Clinical Neurophysiology xxx (2012) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Clinical Neurophysiology
journal homepage: www.elsevier.com/locate/clinph
Invited review
The organization of physiological brain networks
C.J. Stam, E.C.W. van Straaten ⇑
Department of Clinical Neurophysiology, VU University Medical Center, Amsterdam, The Netherlands
a r t i c l e
i n f o
Article history:
Accepted 15 January 2012
Available online xxxx
Keywords:
Brain networks
Graph theory
Small-world networks
Scale-free networks
Functional connectivity
Synchronization
Neurological disease
EEG
MEG
MRI
h i g h l i g h t s
The brain can be represented as a complex network with functionally connected units at several levels
that changes in neurological and psychiatric disease.
Existing clinical neurophysiology techniques and network models to explain network properties are
reviewed.
In addition to the already established network models, we suggest a heuristic model including hierarchical modularity.
a b s t r a c t
One of the central questions in neuroscience is how communication in the brain is organized under normal conditions and how this architecture breaks down in neurological disease. It has become clear that
simple activation studies are no longer sufficient. There is an urgent need to understand the brain as a
complex structural and functional network. Interest in brain network studies has increased strongly with
the advent of modern network theory and increasingly powerful investigative techniques such as ‘‘highdensity EEG’’, MEG, functional and structural MRI. Modern network studies of the brain have demonstrated that healthy brains self-organize towards so-called ‘‘small-world networks’’ characterized by a
combination of dense local connectivity and critical long-distance connections. In addition, normal brain
networks display hierarchical modularity, and a connectivity backbone that consists of interconnected
hub nodes. This complex architecture is believed to arise under genetic control and to underlie cognition
and intelligence. Optimal brain network organization becomes disrupted in neurological disease in characteristic ways. This review gives an overview of modern network theory and its applications to healthy
brain function and neurological disease, in particular using techniques from clinical neurophysiology,
such as EEG and MEG.
Ó 2012 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights
reserved.
Contents
1.
2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The brain as a network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.
Representing the brain as a network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.
Anatomical and neurophysiological evidence for a network perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.
Networks are not enough: the need for a theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.
Networks in relation to complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.
Relevance of brain networks for clinical neurophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Functional interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.
Time series and brain function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.
Functional and effective connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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⇑ Corresponding author. Address: Department of Clinical Neurophysiology, VU
University Medical Center, P.O. Box 7057, 1007 MB Amsterdam, The Netherlands.
Tel.: +31 20 444 0730; fax: +31 20 444 4816.
E-mail address: [email protected] (E.C.W. van Straaten).
1388-2457/$36.00 Ó 2012 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/j.clinph.2012.01.011
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
j.clinph.2012.01.011
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C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
4.
5.
3.3.
Fragile binding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.
Excessive connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.
Disconnection syndromes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.
Strengths and limitations of brain connectivity studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network studies in brain research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.
Modern network theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.
Graph theory, statistical physics and dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2.
Measures of complex networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3.
Representing structural and functional networks as graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.
Healthy subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.
Neuropsychiatric disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.
Epilepsy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.
Dementia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Toward a theory of brain networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.
From local activation to network organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.
A heuristic model of complex brain networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.
Development, aging and neurological disease in network space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.
Future challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
The brain can be seen as a complex anatomical and functional
network. Theories that apply to complex networks in general, such
as modern network theory and complex systems theory, are
increasingly used in neuroscience to understand how normal brain
function arises and what happens in disease. They provide a totally
new perspective leading to some very new insights. This review
gives an overview of the fundamentals and applications of modern
network theory in the study of normal and developing brain function. In addition, we aim to explain how it is possible that various
brain diseases can be understood in terms of failing network characteristics. The focus will be mainly on functional aspects of brain
networks, but structural connectivity, animal studies, and modeling will also be discussed to provide a wider perspective. Since network studies are based upon a fundament of functional
connectivity, the neurophysiological basis of normal and disrupted
synchronization is discussed first. Subsequently, the basic concepts
and tools of modern network theory are introduced. The application of network theory is illustrated by studies in healthy subjects,
and various disorders such as schizophrenia, depression, dementia,
trauma and epilepsy with an emphasis on functional, and in particular neurophysiological approaches. Finally we suggest a general
framework for the study of the organization of physiological brain
networks.
2. The brain as a network
2.1. Representing the brain as a network
In 1906, Ramon Y Cajal and Camillo Golgi shared the Nobel
Prize in Physiology or Medicine. Although they shared the prize,
they did not share each other’s ideas (Jacobson, 1995; Rapport,
2005). According to Golgi, a defender of the reticular theory, the
brain should be viewed as a large syncytium, or conglomerate, of
directly connected neurons. Instead, Cajal, using the silver nitrate
preparation developed by Golgi, interpreted his findings in support
of the neuron theory that maintained that the brain consists of discrete neurons. In a sense, both turned out to be right: the brain is
now considered to be a large network of about 1010 neurons, with
each neuron communicating with about 104 other neurons, mostly
by synapses, but to a small extent also by gap junctions that directly connect the cytoplasm of two adjacent cells and may play
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an important role in physiological and pathological synchronization processes. Alternative views of the brain, stressing its diffuse
and holonomic nature, have been proposed by Lashley (1923), Pribram and Carlton (1986), and more recently by Nunez (1995). Despite these suggestions, the concept of the brain as a large complex
network of interconnected elements, at multiple scales, has obtained a dominant place in modern neuroscience (Nunez, 2010).
2.2. Anatomical and neurophysiological evidence for a network
perspective
The concept of the brain as a complex network is increasingly
supported by a wealth of neuroanatomical and neurophysiological
data. Many different types of neurons have been recognized, and
they connect in characteristic ways, from the level of micro or
minicolumns (the smallest functional unit of the brain containing
about 80–100 neurons) and macrocolumns (consisting of 60–80
minicolumns; for a recent review see: Buxhoeveden and Casanova,
2002) up to the level of local brain regions, lobes and functional
brain systems. Izhikevich and Edelman (2008) included 22 different types of neurons in their model of the whole brain. Advanced
techniques are now being used to dissect the precise anatomy at
the smallest levels of the brain. Recently, using the micro-optical
sectioning tomography (MOST) system, a major breakthrough
was achieved in the detailed description of the connectivity of a
mouse brain (Li et al., 2010). Major efforts to describe the microscopic anatomy of the brain are the Blue Brain Project and the Human Brain Project (Markram, 2006; Perin et al., 2011;
www.humanbrainproject.eu). At higher levels of resolution, classic
neuroanatomic techniques of tracing, successful in showing the
network blueprint of the brain of macaques, and cats, are now replaced by modern imaging techniques, in particular high-field
strength MRI in combination with DTI (diffusion tensor imaging)
and tractography. These methods have revealed the large-scale
patterns of brain connectivity from the level of voxels to the level
of Brodmann areas (Hagmann et al., 2007, 2008; Gong et al.,
2009a).
Since the early nineties of the last century, invasive neurophysiological studies in animals have been able to demonstrate communication, mostly by synchronization in the gamma band, between
widely separated but connected brain regions (Eckhorn et al.,
1988; Engel et al., 1991; Gray et al., 1989; König et al., 1995). Many
of these findings have now been confirmed with EEG and MEG,
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
j.clinph.2012.01.011
C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
demonstrating the existence of functional brain networks, even in
a ‘‘resting-state’’ (Fell and Axmacher, 2011; Uhlhaas and Singer,
2006). In a classic review of neurophysiological synchronization
studies, Francisco Varela referred to the brain as a ‘‘Brainweb’’
(Varela et al., 2001). fMRI BOLD (blood oxygen level dependence)
studies confirm the network character of the brain, and provide
evidence for the existence of multiple resting-state networks, of
which the default mode network has become the most well known
(Damoiseaux et al., 2006). Other resting-state networks or modules
correspond to well-known functional systems devoted to vision,
sensory motor, working memory and attention. To a large extent,
structural and functional studies reveal the same complex and in
particular hierarchical network structure of the brain at multiple
levels (Bassett et al., 2008). From a network perspective, anatomical and physiological studies present an increasingly interrelated
view upon the organization of the brain. Donald Hebb’s notion of
‘‘synchronous cell assemblies’’ has developed from an abstract concept to a neuroanatomical and neurophysiological reality (Hebb,
1949).
2.3. Networks are not enough: the need for a theoretical framework
The concept of the brain as a complex structural and functional
network thus has a relatively long history, and a high face value given what we know about brain anatomy and neurophysiology.
However, the notion of brain networks is too general to be optimally useful in understanding normal and disturbed brain physiology unless it is linked to a specific theory of networks. As indicated
by Carter Butts: ‘‘To represent an empirical phenomenon as a network is a theoretical act.’’ (Butts, 2009). A complex network is
more than the sum of the elements that constitute its building
blocks; it is also more than the sum of all its pair-wise interactions.
Complex systems have emergent properties that require exact
mathematical theories for their proper understanding (Sporns,
2011a). In this respect, the development of modern network theory
in the late nineties of the last century constitutes an important step
forward. The introduction of powerful new mathematical models
of complex networks such as the small-world networks and
scale-free networks gave rise to a widespread and rapidly growing
interest in network studies in many fields of science, ranging from
physics, communication and transport systems to biological and
sociological networks (Boccaletti et al., 2006). The development
of modern network theory has greatly stimulated the interest in
brain networks, and has enabled a more exact, quantitative study
of the determinants of growth, learning, plasticity and failure in
neural networks.
2.4. Networks in relation to complexity theory
While a network perspective is rapidly becoming a promising
approach to understanding the complex nature of the brain, it is related to other scientific fields that deal with complex systems.
Cybernetics, information theory and general system theory constitute early examples of attempts to define general mathematical
frameworks for complex systems including neural networks (Shannon and Weaver, 1949; Simon, 1962; von Bertalanffy, 1969; Wiener, 1948). An excellent introduction to modern complexity
science can be found in Melanie Mitchell’s book Complexity
(Mitchell, 2009). The modern theory of dynamical systems, in particular chaos theory, has had a considerable influence on our
understanding of complex behavior in deterministic but highly
nonlinear systems. These ideas have influenced our concepts of
brain function, in particular with respect to the predictability of
epileptic seizures (Stam, 2005). Perhaps the most fruitful contribution of nonlinear dynamics has been the development of a general
theory of synchronization processes, with considerable relevance
3
for the study of communication between neurons and brain areas
(Boccaletti et al., 2002).
