Investigations into Resting-state Connectivity
using Independent Component Analysis
FMRIB Technical Report TR05CB1
(A related paper has been accepted for publication in
Philosophical Transactions of the Royal Society,
Special Issue on ’Multimodal neuroimaging of brain connectivity’)
Christian F. Beckmann, Marilena DeLuca, Joseph T. Devlin and Stephen M. Smith
Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB),
Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital,
Headley Way, Headington, Oxford, UK
Corresponding author is Christian F. Beckmann: firstname.lastname@example.org
Inferring resting-state connectivity patterns from functional magnetic resonance imaging (FMRI) data is a challenging
task for any analytical technique. In this paper we review a probabilistic independent component analysis (PICA) approach,
optimised for the analysis of FMRI data (Beckmann and Smith, 2004), and discuss the role which this exploratory technique
can take in scientific investigations into the structure of these effects. We apply PICA to FMRI data acquired at rest in order
to characterise the spatiotemporal structure of such data, and demonstrate that this is an effective and robust tool for the
identification of low-frequency resting-state patterns from data acquired at various different spatial and temporal resolutions.
We show that these networks exhibit high spatial consistency across subjects and closely resemble discrete cortical functional
networks such as visual cortical areas or sensory motor cortex.
Keywords: Functional Magnetic Resonance Imaging; brain connectivity; Resting-state fluctuations; Independent Com-
in the human brain. Typical FMRI experiments have focused on the acquisition of T2*-sensitive MR images during periods
of increased oxygen consumption (due to neuronal response to externally controlled experimental conditions) and contrast
the measured image intensities with recordings obtained at ’rest’. Critically, some important quantitative concepts in FMRI
analysis such as the calculation of percent signal change or the interpretation of de-activation implicitly hinge on a suitable
definition of this baseline/rest signal. The baseline ’resting-state’ of the brain itself, however, is a somewhat ill defined and
poorly understood concept.
Of particular interest in this context are certain low-frequency fluctuations of the measured cerebral haemodynamics
(around 0.01–0.1Hz) which exhibit complex spatial structure reminiscent of FMRI ’activation maps’ and which can be iden-
tified in FMRI data taken both under rest condition and under external stimulation. Recently, some attention has been focused
on the characterisation of these maps and the identification of possible origins of slow variations in the measured blood oxygen
level dependent signal. Various researchers have suggested that these signal variations, temporally correlated across the brain,
are of neuronal origin and correspond to functional resting-state networks (RSNs) which jointly characterise the neuronal
baseline activity of the human brain in the absence of deliberate and/or externally stimulated neuronal activity, and may reflect
functionally distinct networks.
Biswal et al. (1995) first demonstrated the feasibility of using FMRI to detect such spatially distributed networks within
primary motor cortex during resting-state by calculating temporal correlations across the brain with the time course from a
seed voxel whose spatial location was chosen from a prior finger-tapping study. The temporal signal from a seed voxel in the
motor cortex was correlated with other motor cortex voxels and uncorrelated with other voxels, with major frequency peaks in
the resting correlations at around 0.02Hz. Lowe et al. (1998) found similar results using both single-slice low time of repetition
(TR of 130ms) and whole-head volumes with longer TR (2000ms) while Xiong et al. (1999) describe functional connectivity
maps that cover additional non-motor areas. Based also on findings from PET studies, the existence of a default mode brain
network involving several regions including the posterior cingulate cortex has been proposed (Raichle et al., 2001; Shulman
et al., 1997; Mazoyer et al., 2001). Using simulteneously acquired EEG and FMRI data under rest, Goldman et al. (2002)
have shown that the variation in Alpha rhythm in EEG (8-12Hz) is correlated with the FMRI measurements. In particular,
the authors report that increased alpha power was correlated with decreased BOLD signal in multiple regions of occipital,
superior temporal, inferior frontal, and cingulate cortex, and with increased signal in the thalamus and insula. These results
have important implications for interpretation of RSNs as they suggest a neuronal cause for these fluctuations.
