Brain-wide connectome inferences using functional connectivity MultiVariate Pattern Analyses (fc-MVPA)

Abstract

Current functional Magnetic Resonance Imaging technology is able to resolve billions of individual functional connections characterizing the human connectome. Classical statistical inferential procedures attempting to make valid inferences across this many measures from a reduced set of observations and from a limited number of subjects can be severely underpowered for any but the largest effect sizes. This manuscript discusses fc-MVPA (functional connectivity Multivariate Pattern Analysis), a novel method using multivariate pattern analysis techniques in the context of brain-wide connectome inferences. The theory behind fc-MVPA is presented, and several of its key concepts are illustrated through examples from a publicly available resting state dataset, including an analysis of gender differences across the entire functional connectome. Finally, Monte Carlo simulations are used to demonstrate the validity and sensitivity of this method. In addition to offering powerful whole-brain inferences, fc-MVPA also provides a meaningful characterization of the heterogeneity in functional connectivity across subjects.

Full text

RESEARCH ARTICLE
Brain-wide connectome inferences using
functional connectivity MultiVariate Pattern
Analyses (fc-MVPA)
Alfonso Nieto-CastanonID
1,2
*
1Department of Speech, Language, and Hearing Sciences, Boston University, Boston, Massachusetts,
United States of America, 2Department of Brain and Cognitive Sciences, Massachusetts Institute of
Technology, Cambridge, Massachusetts, United States of America
*alfnie@bu.edu
Abstract
Current functional Magnetic Resonance Imaging technology is able to resolve billions of
individual functional connections characterizing the human connectome. Classical statistical
inferential procedures attempting to make valid inferences across this many measures from
a reduced set of observations and from a limited number of subjects can be severely under-
powered for any but the largest effect sizes. This manuscript discusses fc-MVPA (functional
connectivity Multivariate Pattern Analysis), a novel method using multivariate pattern analy-
sis techniques in the context of brain-wide connectome inferences. The theory behind fc-
MVPA is presented, and several of its key concepts are illustrated through examples from a
publicly available resting state dataset, including an analysis of gender differences across
the entire functional connectome. Finally, Monte Carlo simulations are used to demonstrate
the validity and sensitivity of this method. In addition to offering powerful whole-brain infer-
ences, fc-MVPA also provides a meaningful characterization of the heterogeneity in func-
tional connectivity across subjects.
Author summary
The human connectome comprises billions of functional connections between distant
brain areas. In recent years, analyses of functional Magnetic Resonance Imaging (fMRI)
data have provided large amounts of information exploring the differences in the human
connectome across individuals, developmental trajectories, or mental states. However, sci-
entists’ ability to derive strong conclusions from the analysis of these data are often hin-
dered by the sheer number of connections analyzed, where only connections that show
exceptionally large effects are able to stand out against that vast background. This leads to
results that tend to overemphasize similarities and mask out differences that are either
weaker or distributed across multiple individual connections, potentially misleading con-
ceptual models of the human connectome. This manuscript discusses a novel method for
the analysis of the human connectome (functional connectivity Multivariate Pattern Anal-
ysis) that addresses these limitations and enables strong conclusions from fMRI data by
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OPEN ACCESS
Citation: Nieto-Castanon A (2022) Brain-wide
connectome inferences using functional
connectivity MultiVariate Pattern Analyses (fc-
MVPA). PLoS Comput Biol 18(11): e1010634.
https://doi.org/10.1371/journal.pcbi.1010634
Editor: Daniele Marinazzo, Ghent University,
BELGIUM
Received: June 30, 2022
Accepted: October 4, 2022
Published: November 15, 2022
Peer Review History: PLOS recognizes the
benefits of transparency in the peer review
process; therefore, we enable the publication of
all of the content of peer review and author
responses alongside final, published articles. The
editorial history of this article is available here:
https://doi.org/10.1371/journal.pcbi.1010634
Copyright: ©2022 Alfonso Nieto-Castanon. This is
an open access article distributed under the terms
of the Creative Commons Attribution License,
which permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: This manuscript
analyses are based on a publicly available resting
state dataset (Cambridge 1000-connectomes
dataset, n=198; www.nitrc.org/projects/fcon_1000;
doi:10.1073/pnas.0911855107), and all methods

combining classical statistics with modern pattern analysis techniques. This technique is
exemplified using a publicly available database of resting state data to characterize some of
the main aspects of the human connectome that differ across individuals, and to identify
specific differences in the human connectome across gender.
This is a PLOS Computational Biology Methods paper.
Introduction
Functional connectivity Magnetic Resonance Imaging (fcMRI) is used to characterize the state
(e.g. during rest or during individual cognitive tasks) of the human connectome, the set of all
functional connections within the brain. In its most basic form, the entire human connectome
state can be represented in a way that is limited only by the spatial resolution of the MRI acqui-
sition sequence as a matrix of voxel-to-voxel functional connectivity values. Human connec-
tome research is often motivated by the attempt to characterize similarities and discrepancies
in these functional connectivity matrices across subjects or across experimental conditions,
performing inferences that extrapolate from the limited data available in a study sample to
properties of the human connectome in a larger population. However, this form of uncon-
strained brain-wide connectome inferences can suffer from a curse of dimensionality. A
mass-univariate approach analyzing each functional connection between every pair of voxels
in the brain may consist of over 60 billion individual statistical tests (the total number of pair-
wise functional connections among approximately 250,000 isotropic 2mm voxels within gray
matter areas). This poses considerable difficulties. First, analytically, as an appropriate correc-
tion for multiple comparisons across this abundance of tests results in exceedingly low sensi-
tivity. For example, simple Bonferroni or False Discovery Rate corrections [1,2] would require
at least one individual connection below a p<10
−12
significance level in order to resolve FWE-
corrected significance at the analysis level, limiting the sensitivity and applicability of these
analyses (e.g. [3]). Second, computationally, as each voxel-to-voxel correlation matrix would
require approximately 400Gb of memory or storage space for each individual subject and
experimental condition of interest, making storing and working with these matrices extraordi-
narily demanding. Third, practically, as the potential wealth of information of voxel-to-voxel
connectivity results makes reporting and interpreting the results of these analyses a significant
challenge in itself.
Existing approaches have addressed these issues by either narrowing the focus of the analy-
ses to connectivity with one or a few a priori seed areas (e.g. connectivity with amygdala) and
then performing seed-based connectivity analyses (SBC), or by limiting the analysis units from
voxels to larger parcels or Regions of Interest (ROIs) and then performing ROI-to-ROI con-
nectivity analyses (RRC). The first approach (SBC) reduces the multiple-comparison problem
by focusing on individual (or linear combinations of) rows of the functional connectivity
matrices, disregarding all other possible effects beyond those involving at least one of these a
priori seed areas. The main advantage of this approach is its simplicity, as it can take advantage
of the same cluster-level inferential procedures that have been proven effective in standard
analyses of functional activation, such as Gaussian Random Field theory inferences [4], per-
mutation/randomization analyses [5] or Threshold Free Cluster Enhancement (TFCE) [6].
The main disadvantage of this approach is a high chance of potential false negatives, as other
effects not involving the chosen seed areas may be missed.
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described in this manuscript are publicly available
as part of CONN (RRID:SCR_009550, www.nitrc.
org/projects/conn).
Funding: This research was supported by the
National Institute on Deafness and Other
Communication Disorders (R01 DC007683, R01
DC002852, R01 DC016270), National Institute of
Neurological Disorders and Stroke (U01
NS117836) and National Institute of Mental Health
(U01 MH108168).
Competing interests: The authors have declared
that no competing interests exist.

The second approach (RRC) is able to perform brain-wide connectome inferences with suf-
ficient sensitivity by drastically reducing the multiple-comparison problem, typically focusing
on no more than a few hundred ROIs, often defining an entire parcellation of the brain (e.g.
Harvard-Oxford atlas). In addition, specialized false positive control approaches, such as Func-
tional Network Connectivity [7], Network Based Statistics [8], Spatial Pairwise Clustering [9],
Threshold Free Cluster Enhancement [6], or Multivariate cNBS [10] can be used to further
increase the sensitivity of these analyses. Nevertheless, ROI-to-ROI analyses suffer from
reduced spatial specificity arising from the a priori selection of parcels of interest, and their
conclusions can be particularly sensitive to the choice of ROIs. Advances in connectivity-
based parcellations (e.g. [11]) or subject-specific functional ROIs (e.g. [12]) can be useful to
partially alleviate these concerns.