Other important contributions have come from statistical physics and the theory of phase transitions. According to the theory of
self-organized criticality, introduced by Per Bak, complex systems
evolve autonomously to a critical state characterized by power
laws (Bak et al., 1987). There is now increasing evidence that such
phenomena may also be observed in the brain (Beggs and Plenz,
2003; Werner, 2007). The plenitude of complexity theories might
suggest the lack of a unifying framework. However, modern network theory is a promising candidate for integrating many of the
emerging concepts in complexity studies. Herbert Simon stressed
that complex systems could be viewed in general as large collections of interacting elements displaying a hierarchical organization
(Simon, 1962). Modern network theory is based upon graph theory
as an exact mathematical formalism for the description of networks, including their hierarchical organization. In addition, it uses
ideas from probability theory and statistical mechanics to deal
with stochastic aspects of large networks, and finally it involves
the theory of dynamical systems and synchronization for the study
of processes taking place upon complex networks (Barrat et al.,
2008). Therefore, modern network theory presents an attractive
and versatile framework for the study of the organization of complex brain networks.
2.5. Relevance of brain networks for clinical neurophysiology
Normal brain functioning and its breakdown in neurological
and neuropsychiatric disease cannot be properly understood without taking the network perspective into account. However, a general, informal notion of brain networks is no longer sufficient.
Modern network theory is beginning to have an impact on our
ideas about the way brain networks develop, how this development is organized and constrained by genetic and geometric factors, how the architecture of brain networks comes to display
universal features such as hierarchical modularity and how this relates to cognitive function and intelligence. In addition, network
theory influences notions of localization of brain function and global effects of local lesions. The idea that such different conditions
as Alzheimer’s disease and epilepsy may both be explained in
terms of ‘‘hub failure’’ clearly illustrates how network science is
beginning to have an impact upon clinical concepts.
Clinical neurophysiology is the medical specialism that deals
with diagnosis, monitoring, and prognosis of clinical neurological
conditions using a variety of neurophysiological techniques. Modern network theory may be particularly relevant for clinical neurophysiology since two of its main techniques to study central
nervous system function, the EEG and MEG, present an enormous
and hardly explored potential to investigate how brain areas communicate, and how this communication fails in neurological disease. The fact that both EEG and MEG measure neural activity
directly, and not indirectly such as is the case with fMRI BOLD, is
a major advantage. Combining functional connectivity with modern network theory presents a unique opportunity for clinical neurophysiology. EEG and MEG are not only the most direct tools to
measure brain function; they also can provide a window on the
brain as a complex evolving network (Nunez, 2010).
3. Functional interactions
3.1. Time series and brain function
To gain an understanding how the brain is organized as a functional network, several elements are required: (i) we need to have
a reliable measurement of the level of activity of the network ele-
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
j.clinph.2012.01.011
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C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
ments; (ii) communication between network elements needs to be
characterized and quantified; (iii) the full pattern of pair wise
interactions between network elements needs to be integrated
and analyzed within the framework of network theory; (iv) the
interdependencies between functional and structural network levels need to be assessed. Here, we focus upon the first element. The
other three aspects will be addressed in subsequent sections.
Neurons are the obvious structural and functional building
blocks of the nervous system. Neuronal function is reflected in
two different types of activity: (i) small fluctuations of the resting
membrane potential of the dendrites induced by excitatory and
inhibitory synapses; (ii) action potentials along the axon. A typical
cortical pyramidal neuron may be covered by up to 1000–10,000
synapses, with the excitatory synapses predominantly located at
the dendritic tree far away from the soma and the inhibitory synapses located closer to the nerve cell body. Activation of an excitatory synapse induces a brief depolarization of the membrane
potential (excitatory post synaptic potential or EPSP) and activation of an inhibitory synapse causes a somewhat larger and longer
lasting hyper polarization of the membrane potential (inhibitory
post synaptic potential or IPSP). If the joint effect of all EPSPs and
IPSPs on the dendritic tree, determined by spatial and temporal
summation, causes the membrane potential near the axon hillock
to exceed the threshold, an action potential is generated that will
be propagated along the axon, but also backwards to the dendritic
tree. Neurons influence other neurons by their action potentials,
therefore the firing rate – number of action potentials per second
– is an important parameter of neuronal activity. A major controversy in neuroscience concerns the question whether the relevant
information is carried only by the firing rate, or also by the exact
timing of action potentials. The fact that the exact timing of firing
of pre and postsynaptic neurons may influence synaptic plasticity
(a concept known as spike timing dependent plasticity or STDP)
suggests that timing information may have to be taken into account (Abbott and Nelson, 2000).
While neuronal spiking is fundamental for inter neuronal communication, the most commonly used techniques to assess human
brain function such as EEG, MEG, and fMRI do not measure action
potentials directly. Both EEG and MEG reflect large-scale summated field potentials that are ultimately caused by EPSPs and
IPSPs of cortical, mainly pyramidal neurons. Characteristically,
the field potentials recorded with EEG and MEG show oscillations
in a wide range of frequency bands, ranging from (sub) delta to
gamma, high-frequency oscillations (HFO) and ripples (Buzsaki
and Draguhn, 2004). The temporal resolution of EEG and MEG is
essentially unlimited, which is of interest since both very low as
well as very high frequency oscillations have been shown to have
neurophysiological significance (Bragin et al., 2010; Van Someren
et al., 2011). The functional significance of oscillatory activity in
different frequency bands has long remained a mystery, and it
has even been suggested that these oscillations might be mere
epiphenomena (Niedermeyer and Schomer, 2011). There is now
increasing evidence, however, that a phase relation exists between
oscillatory field potentials and neuronal actions potentials
(Eckhorn and Obermueller, 1993; Lee et al., 2005). This observation
stresses that oscillatory brain activity contains information that is
relevant for understanding local and interregional communication.
The BOLD signal reflects the presence of excess oxyhemoglobin
in the smallest blood vessels near activated brain regions; it can be
thought of as a low pass, integrated measure of neuronal spiking
with a typical time scale of seconds. Studies combining EEG with
fMRI are now beginning to reveal the complex relations between
oscillatory band power and BOLD signals (Rosenkranz and Lemieux, 2010). In particular, a relation between the envelope of gamma band oscillations and BOLD time series has been suggested
(Logothetis, 2002).
Time series techniques can be used to analyze the recordings of
local brain activity by EEG, MEG and BOLD. The most common
technique is spectral analysis, that represents signal power as a
function of frequency. This approach is especially helpful to determine oscillatory components of the signal, such as a dominant
peak due to the alpha rhythm, and power changes in certain frequency bands. Activation of specific brain regions is often associated with a decrease or increase of power in a particular
frequency band; these changes can be captured with a technique
called event-related desynchronization (ERD) or event-related synchronization (ERS) (Pfurtscheller and Lopes da Silva, 1999). Shortlasting, non-stationary changes in oscillatory brain activity can also
be characterized with evoked or event-related potentials. In contrast to ERD and ERS, that measure induced power changes, evoked
and event-related potentials only measure oscillatory changes that
are exactly phase-locked to some event. Another effective technique to assess non-stationary aspects of oscillatory power is
wavelet analysis. Oscillatory brain activity may show correlations
on very long time scales. These correlations can be characterized
by spectral analysis, the Hurst exponent, or detrended fluctuation
analysis (Benayoun et al., 2010; Linkenkaer-Hansen et al., 2007;
Stam and de Bruin, 2004). Finally, the non-random structure of
Fig. 1. Schematic illustration of time series analysis. Measures of activation can be determined from individual time series using for instance spectral analysis. Measures of
correlation reflect statistical interdependencies between two time series. These correlations can be bidirectional (functional connectivity) or unidirectional (effective
connectivity). Both types of connectivity can be studied with linear techniques (correlation coefficient, coherence) or nonlinear techniques (phase synchronization,
generalized synchronization). Shown are two EEG signals recorded at O2 and O1 positions; vertical bars are second markers.
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
j.clinph.2012.01.011
C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
neurophysiological time-series analysis can also be characterized
with techniques derived from nonlinear dynamics. Here, the time
series is assumed to reflect a trajectory in a high-dimensional state
space of the underlying dynamical system, and this trajectory can
be characterized with measures such as the correlation dimension,
Lyapunov exponents and entropy measures (for review see: Stam,
2005).
3.2. Functional and effective connectivity
In the previous section, we discussed how the activity of neurons or local brain regions can be measured and characterized with
neurophysiological techniques and time series analysis. The next
step is to consider how this information can be used to infer how
communication between neurons and brain regions takes place
(Fig. 1). An important concept for understanding communication
in brain networks is synchronization. Unfortunately, synchronization is often used in neurophysiology in a rather informal way,
referring to a vague notion of the level of cooperation between
neurons. It is often assumed that band power of field potentials
oscillations reflect this cooperation between neurons; however,
band power of an EEG or MEG signal recorded at a particular electrode or sensor may depend upon many factors, and its interpretation in terms of (local) levels of neuronal synchronization is not
straightforward (Daffertshofer and van Wijk, 2011). In physics,
and in particular in dynamical systems theory, the notion of synchronization has been given a far more specific and quantitative
interpretation, that goes back all the way to the groundbreaking
work of the Dutch physicist Christiaan Huygens. Much of the work
on synchronization has been based upon the notion of phase coupling between pairs of harmonic oscillators (Boccaletti et al., 2002;
Rosenblum and Pikovsky, 2003). With the advent of nonlinear
dynamics and chaos theory, the concept of synchronization has
been broadened, and now includes any type of statistical interdependency between dynamical systems, even if they do not display
regular oscillations and even if strict phase coupling is absent: so
called generalized synchronization (Rulkov et al., 1995). These
advances in the physics of synchronization influenced the possibilities for detecting and quantifying synchronization in neurophysiological data.
While synchronization in its various manifestations should be
considered the fundamental physical concept for understanding
correlations between neurophysiological time series, different notions have also become popular in the neuroimaging literature.
Functional connectivity, first introduced by Aertsen et al. (1989),
refers to the existence of any statistical interdependency between
neurophysiological time series. Originally, this concept referred to
recordings of neuronal spiking, but it is now also used for EEG,
MEG and in particular fMRI (Lowe, 2010; Stevens, 2009). While
functional connectivity is a model free concept, the notion of effective connectivity stresses asymmetric causal interactions between
neural systems (Friston, 2002). The notion of effective connectivity
is especially clear in the approach called dynamic causal modeling
(DCM) introduced by Karl Friston (Penny et al., 2004). However, we
should stress that both effective and functional connectivity could
be defined in terms of synchronization theory, that is: in the language of interacting dynamical systems. As we will see, this is of
particular importance if we want to obtain a formal understanding
of communication processes taking place on complex networks
(Arenas et al., 2008).