Alternatively, it has been argued that these effects simply reflect vascular processes unrelated to neuronal function, which
would make RSNs of less interest to neuroscience (though still of potential clinical interest). Physiological noise in the
resting brain and its echo-time and field strength dependencies were investigated by Kruger and Glover (2001) who showed
that physiological noise demonstrates a field strength dependency, exceeds the thermal as well as scanner noise at 3T and is
increased in grey matter (see also Woolrich et al. (2001)). Various researchers have investigated the relation between low-
frequency fluctuations in the measured BOLD signal and other physiological observations: Obrig et al. (2000) reviewed and
studied low-frequency variations in oxygenation, cerebral blood flow (CBF) and metabolism and report significant correlations
with similar fluctuations observed by near infrared spectroscopy (NIRS). More recently, Wise et al. (2004) have investigated
the influence of arterial carbon dioxide fluctuations by using the endtidal level of exhaled carbon dioxide as covariate of
interest in a General Linear Model (GLM) analysis. The most significant changes were concentrated in the occipital, parietal
and temporal lobes as well as in the cingulate cortex, and suggest that vascular processes (unrelated to neuronal function) play
a significant role in the generation of such resting-state patterns.
challenge to analytical techniques. In the majority of existing studies, resting patterns are inferred by a correlation analysis
of the voxel-wise FMRI recodings against a reference time course obtained from secondary recordings (e.g. from EEG,
NIRS or physiologic measurements like the carbon-dioxide concentration) or simply by regressing against a single voxel’s
time course from resting data which is believed to be of functional relevance (seed-voxel based correlation analysis). These
techniques fundamentally test very specific hypotheses about the temporal structure of these effects. Recently, however, In-
dependent Component Analysis has succesfully been applied to the estimation of certain low-frequency patterns (Goldman
and Cohen, 2003; Kiviniemi et al., 2003; Greicius et al., 2004). An important benefit of such exploratory techniques over
more hypothesis-based techniques is the ability to identify various types of signal fluctuations by virtue of their spatial and/or
temporal characteristics without the need to specify an explicit temporal model. Such flexibility in data modelling is essential
in cases where the effects of interest are not very well understood and cannot be predicted acurately.
This paper is organised as follows: in section 2 we review a probabilistic approach to Independent Component Analysis
(PICA) specifically optimised for the analysis of FMRI data (Beckmann and Smith, 2004). Section 3 discusses the constraints
of this exploratory data analysis technique when used for the identification of large-scale noise fluctuations. In particular,
we demonstrate that optimisation for maximally independent spatial sources does not imply an inability to estimate largely
overlapping spatial maps. We demonstrate the ability of PICA to extract resting fluctuations and apply the technique to FMRI
resting data in order to test a set of important hypotheses about the structure of resting-state connectivity in the human brain.
In particular, we will investigate (i) if and how estimated source processes are driven by less interesting physiological effects
such as the cardiac or respiratory cycle, (ii) the spatial characteristics of estimated maps in terms of locality within grey matter
and (iii) the consistency of maps obtained from multiple subjects.
2 Decomposing FMRI data using ICA
Independent Component Analysis (ICA, Comon (1994); Bell and Sejnowski (1995); McKeown et al. (1998)) is a technique
which decomposes a 2-dimensional (time × voxels) data matrix1into a set of time courses and associated spatial maps which
jointly describe the temporal and spatial characteristics of underlying hidden signals (components). A probabilistic ICA model
extends this by assuming that the p-dimensional vectors of observations (time series in the case of FMRI data) is generated
from a set of q(< p) statistically independent non-Gaussian sources (spatial maps) via a linear and instantaneous ’mixing’
process corrupted by additive Gaussian noise η(t):
xi= Asi+ ηi
1Here, we only discuss the case of a decomposition into spatially independent source signals; the reason for this will become apparent later.
Here, xidenotes the individual measurements2at voxel location i, sidenotes the non-Gaussian source signals contained in
the data and ηidenotes Gaussian noise3ηi∼ G(0,σ2Σi).
