Wireless Sensor Network, 2010, 2, 208-217
doi:10.4236/wsn.2010.23028 Published Online March 2010 (http://www.scirp.org/journal/wsn)
Modeling of Circuits within Networks by fMRI
G. de Marco, A. le Pellec
Université Paris X, Laboratoire du contrôle moteur et d’analyse du mouvement, Paris, France
Received November 24, 2009; revised December 19, 2009; accepted January 8, 2010
Abstract
In this review, the authors describe the most recent functional imaging approaches used to explore and iden-
tify circuits within networks and model spatially and anatomically interconnected regions. After defining the
concept of functional and effective connectivity, the authors describe various methods of identification and
modeling of circuits within networks. The description of specific circuits in networks should allow a more
realistic definition of dynamic functioning of the central nervous system which underlies various brain functions.
Keywords: fMRI, CNS, Modeling, Network, Effective Connectivity
1. Introduction
Imaging can be used to locate the brain areas involved in
various forms of motor behavior, attention, vision or
emotion, self-awareness and awareness of others, but
brain network modeling probably remains the greatest
challenge in the field of imaging data analysis [1].
Neuroimaging first allowed researchers to describe the
cortical and subcortical activity of regionally segregated
functional regions during a variety of experimental or
cognitive tasks. More recently, functional integration
studies have described how these functionally special-
ized areas, i.e., areas whose activity is temporally modi-
fied, interact within a highly distributed neural network.
By using functional magnetic resonance imaging (fMRI),
which has become the method most commonly used to
investigate human brain functions and define neural
populations as distributed local networks transiently,
linked by large-scale reciprocal dynamic connections [2].
After defining the concept of functional and effective
connectivity, various approaches to the identification and
modeling of circuits into networks will be presented in
order to more realistically define the dynamics of the
central nervous system which underlies various cerebral
functions. A distinction should be made between meth-
ods that only consider correlations and ignore issues of
causality and influence and methods that attempt to de-
scribe or draw inferences concerning the direction of
influence between regions. Methodological approaches
to the study of connectivity using fMRI data may be
broadly divided into those that are more data-driven and
attempt to map connectivity in the whole brain and those
that use prior knowledge or hypotheses-driven, limited to
a restricted set of regions [3]. These two categories of
analysis are described, as indicated below, as functional
connectivity and effective connectivity, respectively
[4-6]. Techniques in the first group that consider only
correlations between regions include mapping using
seed-voxel correlations. Techniques in the second group
use more elaborate models and additional assumptions
applied to calculate correlations or covariances to ad-
dress questions about directional influences and include
mapping based on structural equation modeling (SEM),
multivariate autoregressive (MAR) modeling, dynamic
causal modeling (DCM).
2. Functional and Effective Connectivity
The dichotomy between local and large-scale networks
serves as a neural basis for the key assumption that brain
functional architecture abides by two principles: func-
tional segregation and functional integration [2,3,7]. A
large-scale brain network can be defined as a set of seg-
regated and integrated regions that share strong ana-
tomical connections and functional interactions. Whether
top-down or bottom-up, connections and interactions are
quintessential aspects of networks [8,9]. Cognitive and
sensorimotor processes depend on complex dynamics of
temporally and spatially segregated brain activities.
While the segregation principle states that some func-
tional processes specifically engage well-localized and
specialized brain regions, it is now thought that brain
functions are most likely to emerge through integration
of information flows across widely distributed regions
[2,10,11]. According to this approach, it is not only iso-
lated brain areas that are presumed to process informa-

G. de Marco ET AL.209
tion but rather a large-scale network, i.e. a set of brain
regions interacting in a coherent and dynamic way.
Hence, according to the functional integration concept,
cortical areas and therefore functions are integrated
within specific dynamic networks.
This concept supposes the existence of a dynamic in-
teraction between interconnected, active areas and that
the brain areas are expressed as networks within inte-
grated systems. In such a system, localized areas are in-
cluded in networks which become dynamic according to
the cognitive task. Brain areas underlie several functions
and can belong successively to several different func-
tional networks. In other words, a given brain area does
not have a single function; its resources can be exploited
in several different cognitive strategies. The principle of
functional integration which is also known in the field of
electrophysiology was used to analyze the event poten-
tials obtained from multielectrode recordings [12]. Thus,
based on the functional integration principle, the rela-
tionships between several brain areas may be examined.