An alternative approach uses Principal Component Analyses (PCA) or Independent Com-
ponent Analyses (ICA) to assess differences in functional networks, or sets of functionally cor-
related areas, across groups [13]. Similar to seed-based approaches, PCA and ICA are able to
drastically reduce the multiple-comparison problem by focusing on individual networks, each
comprising a group of functional-related areas, and then evaluating measures of within- and
between- network connectivity. One advantage of this approach, compared to SBC, is that
these networks do not need to be defined a priori and can be instead estimated directly from
the functional data. In the context of brain-wide connectome inferences, nevertheless, these
methods suffer from similar shortcomings as SBC, namely the potential of false negatives,
where finer functional connectivity differences that are not present at the level of entire net-
works may be missed.
This paper proposes functional connectivity Multivariate Pattern Analysis (fc-MVPA), an
alternative approach to the analysis of the brain-wide connectome at the resolution of individ-
ual voxels, that overcomes the difficulties of brain-wide connectome analyses using multivari-
ate pattern analysis techniques. Like other MVPA approaches in neuroimaging, fc-MVPA
follows a general searchlight procedure (e.g. [14]), but instead of focusing on the pattern of
activation surrounding each voxel, fc-MVPA considers separately for each voxel the entire
multivariate pattern of functional connections between this voxel and the rest of the brain.
Most MVPA methods focus on the relationship between properties of these multivariate pat-
terns, characterizing a subject’s mental state, and either static (e.g. patients vs. controls) or
dynamic properties of the experimental design (e.g. pre- vs. after- intervention, task vs. rest,
etc.), while varying in how exactly these relationships are modeled or analyzed. Classical
MVPA analyses (e.g. [15,16]) attempt to estimate, from these or other searchlight patterns,
properties of the experimental paradigm. These MVPA analyses are often referred to as back-
ward models [17], and typically use machine learning classification models embedded in a
cross-validation framework to decode information about a subject’s mental state from their
activation patterns. In this popular class of MVPA models searchlight patterns act as indepen-
dent/explanatory variables, while known subject or experimental properties act as dependent/
outcome variables. Unlike those forms of MVPA, fc-MVPA instead uses a forward model of
the data, attempting to estimate the shape of these searchlight patterns from known subject or
experimental properties, switching the role of dependent/independent variables. In addition to
being advantageous in terms of the interpretability of model weights, forward models also
enable us to frame brain-wide connectome inferences in the context of the General Linear
Model (GLM), one of the most widely used inferential statistical methods in neuroimaging,
and use powerful multivariate inferences [10] to directly address researchers’ hypotheses. Spe-
cifically, this approach allows us to make statistical inferences about individual voxels in the
brain regarding the shape of their functional connectivity patterns (e.g. is the shape of the func-
tional connectivity pattern between a voxel and the rest of the brain different in patient vs.
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control subjects?), and then repeat these analyses across all voxels within the brain extending
these inferences to the entire connectome (e.g. is the shape of functional connectivity patterns
different in patient vs. control subjects anywhere in the brain?).
Code implementing fc-MVPA, as described in this manuscript, is available in SPM’s
CONN functional connectivity toolbox [18,19] and it has already been evaluated and shown to
provide valuable insights on a variety of research topics (e.g. [20–45]). Despite this, a rigorous
discussion of this method’s approach, validity, and applications has been missing. This manu-
script aims to correct that record by presenting a detailed mathematical description of the fc-
MVPA method, highlighting its relationship to other multivariate statistical inferential proce-
dures, illustrating some of fc-MVPA key concepts and applications through examples, and
demonstrating the method’s validity and sensitivity through Monte Carlo simulations.
In the first section, a general framework to perform univariate and multivariate statistical
inferences in the context of functional connectivity data is presented. This is followed by a sec-
tion describing the specifics of the fc-MVPA approach, where some of its central applications,
including brain-wide connectome inferences as well as the characterization of intersubject het-
erogeneity, are further illustrated with examples. Finally, the last section presents simulation
results demonstrating the method’s validity, and discussing some of the main factors affecting
sensitivity. All analysis examples in this manuscript are based on a publicly available resting
state dataset (Cambridge 1000-connectomes dataset, n = 198; www.nitrc.org/projects/fcon_
1000), and all methods are publicly available in SPM12 [46] (RRID:SCR_007037, www.fil.ion.
ucl.ac.uk/spm/software/spm12/) and CONN [19] (RRID:SCR_009550, www.nitrc.org/
projects/conn).
Methods
Definition of functional-connectivity univariate and multivariate analyses
(fc-MUA, fc-MVA)
For any subject nin a study, and any pair of voxels xand y, we consider the values r
n
(x,y) char-
acterizing the functional connectivity between these two voxels for this subject. Without loss
of generality, we are going to consider r
n
(x,y) to represent Pearson correlation coefficients
between these two voxels BOLD timeseries (but all of the following descriptions would equally
apply to any arbitrary connectivity or similarity measure between pairs of elements in any
high-dimensional set). In a study we desire to make inferences regarding the properties of
these connectivity measures in the population from which the study subjects are being drawn.
For example, we may ask, based on our current study data, whether there are any connectivity
differences when comparing patients to control subjects, or whether functional connectivity
strength correlates with age, or whether it is modulated by some experimental condition. To
that end we could use a forward model of the data defining for each individual connection a
separate General Linear Model (GLM) of the form:
functional connectivity Mass-Univariate Analyses (fc-MUA)
8x;y rnðx;yÞ ¼ gnbðx;yÞ þ εnðx;yÞ  sðx;yÞ
Null hypothesis : Cbðx;yÞ ¼ 0ð1Þ
(note on notation: in this document’s equations we use regular fonts to refer to scalars,
bolded lower case fonts for vectors, and bolded capital fonts for matrices; see glossary of terms
in the supplementary materials S1 Table for details about the interpretation and dimensional-
ity of all terms in these equations.) In Eq (1)g
n
is a predictor vector for each subject n
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characterizing known factors in our experimental design, such as group membership or behav-
ioral measures (also known as the model design matrix), b(x,y) is an unknown vector of regres-
sion coefficients, estimated from the data and characterizing the effect of each modeled
predictor on the outcome functional connectivity measures (e.g. the average connectivity
strength within each group), ε
n
(x,y) represents an error term sampled independently for each
subject from a random Gaussian field with zero mean and unit variance (GLM asymptotic
normality assumption), and σ(x,y) is an intersubject variance term that depends on position x
but is otherwise constant across subjects (GLM homoscedasticity assumption). The General
Linear Model uses Ordinary Least Squares (OLS) to computed an unbiased estimate of the vec-
tor bfrom the data (Gauss-Markov theorem). After estimating these regressor coefficients, we
can specify a null-hypothesis of the form Cb(x,y) = 0 for any given between-subjects contrast
C(e.g. a null hypothesis might evaluate whether functional connectivity differs between
patients and controls) and use a classical hypothesis testing framework to evaluate this hypoth-
esis. Hypothesis testing extrapolates from the observed properties of b, estimated only from
our sample data, to the effects of the associated predictor terms in the larger population, allow-
ing us to make valid inferences about any hypothesis in this larger population. Null-hypotheses
are generally evaluated using a Likelihood Ratio Test (LRT) based on a Wilks Lambda distribu-
tion and associated T- and F- statistics [47]. These hypotheses are tested separately for each
pair of voxels xand y, resulting in a statistical parametric map F(x,y) of T- or F- statistics and
associated p- values, characterizing the likelihood of our observations under the null hypothe-
sis for every individual seed and target voxels.
We refer to this approach as mass-univariate (fc-MUA) because it is based on a separate
univariate test for each connection (for each pair of voxels xand y) in the entire brain-wide
connectome. As mentioned before, one of the main difficulties with fc-MUA in the context of
brain-wide connectome inferences is the extremely large number of connections evaluated
(one for every pair of voxels) leading to the need for very strong multiple comparison correc-
tions and reduced sensitivity to detect anything but the largest effects.