Many time series analysis techniques have been developed to
characterize statistical interdependencies between two or more
time series of neurophysiological activity. An excellent overview
is the review by Pereda et al. (2005). A useful distinction is that between linear and nonlinear techniques. The most important linear
measure of correlation between time series is coherence. It can be
5
considered a generalization of the (time-lagged) correlation between time series. Coherence describes the strength of the correlation between two time series as a function of frequency; it depends
upon the consistency of the phase difference as well as the power
of the two time series (Nunez et al., 1997). To track correlations in
non-stationary signals, coherence can also be based upon wavelet
analysis (Lachaux et al., 2002). At least three categories of nonlinear synchronization measures can be distinguished. The first group
consists of measures that quantify the consistency of the phase difference between two signals, discarding the influence of signal
amplitude (Rosenblum et al., 1996). Phase synchronization analysis can be based upon a Hilbert transform of the data (Mormann
et al., 2000) as well as wavelet analysis (Lachaux et al., 1999); both
approaches are equivalent (Burns, 2004). A second group of measures is based upon nonlinear dynamics and quantifies generalized
synchronization between attractors reconstructed from the time
series (Arnhold et al., 1999; Hu and Nenov, 2004; Pecora and Carroll, 2000; Schiff et al., 1996; Schmitz, 2000). The synchronization
likelihood is a relatively simple unbiased estimator of generalized
synchronization (Stam and van Dijk, 2002; Montez et al., 2006). Finally, there is a group of nonlinear coupling measures such as
event-synchronization and h2 that cannot be easily characterized
(Quian Quiroga et al., 2002b; Lopes da Silva et al., 1989).
The linear and nonlinear coupling measures mentioned above
are generally symmetric, and do not provide information on causality or the direction of the influence. One approach to determine
effective connectivity is based upon Granger causality, originally
developed in economics (Granger, 1969). In the case of Granger
causality, the future of time series x can be predicted better, if
not only its own past, but also the past of another time series y
is taken into account. If including the information of y improves
the prediction of x, y is said to have a causal effect on x. An important technique to apply Granger causality in a frequency dependent way is the directed transfer function (DTF) (Blinowska,
2011). The direction of coupling can also be determined with techniques based upon phase synchronization. An important example
is the phase slope index (Nolte et al., 2008). Measures of causal
interactions have also been derived in a nonlinear dynamical systems context (Arnhold et al., 1999; Hu and Nenov, 2004; Pecora
and Carroll, 2000; Schiff et al., 1996; Schmitz, 2000). Finally, the
nonlinear correlation coefficient h2 has also been used to derive a
nonlinear directed coupling measure (Wendling et al., 2005).
In contrast to fMRI, neurophysiological techniques for determining the coupling between different brain regions suffer from
the common source bias. If the activity recorded at two different
electrodes or sensors is influenced by a single underlying source,
the estimated correlations do not reflect true interactions between
different neural systems. In the case of EEG, the activity recorded at
the reference electrode can also influence the estimated correlations between different EEG channels (Nunez et al., 1997). These
are serious methodological problems that cannot simply be solved
by performing the connectivity analysis in source space instead of
signal space. While source or Laplacian derivations may diminish
these influences, they do not solve the problem completely, and
they may obscure true long-distance interactions. Nunez et al.
(1997) proposed a corrected version of the coherence to account
for the influence of volume conduction in EEG. Nolte et al. (2004)
suggested to use the imaginary component of coherence as a measure of coupling, that is not influenced by volume conduction. A
disadvantage of the imaginary coherence is the fact that its magnitude is influenced by signal power as well as the magnitude of
phase lag between the channels; therefore, increases or decreases
cannot be interpreted easily. The only conclusion that is certain,
is that the existence of a non-zero imaginary coherence cannot
be explained by volume conduction. The phase lag index is based
upon the idea of the imaginary coherence but it is not influenced
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by the signal amplitude nor by the magnitude of the phase lag
(Stam et al., 2007b). The recently proposed weighted phase lag index WPLI is claimed to be less sensitive to noise, but re-introduces
the dependency upon the magnitude of the phase difference (Vinck
et al., 2011). Interestingly, coherence, imaginary coherence, phase
coherence, WPLI and PLI are all intimately related to each other
since they can all be computed from the Hilbert transform of the
data (Stam et al., 2007b).
The availability of so many different measures of synchronization raises the question whether a rational choice can be made between them in terms of performance on real data. A few studies
have attempted to compare a subset of measures with respect to
their ability to correctly detect the presence of synchronization
in a multivariate data set (Ansari-Asl et al., 2006; David et al.,
2004; Quian Quiroga et al., 2002a). Unfortunately the results are
not unequivocal. Performance of the measures seems to depend
upon characteristics of the data set as well as characteristics of
the method itself. In the case of complex synchronization measures
based upon nonlinear dynamical systems theory, many parameters
have to be taken into account (Montez et al., 2006). Performing the
analysis in source space does not simplify matters, since each
source reconstruction algorithm has its own assumptions that
may influence correlations between source-based signals in unknown ways. At present, the most rational approach may be to
use more than one technique, and to search for results that are
as consistent as possible, irrespective of the technique that was
used.
3.3. Fragile binding
Interregional synchronization, or functional/effective connectivity, conveys important information about healthy brain functioning. Here we briefly indicate what connectivity studies have
shown even without a formal network perspective; below we will
continue this discussion in the context of modern network theory.
In healthy subjects the strength of interregional synchronization
depends upon age. Long distance synchronization is relatively
low at birth, and increases during development, possibly due to
maturation and myelinization of long distance association pathways (Barry et al., 2004; Gmehlin et al., 2011; González et al.,
2011; Thatcher et al., 2008). The strength of synchronization depends not only upon maturation and age, but is also strongly influenced by genetic factors (Chorlian et al., 2007; Posthuma et al.,
2005; Van Beijsterveldt et al., 1998). How genetic factors can determine the strength of functional interactions between brain regions
is not yet understood.
The level of interregional functional connectivity is different for
different frequency bands and seems to be related to local band
power as well as distance. According to some authors, long distance communication is mainly supported by synchronization in
low frequency bands, while short distance local communication
depends upon synchronization in beta and gamma frequency
bands (Von Stein and Sarnthein, 2000). The strength of synchronization between different brain regions shows characteristic fluctuations that may not simply reflect noise effects, but could also be
due to the rapid formation and dissolution of functional connections. This dynamic nature of synchronization, even during a resting-state, has been referred to as ‘‘fragile binding’’ (Stam and de
Bruin, 2004). Two different models of synchronization dynamics
have been suggested. According to the idea of microstates the brain
manifests a succession of relatively stable states of a few hundred
milliseconds duration connected by abrupt transitions (Lehmann
et al., 2006). However, empirical studies show that the distribution
of synchronization and desynchronization times is rather complex,
pointing in the direction of a process with (self-organized) criticality (Beggs and Plenz, 2003; Breakspear et al., 2004; Stam and de
Bruin, 2004; Gong et al., 2007). Interregional synchronization is
also clearly influenced by behavioral state and cognition. During
sleep, characteristic changes in synchronization occur in different
sleep stages (Ferri et al., 2005). Tasks that involve working memory
are associated with increased synchronization, especially in the
theta band, and possibly also in the alpha band (Jensen Axmacher
et al., 2008; Fell and Axmacher, 2011; Jensen et al., 2002; Klimesch,
1997; Sarnthein et al., 1998; Tesche and Karhu, 2000). Gamma
band synchronization is now believed to be an important neurophysiological mechanism underlying binding, attention and even
consciousness (Dehaene and Changeux, 2011; Fries et al., 2007;
Jensen et al., 2007; Uhlhaas et al., 2011).
3.4. Excessive connectivity
Synchronization in the healthy brain is a delicate process characterized by the rapid formation and disappearance of functional
connections between different brain regions. This ‘‘fragile binding’’
can become disturbed in two different ways: excessive connectivity or disconnection. The most obvious example of a syndrome
characterized by excessive synchronization would seem to be epilepsy (Lehnertz et al., 2009). At the neuronal level, epileptic phenomena are associated with paroxysmal depolarization shifts
(Gorji and Speckmann, 2009). Recently, the significance of gap
junctions as an important cause of hypersynchronization in epilepsy has also been pointed out (Volman et al., 2011). The classic
interictal and ictal EEG patterns of epilepsy, such as spikes and
generalized spike-wave discharges, are usually assumed to reflect
excessive synchronization at the neuronal as well as the macroscopic level (Gorji and Speckmann, 2009). However, recent observations of the activity of individual neurons during seizures in
humans suggest that a hypersynchronization-only concept of epileptic activity may be too simple (Truccolo et al., 2011).
Although some types of seizures, notably absence seizures characterized by generalized 3-Hz spike-wave discharges, may indeed
reflect excessive synchronization compared to pre- and postictal
states, this pattern may not hold for all seizures, and in particular
not for partial seizures. Partial seizures may represent a complex
sequence of increases as well as decreases of synchronization in
different frequency bands during different stages of the seizure.
In fact, the highest level of synchronization during partial seizures
may be reached only at the very end of the seizure (Schindler et al.,
2007). Pre-ictal changes in EEG synchronization have also been the
topic of intense research, especially since papers by German and
French groups suggested the possibility of seizure prediction
(Lehnertz and Elger, 1998; Le van Quyen et al., 2001a; Martinerie
et al., 1998). Of all EEG measures that have been tested for their
performance in predicting seizures, measures of interregional synchronization turn out to be the most successful (Mormann et al.,
2005). Surprisingly, the pre-ictal state may be characterized by a
decrease rather than an increase in synchronization (Mormann
et al., 2003). It has been suggested that this pre-ictal drop in synchronization might reflect a release of the ictal focus from the
inhibitory influence of other brain regions (Le van Quyen et al.,
2001b). In this respect, the interictal state should be distinguished
from the pre-ictal state. There are indications that synchronization
levels during the inter ictal state are abnormally high, not only
compared to the pre-ictal state, but also to synchronization levels
in healthy subjects. Increased synchronization in the theta band
has been described in patients with different types of epilepsy,
and may predict the risk of epilepsy even independently of the
occurrence of epileptic spikes (Douw et al., 2010b).
Increased levels of synchronization are not limited to epilepsy.
Although Parkinson’s disease is considered to be mainly a basal
ganglia disorder, increased interregional synchronization in the alpha band has been found in early, untreated Parkinson’s disease
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
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(Stoffers et al., 2008a). In later phases of the disorder, this hyper
synchronization extends to lower as well as higher frequency
bands (Berendse and Stam, 2007; Moazami-Goudarzi et al.,
2008). The excessive synchronization is influenced by treatment
(Silberstein et al., 2005; Stoffers et al., 2008b). In healthy children
with obesitas, excessive synchronization during a no-task restingstate has been shown in very low and very high frequency bands
(Olde Dubbelink et al., 2008). Of interest, MRI has shown excessive
white matter volume in subjects with obesitas. In patients with
diabetes, excessive synchronization has also been demonstrated
(van Duinkerken et al., 2009) but a decrease of connectivity has
also been observed (Cooray et al., 2011). In subjects with mild cognitive impairment (MCI), a precursor of dementia in a substantial
proportion of cases, increased levels of synchronization have also
been reported, either at rest, or during the performance of cognitive tasks (Buldú et al., 2011; Jiang, 2005; Pijnenburg et al., 2004).