Effective connectivity, closer to the intuitive notion of
a connection, can be defined as the influence that one
neural system exerts over another, either at a synaptic
level (synaptic efficacy) or a cortical level [13,14]. This
approach emphasizes that determining effective connec-
tivity requires a causal model of the interactions between
the elements of the neural system of interest. In electro-
physiology, there is a close relationship between effec-
tive connectivity and synaptic efficacy [15]. Effective
connectivity can be estimated from linear models to test
whether a theoretical model seeking to explain a network
of relationships can actually fit the relationships esti-
mated from the observed data. In the case of fMRI, the
theoretical model is an anatomically constrained model
and the data are interregional covariances of activity
[16].
Consequently, effective connectivity represents the
dynamic influence that cortical and subcortical regions
exert on each other via a putative network of interde-
pendent areas [5,12]. This approach might be based on
linear time-invariant models that relate the time-course
of experimentally controlled manipulations to BOLD
signals in a voxel-specific fashion. Although various
statistical models have been proposed [17], these stan-
dard models treat the voxels throughout the brain as iso-
lated black boxes, whose input-output functions are
characterized by BOLD responses evoked by various
experimental conditions [18]. fMRI provides simultane-
ous recordings of activity throughout the brain evoked by
cognitive and sensorimotor challenges, but at the ex-
pense of ignoring temporal information, i.e., the history
of the experimental task (input) or physiologic variable
(signal). This is important, as interactions within the
brain, whether over short or long distances, take time and
are not instantaneous which is implicit within regression
models. Furthermore, the instantaneous state of any brain
system that conforms to a dynamic system will depend
on the history of its input.
3. Data-Driven Approaches
The first category of methods includes seed-voxel corre-
lations, Granger causality derived autoregressive models
[19], fuzzy clustering which assumes that brain voxels
can be grouped into clusters sharing similar activity pat-
terns [20–22], hierarchical clustering [23,24], psycho-
physiologic interactions which test for changes in the
regression slope of activity at every voxel on a seed
voxel that are induced by an experimental manipulation
[25], and spectral analysis [26–28]. Other techniques,
such as principal component analysis [29–31] and inde-
pendent component analysis (ICA) [32–35], suppose that
fMRI data are a linear mixing of a given number of
temporal factors with an associated factor-specific spatial
distribution. Among all of these methods, we propose to
briefly describe the ICA method (time analysis of the
BOLD response) and the spectral method (frequency
analysis of the time response) that are two interesting
methods to spatially identify circuits within networks in
the brain.
3.1. Independent Component Analysis
Independent component analysis (ICA) is a data-based
multivariate statistical technique that uses higher order
statistics to perform decomposition of linearly combined
statistically independent sources [36]. Each statistically
independent component represents a hemodynamic map
of the whole brain. Each independent component is sup-
posed to describe a particular functional activity of the
brain with its deployment over time [37–39]. Each inde-
pendent component extracted by applying a spatial ICA
is spatially independent of all other independent compo-
nents [35]. Therefore, the contribution of a spatial inde-
pendent component to each voxel is given by the inde-
pendent component magnitude at that point modulated
over time by the associated time-course. The main ad-
vantage of ICA is that it requires little knowledge about
the nature of the data. The only necessary hypothesis
concerns the presence of a sufficient amount of inde-
pendent sources (temporal or spatial), which are linearly
mixed. Conversely, one of the main drawbacks of ICA is
the large amount of brain activations resulting from this
kind of decomposition [40]. At some point, hypotheses
are necessary to select relevant from spurious activa-
tions.
For this reason, ICA can be used in conjunction with
other well-established techniques [41] or further infor-
mation may be associated with the reference time-course,
such as the spatial localization of activities [42] and the
covariate relation of independent component time-course

G. de Marco ET AL. 210
[43]. ICA could be combined with SEM to extend the
explanatory power of each technique. SEM is a well de-
veloped, computationally minimally intensive connec-
tivity analysis technique suitable for neuroimaging data,
especially when it is combined with other data-driven
methods such as ICA. In this case, SEM coupled with
ICA is capable to handle data from a large number of
subjects [32]. The biological relevance and cortical con-
nections of the SEM models have also been evaluated
with reference to available knowledge based on animal
and human circuitries. The main advantage of spatial
group ICA is its ability to identify the distinct functional
elements involved in the circuitry [33]. Functionally
connected brain regions encompassed in each independ-
ent component are active at the same time, suggesting
that one or more anatomical connections are in use dur-
ing performance of the task. Although this reasoning is
more in line with the “connectionist” approach to brain
functions based on parallel processing mechanisms per-
formed by a group of connected functional elements, the
ICA approach lacks a statistical method to model the
functional connections assumed to exist between regions.
The addition of ICA to SEM can address this issue. Each
ICA map or part of the map corresponds to one compo-
nent in an SEM.