Compared to this mass-univariate approach, functional connectivity MultiVariate Analyses
(fc-MVA) use a searchlight approach where each individual analysis focuses on one individual
voxel-of-interest x, and perform an omnibus test of the connectivity between this voxel and
the rest of the brain using a multivariate GLM analysis of the form:
functional connectivity MultiVariate Analyses (fc-MVA)
8xrnðxÞ ¼ gnBðxÞ þ εnðxÞ  ΣðxÞ
Null hypothesis : CBðxÞ  PðxÞ ¼ 0ð2Þ
The term r
n
(x,y) in Eq 2 contains an entire map of connectivity values defined in vector
form (each element of this vector contains the connectivity value r
n
(x,y) for a different target
voxel y), fully characterizing the functional connectivity pattern for subject nbetween the
seed-voxel xand the rest of the brain. B(x) is now an unknown predictors-by- voxels matrix of
regression coefficients, ε
n
(x) is a residual error vector sampled from a random multivariate
Gaussian distribution with zero mean and unit variance, and S(x) is a voxels-by-voxels semi-
positive definite matrix characterizing the spatial covariance in functional connectivity pat-
terns, which again may depend on position xbut is otherwise constant across subjects. As
before, the General Linear Model uses Ordinary Least Squares (OLS) to estimate the matrix of
regressor coefficients Bfrom the data. In the context of hypothesis testing, Cand P(x) repre-
sent the between-subjects and between-voxels contrast matrices, respectively, characterizing
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which aspects of the matrix of regression coefficients we would like to evaluate. Any arbitrary
hypothesis of the form C
t
B(x)P(x) = 0, may be evaluated separately for each searchlight voxel
xusing a statistical parametric map F(x), computed using a Satterthwaite approximation [48]:
F xð Þ ¼ b
c
trðHðxÞÞ
trðWðxÞÞ Fkc;kb ð3Þ
bNrankðGÞ
crankðGCtÞ
ktrðWðxÞÞ2
trðW2ðxÞÞ
where band care error and hypothesis degrees of freedom, respectively, and the matrices W
and Hare the error and hypothesis sum of squares and products, respectively:
WðxÞ ¼ PtðxÞðRðxÞ  GBðxÞÞtðRðxÞ  GBðxÞÞPðxÞ ð4Þ
HðxÞ ¼ PtðxÞBtðxÞCtðCðGtGÞ1CtÞ1CBðxÞPðxÞ
BðxÞ ¼ ðGtGÞ1GtRðxÞ
RðxÞ  ½rt
1ðxÞrt
2ðxÞ...rt
NðxÞt
G ½gt
1gt
2. . . gt
Nt
Eq 3 results in a statistical parametric map F(x) with values that follow, under the null
hypothesis, a standard Fdistribution with kc and kb degrees of freedom. This allows us to com-
pute associated p- values characterizing the likelihood of our observations under the null
hypothesis for every individual searchlight voxel.
The between-voxels contrast matrix P(x) in Eq 2 serves to focus the analyses on a particular
subspace of interest characterizing specific features of the functional connectivity maps r
n
(x).
The choice of P(x) affords great flexibility in the specific form of fc-MVA analyses that can
possibly be implemented. For example, in the simplest possible scenario, we may choose P(x)
to be a constant one-dimensional projector, such as a unit-norm vector with positive weights
over a single voxel or a small area, which would allow us to focus only on the connectivity with
one a priori voxel or region of interest. Interestingly, in this scenario fc-MVA reduces exactly
to a standard seed-based connectivity (SBC) analysis, producing statistical parametric maps F
(x) that characterize the connectivity between the chosen voxel or area and the rest of the
brain. In contrast, in perhaps the most general scenario, we may instead choose P(x) to also be
constant but now equal to the identity matrix, jointly and equally considering all target voxels.
This allows us to simultaneously estimate and evaluate any/all aspects of the functional con-
nectivity maps r
n
(x).
Between these two extrema, there are many reasonable alternatives. For example, a spatial
basis P(x) that would focus on low spatial-frequency components of connectivity profiles (e.g.
[49]), one that would focus only on local connectivity with neighboring areas (e.g. a multivari-
ate Local Connectivity measure), or one that would focus only on connectivity with all voxels
within a fixed area (masked fc-MVA, e.g. connectivity with the cerebellum or any other large/
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heterogeneous area). In the next section we will discuss one particular form of fc-MVA analy-
ses that is based on a data-driven choice of spatial basis P(x) focusing on rich low-dimensional
representations of arbitrary functional connectivity patterns.
Definition of functional-connectivity multivariate pattern analyses (fc-
MVPA)
Functional connectivity multivariate pattern analyses (fc-MVPA) can be considered a particu-
lar case of functional connectivity multivariate analyses (fc-MVA), where the choice of spatial
basis P(x) attempts to achieve a balance between retaining high sensitivity to unknown or arbi-
trary effects while maintaining a good level of specificity to those features more representative
of the data at hand. In particular, representative features in fc-MVPA are chosen to have maxi-
mal intersubject variability and minimal overlap (i.e. orthogonal features). This is achieved by
first constructing the matrix R(x) by concatenating all of the maps r
n
(x) for a given seed-voxel
xacross all subjects, and then defining P(x) implicitly as the right- orthogonal basis from a Sin-
gular Value Decomposition (SVD) factorization of the connectivity matrix R(x):
SðxÞ  DðxÞ  PtðxÞ ¼ RðxÞ ð5Þ
where S(x) and P(x) are orthogonal matrices of left- and right- singular vectors of R(x), respec-
tively, and D(x) is a diagonal matrix containing the positive singular values of Rsorted in
decreasing order. The total number of singular vectors and values in Eq 5 is equal to the num-
ber of subjects N, but typically this dimensionality can be further reduced to a lower value
kNto only include the first few singular values and vectors that achieve a predefined predic-
tive or descriptive target (e.g. those dimensions capturing on average 50% or more of the total
covariance in the patterns of functional connectivity across subjects).
Conceptually, this particular choice of basis in fc-MVPA has a very important benefit, as
the resulting eigenpatterns, defined as the columns of the resulting matrix P(x), have a mean-
ingful interpretation as those patterns that best characterize the observed heterogeneity across
subjects in functional connectivity with an individual seed voxel. In particular the squared
eigenvalues
ξðxÞ  diagðD2ðxÞÞ=traceðRðxÞ  RtðxÞÞ ð6Þ
represent the portion of the total intersubject covariance R(x)R
t
(x) in connectivity maps that
lies within the dimensions characterized by each individual eigenpattern, and by the nature of
SVD these values are maximal (i.e. there is no other k-dimensional subspace containing a
larger percentage of the total covariance of the data than the subspace spanned by the first k
eigenpatterns, for any value k).
In this context, the values s
n
(x), which we will refer to in this manuscript as eigenpattern
scores, and which are defined as the rows of the left-singular matrix S(x), define an optimal lin-
ear low-dimensional representation of the original data r
n
(x) for each subject, meaning that we
can always linearly reconstruct the high-dimensional data r
n
(x) from its low-dimensional
representation s
n
(x) with minimal error.
Mathematically, this approach is similar to functional PCA [50] or to the group-level
dimensionality reduction step in ICA [13] which helps reduce noise, simplify the analyses, and
increase the interpretability of the resulting ICA components, but the main difference is that
in fc-MVPA dimensionality reduction is performed separately for each individual seed-voxel
x. Because the dimensionality reduction step in fc-MVPA is only tasked with characterizing
the heterogeneity in functional connectivity patterns between one individual voxel xand the
rest of the brain, while in PCA/ICA the dimensionality reduction step is tasked with
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simultaneously characterizing the heterogeneity in functional connectivity patterns between
all pairs of voxels, the former can achieve a considerably more compact representation, where
fewer components will explain a larger portion of that heterogeneity (as will be illustrated in
the next section). In addition, the ability to obtain such low-dimensional characterization in a
way that is specific to each anatomical location offers a considerably richer representation of
intersubject heterogeneity compared to other similar but global approaches, such as PCA or
ICA.