3.5. Disconnection syndromes
While abnormally increased levels of synchronization in neurological disease may seem a bit counterintuitive, except in the context of epilepsy, the notion that destruction or loss of anatomical
connections between brain regions would be reflected in diminished functional connectivity would seem more natural. A nice
illustration of this idea is the loss of inter hemispheric EEG coherence in children with agenesis of the corpus callosum (Koeda et al.,
1995). The most obvious example of this idea is Alzheimer’s disease, a degenerative type of dementia characterized by a progressive and severe loss of neurons and connections between brain
regions. An important concept is the idea that at least some types
of cognitive dysfunction should be explained directly in terms of
lost connectivity. Delbeuck et al. (2003) suggested referring to Alzheimer’s disease as a disconnection syndrome. Evidence for a loss
of interregional synchronization comes from a large number of
EEG, MEG, and fMRI studies (Babiloni et al., 2004; Dickerson and
Sperling, 2009; Rombouts et al., 2005; Sorg et al., 2009). Diminished connectivity has been reported mostly during no-task resting-states, but may also be present during the performance of
cognitive tasks (Sperling et al., 2010). Loss of functional connectivity in Alzheimer’s dementia correlates both with structural
changes as well as cognitive impairments, and might improve
the prediction of progression from MCI to AD (Rossini et al.,
2006; Smits et al., 2011; Stam et al., 2006). A particularly nice illustration of functional, structural, and cognitive correlations is
shown in the study of Pogarell et al. (2005), where decreased inter
7
hemispheric EEG coherence correlates with atrophy of the corresponding part of the corpus callosum on MRI. Of interest, dementia
in Parkinson’s disease may also result in a loss of connectivity, in
particular between frontal and temporal areas (Bosboom et al.,
2009). In non-demented Parkinson patients, EEG changes may
accurately predict the later development of dementia (Klassen
et al., 2011).
Although the concept of a disconnection syndrome in dementia
disorders seems appealing, there are some important caveats. First,
even in diffuse dementia disorders increases of synchronization in
certain frequency bands can be observed in the same patients who
also display decreases of connectivity in other bands. In particular,
loss of connectivity in the alpha band may be accompanied by increased connectivity in theta and beta/gamma bands (Stam et al.,
2006; Bosboom et al., 2009). Furthermore, as discussed in the previous section, MCI and other diffuse neurological disorders with
mild cognitive dysfunction are sometimes characterized by increased rather than decreased synchronization. Such paradoxical
findings are often explained in terms of putative compensation
mechanisms, but really present a serious challenge to any simple
interpretation of connectivity changes in neurological disease.
Another challenge comes from findings in local brain disease.
Bartolomei et al. (2006a) showed that brain tumors may cause
widespread changes in resting-state functional connectivity, also
in the unaffected hemisphere. To complicate matters even further,
connectivity in different frequency bands may be increased as well
as decreased following local lesions (Bartolomei et al., 2006b).
These findings have been replicated in several other studies (Guggisberg et al., 2008; Martino et al., 2011; Zaveri et al., 2009). Connectivity changes are also related to cognitive changes in these
patients (Bosma et al., 2008). The most consistent finding seems
to be an increase in theta band connectivity; this increase is partly
reversed after surgery and appears to be related to the severity of
the accompanying epilepsy (Douw et al., 2008, 2010b). During the
Wada procedure, where the activity of one hemisphere is temporarily depressed by amobarbital, similar patterns have also been
observed: increased low frequency connectivity ipsilaterally and
decreased in the opposite hemisphere, and widespread increase
of connectivity in higher frequency bands (Douw et al., 2009).
3.6. Strengths and limitations of brain connectivity studies
Clearly, studies of functional interactions between brain regions
can add important information to observations derived from local
changes in brain activity only. As an obvious example, levodopa
Fig. 2. Graph and adjacency matrix. Networks can be represented mathematically as graphs that consist of nodes (vertices) and connections (edges). In the left panel black
dots correspond to nodes and lines between dots correspond to connections. This is an example of an undirected, unweighted graph. In the right panel the same graph is
represented as an adjacency matrix. This is a square matrix with the same number of rows and columns as the number of nodes in the graph. Each matrix cell provides
information on the presence (value 1; indicated by color red) or absence (value 0; indicated by color blue) of an edge between two vertices. Since this matrix corresponds to
an unweighted, binary graph it is symmetric.
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
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C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
Fig. 3. Schema of small-world networks. The network is represented on a ring and consists of 32 vertices. In the case of a regular or lattice network (shown on the left) each
vertex is connected to six neighbors, three on each side. Subsequently, edges are picked randomly and rewired to a randomly chosen other vertex with a probability p
(rewiring probability). The results for small values of p is a small-world network that combines the high clustering of the lattice with short path lengths caused by the few
random connections. For p = 1 the network is completely random with a low clustering coefficient and a short path length.
changes functional connectivity but does not affect local band
power in Parkinson’s disease (Stoffers et al., 2008a,b). Neurophysiological assessment of functional connectivity has the advantage of
a high time resolution and a direct measurement of neural activity.
Although functional interactions are constrained by anatomical
connectivity, they are not identical to it (Adachi et al., 2011; Honey
et al., 2009). Synchronization studies show fragile binding in
healthy subjects and patterns of hyperconnectivity or disconnection in various neurological disorders. However, the patterns of
synchronization changes in neurological disease are difficult to
interpret. In particular the occurrence of increased as well as decreased synchronization (in different frequency bands and different brain areas) challenges our understanding of the underlying
mechanisms, and demonstrates the need for a more consistent theoretical framework to study changes in network organization.
4. Network studies in brain research
4.1. Modern network theory
4.1.1. Graph theory, statistical physics and dynamical systems
While connectivity studies constitute a useful extension of local
activation studies, the complexity of the data presents a challenge
Fig. 4. Example of scale-free network. This graph consists of 200 vertices and an
average degree of 7.72. Vertices are displayed closer to the middle if their degree is
higher. A few vertices (the hubs) have very high degrees, whereas most nodes are
more in the periphery of the graph. The high diversity of node degrees is
characteristic of scale-free graphs and is absent in lattice, small-world and random
graphs. However, scale-free graphs have a vanishingly small clustering coefficient.
that is hard to face without a proper theoretical framework. One
attempt to develop such a framework is modern network theory,
also referred to as graph theory (Newman, 2010). While network
theory and graph theory are sometimes referred to as if they were
synonyms, they are not, and understanding how they are related
may help to obtain a better idea what network theory is about,
and how it might be helpful in dealing with connectivity data in
neuroscience. Modern network theory is a highly interdisciplinary
field of science, devoted to the study of complex networks. It is
rooted in at least three different but connected branches of mathematics and physics: (i) graph theory; (ii) statistical mechanics of
networks; (iii) dynamical systems theory.
Graph theory is a relatively old branch of discrete mathematics
that deals with networks (Gross and Yellen, 2006). It originated
with the work of Euler, in particular with his solution of the problem of the seven bridges of Konigsberg. Typically, graph theory
deals with relatively small networks that are represented as sets
of network nodes (vertices) and network connections (edges)
(Fig. 2). A network represented as a set of vertices and edges is
called a graph. Different categories of graphs can be distinguished.
Graphs where edges are either present or absent are called binary
or unweighted; when a weight representing for instance the
strength, length or importance of a connection is assigned to each
edge the resulting graph is called weighted. When edges only indicate that a connection or relation exists between two vertices the
graph is undirected. When edges indicate the direction of influence
the graph is directed or a digraph. Many important theorems have
been proven for graphs, in particular in relation to special cases
such as fully connected graphs, Harary graphs, wheel graphs or
trees, but this mathematical knowledge is mainly of interest in
specific fields of science such as computer and communication science (Gross and Yellen, 2006). Classic graph theory could not deal
with large networks with stochastic features.
The scope of graph theory was extended to realistic complex
networks when models were introduced that aided the description
and analysis of large networks with random components. The first
example of such a model was the random graph model proposed
by Erdös and Rényi (1959) (ER). In this model, an edge exists between any randomly chosen pair of vertices with a fixed probability p. While this may seem a very simple model, it turns out to have
many surprising and important mathematical properties, such as
the sudden emergence of a giant connected component for increasing values of p and short path length. The ER random graph is not a
realistic model of many real complex networks, but is often used as
a null model for comparison. The breakthrough in the statistical
approach to networks occurred with the publication of two papers
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9
Table 1
Explanation of basic network terminology and definitions of most important network concepts and measures.
Symbol
Concept
Explanation
G
V
E
Aij
Wij
k
Graph
Vertex
Edge
Adjacency matrix
Weight
Degree
Mathematical representation of a network
A vertex is a node of the network. The capital V refers to the set of all vertices in a graph
An edge is a connection or relation between two vertices. The capital E refers to the set of all edges in a graph
Binary square matrix. If two vertices i and j are connected the matrix cell aij will have value 1; otherwise it will have value 0
Edges in a network can be assigned a weight w that may contain information on the relative importance of the edges
The degree k of a vertex is the number of edges incident with (‘‘connected to’’) this vertex. In a weighted graph the equivalent measure
is the strength of a vertex (sum of weights of all its incident edges)
Two vertices that are connected by a common edge are neighbors
Number of edges in a graph expressed as fraction of all possible edges
The probability distribution P(k) is the probability that a randomly chosen vertex of the graph will have degree k
Correlation coefficient of degrees of pairs of neighbors. Can be defined for unweighted as well as weighted graphs
Measure of the broadness of the degree distribution. Related to the spectral radius and the threshold for the onset of synchronization
The clustering coefficient C is a measure of the local connectedness of a graph. It can be defined as the number of edges present
between the neighbors of a vertex divided by the total possible number of edges between the neighbors
Number of edges in the shortest path between two vertices. In weighted graph: path with lowest sum of edge weights between two
vertices (note that this is not necessarily the path with the smallest number of edges)
Longest shortest path of a graph
Average of the inverse of the shortest paths by setting the value to zero for disconnected vertices
Efficiency computed from the set of neighbors of a vertex
Small characteristic subgraph. A triangle (three connected vertices) is an example of a motif. The prevalence of different types of
motifs is indicated by a motif count
Modules are subgraphs (typically larger than motifs) such that vertices within a module are more strongly connected than vertices of
different modules. For a given partitioning of a graph Newmans modularity index Q indicates the quality of the division into modules
A graph with a hierarchical structure is characterized by a power law relation between C(k) and k with an exponent about 1
Fraction of all shortest paths that pass through a particular vertex or edge
P(k)
r
j
C
L
Neighbor
Cost
Degree distribution
Degree correlation
Degree divergence
Clustering
coefficient
Path length
D
Eglobal
Elocal
Diameter
Global efficiency
Local efficiency
Motif
Q
Modularity
b
k1
Hierarchy
Betweenness
centrality
Closeness centrality
Eccentricity
Hub
Spectral radius
EC
Eigenvector
centrality
Spectral gap
lN1
Algebraic
connectivity
Synchronizability
S
R
GIC
Effective Graph
Resistance
Graph index
complexity
Average shortest path length from a reference vertex to the rest of the graph
Longest shortest path from reference vertex to any other vertex of the graph
Vertex with high centrality. This centrality can be defined in terms of degree, or (combinations of) any of the other centrality measures
Largest eigenvalue of adjacency matrix. Inverse is measure of synchronization threshold. Spectral radius is also related to shortest
path, overall connectedness and degree diversity of a graph
Value of the ith coordinate of the eigenvector corresponding to the largest eigenvector the adjacency matrix. Measure of the
connectedness/centrality of the ith vertex
Difference between largest and second largest eigenvalue of adjacency matrix. Index how rapidly dynamics on the graph will converge
to the synchronized state
Second smallest eigenvalue of Laplacian matrix
Ratio of largest to second smallest eigenvalue of Laplacian matrix. Its inverse is a measure of the stability of the synchronized state of a
dynamic process on the graph
Measure of the resistance to flow on a graph analogous to Kirchoffs law for electronic circuits (Van Mieghem et al., 2010)
Index of the complexity of a graph’s structure (Kim and Wilhelm, 2008)
in 1998 and 1999 (Barabasi and Albert, 1999; Watts and Strogatz,
1998). Watts and Strogatz (1998) introduced a model (WS) of a
regular graph on a ring where each vertex is connected to its k
neighbors (Fig. 3). With a rewiring probability p, an edge is chosen
and re connected to another randomly chosen vertex. For p = 0 this
model corresponds to a regular graph or lattice characterized by
high clustering (each vertex being connected to a few other vertices most of which are also connected to each other). In such a regular graph, it takes on average many steps to travel from one vertex
to any other vertex on the graph, that is: a regular graph has a long
path length. When p = 1 the graph is in fact a random graph, like
the ER random graph. Such a random graph has a low clustering
coefficient, but the average path length is now also very short.