3.2. Spectral Analysis
The description of a correlation structure in the fre-
quency domain can be a promising approach to investi-
gate interregional strengths of interactions of a functional
network. As time-dependent correlations may vary be-
tween fMRI signals and across the space independently
of the underlying neural dynamics, a method of analysis
of frequency-dependent correlations would be one way
to overcome this interregional variability of the BOLD
response and would also be crucial for extracting the fine
detail of information hidden within the fMRI signal.
Functional connectivity analysis in the presence of major
physiologic noise sources is a pitfall especially when the
correlation (or covariance) between BOLD signals is
performed in the time domain. In this case, these noise
sources may artificially increase the magnitude of
cross-correlation. Estimation of coherence between pairs
of voxels at a specific frequency or at a limited range of
frequencies can therefore be one way to deal with nox-
ious physiologic noise.
The frequency domain approach can be used to ana-
lyze a limited range of linear relationships within a re-
stricted frequency band [44]. Consequently, measure-
ment of the correlation between fMRI data can be en-
hanced and can help to resolve the problem of false con-
nectivity derived from cardiac and respiratory cycles
and/or vascular differences. This approach can be per-
formed by using spectral analysis, which allows exami-
nation of the structure of covariance and provides certain
voxel-based parameters such as coherence which as-
sesses the dependence between voxel signals [26].
The spectral theory for multivariate time series has al-
ready been used in several fMRI studies [27,28,45]. By
using fMRI signals, these authors demonstrated that time
domain approaches may be sufficiently susceptible to
substantially high frequency artefacts, whereas the spec-
tral domain is essentially resistant to these artefacts.
They also demonstrated that the frequency-dependent
correlation is higher than that measured in the temporal
domain. In other fields of neuroscience, for instance in
electroencephalography (EEG), coherence analysis is
widely used to investigate correlated oscillatory activities
between various areas in the brain [46–49]. In magne-
toencephalography (MEG), coherence analysis has also
been demonstrated to be a useful technique in clinical
studies for discriminating different rhythmic behaviors in
various brain regions [50–52]. However, although the
relationship between neuronal currents and hemody-
namic response is poorly understood, simultaneous in-
tracortical neural recordings and fMRI signals acquired
in animals recently revealed a significant correlation
between local field potential and vascular response [53].
The feasibility of a correlation between the synchrony of
low frequency BOLD fluctuations in functionally related
brain regions and neuronal connections that facilitate
coordinated activities has been demonstrated in various
applications [54,55].
4. Hypothesis-Driven Approaches
The alternative to data-based approaches is to use a
model that attempts to describe the relationships between
a set of selected regions, in which region-specific meas-
urements such as BOLD time series are extracted from
whole-brain data prior to the connectivity modeling stage.
This category includes structural equation modeling
(SEM) [56–62], multivariate autoregressive (MAR)
modeling [63,64], dynamic causal modeling (DCM)
[65–67], generative models including neural mass mod-
els [68,69] and large-scale neural models [70–72].
4.1. Structural Equation Modelling
Path analysis, also referred to as structural equation
modeling (SEM), was originally developed in the early
1970s by Jöreskog, Keesling, and Wiley, when they
combined factor analysis with econometric simultaneous
equation models [73–76]. In the early 1990s, McIntosh
introduced SEM to neuroimaging [56,59,77–79] for
modeling, testing, and comparison of directional effec-
tive connectivity of the brain. SEM rapidly became
popular in this field [31,57,80–86]. Structural models can
be used to analyze linear relationships between variables

G. de Marco ET AL.211
from analysis of the covariance among the variables.
Structural models were developed from two principal
methods of analyses: factorial analysis (for a review:
[75]) and multiple regression or causal path analysis (a
method developed in the 1930s by Wright e.g., (for a
review: [87]). Structural models examine multiple
sources of influence on the dependent variable in an ex-
periment [88,89].
Structural Equation Modeling (SEM) is a hypothe-
sis-based multivariate statistical technique of data analy-
sis that can be used with neuroimaging data. An increas-
ing number of PET, fMRI and transcranial magnetic
stimulation (TMS) studies have used SEM to investigate
large-scale functional brain networks [90–93] and show
specific networks involved in either working memory
[94–100], attentional processes [64,101–103], face per-
ception [104–106], motor movement processing [61,
107–112], language [32,113,114] or processing of pain-
ful stimuli [62].