In the context of brain-wide connectome analyses, using a simple change of basis allows us
to simplify the fc-MVA multivariate general linear model and null hypothesis in Eq 2. In par-
ticular, by right-multiplying Eq 2 by the matrix P(x)D
−1
(x) we develop an equivalent lower-
dimensional fc-MVPA general linear model and hypothesis of the form:
functional connectivity MultiVariate Pattern Analyses (fc-MVPA)
8xsnðxÞ ¼ gn~
BðxÞ þ εnðxÞ  ~
ΣðxÞ
Null hypothesis : C~
BðxÞ ¼ 0ð7Þ
This is exactly the same model as in Eq 2 but expressed only within a lower-dimensional
subspace represented by the eigenpattern scores s
n
(x), instead of the original higher-dimen-
sional connectivity maps r
n
(x). In this context, the eigenpattern scores s
n
(x) represent what has
also been referred to as Multivariate Connectivity maps (MCOR) [51], a voxel-specific low-
dimensional multivariate representation of the pattern of functional connectivity between a
voxel and the rest of the brain. Similarly, ~
BðxÞand ~
ΣðxÞin Eq 7 are also equal to their Eq 2
counterparts simply projected onto the subspace defined by P(x). The reduced dimensionality
allows us to simplify the computational implementation of these analyses considerably. For
example, the eigenpattern scores s
n
(x) can now be simply stored as multiple whole-brain vol-
umes, with one volume or image per component and per subject, and shared across multiple
second-level analyses. This is in contrast with the considerably larger vectors r
n
(x)which can-
not be easily stored (e.g. one whole-brain volume per subject and per target voxel y). In addi-
tion, the eigenpattern scores s
n
(x) are defined independently of the predictor vectors g
n
, so
they not only offer a model-free characterization of the heterogeneity in the data, but the same
eigenpattern scores can also be used in multiple different group-level analyses of the same
data. Last, the reduced dimensionality of Eq 7 also allows the covariance ~
Σ2ðxÞacross eigenpat-
tern scores to be fully estimable from a limited number of samples, whereas the original covari-
ance S(x) across voxels in Eq 2 could very rarely be so estimated with full rank. Because of this,
the effect of the within-subjects covariance in the resulting null hypothesis Fstatistics at each
individual searchlight voxel does not need to be approximated (e.g. using Satterthwaite
approximation as in Eq 3), allowing us to rely instead on a more sensitive Likelihood Ratio sta-
tistic (LRT) of the form [52]:
F xð Þ ¼ d
ac 1l1=e
l1=eFac;dð8Þ
l¼jWj
jWþHj
arankðPðxÞÞ
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Brain-wide connectome inferences using fc-MVPA
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bNrankðGÞ
crankðGCtÞ
dbacþ1
2
 eac
2þ1
effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2c24
a2þc25
r
where λis Wilks’ Lambda statistic, ais typically equal to k, the number of selected eigenpat-
terns (1<a<b), bis the error degrees of freedom, cis the hypothesis degrees of freedom, and
Wand Hare the same error and hypothesis sum of square matrices as in Eq 4 computed over
the subspace P(x). As before, the resulting F(x) values follow, under the null hypothesis, a stan-
dard Fdistribution with ac and bdegrees of freedom, for each individual searchlight voxel x.
To summarize, Fig 1 illustrates the idealized fc-MVPA procedure. For every searchlight
voxel xwe first compute the functional connectivity maps r
n
(x) between this voxel and the rest
of the brain for every individual subject (Fig 1 top left), and use Eq 5 to compute a reduced set
of eigenpattern scores s
n
(x) best characterizing relevant spatial features of these maps across
subjects (represented in Fig 1 top right). Once each subject’s functional connectivity profiles
are represented in terms of their lower-dimensional associated eigenpattern scores s
n
(x),
group-level functional connectivity analyses proceed normally by entering these scores into a
standard General Linear Model (Eq 7). This model evaluates at this searchlight location xany
hypothesis of the form C~
BðxÞ ¼ 0using LRT (Eq 8), allowing us to make inferences about
the shape of the functional connectivity maps that these scores represent. Last, this procedure
is then simply repeated for each searchlight voxel x, sequentially constructing a statistical
parametric map F(x) characterizing the results of this inferential procedure across the entire
brain.
The general fc-MVPA procedure may be seen as computationally prohibitive, particularly
for whole-brain analyses using relatively small voxel sizes (e.g. isotropic 2mm voxels), since the
computational load scales quadratically with the total number of voxels under consideration, it
appears to require the computation of entire voxel-to-voxel connectivity matrices, and it effec-
tively performs close to 200,000 whole-brain PCA analyses (one per seed voxel) characterizing
the intersubject heterogeneity of seed-based connectivity maps. Despite this, there are several
mathematical tricks that can be used to reduce the complexity of the necessary computations
by several orders of magnitude. In particular, in S1 Appendix we describe how to more effi-
ciently compute the eigenpattern scores s
n
(x) directly from the original BOLD timeseries in a
way that will instead scale only linearly with the number of voxels, and without the need at any
point to compute or store entire voxel-to-voxel connectivity matrices. In the analysis examples
below we use this approach to efficiently compute fc-MVPA analyses on hundreds of subjects
with minimal computation effort.
Results and discussion
Fc-MVPA brain-wide connectome inferences: interpretation and examples
Group-level analyses of the eigenpattern scores s
n
(x) enable statistical inferences at the level of
individual searchlight voxels evaluating the form or shape of the connectivity patterns with
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each voxel. In particular, for any individual hypothesis (e.g. group A = group B) the fc-MVPA
procedure will produce a statistical parametric map F(x) evaluating that hypothesis separately
at each individual searchlight voxel x. In order to test brain-wide connectome hypotheses it is
still necessary to control the resulting maps F(x) for multiple comparisons across the total
number of tests evaluated (one test per voxel). Fortunately, this can be done using the same
nonparametric cluster-level inferential procedures that are common in standard analyses of
Fig 1. Schematic representation of functional connectivity multivariate pattern analyses (fc-MVPA). For each
voxel, fc-MVPA analyses compute the functional connectivity maps between this seed/source voxel and the entire
brain (Top-left; r
n
(x) in Eq 2) separately for each individual subject. Each subject functional connectivity map is then
characterized by a lower dimensional eigenpattern scores (dots in top-right graph; s
n
(x) in Eq 5). This representation is
chosen in a way that captures as well as possible the observed voxel-specific variability in functional connectivity maps
across subjects. A multivariate test is then performed on the resulting lower-dimensional eigenpattern scores to
ascertain potential between- or within- subjects effects of interest (e.g. differences between subjects or between
conditions in functional connectivity at the original seed/source voxel). This process is then repeated for every source
voxel to identify regions that show brain-wide between- or within- subjects differences in functional connectivity.
https://doi.org/10.1371/journal.pcbi.1010634.g001
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Brain-wide connectome inferences using fc-MVPA
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functional activation, such as cluster-mass or TFCE statistics based on permutation / randomi-
zation analyses [5,6,53]. These approaches allow us to compute statistics as well as associated
familywise error corrected p-values for individual clusters of contiguous searchlight voxels in
the statistical parametric map F(x), supporting cluster-level inferences with meaningful false
positive control (e.g. controlling the likelihood of observing or more false positive clusters
across the entire brain below 5%, for a familywise control procedure, or controlling the rate of
false positive clusters below 5% among all significant clusters, for a False Discovery Rate con-
trol procedure).
Another important choice that remains when using fc-MVPA in the context of brain-wide
connectome inferences is to select k, the dimensionality of s
n
(x) or the number of eigenpattern
scores used to represent functional connectivity at each voxel. As it will be discussed in more
detail in the Simulations section below, there is no “correct” choice of this parameter, and fc-
MVPA inferences remain valid for all possible values of this parameter. Nevertheless it is
important that this choice is made a priori and justified (e.g. from prior literature), and, if dif-
ferent values are tested/evaluated, the results of all these different evaluations should be
reported (rather than reporting only the value that produces the best results for a particular
analysis, which would inflate the chance of false positives). Regarding the sensitivity of fc-
MVPA inferences, choosing a low value of kcan be expected to improve sensitivity to detect
relatively large or widespread effects of interest such as inter-network connectivity differences,
while choosing a higher value of kcan improve our ability to detect relatively smaller or mar-
ginal effects such as connectivity with smaller areas or subnetworks. In the absence of assump-
tions about the extent of the expected effects, a reasonable balance is to scale the choice of k
with the dataset size (e.g. suggested 5:1, 10:1, or 20:1 ratio between N:k, the number of subjects
in the analysis and the number of eigenpattern scores retained [54–56]), in order to maintain a
reasonable sensitivity to identify large effects in small samples, and comparatively finer details
in the analysis of larger samples. As with any other preprocessing or analysis choices, as long
as the choice of kis made a priori, statistical inferences will remain valid. If, on the other hand,
the value of kis selected a posteriori as the one that produced the “best” results among several
possible values evaluated, an appropriate multiple-comparison correction should be used for
statistical inferences (e.g. using a Bonferroni corrected cluster-level threshold p-FWE<0.05/10
if the results were selected among 10 different choices of kvalues, or using split or cross-valida-
tion procedures such as choosing kas the value that produces the “best” results in one half of
the subjects and then basing statistical inferences on the analysis of the other half using the
selected kvalue). For new analyses, and in the absence of any other rationale (e.g. based on
prior literature, N:k ratios, or expected extent of effects) we recommend using a value of
k= 10, as that seems to suffice to cover a large proportion of the intersubject variability in
functional connectivity profiles (e.g. see fc-MVPA eigenpatterns section below). In all cases we
encourage researchers to evaluate and report the robustness of their results to different choices
of this parameter (e.g. as exploratory post-hoc analyses, without a need for additional multiple
comparison corrections), as that will help other researchers build upon those results in future
analyses and the field converge toward useful conventions.