The really interesting behavior is seen for small but non-zero values of p: rewiring even a very small number of edges does not affect the high clustering, but results in a dramatic decrease of
average path length. Such intermediate networks that combine
high clustering with short path lengths are called small-world
networks.
The strength of the small-world model is that it explains the
combination of high clustering and short paths encountered in
many real complex networks. However, it does not explain another
property of realistic networks. In ER and WS graphs the vertices all
have about the same number of edges, that is: they have a narrow
degree distribution. The degree distribution P(k) = k is defined as
the probability that a randomly chosen vertex will have degree k.
Surprisingly in many complex networks in nature this distribution
corresponds to a so-called power law: P(k) kc, with c a scaling
exponent between 2 and 3. Barabasi and Albert (1999) were the
first to suggest a simple algorithm for growing networks, where
newly added vertices connect preferentially to existing edges with
higher degrees (Fig. 4). The scale-free network of Barabasi and Albert (BA) is now considered one of the most important models to
explain the presence of highly skewed power law degree distributions in many complex networks, such as the Internet and the
World Wide Web (Barabasi, 2009). An important implication of
such a degree distribution is the presence of so-called hubs, vertices with an exceptionally high number of connections.
Graph theory and statistical physics present two of the major
pillars of modern network theory; the third is dynamical systems
theory. Networks in general, and complex networks in particular,
are mainly important because we are interested in the processes
that take place on these networks (Barrat et al., 2008). For instance,
we could assume that each vertex corresponds to a dynamical system, perhaps an oscillator, and each edge indicates which oscillators interact and how strong this interaction is. More generally,
we can imagine that some transport of traffic, goods, or information is taking place on the graph. The important questions here
are: how does the topology of the graph influence the dynamical
processes on the graph? In particular, do topological properties
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Fig. 5. Overview of different graph measures. In the upper row on the left the two building blocks of any graph are shown: vertices (indicated by a black dot) and edges
(indicated by a line connecting two dots). With these two building blocks all topological network patterns and their corresponding measures can be constructed. The degree is
the number of edges connected to (‘‘incident on’’) a vertex. The red vertex has a degree of three. The clustering coefficient of a vertex is the fraction of its neighbors that are
also connected to each other. The red vertex has three neighbors, and between these neighbors two out of three possible edges are present. The clustering coefficient of the
red vertex is thus 2/3. A minimum of four edges has to be traveled to get from the red vertex to the blue vertex; this corresponds to a path length of four. Betweenness
centrality is the fraction of all shortest paths that pass through a vertex or edge. The central red vertex of the star graph on the left in the middle row has a betweenness
centrality of 1 (all shortest paths have to pass through this vertex). The degree correlation of two connected vertices is the correlation between their corresponding degrees. In
the left pair of red and blue vertices these degrees are four and two; in the right pair these degrees are both two. The left case would correspond with a lower degree
correlation, the left with a higher degree correlation. Modularity refers to the existence of subgraphs that consist of vertices with more connections to other vertices within
the same module than to other vertices outside the module. The graph at the bottom consists of four different modules, each indicated by a different color of the vertices.
predict the onset of synchronization, or the efficiency of information transport in the network? Of interest, Watts and Strogatz were
investigating such questions about oscillator synchronization
when they discovered their small-world model. Finally, perhaps
the most challenging questions of all relate to mutual interactions
between dynamics and topology: how do dynamical processes
change the architecture of the networks, and how does altered network structure constrain dynamics?
4.1.2. Measures of complex networks
Modern network science, with its roots in graph theory, statistical physics, and nonlinear dynamics, has developed a large and
still increasing number of measures that can be used to characterize large and complex networks (Table 1 and Fig. 5). We have already encountered a few of these measures, such as the
clustering coefficient, the average shortest path, and the degree
distribution. Often, new measures are developed when new models or new observations on real networks suggest that a particular
feature of networks may be of interest and deserves exact quantification. An overview of currently used network measures can be
found in several review papers and textbooks (Kaiser, 2011; Newman, 2010; Rubinov and Sporns, 2010). Here we briefly mention
the most important categories.
The first category involves measures that characterize the presence and nature of structural building blocks and sub networks.
The simplest example is in fact the clustering coefficient, which reflects the presence of triangles (complete subgraphs of three vertices) in networks. Triangles are important since they are directly
related to the robustness and error tolerance of the network (Boccaletti et al., 2006). Triangles are the simplest example of motifs.
Motifs are small subgraphs consisting of a few vertices connected
in a particular way and possibly associated with a specific function.
Motif analysis involves counting the prevalence of different motifs
in a network and comparing the results with those for random networks (Milo et al., 2002). Exponential random graph analysis is a
sophisticated statistical technique popular in the social network
research, that also involves reconstructing complex networks from
a set of simple motif-like building blocks (Simpson et al., 2011).
One level up, it may be possible to distinguish clusters or modules,
sub graphs with vertices that are more strongly connected to each
other than to vertices outside their own module (Newman, 2006).
Finally, modules can sometimes combine to form higher order
structures, giving rise to a true hierarchical modularity (Ravasz,
2009).
A second category involves measures sensitive to the level of
integration in a network. Global integration can be assessed with
the average shortest path, the diameter (the longest shortest path),
global efficiency (Latora and Marchiori, 2001, 2003) and the spectral radius (the largest eigenvalue of the adjacency matrix of the
network) (Van Mieghem, 2011).
A third category involves measures of relative vertex importance. The most basic measure is the degree of vertex (the number
of edges connected to a vertex), and the degree distribution P(k). As
we have seen, different types of complex networks may be distinguished on the basis of their degree distribution. Degree is thus a
measure of the importance of a vertex in a network. This concept
is referred to as centrality. The degree correlation indicates
whether high degree vertices tend to connect preferentially to
other high degree vertices (assortative mixing) or to low degree
vertices (disassortative mixing) (Newman, 2003). Centrality is
not always the same as degree, and other more sophisticated centrality measures have been developed as well. Betweenness centrality of a vertex is the number of shortest paths through that
vertex divided by the total number of shortest paths (in a similar
way it is also possible to define edge betweenness) (Wang et al.,
2008). The eccentricity of a vertex is the length of its longest shortest path to any other vertex; the lower the eccentricity, the more
central a vertex is in a network. Eigenvalue centrality determines
the importance of a vertex on the basis of its connections to other
vertices, but also with respect to how those other vertices are
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Fig. 6. Representing multi channel time series as a graph. First, all pair-wise
correlations (using any of the synchronization measures shown in Fig. 1) between
all time series are computed. The results constitute a matrix, with the cell values
indicating the strength of the correlation between two time series. Next, all cell
values below a threshold are set to zero, and all cell values above the threshold are
set to 1. This converts the weighted matrix to an unweighted binary matrix (the
binary matrix and unweighted graph representations are equivalent). From this
matrix all network measures of interest (see Fig. 3) can be computed. As a control
procedure, a large ensemble (indicated by the open arrow) of random networks is
generated from the unweighted matrix, preserving the number of nodes, edges and
usually also the degree distribution. The network measures are also computed for
all the random graphs. Combining the network measures from the original network
and the average values obtained from the ensemble of random graphs normalized
measures can be computed. As an alternative procedure (indicated by interrupted
lines), the computation of network measures can also be done directly on the
weighted matrix skipping the threshold and binarization steps.
connected (and so on); this is also the idea behind the PageRank
algorithm used by Google to determine the importance of webpages (Lohmann et al., 2010).
The final and fourth category we want to mention consists of
measures that relate network topology to the dynamics of processes on the network, in particular the synchronization threshold
and the stability of the synchronous state (Arenas et al., 2008).
Topological features that are associated with a lower synchronization threshold are the inverse of the spectral radius (the largest
eigenvalue of the adjacency matrix of a graph) and a measure of
2
degree diversity called kappa (hk i=hki). Kappa is especially high
in networks with a scale-free degree distribution and high-degree
hubs. These concepts are based upon the assumption that the
dynamics can be described by non-identical Kuramoto oscillators
(Restrepo et al., 2005). If we assume identical oscillators, synchronization can be analyzed with the so-called master stability equation (Barahona and Pecora, 2002; Chen and Duann, 2008;
Nishikawa et al., 2003). This suggests that the stability (not the
threshold) of the synchronous state depends upon the ratio of
the second smallest to the largest eigenvalue of the Laplacian of
the adjacency matrix of the graph (Arenas et al., 2008). The second
smallest Laplacian eigenvalue, referred to as the algebraic connectivity, is of special interest: it is only >0 if the graph is connected,
and it is smaller than or equal to the smallest degree. The magnitude of the algebraic connectivity is not only important for synchronization, but also reflects network robustness (resilience
against damage).