SEM methods, in comparison with classical ap-
proaches such as linear regression, allow simultaneous
analysis of several types of interrelationships between
variables in an experiment [13,115–117]. The nature of
the relationship between variables is given by the regres-
sion coefficient; it describes how much the dependent
variable changes when an independent variable changes
by one unit. SEM directly integrates measurement errors
into a statistical model, so that estimates of regression
coefficients are consequently more precise than with
classical methods such as multiple regression, factorial
analysis, or analysis of variance. The older methods ex-
amine only one linear relationship at the same time be-
tween independent and dependant variables and only
within a range of values set by the investigator [14]. In
contrast with classical methods, SEM analyzes a struc-
ture of variances and covariances in a dataset of observed
variables and can be used to predict dependences be-
tween variables. In other words, SEM seeks to explain as
much of the variance in dependant variables as it can
from simultaneous measurement of the variances of the
independent variables included in the model. Similarly,
SEM incorporates measurement errors of the independ-
ent variables into calculation of the estimate, which re-
inforces the statistical power of the method and provides
more precise estimates of regression coefficients. A
model of measurement can therefore be validated from a
theoretical model or empirical data [99]. The objective of
effective connectivity analysis is to estimate parameters
that represent influences between regions that may
change over time and with respect to experimental tasks.
In order to describe a functional network, network
nodes and anatomical connections must therefore be
proposed in conjunction with a SEM model to explain
interregional covariances and determine the intensity of
the connections. When applied to PET or fMRI data,
SEM allows modeling of connection pathways between
cortical or subcortical areas and reveals relationships,
interdependencies and covariance between the various
areas. In a given anatomical model, SEM shows the ef-
fects of an experimental task on a specific network of
connections [14,118–120]. In this type of statistical
analysis, normalized variables are considered in terms of
the structure of their covariances. SEM therefore allows
inference of interregional dependencies between various
cerebral cortical areas.
SEM is a simple and pragmatic approach to effective
connectivity when dynamic aspects can be disregarded.
A linear model is sufficient and the observed variables
can be measured precisely, the input is unknown but
stochastic and stationary. SEM comprises a set of regions
and a set of directed connections. Importantly, a causal
relationship is ascribed to these connections. Causal rela-
tionships are therefore not inferred from the data, but are
assumed a priori. The strengths of connections can
therefore be set so as to minimize the discrepancy be-
tween observed and implied correlations and thereby fit a
model to the data. Changes in connectivity can be attrib-
uted to experimental manipulation by partitioning the
data set. If, for example, a given fMRI data set is parti-
tioned into those scans obtained for different levels of an
experimental factor, differences in connectivity can then
be attributed to that factor leading to the conclusion that
a pathway has been activated. An SEM with particular
connection strengths implies a particular set of instanta-
neous correlations between regions. Structural equation
models posit a set of theoretical causal relationships be-
tween variables and model instantaneous correlations i.e.,
correlations between regions at the same time-point. In-
stantaneous activity is assumed to be the result of local
dynamics and connections between regions.
4.2. Multivariate Autoregressive (MAR) Models
To overcome the difficulties of SEM, Harrison et al. pro-
posed the use of multivariate autoregressive (MAR)
models for the analysis of fMRI data [63]. They were the
first to introduce multivariate autoregressive (MAR)
models into brain pathway analyses to characterize in-
terregional dependence. MAR models are time-series
models and consequently model temporal order within
measured brain activity. Goebel et al. [19] and Roe-
broeck et al. [121] subsequently generalized the MAR
approach by incorporating Granger causality between
two time series. MAR models posit a set of causal relation-
ships between variables; they incorporate cross-covariances
between regions (covariances at multiple lags) and ex-
ploit temporal relationships between different scans to
allow conclusions about predominant directions of in-
fluence between regions as well as their strength [18,
122,123].

G. de Marco ET AL. 212
An autoregressive approach is used to characterize a
structure in a time series, whereby the current value of a
time series is modeled as a weighted linear sum of pre-
vious values. Consecutive measurements within a given
time series contain information about the process that
generated this series. This is an autoregressive process
and is a very simple, yet effective, approach to time se-
ries characterization. This is distinct from regression
techniques that quantify instantaneous correlations, but is
similar to the SEM model in that it estimates the relative
influences over time. Autoregressive models of fMRI
data address the temporal aspect of causality in a BOLD
time series, focusing on the causal dependence of the
present on the past. Each data point of a time series is
explained as a linear combination of past data points.
This approach contrasts with SEM regression-based
models in which the time series can be permuted without
changing the results. MAR models contain directed in-
fluences among a set of regions whose causal interac-
tions, expressed at the BOLD level, are inferred via their
mutual predictability from past time points.