Last, regarding the interpretation of fc-MVPA results, when reporting statistical inferences
from fc-MVPA brain-wide connectome analyses, those inferences should be if possible framed
regarding the patterns of connectivity between each voxel or cluster and the rest of the brain.
For example, when using fc-MVPA to evaluate the difference in connectivity between two
groups of subjects, if the fc-MVPA procedure above produces one supra-threshold cluster
with corrected significance level below p <.05 that should be interpreted as indicating that the
pattern of connectivity between this cluster and the rest of the brain is (significantly) different
between the two groups. The fc-MVPA method does not afford further spatial specificity in
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the resulting statistical inferences, but it is still useful to report measures of effect-size charac-
terizing the patterns of connectivity within each individual significant cluster, as a way to sug-
gest possible interpretations and future analyses.
Effect-sizes in GLM analyses are typically represented by linear combinations of the esti-
mated regressor coefficients B, and, specifically in the context of hypothesis testing, linear
combinations of the form cB, as these measures quantify the extent of the observed departures
from the null hypothesis (cB= 0). The interpretation of effect-sizes in GLM depends naturally
on the choice of hypothesis being evaluated. For example, for a GLM two-sample t-test com-
paring connectivity between two groups, effect-sizes of the form cBwould represent the dif-
ference in connectivity (e.g. average differences in r- values) between the two groups, while for
a GLM regression analysis evaluating the association between some behavioral measure and
connectivity strength, effect-sizes of the form cBwould represent the slope of the regression
line approximating the observed associations. Note that in both cases an effect-size of zero
would represent the null hypothesis (of no differences or no associations, respectively). In the
analysis of the statistical parametric map F(x), for any significant cluster O(group of contigu-
ous suprathreshold voxels with a cluster-level corrected p-value below the chosen family-wise
error threshold), we recommend reporting the effect-sizes hcj~
Bseparately for each mean-
ingful between-subjects contrast c
j
(e.g. individual rows of the contrast matrix C). Effect sizes
can be reported as a vector of eigenpattern weights (h
eig
(O)), or as a whole-brain projected
map (h
map
(O)):
fc-MVPA effect-sizes at location O
heigðOÞ  Xx2Ocj~
BðxÞ ð9Þ
hscoresðOÞ  X
x2O
cj~
BðxÞ  StðxÞ ¼ X
x2O
heigðxÞ  StðxÞ
hmapðOÞ  X
x2O
cj~
BðxÞ  DðxÞ  PtðxÞ ¼ X
x2O
hscoresðxÞ  RðxÞ
The effect-size measure h
eig
(O) is a vector, with one element per eigenpattern, estimated
separately at each location O. It represents the effect-size of a group-level analysis contrast of
interest cj~
BðxÞevaluated separately for each individual eigenpattern (columns of ~
B). For
example, if the group-level analysis was a two-sample t-test comparing connectivity between
two subject groups, then the k-th element in the h
eig
(O) effect-size vector will evaluate what is
the difference in the k-th eigenpattern scores at location Obetween these two groups.
Similarly, and perhaps more directly interpretable, the effect-size measure h
map
(O) repre-
sents the same contrast but is now evaluated separately at each individual voxel. In the example
above, the value of h
map
(O) at a particular voxel will represent the difference in functional con-
nectivity between Oand this voxel between the two subject groups analyzed. It should be
noted that a very similar whole-brain projected map of effect-sizes h
map
(O) can also be com-
puted from the voxel-level effect-sizes of a post hoc analysis that would evaluate the same
group-level model as in Eq 7 but this time focusing on the seed-based connectivity maps (SBC)
associated with seeds defined from each individual significant cluster O. As in any post-hoc
analysis, p-values derived from these SBC post-hoc analyses will be partially inflated due to
selection bias and should not to be used to make secondary inferences regarding individual
connections within the reported patterns. Despite this limitation, post-hoc SBC analyses on
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the same dataset offer a simple and perfectly valid alternative approach for reporting fc-MVPA
effect sizes within each significant cluster, while, when performed on an independent dataset,
they also offer a natural method to further probe specific aspects of these connectivity patterns.
For those interested, the resulting h
map
(O) effect-sizes following this approach can be shown
to be equal to those derived from Eq 9 in the limit when the number of eigenpatterns retained
equals the total number of subjects in the study, simplifying the variable h
scores
(O) to a constant
vector independent of the location O:
h1
scoresðOÞ ¼ cj ðGtGÞ1Gtð10Þ
Effect-size measures such as those described in Eqs 9and 10 represent post-hoc estimates,
and as such they should always be understood to contain a certain amount of bias. Their appli-
cation is mainly for interpretation purposes and for hypothesis building. In other contexts,
when the accuracy of these estimates may be essential, a cross-validation approach may be
used where, for example, the clusters Omay be computed from an initial General Linear
Model (Eqs 7and 8) that includes only data from a subset of subjects, while the effect-size esti-
mates (Eqs 9and 10) may be computed using a second GLM that includes data from a sepa-
rate/independent subset of subjects.
As an illustration of fc-MVPA brain-wide connectome inferences, we analyzed gender dif-
ferences in resting state functional connectivity using the Cambridge 1000-connectomes data-
set (n = 198, see S2 Appendix for a description of this dataset preprocessing and fc-MVPA
analysis methods [57–66]). The question that these analyses ask is whether there are any differ-
ences across the entire voxel-to-voxel functional connectome between male and female sub-
jects. To answer this question, we performed fc-MVPA analyses focusing on the first 10
eigenpatterns (an approximate 20:1 subjects to eigenpattern ratio), entering the corresponding
eigenpattern scores into a second-level group analysis evaluating a multivariate ANCOVA test
with gender as a between-subjects factor and subject motion (average framewise displacement)
as a control variable. The resulting statistical parametric maps were thresholded using Thresh-
old Free Cluster Enhancement [6] (TFCE, with default H = 1, E = 0.5 values) at a familywise
error corrected 5% false positive level.
The results, shown in Fig 2 show a large number of areas with significant gender-related dif-
ferences in connectivity (p-FWE <.05, shown as yellow and black areas in the center image).
Given the abundance of areas showing significant gender effects, for illustration purposes we
focused our description only on a subset of cortical regions showing some of the strongest
effects (TFCE>200; p-FWE <.001, highlighted in black in Fig 2 center image). For each clus-
ter in this reduced subset, we computed effect-size maps h
map
(O) characterizing the pattern of
gender-related differences in connectivity with each cluster (displayed in Fig 2 as a circular
array of brain displays), with yellow indicating higher connectivity with this cluster in male
compared to female subjects, and blue indicating higher connectivity in female compared to
male subjects.
Some of the strongest effects were visible in the bilateral Occipital Pole visual areas. A left
hemisphere cluster centered at MNI coordinates (-22,-94,+4) mm showed a pattern of
increased connectivity with Default Mode Network (DMN) and increased anticorrelations
with Salience Network (SN) areas in male subjects (see Fig 2 Occipital Pole plot). A similar pat-
tern (not shown) was present in another cluster in right hemisphere Occipital Pole areas
(+28,-82,+2). Similarly, there were significant gender effects in several DMN areas, such as
Medial Prefrontal Cortex (+6,+54,-12) and Precuneus (+18,-72,+32), showing a similar pattern
of stronger connectivity with visual and sensorimotor areas (shown in yellow in Fig 2 Medial
Frontal Cortex and Precuneus plots) in male subjects compared to stronger connectivity
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(weaker anticorrelations) with SN and attention areas in female subjects (shown in blue in
same plots).
In the left hemisphere, Inferior Frontal Gyrus pars triangularis (-54,+22,+14) showed a pat-
tern of stronger connectivity in female subjects with frontoparietal network areas and with
Inferior Temporal Cortex (shown in blue in Fig 2 Inferior Frontal Gyrus plot).