4.1.3. Representing structural and functional networks as graphs
To be able to use the concepts and measures of network theory
for a better understanding of the organization of brain networks, it
is important to have a strategy to represent neuroanatomical and
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neurophysiological data in the form of graphs. For structural data,
this is relatively easier than for physiological data. Since the full
information on the connectivity of the neuronal network of Caenorhabditis elegans is known, Watts and Strogatz were able to represent this nervous system as a graph by taking the neurons as
vertices and connections between neurons as edges (Watts and
Strogatz, 1998). For several animals, maps are available of the
structural connections between macroscopic brain areas; this allows a representation as graphs by taking brain areas instead of
neurons as vertices (Sporns et al., 2000; Stephan et al., 2000). Modern MRI studies in humans present two possibilities to models anatomical connectivity as graphs. Matrices of correlations in cortical
thickness can be taken as surrogate measures of anatomical connectivity (He et al., 2007). More sophisticated techniques allow
the reconstruction of anatomical networks in individuals (Tijms
et al., 2011). Alternatively, DTI and tractography can be used to
construct connectivity matrices at various levels of anatomical resolution (Hagmann et al., 2007, 2008; Iturria-Medina et al., 2007).
Representing neurophysiological data as a graph is less straightforward. We first describe the basic approach (Fig. 6). Then we
mention some alternative possibilities and point out methodological caveats. Assume we have a recording of n channels of ‘‘highdensity EEG’’ or MEG. After preprocessing, artefact removal and filtering we can compute the strength of synchronization between all
possible pairs of channels or sensors, using one of the techniques
described in Section 3.2. The resulting information can be represented in the form of an n by n matrix, with each ni,j matrix cell
containing a real number reflecting the strength of synchronization
between channel i and j. In graph theory, such a matrix is referred
to as the adjacency matrix A. To represent this information in the
form of a graph, we consider each of the n channels to be a vertex.
We choose a threshold T such that if ni,j > T, an edge is present between vertex i and j. If ni,j < T, no edge is present between vertices i
and j. If we indicate the presence of an edge by 1, and the absence
by 0, we have converted the n channel EEG or MEG data into a binary graph. Once the data are represented as a graph (with its equivalent adjacency matrix), in principle all measures available in
network theory can be applied to the network. Clearly, this approach opens up new and potentially interesting ways to analyze
functional as well as structural network data.
There are a number of variations to this basic scheme, and a few
methodological problems. First, a strong point of the approach is
that it can be applied to all sorts of data, structural, functional
(EEG, MEG, fMRI BOLD time series, correlations in spike trains or
calcium fluctuations). However, it is not always clear what should
be considered a vertex, and what is an edge, especially in the case
of functional data (Kaiser, 2011). EEG and MEG recorded in signal
space do not have a simple relation to underlying sources, and
therefore according to some authors might not qualify as bona fide
vertices (Antiqueira et al., 2010). The alternative would be to do
network analysis in source space rather than signal space; however, even this will not solve problems related to volume conduction and spatial sampling that might interfere with properties of
the reconstructed networks (Bialonski et al., 2010). A further problem concerns the choice of a threshold. This can be based upon the
significance of the synchronization, or a requirement that the
resulting graph is connected. The same threshold can be used for
all networks to be compared, or it can be adjusted individually to
obtain networks with the same degree for all subjects and conditions (Stam et al., 2007a). In many cases, a range of thresholds is
studied to avoid the rather arbitrary choice of a single threshold.
Clustering coefficients, motif prevalence, and path lengths are often normalized by comparing the results with those of random
networks with the same number of nodes, edges, and degree distribution (Achard and Bullmore, 2007). Unfortunately, comparison of
networks with different numbers of vertices and edges remains
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problematic, even after normalization by random network data
(van Wijk et al., 2010).
4.2. Healthy subjects
Network studies of structural and functional brain networks
have revealed a number of consistent characteristics. First, there
is considerable evidence that brain networks are characterized by
a combination of high clustering (the fraction of neighbors of a
node that are connected to each other) and short average path
lengths (the minimum number of edges to be travelled to get from
one vertex to another). This has first been shown for functional
networks with MEG and fMRI (Eguíluz et al., 2005; Stam, 2004;
Stephan et al., 2000). Similar results have been obtained with
structural MRI techniques, in particular cortical thickness correlations and DTI tractography (He et al., 2007; Hagmann et al.,
2007,2008; Iturria-Medina et al., 2007; Gong et al., 2009a). Human
brain networks should thus be classified as small-world networks.
This is important since the strong clustering of small-world networks might support the segregation, and short path length might
support the integration that are both assumed to be necessary for
optimal information processing in networks (Sporns, 2011b; Stam,
2010). In addition to clustering, human brain networks also display
structure at the larger scales of network motifs, and in particular
network modules (Meunier et al., 2010; Sporns and Kötter,
2004). Structural and functional MRI studies have shown that human brain networks consist of around five to six modules, each
corresponding with known functional subsystems (Gong et al.,
2009a; He and Evans, 2010). These large-scale modules may be
composed of smaller sub modules, suggesting a hierarchical organization of brain networks (Ferrarini et al., 2009; Gleiser and
Spoormaker, 2010; Meunier et al., 2010). It is interesting to see that
the hierarchical organization of human brain networks is exactly
the kind of organization that Herbert Simon predicted as optimal
for any complex system (Simon, 1962).
Not all nodes or vertices in brain networks have the same significance. There is accumulating evidence that the degree distribution
of human brain networks is highly skewed, with a relatively high
prevalence of high-degree nodes or hubs (Eguíluz et al., 2005;
Van den Heuvel et al., 2008). There is still some controversy
whether human brain networks are truly scale-free with a
power-law degree distribution. Some studies suggest that the degree distribution might be more properly modeled by an exponentially truncated power-law (Achard et al., 2006). This controversy
is probably due, at least to some extent, to methodological issues,
such as the as the level of resolution (ROIs versus voxels) used for
the network analysis (Fornito et al., 2010). There is little doubt that
the so-called default mode network, consisting of medial frontal,
anterior and posterior cingulate cortex, precuneus and parietal cortex, constitutes a connectivity backbone that links the hubs with
the highest degrees (Hagmann et al., 2008). The highly connected
hubs of the default network are strongly correlated with intelligence, and may also be of central importance in consciousness
(Douw et al., 2011; Li et al., 2009; Van den Heuvel et al., 2009; Soddu et al., 2011).
Network changes have also been described in relation to the
physiological changes of consciousness related to sleep (Ferri
et al., 2007, 2008). As we will discuss below, brain network hubs
may not only play a central role in normal brain function, but also
in some types of brain disease. The fact that high degree nodes in
brain networks preferentially connect to each other suggests a positive degree correlation or assortative mixing (De Haan et al.,
2009; Hagmann et al., 2008). A similar type of degree correlation
is also found in social networks, whereas most other networks in
nature tend to be disassortative (Newman, 2003). However, at
the neuronal level, brain networks are disassortative rather than
assortative (Bettencourt et al., 2007). This implies that the human
brain is the only type of complex network known that displays
opposite patterns of mixing at different spatial scales. At the cellular level, the brain is organized as a communication network, and
at the macroscopic level the brain is organized as a social network.
The origin and significance of this type of organization of brain networks are currently unknown.
The complex architecture of human brain networks arises during development and seems to be under strong genetic control. A
large EEG study of children investigated at age 5 and 7 showed
an evolution of a relatively more random to a more small-world
like topology of functional brain networks (Boersma et al., 2011;
Micheloyannis et al., 2009). In a follow up study, it was shown that
the optimal small-world pattern of adult age is gradually replaced
by a more random topology at higher ages (Smit et al., 2010). Thus,
the evolution of brain network topology with respect to clustering
and path length seems to follow an inverted u-curve. In addition,
clustering coefficients and path lengths were shown to have a high
heritability in twin studies (Smit et al., 2008). The evolution from
random to small-world topology has been replicated in model
studies (Siri et al., 2007). Another interesting finding is the difference in brain network topology in males and females, with higher
connectivity and shorter path lengths in females (Boersma et al.,
2011; Gong et al., 2009b; Schoonheim et al., 2011a).
4.3. Neuropsychiatric disease
Neuropsychiatric disease is a broad category of disorders, characterized by cognitive and psychiatric symptoms and a suspected
but generally unknown neurobiological substrate. The need for
objective diagnostic findings and the failure of structural imaging
methods, in particular MRI, to show consistent patterns of abnormalities in neuropsychiatric disease presents a challenge for a
more physiological and network based approach. One of the most
prevalent and disabling psychiatric conditions is schizophrenia.
Schizophrenia is not a simple Mendelian disorder, and complex genetic and developmental factors are assumed to play an important
role (Kim et al., 2011). In particular, there is a suspicion that the
normal development of long-range fronto-temporal connections
is disrupted. Network studies provide some support for this thesis.
In an early EEG study, functional brain networks of schizophrenic
patients were shown to have lower normalized clustering and path
length compared to healthy controls (Micheloyannis et al., 2006).
An EEG study based upon network analysis of the level of nonlinear
coupling also showed more random network topology in schizophrenic patients (Rubinov et al., 2009a). A few MRI studies showed
rather subtle network changes in schizophrenia. Alexander-Bloch
et al. (2010) reported a disruption of modular network structure
in schizophrenia, with a preservation of global clustering and path
length. Abnormal modular structure in schizophrenia has also been
detected with EEG (Jalili and Knyazeva, 2011). Van den Heuvel
et al. (2010) demonstrated very local network changes in frontal
and temporal areas, with a preservation of global network properties. An overview of modern network research in schizophrenia is
given in the review by van den Heuvel and Hulshoff Pol (2010).
As we have discussed above, brain networks undergo systematic
changes during development and such changes depend upon
strong genetic influences as well as experience. If schizophrenia
is indeed – at least to some extent – a genetic disorder of brain
development subtle abnormalities of brain network topology in
patients should be expected. These may be characterized by a relatively local fronto-temporal disturbance of normal network
architecture.
Other neuropsychiatric conditions have not been studied extensively. In acutely depressed patients, global connectivity assessed
with synchronization likelihood was lower and EEG functional
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network topology during sleep was more random compared to
healthy controls (Leistedt et al., 2009). In a large fMRI study, depressed patients had lower normalized path length and increased
node centrality of their functional brain networks compared to
controls (Zhang et al., 2011). These changes are also consistent
with a randomization of network topology in neuropsychiatric disease. Of interest, some of the altered node centrality measures correlated with disease duration and severity.
Structural and functional network changes, possibly under genetic control, have also been implicated in attention deficit hyperactivity disorder or ADHD (Konrad and Eickhoff, 2010). In a restingstate fMRI study, an increase of local and a decrease of global efficiency, suggesting abnormally regular networks, was reported in
ADHD patients (Wang et al., 2009). Local changes in nodal efficiency mainly involved frontal, temporal and occipital areas. Community analysis of EEG networks, based upon a fuzzy
synchronization likelihood, has been shown to be able to distinguish between ADHD patients and healthy controls (Ahmadlou
and Adeli, 2011).
Functional network analysis of delta band EEG in autism spectrum disorder (ASD) revealed increased short-range fronto-lateral
connectivity and a loss of long-range fronto-occipital connections
(Barttfeld et al., 2011). These changes were accompanied by a lower clustering coefficient, longer path length, and increased modularity in ASD patients. An earlier EEG study also reported loss of
global (alpha band) coherence and an increase in local (theta band)
connectivity in ASD (Murias et al., 2007). In this study, no formal
network analysis was applied. Graph theoretical analysis of structural brain networks in subjects with grapheme-color synesthesia
showed a shift from global to more local processing (Hänggi
et al., 2011).