4.3. Dynamic Causal Modeling
A major criticism of SEM or MAR with regard to
neuroimaging data is that they model effective connec-
tivity changes at the “hemodynamic level” rather than
the “neuronal level”. This is a serious problem because
the causal architecture of the system that we want to
identify is expressed in terms of neuronal dynamics,
which are not directly observed using noninvasive tech-
niques. In the case of fMRI data, previous models of
effective connectivity have been fitted to the measured
time series which result from a hemodynamic convolu-
tion of the underlying neural activity. Since classical
statistical models do not include the forward model link-
ing neuronal activity to the measured hemodynamic data,
analyses of interregional connectivity performed on
hemodynamic responses are problematic. For example,
different brain regions can exhibit marked differences in
neurovascular coupling, and these differences, expressed
in different latencies (see above) may lead to false infer-
ences about connectivity [124].
Dynamical Causal Modeling (DCM) has recently been
developed as a generalization of both convolution mod-
els and SEM [66,67]. As described in Penny et al. [66],
SEM can be shown to be a simplified version of DCM
which also depends on the definition of a structural
model. DCM model assumes a dynamic neuronal model
of interacting brain regions, whereby neuronal activity in
a given brain region causes changes in neuronal activity
in other regions according to the structural model. This
neuronal model is then supplemented with a forward
model of how neuronal activity generates a measured
BOLD response through the balloon model which was
initially formulated by Buxton et al. [125] and later ex-
tended by Friston et al. [126]. A Bayesian inference
scheme is devised to infer the model parameters from the
data. The mathematical framework of DCM takes into
account nonlinearities and temporal correlations. It also
quantifies the interaction strength that one brain region
exerts on another brain region at the neuronal level,
whereas SEM only concerns the observed BOLD signal.
DCM is suspected to be less sensitive than SEM to the
number of degrees of freedom. Unlike SEM, DCM also
models the effect of experimental, external, and modula-
tory inputs on network dynamics. Since DCM models
neurobiologically plausible neural activities and takes
into account dynamics and modulations, this mathemati-
cal framework would appear to be more advantageous
than SEM.
4.4. Diffusion Tensor Imaging
While fMRI provides detailed information about the spa-
tial location of functionally active cortical areas, the
question of anatomical interdependency between cortical
areas remains elusive. A key tool to assess the validity of
large-scale distributed networks in fMRI is knowledge of
the underlying anatomical connections. The original idea
behind SEM and functional neuroimaging was to com-
bine two data sets: a functional set with an anatomical set
(connections between regions), based on the assumption
that anatomy was the source of spatial causal relation-
ships. Our understanding of the connections between
regions is limited, but since the advent of newer tracto-
graphy methods, the main white matter tracts can be de-
scribed. Diffusion Tensor Imaging (DTI) is a powerful
MRI technique [127,128] that can be used to translate
self-diffusion, or microscopic motion of water molecules
in tissue into a MRI measure of tissue integrity and
structure (white matter fibers). Data from diffusion ten-
sor imaging (DTI) and fMRI have been combined in a
few previous studies [129–131]. These studies showed
that a combination of techniques can give additional in-
formation about brain organization which may give more
specific information about organization of brain func-
tions and brain injuries. In this latter case, a DTI-driven
SEM would integrate information about white matter
changes (e.g. maturation, aging) [100,132]. The prospect
of using information derived from tractography could be
used to constrain structural models. DTI and fMRI com-
binations will be essential to discover to what extent the
brain functional organization as investigated with fMRI
reflects structural features of the brain and, hence, to
more accurately assess the relevance of fMRI to examine
the relationship between functional and large-scale ana-
tomical networks. However, more studies are still needed
to investigate anatomical correlates which would be re-
lated to effective connectivity.

G. de Marco ET AL.213
5. Conclusions
This article describes the most recent imaging ap-
proaches used to explore and identify circuits within
networks and to spatially and anatomically model inter-
connected regions. Structural equation modeling is the
most widely used method to model effective connectivity
[56,82,133]. The relevance of applying SEM to fMRI
neuroimaging data has been discussed in detail elsewhere
[58,66,82,134]. SEM allows one to start with simpler
models and then progress to more complex models by
repeatedly testing the model fit to real data. SEM is use-
ful when some information is available, such as a small
set of potential structural models or partial information
concerning connectivity. Newer, more sophisticated ef-
fective connectivity analysis methods such as Dynamic
Causal Modeling might circumvent the drawbacks of
SEM and may shed more insight into how brain regions
interact in information processing. Nevertheless, SEM is
a well developed, computationally less intensive connec-
tivity analysis technique suitable for neuroimaging data
especially for block designs and combined with other
methods such as independent component analysis, partial
correlation or DTI. The use of SEM may be justified by
the fact that, unlike DCM, the statistical model underly-
ing SEM is quite simple and not computationally de-
manding.
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