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Conn : A Functional Connectivity Toolbox for Correlated and Anticorrelated Brain Networks

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Resting state functional connectivity reveals intrinsic, spontaneous networks that elucidate the functional architecture of the human brain. However, valid statistical analysis used to identify such networks must address sources 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 reduction, additional removal of movement, and temporal covariates, temporal filtering and windowing of the residual 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 connectivity. 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.
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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
jGj1jGnjC(G)=1
jGjP
neG
Cn(G)
Global efficiency Eglobal
n(G)=1
jGj1P
mn2G
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
dnk(=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
Article
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