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There is growing empirical support for the idea that changes in
brain network topology might play a crucial role in epilepsy, as
suggested by the model studies. Analysis of EEG depth recordings
in patients suffering from temporal lobe seizures showed a transition from a more random to a more regular topology of functional
brain networks during seizures (Ponten et al., 2007). The observation of abnormal regularization of functional brain networks during seizures has been confirmed by other studies, based upon
electro corticographic recordings (Kramer et al., 2008; Schindler
et al., 2008). A similar pattern has been demonstrated with surface
EEG and MEG during absence seizures (Gupta et al., 2011; Ponten
et al., 2009). Several studies have shown that interictal functional
networks in epilepsy patients may be characterized by increased
connectivity, especially in the theta band, a shift to a more regular
topology, changes in modular structure and prominent hub-like regions (Chavez et al., 2010; Douw et al., 2010b; Horstmann et al.,
2010; Wilke et al., 2011). Inter-ictal network changes as well as
cognitive impairment may also be related to the duration of the
epileptic condition (van Dellen et al., 2009; Vlooswijk et al.,
2011). MRI studies have also demonstrated abnormal brain network organization in epilepsy (Bernhardt et al., 2011; Liao et al.,
2010; Vlooswijk et al., 2011; Zhang et al., 2011). These observations are clinically relevant since it has been shown, for instance,
that removal of hub regions, identified with corticography during
epilepsy surgery, is associated with a better outcome (Ortega
et al., 2008; Wilke et al., 2011). Of interest, in the study of Ortega
et al., the minimum spanning tree, based upon the corticographic
time series, was used to identify the hub nodes. Characterization
of interictal changes in connectivity could also improve the diagnostic yield of routine EEG (Douw et al., 2010a).
4.5. Dementia
4.4. Epilepsy
Epilepsy refers to a group of neurological conditions, characterized by the unexpected and unpredictable emergence of an abnormal dynamic state of the brain with excessive neuronal firing and
synchronization. Epilepsy is one of the most prevalent and important neurological conditions. Because of its highly dynamic character, proper investigation of epilepsy requires tools with the highest
possible temporal resolution such as EEG and MEG. How and why
epileptic seizures arise in the brain is still essentially an enigma.
Network theory can be of help here by studying changes in the
organization of brain networks, that might lower the threshold
for pathological synchronization and facilitate the spread of abnormal activity throughout the brain.
The first indications that network analysis might be helpful in
understanding epilepsy, came from model studies. One study
showed that changing the topology of simulated hippocampal neuronal networks from regular to small-world and then to random
lowers the threshold for synchronization and gives rise to seizure-like activity and ultimately to epileptic bursting (Netoff
et al., 2004). Percha et al. (2005) also suggested a relation between
small-world topology and epileptic seizures. Another topological
feature, that may be relevant for understanding seizures, is the degree-distribution, in particular the presence of hubs. Morgan and
Soltesz (2008) showed that scale-free networks with hub nodes
had the lowest threshold for the emergence and spreading of seizures. Healthy brain networks are probably characterized by a
hierarchical modular structure, that could arise naturally during
development and might subserve functional specialization (Rubinov et al., 2009b; Stam et al., 2010; Wang et al., 2011). Interestingly, modular structure might actually present a threshold for
system-wide synchronization. Epileptic seizures could arise when
the normal modular structure breaks down in brain disease (Kaiser
and Hilgetag, 2010).
Alzheimer’s disease is one of the most common causes of
dementia in the aging population. It was also one of the first neurological conditions to be studied with the tools of modern network theory (Pievani et al., 2011). Functional network analysis of
EEG filtered in the beta band showed changes in the path length
in Alzheimer patients (Stam et al., 2007a). These changes correlated with a lower mini mental state score in the patient group.
This study also showed the methodological problems associated
with different ways for reconstructing networks from multi-channel EEG data, in particular using a fixed threshold or a fixed degree
for all networks. In a later MEG study, resting-state functional
brain networks in Alzheimer patients were characterized by a lower normalized clustering coefficient and path-length, especially in
the lower alpha band (Stam et al., 2009). The network changes in
Alzheimer patients could be replicated with a simple damage model that assumed preferential damage of hub-like network connections. A decrease of the normalized clustering coefficient and
path-length in Alzheimer’s disease could be confirmed in a more
recent EEG study (De Haan et al., 2009). This study revealed an
opposite pattern of network changes in frontal lobe dementia
(Fig. 7). All patients as well as healthy control subjects displayed
positive degree correlations, but the degree correlations were significantly lower in Alzheimer’s disease and frontal lobe dementia.
Functional connectivity of MEG during a working memory study
was investigated in patients with mild cognitive impairment (Buldú et al., 2011). The average level of connectivity and a so-called
outreach parameter were increased in the MCI group. These
changes were interpreted in terms of higher energy expenditure
and a tendency toward random structure in MCI patients. The increased synchronization likelihood during a working memory task
in MCI is in line with the EEG study of Pijnenburg et al. (2004).
Network changes in dementia have also been shown with structural and functional MRI studies (Bokde et al., 2009; Filippi and
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j.clinph.2012.01.011
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Fig. 7. Unweighted graphs of the lower alpha band (8–10 Hz) for different patient groups and different fixed average degrees (K), from EEG with 17-channel average reference
montage. For the Alzheimer’s disease (AD), frontotemporal dementia (FTLD) and control groups, the functional connectivity (synchronization likelihood, SL) based graphs are
shown as headplots for different values of K. Lower K values (higher threshold) result in a sparser network. On visual inspection, it is obvious that there are inter-group
differences. From: De Haan W et al. Functional neural network analysis in frontotemporal dementia and Alzheimer’s disease using EEG and graph theory. BMC Neurosci
2009;10:101. (Copyright by the authors under the license agreement that the work can be freely distributed when fully cited.)
Fig. 8. Synchronization differences between Alzheimer’s disease (AD) and controls. A subset of connectional differences of synchronization between the groups (2-tailed ttest, p < 0.05 uncorrected) are plotted at three superior-to-inferior levels through the AAL brain template: (b) = z53; (c) = z73; (d) = z111. Lines depict synchronization
between pairs of regions: solid lines = enhanced synchronization; dashed lines = reduced synchronization. Note the pattern of generalized posterior (parietal and occipital)
synchronization reductions and increased frontal synchronization. Adapted from: Sanz-Arigita EJ et al. Loss of ‘small-world’ networks in Alzheimer’s disease: graph analysis of
FMRI resting-state functional connectivity. PLoS One 2010;5:e13788. (Copyright by the authors under the license agreement that the work can be freely distributed when
fully cited.)
Agosta, 2011). The first resting-state fMRI network study in Alzheimer’s disease reported a significant decrease of the clustering coefficient, interpreted as a signature of local connection loss (Supekar
et al., 2008). In contrast, a more recent study showed normal
clustering coefficients but decreased normalized path length
(Sanz-Arigita et al., 2010) (Fig. 8). In this study, the average level
of connectivity, quantified as the synchronization likelihood of
the resting-state BOLD time series, was the same in both groups,
but Alzheimer patients had a relatively high connectivity level in
frontal areas, and a relatively lower connectivity level in parietal
and occipital areas. This anterior to posterior disconnection was
also observed in another fMRI study (Wang et al., 2007).
There is increasing evidence from these studies, that different
types of dementia might be characterized by specific patterns of
network changes (Zhou et al., 2010). This observation suggests that
network analysis could be of clinical use in the differential diagnosis of dementia (Pievani et al., 2011; Zamrini et al., 2011).
One of the most challenging questions at this moment is
whether the network changes in Alzheimer’s disease can be related
to specific underlying mechanisms. One of the classic neuropathological hallmarks of Alzheimer’s disease is the deposition of amyloid beta. Deposition of amyloid beta in the brain is not random
but follows a specific pattern with the highest concentrations in
multimodal association areas. There is a considerable overlap be-
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Fig. 9. Local activation, pair-wise interactions, networks organization. Brain function can be studied from different viewpoints depending on the way in which interactions
are taken into account. Local activations studies indicated on the left focus on determining the level of activation of individual functional units (neurons, small or large brain
areas); this corresponds to a ‘‘zero order’’ model of interaction. At the next level (middle panel) pair-wise interactions are taken into account. Many studies of functional
connectivity explicitly or implicitly are based on this ‘‘first order’’ model of interaction. Finally (right panel) all higher order interactions (interactions of interactions) can also
be taken into account. This ‘‘second and higher order’’ or network model allows the definition and quantification of concepts such as clustering, path length, modularity that
cannot be handled by lower order models.
5. Toward a theory of brain networks
5.1. From local activation to network organization
Fig. 10. A heuristic model of complex brain networks. The model consists of three
axes and three corners. The lower horizontal axis, corresponding to the degree of
‘‘order’’, connects the lattice network on the left to the random ER network on the
right. Intermediate networks on this axis are small-world networks. The vertical
axis indicating the level of degree diversity connects the random network below to
the scale-free network in the upper right corner. Finally, the diagonal axis that
reflects the level of hierarchy connects the scale-free network with the lattice.
Intermediate networks on this axis correspond to hierarchical modular networks
(HMN). Interrupted arrows suggest that networks have a natural tendency due to
‘‘multiconstraint optimization’’ to evolve toward HMNs. The lower left corner
corresponds to maximal segregation, the upper right corner to maximal integration,
the lower right corner to maximal entropy or randomness.
tween these areas and the default mode network known from resting-state fMRI studies (Buckner et al., 2008; Damoiseaux and Greicius, 2009; Greicius et al., 2009; Van den Heuvel and Hulshoff Pol,
2010). These areas show the highest levels of activity in the resting-state, and also have the highest level of connectivity (Hagmann
et al., 2007,2008; van den Heuvel et al., 2008). In a combined PET
and fMRI study, an intriguing correlation was demonstrated between brain network node degree and amyloid deposition: areas
with high degree hubs were also areas with the largest concentration of amyloid (Buckner et al., 2009). Amyloid deposition and hub
connectivity may even be correlated in non-demented subjects
(Drzezga et al., 2011). Simulations on the basis of MEG in Alzheimer’s disease also pointed to a specific vulnerability of hub-like
brain regions (Stam et al., 2009). Network theory predicts that
the distribution of traffic in communication networks depends
upon the degree distribution, with high-degree hubs receiving
most of the traffic. If the traffic load becomes excessive, hubs will
be the most likely network nodes to fail. If network traffic can be
somehow related to neuronal firing, the missing link between
amyloid deposition and high degree hubs might be excessive synaptic activation (Bero et al., 2011).