In the right hemisphere, there was a cluster of regions in the Temporal Parietal Occipital
Junction that also showed strong gender-related differences in connectivity. Lateral superior
Postcentral Gyrus (+36,-32,+48) showed a mixed pattern of increased integration with other
Fig 2. Fc-MVPA results evaluating gender-related differences in connectivity. Central figure shows left- and right- hemisphere medial (bottom) and lateral
(top) views of the main fc-MVPA results showing areas with significant gender-related differences in functional connectivity (highlighted in yellow and black,
TFCE statistics p-FWE<0.05). Among all significant results a reduced subset showing some of the strongest effects are highlighted in black, and the effect-sizes
within these areas (pattern of differences in connectivity with each area between male and female subjects) are shown in the additional circular plots (yellow
indicating higher connectivity in male compared to female subjects, and blue indicating higher connectivity in female compared to male subjects).
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Dorsal Attention areas in female subjects, compared to increased connectivity with Central
Sulcus, including somatosensory and motor areas in male subjects. A cluster in superior Angu-
lar Gyrus centered at coordinates (+50,-54,+42) mm showed increased integration with medial
prefrontal and posterior cingulate areas in female subjects, and increased integration with lat-
eral prefrontal and reduced anticorrelations with insular areas in male subjects. Relatedly, a
cluster in the mid Insular Cortex (+38,+10,+2) showed a similar pattern of higher connectivity
with angular Gyrus and other DMN areas in male subjects. Another relatively proximal cluster
in the Anterior Supramarginal Gyrus (+60,-26,+30) also showed increased local associations
with superior postcentral areas in male subjects. Posterior Superior Temporal Gyrus (+48,-26,-
2) showed a pattern of higher connectivity (mixed with reduced anticorrelations) with fronto-
parietal areas in female subjects (shown in blue in Fig 2 Superior Temporal Gyrus plot), com-
pared to a similar pattern of stronger local connectivity in male subjects (shown in yellow in
same plot).
Last, Medial Precentral Gyrus areas (+16,-26,+40) showed relatively higher integration with
SN or ventral attention network in female subjects. In contrast, lateral Precentral Gyrus areas
(+54,-4,+22) showed higher integration with the same networks in male subjects, while in
female subjects this area showed stronger local correlations (shown in blue in Fig 2 Precentral
Gyrus plot).
From a validation and generalization perspective, it is interesting to question whether the
same or similar results would have been observed if, instead of using 10 eigenpattern scores,
based on a conservative suggestion to maintain an approximately 20:1 ratio between subjects
and eigenpatterns, we would have chosen a different number. To that end we repeated the pre-
vious group-level analyses evaluating gender differences in connectivity but now using differ-
ent number of eigenpattern scores, ranging from 1 to 100, and compared the resulting fc-
MVPA statistic parametric maps F(x).
The results (Fig 3 top) show very similar F(x) statistics when varying the number of eigen-
patterns around the k = 10 value selected for our original analyses. In addition, the distribution
of resulting statistics across the entire brain (Fig 3 bottom) shows high sensitivity across the
entire range of evaluated k values, consistent with the sensitivity simulations in the sections
below, and with average sensitivity peaking at k = 50 (approximately a 4:1 ratio in subjects to
eigenpatterns) for detecting gender effects in this dataset. While there were several areas like
superior Postcentral Gyrus where the statistics peaked at relatively low number of eigenpat-
terns, suggesting that the effects in these areas may be best represented by the first few fc-
MVPA eigenpatterns (i.e. they may be better described in terms of common/large sources of
variability across subjects), there were also many areas where the F(x) statistics peaked when
using a large number of eigenpatterns (e.g. 50 or above), suggesting that there may still be
widespread gender differences in functional connectivity beyond those highlighted in our orig-
inal analyses and described in Fig 2 that are better expressed in some of the higher-order fc-
MVPA eigenpatterns (i.e. representing more subtle or less common sources of intersubject
variability).
Fc-MVPA eigenpatterns P(x): interpretation and examples
In addition to enabling brain-wide connectivity inferences, fc-MVPA estimates a model-free
representation of the observed intersubject variability in functional connectivity in terms of
the resulting eigenpatterns P(x), which can be useful on its own. In this context, the eigenpat-
terns, defined as the columns of the voxel-specific matrix P(x), represent a set of mutually
orthogonal spatial patterns, different for each voxel, that best characterize the diversity across
subjects in functional connectivity between this voxel and the rest of the brain. By convention
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they are sorted in descending order based on the proportion of the total intersubject covari-
ance explained by each eigenpattern.
In practice, while it is straightforward to compute and store the eigenpattern scores s
n
(x),
storing the entire set of eigenpatterns P(x) can be particularly demanding as it consists of a set
of orthogonal whole-brain maps for each individual voxel. Luckily it is simple to define P(x)
analytically at any individual voxel xfrom its associated eigenpattern scores as:
PðxÞ / Xnrt
nðxÞ  snðxÞ ð11Þ
which can be generalized to define characteristic eigenpatterns over small homogeneous areas
by integrating the corresponding voxel-specific eigenpatterns:
fc-MVPA eigenpatterns at location O
PðOÞ / X
nX
x2O
rt
nðxÞ  snðxÞ ð12Þ
This can also be useful in the context of a cross-validation framework where it may be nec-
essary to compute eigenpattern scores s
n
(x) for a new, yet unseen, set of subjects. This can be
done using Eq 12 to first characterize P(x) from the original data, and then Eq 5b to compute
s
n
(x) from the new subjects’ connectivity maps.
In general, reporting and describing the fc-MVPA eigenpatterns in Eq 12 over a small area,
along with the proportion of the total covariance explained by each eigenpattern at this area,
allows to gain a better understanding of the main factors affecting intersubject heterogeneity
in functional connectivity between this area and the rest of the brain.
For example, from the analysis of the same resting state data of 198 subjects in the Cam-
bridge dataset, the map ξ
1
(x) shown in Fig 4 (bottom) describes the proportion of the total
Fig 3. Selecting different number of fc-MVPA eigenpatterns. Difference in fc-MVPA statistic parametric maps evaluating gender
differences in connectivity, when varying k, the number of fc-MVPA eigenpatterns used in the analysis, from k = 1 (left) to k = 100
(right). For reference, the original results shown in Fig 2 used k = 10 (highlighted here inside black box). Top: Statistic parametric maps
with color coding showing voxel-level -log
10
(p) values for four different choices of k (from 5 to 20). The results show consistent statistic
parametric maps across different kvalues. Bottom: Distribution of fc-MVPA statistics across all gray matter voxels with kranging from 1
to 100, compared to null hypothesis distribution (shown in leftmost ‘null’ histogram). The results indicate high sensitivity across the
entire range of evaluated kvalues, with sensitivity peaking at around k = 50 (close to a 4:1 ratio in subjects to eigenpatterns) for detecting
widespread gender effects in this dataset.
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intersubject covariance explained by each voxel’s first eigenpattern. In this sample, ξ
1
(x) values
range between 5% and 42% across different voxels. The values ξ
1
(x) associated with the first
eigenpattern are particularly interesting because they provide a simple measure of the overall
intersubject homogeneity of the connectivity maps at each voxel (with higher values indicating
higher homogeneity), as the first eigenpattern can often be expected to lie in the direction of
the average pattern of connectivity with that voxel (as illustrated in the analyses below). Over-
all, this sample ξ
1
(x) map shows high contrast between gray matter areas and other tissue clas-
ses, with higher values within gray matter areas, particularly those located along cortical gyra,
and relatively lower values for areas located deeper into cortical sulci. Some of the regions that
show the highest homogeneity include Medial Prefrontal, Posterior Cingulate, or Lateral Parie-
tal areas, part of the Default Mode Network (DMN), as well as anterior Insula and other
Salience Network (SN) areas. In contrast, cerebellar, subcortical, and Limbic Network areas
are some of the regions that show the most heterogeneous functional connectivity profiles
across subjects (lowest ξ
1
(x) values).
Looking at the contribution of additional eigenpatterns beyond the first one, Fig 4 (top)
shows histograms of the values ξ
k
(x), the percentage of the total intersubject covariance
Fig 4. Percentage of total covariance associated with each fc-MVPA eigenpattern. Top: histogram of ξ
k
(x)|1i100 values, percentage of
the total covariance explained by each of the first 100 eigenpatterns. Histograms are further broken down by the most likely tissue class (gray
matter in black, CSF areas in grey, and white matter in white) at each individual voxel as defined by SPM’s tissue probability map templates.
Bottom: spatial map ξ
1
(x) showing the proportion of the total intersubject covariance explained by each voxel’s first eigenpattern (a measure
of the overall intersubject homogeneity in functional connectivity patterns at each voxel; seetext for details).