The p × q dimensional mixing matrix A is assumed to be non-degenerate, i.e. of rank q. Solving the blind separation
problem requires finding a linear ’unmixing’ matrix W of dimension q × p such that
? s = Wx
The PICA model is similar to the standard GLM with the difference that, unlike the design matrix in the GLM, the mixing
matrix A is no longer pre-specified prior to model fitting but will be estimated from the data. The spatial source signals
correspond to parameter estimate images in the GLM with the additional constraint of being statistically independent of each
is a good approximation to the true source signals s.
2.1 Parameter estimation
Without loss of generality we can assume that the source signals have unit variance. If the noise covariance Σiis known,
we can pre-whiten the data and obtain a new representation ¯ xi =¯Asi+ ¯ ηi, where ¯ ηi∼ G(0,σ2I), i.e. where the noise
covariance is isotropic at every voxel location. To simplify notation, we will henceforth assume isotropic noise and drop the
Noise and signal are uncorrelated, so the data covariance matrix Rx= ?xixt
matrix A can be estimated as the matrix square root of Rx−σ2I: let X be a p×N matrix containing all N different FMRI
time series in its columns and let X = U(NΛ)
i? = AAt+ σ2I, i.e. the unknown mixing
2V be the singular value decomposition of X. Then
?AML= Uq(Λq− σ2Iq)
where Uqand Λqcontain the first q Eigenvectors and Eigenvalues. The matrix Q denotes a q × q orthogonal rotation matrix,
i.e. a matrix with QQt= I. This matrix is not directly identifyable from the data covariance matrix since Rxis invariant
under post-multiplication of A by any orthogonal rotation¯Q given that (A¯Q)(A¯Q)t= A¯Q¯QtAt= AAt= Rx− σ2I.
Estimating the mixing matrix A, however, reduces to identifying the square matrix Q after whitening the data with respect
to the noise covariance Σiand projecting the temporally whitened observations onto the space spanned by the q Eigenvectors
of Rxwith largest Eigenvalues. The maximum likelihood estimates of sources and σ are obtained using generalised least
Solving the model in the case of an unknown noise covariance can be achieved by iterating estimates of the mixing matrix
and the sources and re-estimating the noise covariances from the residuals ? η. The form of Σitypically is constrained by a
et al., 2001), and restrict the structure to autoregressive noise. However, since the exploratory approach allows modelling of
various sources of variability, e.g. temporally consistent physiological noise, as part of the signal in equation 1, the noise
model itself can actually be quite simplistic.
A consequence of the isotropic noise model is that as an initial pre-processing step we will modify the original data time
courses to be normalised to zero mean and unit variance. This pre-conditions the data under the null hypothesis of no signal:
the data matrix X is identical (up to second order statistics) to a simple set of realisations from a G(0,I) noise process. Any
signal will have to reveal itself via its deviation from Gaussianity.
The maximum likelihood estimators depend on knowledge of the number of underlying sources q. In the noise free case
this quantity can easily be deduced from the rank of the covariance of the observations Rxwhich is of rank q. In the presence
of isotropic noise, however, the covariance matrix will be of full rank where the additional noise has the effect of raising the
Eigenvalues of the covariance matrix by σ2(Roberts and Everson, 2001). Inferring the number of estimable source processes
amounts to testing for sphericity of Eigenspaces beyond a given threshold level (Beckmann and Smith, 2004). Simplistic
criteria like the reconstruction error or predictive likelihood will naturally predict that the accuracy steadily increases with
increased dimensionality. Thus, criteria like retaining 99.9% of the variability result in arbitrary threshold levels (Beckmann
W = (?A
p − q
suitable parameterisation; here we use the common approaches to FMRI noise modelling (Bullmore et al., 1996; Woolrich
2For simplicity we assume de-meaned data.
3The covariance of the noise is allowed to be voxel dependent in order to encode the vastly different noise covariance observed within different tissue
types (Woolrich et al., 2001).
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