In this review, we have argued for the need and potential usefulness of studying the brain from the perspective of a complex organized network (Fig. 9). The underlying question is simple and
daunting at the same time: why is our brain so complex? If we assume that our brain is the result of a long evolutionary process we
can expect that its complex organization is not random but rather
reflects some kind of optimal solution to multiple, possibly conflicting, constraints. The question then becomes whether a network perspective can provide an explanation, even if only
tentatively, of the origins and functional significance of the organization of brain networks.
The study of the function of elementary units of the brain, from
the level of neurons and microcolumns all the way up to macroscopic brain regions, is a necessary first step for understanding
brain function (Section 3.1). Much of the progress in neuroscience
in the past few decades is related to a better understanding of the
conditions underlying local activation of functional brain units.
However, it is also obvious that the activity of functional units of
the brain makes little sense (literally as well as metaphorically) if
it cannot be communicated to other units of the brain. The function
of the brain is more than the sum of the firing rates of its neurons.
Thus, it is of the utmost importance to understand how pairs of
functional units interact and exchange information. This realization underlies the rapidly increasing interest in studies of ‘‘functional connectivity’’ and related concepts (Section 3.2). A major
focus in this kind of studies is the development of reliable measures of pair-wise interactions. However, when this approach is applied to the study of interactions in systems with large numbers of
elements, its limitations become apparent. Often, it is very difficult
to make sense out of the multitude of increases and decreases of
functional connectivity levels that are observed in different behavioral states and neurological disorders. The point is, that the functional architecture of the brain cannot be understood as the sum of
a large number of pair-wise interactions (Section 3.6).
The point of modern network theory is that a kind of systems
perspective is needed to understand how communication is organized in a brain, consisting of a large number of interacting elements (von Bertalanffy, 1969). A network perspective introduces
concepts such as clustering, modularity, path length and degree
distribution, that do not make sense, and in fact do not even exist,
at the level of pair-wise interacting brain regions or neurons (Section 4.1). Such high level concepts and measures, based upon these
concepts, can help to identify organizational themes in brain net-
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
j.clinph.2012.01.011
16
C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
works, and common patterns in the way these networks are disrupted in disease.
5.2. A heuristic model of complex brain networks
The concept of a graph provides a general framework for dealing with all sorts of networks and allows studying the organization
of the brain using novel concepts and measures (Fig. 10). The concepts and measures derive their meaning from the fact that they
are defined in the context of specific structural models of complex
networks. The introduction of random graphs bridged the gap between small, deterministic graphs, and large networks that require
a statistical description (Erdös and Rényi, 1959). The ER model explains some important aspects, but fails to explain the phenomenon of clustering. The small-world model solves this problem but
cannot explain the diversity of node properties, observed in real
networks. The scale-free model of Barabasi and Albert does explain
node diversity and the existence of hubs or highly connected
nodes, and has a very short path length, but unfortunately also a
vanishing clustering coefficient.
The problems we are faced with can be described as follows:
currently, we have two powerful models, the WS model (that contains the ER model as a special case) and the SF (scale-free) model,
both of which explain some of the properties of real brain networks
(notably: clustering, short paths and degree diversity). The first
problem is that of the relationship between the two models. Neither model can be reduced to the other; some properties can only
be explained by one model, and not by the other. In other words:
we need them both, but they do not go well together. The second
problem is that a number of network properties have been detected that are clearly present in structural and anatomical networks, but cannot be explained by either model. The two most
obvious examples are (hierarchical) modularity and degree correlations. These two problems present an important challenge for a
network theory of the brain.
To overcome these problems, and to guide future studies, we
present a heuristic model of complex brain networks (Fig. 10).
The heuristic model assumes an abstract ‘‘network space’’ defined
by three corners and three axes. The first, horizontal axis, is a representation of the WS model. The left corner is occupied by the regular lattice network, the right corner by the random ER network,
and intermediate networks between the lattice and random network correspond to small-world configurations. The horizontal
axis indicates the balance between order and randomness. The vertical axis with the SF network in the upper right corner indicates
the level of node diversity. While interpolation between ER and
SF networks is not commonly used, it is easy to see how this could
be done, for instance by randomly rewiring the edges of the SF net2
work with a probability p. The degree diversity kappa (hk i=hki)
could be a useful measure to locate a network between the extremes of ER and SF. The third diagonal axis connects the lattice
in the lower left corner to the SF network in the upper right corner.
According to Ravasz and Barabási (2003), networks that are intermediate between a lattice and a scale-free network display hierarchical modularity (hierarchical modular network, HMN). This can
be quantified by the exponent of the power law relation between
C(k) and k (with C the clustering coefficient and k the degree). In
networks with hierarchical modularity, the clustering coefficient
is inversely correlated with the degree.
The heuristic model of ‘‘network space’’ suggests that the HMN
presents a very special case. This type of network is not only intermediate between the lattice and the SF network, but, if we project
its position on the X axis, also between the lattice and the ER model, and if we project it on the Y axis, between the ER and the SF network. In fact, the HMN combines properties of a lattice, a random
and a scale-free network as a kind of compromise. It combines high
clustering with short path lengths and high degree diversity. In
addition, it displays hierarchical modularity and could be linked
to positive degree correlations (Newman, 2003, 2006).
The concept of HMN is very attractive, since it suggests how the
fundamental types of network models might be related. Furthermore, HMNs are the only network model that explain all major
properties (clustering, short paths, degree diversity, modularity
and assortativity) observed in real structural and functional networks. There are indications that modular structure can evolve
naturally in networks that use optimization algorithms such as
simulated annealing (Kirkpatrick et al., 1983) to deal with multiple
conflicting constraints (Pan and Sinha, 2007; Rubinov et al., 2009b;
Stam et al., 2010; Yuan and Zhou, 2011). Furthermore, hierarchical
modularity has been shown to explain critical dynamics as well as
oscillations of neural activity, both of which are assumed to play a
key role in neuronal information processing (Kaiser et al., 2010;
Rubinov et al., 2011; Wang et al., 2011). Since HMNs present a
compromise between many network properties and may evolve
naturally in networks dealing with multiple constraints, it is
seductive to conceive of HMNs as optimal configurations functioning as attractors in ‘‘network space’’. However, it should be stressed
that at present, in contrast to the ER, WS and SF models, we do not
have a simple, universal mathematical model of HMNs. The discovery of such a model could offer substantial support for the heuristic
model presented here.
5.3. Development, aging and neurological disease in network space
Can we use our heuristic model of complex brain networks to
make sense out of the many network studies described in this review? We will address this question with respect to development,
aging and various types of neurological disease. The central idea of
the heuristic model is that hierarchical modular networks present
an optimal state characterized by high clustering, shorts paths, degree diversity, hierarchical modularity and assortativity. In this
view, network changes are either towards or away from this optimal state. Furthermore, the three corners in ‘‘network space’’ can
characterize distinct patterns of network changes.
In general, network studies of brain development are in agreement with the notion of a movement toward an optimal HMN organization (Boersma et al., 2011; Fair et al., 2009; Micheloyannis et al.,
2009; Yap et al., 2011). Here we should note that changes from
either a lattice or a random network to a small-world network
could be translated to changes in the direction of a HMN network;
in fact, the heuristic model predicts that node diversity should increase at the same time. Studies differ with respect to the starting
point of the network trajectory. EEG studies suggest that network
development may start in the lower right corner, whereas MRIbased studies indicate that networks may be more lattice-like early
in development (Boersma et al., 2011; Fair et al., 2009; Yap et al.,
2011). Evolution toward a hierarchical modular structure, irrespective of the initial state, is supported by model studies (Siri et al.,
2007). Network changes in aging may be explained by a movement
away from the HMN pattern toward either a more random or a
more lattice like structure (Smit et al., 2010). Loss of modular structure has also been observed in aging (Meunier et al., 2009).
The heuristic model suggests that brain disease at the network
level always implies a movement away from the optimal HMN
state. Depending upon the trajectory through ‘‘network space’’, different patterns may be discerned. First, network disease could result in a straightforward movement to the lower right corner.
This network randomization is a pattern that has been observed
in many different types of brain disease, ranging from Alzheimer’s
disease, brain tumors, depression to schizophrenia (De Haan et al.,
2009; Leistedt et al., 2009; Micheloyannis et al., 2006; Rubinov
et al., 2009a; Stam et al., 2009). Network randomization seems to
Please cite this article in press as: Stam CJ, van Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/
j.clinph.2012.01.011
C.J. Stam, E.C.W. van Straaten / Clinical Neurophysiology xxx (2012) xxx–xxx
be a characteristic of relatively severe and advanced brain disease.
Second, in some other conditions brain networks seem to move
away from the HMN toward the lower left corner. This type of
change would be characterized by a shift from global to local connectivity, and a pathological increase in ‘‘network regularity’’. This
pattern of brain network changes has been observed in developmental disorders (Barttfeld et al., 2011; Hänggi et al., 2011; Wang
et al., 2009) as well as in the early stages of various neuropsychiatric conditions (Buldú et al., 2011; Pijnenburg et al., 2004; Schoonheim et al., 2011b). Third, in some situations brain disease may be
characterized by the abnormal activation of hubs, or the emergence of new pathological hub-like brain areas. This would correspond to a movement to the upper right corner. Excessive
activation of hub nodes has been associated with epileptic brain regions (Chavez et al., 2010; Douw et al., 2010b; Horstmann et al.,
2010; Wilke et al., 2011; Zhang et al., 2011). In addition, excessive
hub activation could be a temporary state in other neurological
disorders, followed by a more general network randomization.
While different trajectories in network space might characterize
different stages or patterns of disease, if the disease is long lasting
and severe, the natural endpoint seems to be the random network
in the lower right corner. This suggests that the random network
could be thought of as an ‘‘attractor’’ for pathological networks,
just like the HMN could be an attractor for healthy brain networks.
5.4. Future challenges
The additive value of network analysis of the brain is that it allows us to go beyond local activation and pair-wise interaction
studies, and define higher order concepts relevant for normal and
abnormal brain function. One challenge is to develop better measures of brain network characteristics, and use these measures to
integrate information from multiple imaging modalities such as
EEG, MEG, structural and functional MRI. However, even more urgent is the need for new and better models that can fill the gap between the existing small-world and scale-free models. A simple
mathematical model of hierarchical networks could be an important step forward. A deeper understanding of complex brain networks also requires the identification of the most relevant
constraints that confront developing brain networks, as well as
the algorithms that are used to deal with them. In particular, the
role of space and geometric constraints may be important factors
(Henderson and Robinson, 2011; Kaiser and Hilgetag, 2004). It
may be helpful to think of the complex architecture of brain networks in terms of solutions; the main task then becomes to identify
the problems these networks are trying to solve. Finally, the ultimate challenge for network studies in a clinical context, and in particular in clinical neurophysiology is whether this approach will
have an impact not only on our understanding of pathophysiological mechanisms, but also on clinical diagnosis and treatment.
Acknowledgements
We thank Alexandra Linger for her secretary support.
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