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explained by each of the first 100 eigenpatterns. The distribution shows a strongly anisotropic
covariance (for reference, the dashed line simulates the expected percent covariance values
associated with each eigenpattern if the intersubject covariance was isotropic, with equal
covariance along all subject dimensions and approximate 100-voxel spatial resels). In general,
approximately 20 eigenpatterns are associated with higher-than-average covariance values.
The first 5 eigenpatterns combined explain a range between 18% and 51%, 10 eigenpatterns
between 28% and 57%, and 20 eigenpatterns between 39% and 64%, of the total intersubject
covariance (from a maximum of 197 eigenpatterns that could be theoretically computed from
this sample).
While ξ
k
(x) maps allow us to explore how the degree of anisotropy varies across different
areas, it is often also of interest to display the actual eigenpatterns P(x) at some particular rep-
resentative locations, in order to better understand the shape of that intersubject covariance.
Fig 5 shows the first 5 eigenpatterns at 14 example locations. These 14 locations were manually
chosen to illustrate some of the similarities and differences across different locations in inter-
subject variations of functional connectivity patterns. They were selected among the set of all
local maxima of the cumulative P5
k¼1xkðxÞmaps (shown in Fig 5 center image for reference)
trying to cover most of the larger clusters observed there. As this figure illustrates, the first
eigenpattern across different locations (shown in the leftmost portion of each individual-
region display in Fig 5) often reflects a pattern mimicking the average connectivity between
each location and the rest of the brain. For example, the first eigenpattern at the Posterior Cin-
gulate gyrus, a region part of the Default Mode Network (DMN), reflects an arrangement very
similar to the expected pattern of positive and negative associations with the DMN, and the
same arrangement appears in the first eigenpattern at other distant but related locations, such
as Frontal Medial Cortex. Similarly, the first eigenpattern at Anterior Insula or Anterior Cin-
gulate also shows similar profiles mimicking Salience or Ventral Attention Network connectiv-
ity. In contrast, second- and higher- order eigenpatterns, even from regions that are part of the
same network, show noticeable differences in their profiles, possibly indicating nonoverlap-
ping sources of intersubject variability beyond simple within-network connectivity variations.
Other regions, in contrast, show eigenpatterns that reflect perhaps competing contributions.
For example, in Paracingulate Gyrus the first eigenpattern reflects sources of variability in con-
nectivity with nearby DMN areas while the second eigenpattern possibly reflects variability in
connectivity with Anterior Cingulate and Medial Prefrontal regions.
From a validation perspective, it is interesting to note that the covariance explained by the
first few fc-MVPA eigenpatterns and represented by the ξ
k
(x) values is always, by nature of the
fc-MVPA SVD procedure, larger than what could be achieved by any other alternatively-
defined eigenpatterns. In particular, it is larger than the spatial patterns that would result from
a standard ICA or PCA decomposition of the same functional data. In order to highlight this,
we computed on this dataset a Principal Component Analysis in CONN using the same conca-
tenative approach and dimensionality reduction steps as in Calhoun et al. [13] to produce a set
of representative components sorted by decreasing explanatory power (shown in Fig 6 bot-
tom). We then computed, for each of these components, the covariance in functional connec-
tivity with each individual voxel along those dimensions, and plotted histograms of the
resulting cumulative variance as a function of the number of components retained (Fig 6 top).
As expected, the covariance explained cumulatively by the first kfc-MVPA eigenpatterns at
each individual voxel (shown in gray in Fig 6 top) is always equal to or larger than the covari-
ance explained cumulatively by the same number of PCA components (shown in black in the
same plots). While this is a necessary consequence of the SVD properties as used in the context
of fc-MVPA, it is important to note that in particular this implies that if we would like to
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characterize the functional connectivity pattern at each voxel using a reduced fixed number of
scores, then the representation produced by the fc-MVPA eigenpattern scores would always be
more efficient (it would better approximate the functional connectivity data) than an equally-
sized multivariate representation produced by characterizing each voxel connectivity in terms
of network-level properties (at least for the general class of linear transformations projecting
each connectivity pattern onto multiple networks, including those resulting from Principal
Fig 5. First 5 fc-MVPA eigenpatterns, characterizing the principal components of the localintersubject heterogeneity in functional connectivity maps. The
central display shows the cumulative total covariance in functional connectivity patterns explained by the first 5 eigenpatterns at each voxel (colormap values range
between 22%/black to 50%/white). The first five eigenpatterns at 14 manually-defined example locations are shown in a circular display. In each of these plots,
eigenpatterns range from first/left to fifth/right, and each eigenpattern is shown projected to a left hemisphere lateral (top plot) and medial (bottom plot) views, on a
relative color scale ranging from blue (highest negative values for each eigenpattern) to yellow (highest positive values).
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Fig 6. Comparison between PCA and MVPA components. Top: Median (dots) and 25%-75% percentile range (vertical lines) of the total
covariance in functional connectivity patterns at each voxel explained cumulatively by the first kcomponents from a functional connectivity
Principal Component Analysis (black dots and lines), and by the first kfc-MVPA eigenpatterns (light gray dots and lines), from the analysis
of the same sample dataset (Cambridge, n = 198 dataset). Bottom: First five principal components from PCA (first row) and from fc-MVPA
(second and third row, first five eigenvariates shown only at two sample locations: posterior cingulate and anterior insula). Each row shows
individual components sorted from first/left to fifth/right, projected to a left hemisphere lateral view (top image) and medial view (bottom
image), on a relative color scale ranging from blue (highest negative values for each component) to yellow (highest positive values). Larger
explanatory power of fc-MVPA components compared to PCA (shown on top figure) stems largely from the ability of fc-MVPA
components to adapt to the specificity of the functional connectivity patterns at each individual location (as exemplified in the bottom
figures by the differences and commonalities between the components describing posterior cingulate vs. anterior insula connectivity
patterns).
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Component or Independent Component Analyses of the same data). This, naturally, also sup-
ports the use of fc-MVPA eigenpattern scores in the context of brain-wide connectome infer-
ences as a rich low-dimensional representation of the functional connectivity patterns for each
subject.
Simulations: validity and sensitivity of fc-MVPA statistics
In order to evaluate the validity and sensitivity of the general fc-MVPA inferential approach,
we constructed a set of simplified simulations. All of the simulations consider a dataset with 50
subjects. Each subject’s BOLD data encompass 50 timepoints and 1,000 voxels. For each voxel,
the simulated BOLD timeseries contained a mixture of noise (independent samples from a
Gaussian distribution for each timepoint, computed separately for each voxel and subject, and
spatially convolved with a Gaussian filter with FWHM 10 voxels) and signal (independent
samples from a Gaussian distribution for each timepoint, computed separately for each subject
and shared across all voxels where the signal was present). The signal was only present in one
half of the subjects and, among those subjects, only within 10% of all contiguous voxels. For
each individual simulation Eq 5 was used to estimate the projection matrix P(x) at each indi-
vidual voxel, and Eqs 7&8were used to evaluate between-group differences in the patterns of
connectivity between this voxel and all other voxels, using a 50-by-2 design matrix Gcharac-
terizing the two groups of subjects and a between-subjects contrast vector C= [–1 1] evaluat-
ing the difference in functional connectivity between the two groups.
For each individual simulation, we computed the statistical parameter map F(x) and the
associated map of raw/uncorrected voxel-level p-values evaluating the null hypothesis sepa-
rately for two different voxels: one where the signal was present (so the connectivity between
that voxel and all of the other voxels is expected to differ between the two subject groups), and
one where the signal was not present (so the connectivity between that voxel and all other vox-
els is not expected to differ between the two subject groups). We run 10,000 simulations, and
from their results, we computed summary Receiver Operating Characteristic (ROC) curves
describing the true positive rate (probability of a voxel showing a significant between-group
difference in connectivity) as a function of different prescribed false positive rates (p-value
threshold used to determine significance), for each of these two representative voxels. The
results from the first voxel, where the signal was present, were used to obtain estimates of the
sensitivity of voxel-level fc-MVPA connectome inferences (sensitivity analyses), and the results
from the second voxel, where the signal was not present, were used to obtain estimates of the
validity of this inferential procedure (validity analyses).
