of the

time series s(x,t) limited to within ROI voxels [and after an ini-

tial orthogonalization with respect to the known c

n

(t) factors].

e(x,t)=s(x,t)X

N

n=1

^

an(x)cn(t)

dk1(t)=1

jOkjX

x2Ok

e(x,t)

dkn(t)(n=2...Mk)j1

jOkjX

x2Ok

(e(x,u)dk1(u)) (e(x,)dk1())

=X

T

n=2

dkn(u)dkn ()

X

T

t=1

dkn(t)dkm (t)=081<n<m

X

T

t=1

d2

kn(t)qX

T

t=1

d2

km(t)81<n<m

Where d

k1

(t) represents the average residual BOLD time se-

ries within the ROI voxels W

k

, and d

kn

(t) (for 1 <n£M

k

) repre-

sents the ﬁrst M

k

–1 principal components of the temporal

covariance matrix within the same voxels. The initial orthog-

onalization in e*(x, t) guarantees that the noise ROI time se-

ries d

kn

(t) are in turn orthogonal to the confounding factor

time series c

n

(t), making the resulting model (1) maximally

predictive.

This approach assumes that the BOLD signal of interest e(x,t)

is orthogonal to each confounding factors c

n

(t)andd

kn

(t).

While this can lead to decreased sensitivity in those cases

where the signal of interest is in fact correlated with some of

the confounding factors, we believe that the increased robust-

ness and validity of the resulting functional connectivity mea-

sures compensates the decreased sensitivity in these cases.

Appendix A.2: Computing Voxel-Wise Linear

Functional Connectivity Measures Singular

Value Decomposition

We represent the raw BOLD signal at voxel xand time tas

s(x,t). We are interested in computing the functional connec-

tivity matrix R, characterizing the temporal correlation be-

tween the BOLD signal at two arbitrary voxels x

1

and x

2

:

R: r(x1,x2)X

t

~

s(x1,t)~

s(x2,t)(A:2:1)

where ~

s(x,t) represents the normalized BOLD time series

(after subtracting its mean, and dividing by its standard devi-

ation). Typically the number of voxels is considerably larger

than the number of time points (scans). Because of this it is

more efﬁcient to compute the crosscovariance matrix (time-

by-time covariance matrix, aggregated across all brain voxels

W), and to perform a singular value decomposition as follows:

C: c(t1,t2)X

x2O

~

s(x,t1)~

s(x,t2)=X

n

dnqn(t1)qn(t2)

From this decomposition we can deﬁne a new set of maps

b

n

as:

bn:bn(x)X

t

qn(t)~

s(x,t)

This decomposition allows a simple reconstruction of the

voxel-to-voxel BOLD signal temporal correlation matrix Ras:

R: r(x1,x2)=X

n

bn(x1)bn(x2)(A:2:2)

In this way the b

n

maps (eigenvectors) and associated d

n

values (eigenvalues) implicitly characterize the correlation

matrix R. In the presence of band-pass ﬁltering the number

of independent eigenvectors nis signiﬁcantly smaller than

the number of independent time points (scans), so storing

the b

n

(x) maps requires always less storage than a copy of

the original functional data s(x,t). In addition, many measures

derived from the matrix Rcan easily be computed without

ever requiring to explicitly estimate the elements of this ma-

trix as the following section will illustrate.

Derived Measures

It can be shown that the b

n

maps form an orthogonal basis

{

for the entire set of possible connectivity maps (the row or col-

umn space of R). For example, an entire voxel-to-voxel con-

nectivity map with voxel x

1

as seed (one row of R) can be

computed as follows:

r(x1)=X

n

bn(x1)bn

And the average of several connectivity maps (averaging sev-

eral rows of R, e.g., those corresponding to voxels within a

given seed ROI X) can be computed as follows:

r(X)Ær(x)æx2X=X

n

bn(X)bn

where

bn(X) represents the average value of the b

n

map at the

voxels within the ROI X.

The norm of these connectivity maps, a measure of the over-

all strength of each connectivity map averaged across all target

voxels (corresponding to the norm of one row of R), can be com-

puted, respectively, and with minimal computation as follows:

kr(x1)k2X

x22O

r2(x1,x2)=X

n

dnb2

n(x1)

kr(X)k2X

x22O

Ær(x1,x2)æ2

x12X=X

n

dn

b2

n(X)

The average voxel-to-voxel connectivity between two

ROIs x

1

and x

2

(averaging the values within a submatrix

of R) can also be computed with minimal computation

{

as

follows:

{

Not orthonormal, as the squared-norm of each b

n

map equals d

n

.

{

For example, we can consider the costs associated with computing

the average connectivity between any two arbitrary pairs of ROIs in

one atlas encompassing the entire set of brain voxels. Using (1) we

would need to compute all voxel-to-voxel connectivity values ﬁrst

and then average across the desired ROIs. This computation scales

quadratically with the number of voxels, which is usually

prohibitive both in terms of time and required memory storage.

Using (2) instead, this computation scales linearly with the number

of voxels, as it only requires computing the average values of the b

maps within each ROI (no voxel-wise cross-products involved).

140 WHITFIELD-GABRIELI AND NIETO-CASTANON

r(X1,X2)Ær(x1,x2)æx12X1,x22X2=X

n

bn(X1)

bn(X2)

In addition, several complex measures derived from voxel-

wise connectivity measures can also be computed with

reduced computational cost. For example, the map of differ-

ential connectivity between two voxels x

1

and x

2

can be

computed as follows:

r(x1)r(x2)=X

n

(bn(x1)bn(x2)) bn

The overall strength of this differential connectivity map, a

measure of the difference between the two individual connec-

tivity maps (similar to g

2

measure in Cohen et al., 2008), can

be computed as follows:

kr(x1)r(x2)k2X

x2O

(r(x1,x)r(x2,x))2

=X

n

dn(bn(x1)bn(x2))2

Similarly, the functional similarity measure between voxels

x

1

and x

2

(Kim et al., 2010), another way to characterize the

difference between two individual connectivity maps, can

be computed as follows:

S1(x1,x2)X

x2O

r(x1,x)r(x2,x)=X

n

dnbn(x1)bn(x2)

The integrated local correlation measure (Deshpande et al.,

2007), characterizing the average connectivity between a

voxel and its neighbors (where the neighborhood is deﬁned

by a spatial convolution kernel h), can be computed as

follows:

ILC(x1)Ær(x1,x2)æx22D(x1)=X

n

bn(x1)(hbn)(x1)

Similarly, the radial correlation contrast vector measure

(Goelman, 2004) can also be computed using multiple convo-

lution kernels (one for each spatial dimension), jointly deﬁn-

ing the difference vector for each neighboring voxel.

RCC(x1)=X

n

bn(x1)º(hibn)(x1)i

!þ(hjbn)(x1)j

!

þ(hkbn)(x1)k

!ß

Last, the norm of the local spatial gradient of a connectivity

map (a measure of the similarity between the global connec-

tivity patterns of neighboring voxels) can be computed as

follows:

k(=r)(x1)k2=X

n

dnk(=bn)(x1)k2

=X

n

dn((qibn)2(x1)þ(qjbn)2(x1)þ(qkbn)2(x1))

Comparing Connectivity Patterns Across Conditions

In an experimental design with multiple conditions

(e.g., block design) we might wish to compute the task- or

condition-speciﬁc connectivity matrices.

RA:rA(x1,x2)X

t2A

~

s(x1,t)~

s(x2,t)

RB:rB(x1,x2)X

t2B

~

s(x1,t)~

s(x2,t)

where Aand Brepresent the time points associated with

two conditions of interest. For any given seed voxel x

1

, the

between-conditions correlation, a measure of the similarity

between the connectivity patterns during two different condi-

tions, separately for each seed voxel and for each subject, can

be computed as follows:

corr(rA(x1), rB(x1))

=P

m,n

dA,B

m,nN

bA

m

bB

n

bA

m(x1)bB

n(x1)

P

n

dA

nN

bA2

n

bA2

n(x1)P

n

dB

nN

bB2

n

bB2

n(x1)

1=2

where Nis the total number of voxels,

bA

nrepresents the aver-

age (across all voxels) of the bA

ncomponent map, and the

matrix D

A,B

represents the between-conditions covariance

matrix:

DA,B:dA,B

m,nX

x2O

bA

m(x)bB

n(x)note :dA,A

m,n=dA

ndmn

One possible application of this between-conditions corre-

lation measure is exempliﬁed in this article in the voxel-to-

voxel analysis subsection of the results.

CONN: A RS-FCMRI TOOLBOX 141