Each of the above sets of 10,000 simulations was repeated 50 times, each time using a differ-
ent number of eigenpatterns retained (ranging between 1 and 50) in Eq 5. In addition, all of
the above simulations were repeated under six different scenarios in order to further evaluate
the robustness of the obtained sensitivity and validity estimates in the presence of: a) varying
levels of spatial cross-correlation of BOLD noise (FWHM set to 1, and 25 voxels); b) varying
numbers of timepoints in BOLD scanning sessions (10, and 100 samples); and c) varying num-
bers of subjects in the study (10, and 100 subjects).
The results of the validity analyses are shown in Fig 7. The reported voxel-level p-values
(shown in the x-axis labeled as false positive rate) from fc-MVPA inferences matched very pre-
cisely the empirically observed false positive rates (shown in z-axis labeled as positive rate),
with all tested conditions showing accurate diagonal ROC curves. Differences between
reported voxel-level p- values and observed false positive rates were below ±0.22% in 50% of
all simulations, and below ±0.98% in 99% of all simulations. When controlling voxel-level
false positives at a 5% level, and across a total of 386 sets of different conditions evaluated, the
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empirical false positive rate observed across the 10,000 fc-MVPA analyses within each set ran-
ged between 4.5% and 5.4% (Fig 7 top right). Statistics remained valid across the entire range
of eigenpatterns tested up to the point where the number of eigenpatterns (ain Eq 8) equals
the error degrees of freedom (bin Eq 8, equal to the number of subjects minus the number of
Fig 7. Validation of fc-MVPA voxel-level inferences. Analysis of Receiver Operating Characteristic curves evaluating between-group
differences in functional connectivity under the null hypothesis (when there are no true differences in the population). Top left: surfaces,
and highlighted thick black lines, show, for a chosen combination of false positive threshold (false positive rate x-axis) and number of
eigenpatterns (ky-axis), the resulting proportion of false positive results (positive rate z-axis), where the fc-MVPA procedure would falsely
conclude there is a significant difference in connectivity between the groups. The red line marks the observed rate of false positives when
fixing the prescribed false positive rate threshold at a fixed 5% level (graphically, the intersection of each ROC surface and a vertical plane
with constant false positive rate = 0.05), matching the expected 5% level. Top Right: Observed false positive rates (y-axis) when using fc-
MVPA statistical analyses controlled at a p <.05 level across the reference simulations (‘reference’) and simulations evaluating different
conditions (FWHM = 0, FHWM-25, N = 10, N = 100, Nt = 10, Nt = 100). The average (black dots) and histogram (gray surfaces) of the
observed false positive rates across these simulations all indicate an appropriate match to the expected/prescribed false positive level (5%).
Bottom: evaluating validity under different conditions: (A) low spatial autocorrelation (FWHM = 0); (B) large spatial autocorrelation
(FWHM = 25 voxels); (C) low number of subjects (N = 10); (D) high number of subjects (N = 100); (E) short scanning session (Nt = 10);
(F) long scanning session (Nt = 100).
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model regressors, or 48 in our simulations) where the data covariance becomes rank deficient
and the likelihood ratio test assumptions no longer hold.
The results of the sensitivity analyses are shown in Fig 8. Generally, sensitivity was large
across the entire range of eigenpatterns tested, only decreasing markedly as the number of
Fig 8. Sensitivity of fc-MVPA voxel-level inferences. Analysis of Receiver Operating Characteristic curves evaluating between-group
differences in functional connectivity. Top Left: surfaces, and highlighted thick black lines, show, for a chosen combination of false positive
threshold (false positive rate x-axis) and number of eigenpatterns (ky-axis), the resulting proportion of true positive results (positive rate z-
axis), where the fc-MVPA procedure would correctly conclude there is a significant difference in connectivity between the groups in our
reference simulations. Top Right: Observed true positive rates (y-axis) when using fc-MVPA statistical analyses controlled at a p <.05 level
across the reference simulations (‘reference’) and simulations evaluating different conditions (FWHM = 0, FHWM = 25, N = 10, N = 100,
Nt = 10, Nt = 100). The average (black dots) and histogram (gray surfaces) of the observed true positive rates, or proportion of significant
results, across these simulations indicate that sensitivity is typically higher when using low or intermediate numbers of eigenpatterns, with
poorer sensitivity when the number of timepoints for functional connectivity estimation is low (Nt = 10), or when the number of subjects
included in the analysis is low (N = 10). Bottom: evaluating sensitivity under different conditions: (A) no spatial autocorrelation
(FWHM = 0); (B) large spatial autocorrelation (FWHM = 25 voxels); (C) low number of subjects (N = 10); (D) high number of subjects
(N = 100); (E) short scanning session (Nt = 10); (F) long scanning session (Nt = 100).
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eigenpatterns approached their maximum possible value. For example, sensitivity at a p <.05
level was above 80% in the main simulations (with 50 subjects) as long as the number of eigen-
patterns was kept below 42, below 92 in the simulations with 100 subjects, and below 4 in the
simulations with 10 subjects. While optimal sensitivity will naturally vary on multiple factors,
including the size and nature of the effects that we are trying to evaluate, several trends in sen-
sitivity were apparent from different scenarios evaluated. In particular, the degree of spatial
autocorrelation in the functional data (simulations A-B in Fig 8) appeared to have a relatively
small effect on sensitivity, while the number of subjects in the study (simulations C-D) and the
duration of the scanning session (simulations E-F) both had a larger impact. For example,
when fixing the number of eigenvariates to 5, sensitivity to detect a group effect was above
99% at a p <.05 level in the simulations with 50 or 100 subjects, but sensitivity dropped to
56% when the number of subjects was only 10. Similarly, sensitivity at a p <.05 level was
above 99% when the number of simulated timepoints (or equivalently, the number of effective
degrees of freedom of the BOLD timeseries in a study after the denoising and bandpass filter-
ing procedure) was above 50, but it dropped to 92% when the number of simulated timepoints
was only 10.
Conclusions
This manuscript presented the theory and motivation behind functional connectivity Multi-
variate Pattern Analyses (fc-MVPA), both in the context of brain-wide connectome inferences,
as well as a model-free characterization of the heterogeneity in functional connectivity across
subjects. Fc-MVPA extends or complements other MVPA approaches commonly used in neu-
roimaging in three different ways: first, to characterize a subject’s mental state, instead of the
patterns of activation surrounding each voxel considered by many MVPA applications, fc-
MVPA considers the patterns of connectivity between each voxel and the rest of the brain; sec-
ond, instead of a backward model focusing on decoding known properties of a subject or of
the experimental paradigm, fc-MVPA uses a forward model focusing on testing a researcher’s
hypothesis about the subject’s connectivity state across the entire connectome (brain-wide
connectome inferences); and third, in addition to the above inferential framework, fc-MVPA
also provides a model-free characterization of the sources of intersubject heterogeneity in con-
nectivity patterns.
Monte Carlo simulations showed that fc-MVPA inferences remain valid for the entire
range of evaluated scenarios, including using any arbitrary number of eigenpattern scores, dif-
ferent sample sizes, and scanning session durations. Simulations and example analyses of gen-
der-related differences in functional connectivity illustrated the high sensitivity of fc-MVPA
inferential statistics to detect meaningful effects across the entire human connectome. In addi-
tion, an example analysis of fc-MVPA eigenpatterns in functional connectivity during resting
state showed rich and varied sources of intersubject heterogeneity in functional connectivity.
One of the main practical advantages of fc-MVPA in the context of brain-wide connectome
inferences, is that it combines the benefits of pattern analysis techniques, such as the increased
interpretability and reduced noise of lower-dimensional projections, with the benefits of a clas-
sical statistical framework, such as the ability to use popular approaches to group-level analyses
(e.g. GLM’s ANOVA and regression framework), novel multiple comparison techniques (e.g.
TFCE), and well understood statistical control procedures (e.g. ANCOVA in this manuscript
example analyses). Similarly, fc-MVPA eigenpattern representations offer a natural way to
extend common multidimensional reduction approaches in neuroimaging, such as ICA or
PCA, to begin considering the specificity of these lower-dimensional representations across
different brain areas. From its theoretical and practical advantages, we believe that fc-MVPA
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can be a powerful and hopefully useful tool for researchers to further explore the complexities
of the human connectome.
Supporting information
S1 Appendix. Efficient computation of multivariate patterns.
(DOCX)
S2 Appendix. Preprocessing and analysis of resting state functional data.
(DOCX)
S1 Table. Glossary of terms in manuscript equations.
(DOCX)
Author Contributions
Conceptualization: Alfonso Nieto-Castanon.
Formal analysis: Alfonso Nieto-Castanon.
Methodology: Alfonso Nieto-Castanon.
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