the organization of physiological brain networks · invited review the organization of...

Clinical Neurophysiology xxx (2012) xxx–xxx

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Clinical Neurophysiology

journal homepage: www.elsevier .com/locate /c l inph

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 h i g h l i g h t s

Article history:Accepted 15 January 2012Available online xxxx

Keywords:Brain networksGraph theorySmall-world networksScale-free networksFunctional connectivitySynchronizationNeurological diseaseEEGMEGMRI

1388-2457/$36.00 � 2012 International Federation odoi:10.1016/j.clinph.2012.01.011

⇑ Corresponding author. Address: Department of CUniversity Medical Center, P.O. Box 7057, 1007 MB ATel.: +31 20 444 0730; fax: +31 20 444 4816.

E-mail address: [email protected] (E.C.W. van

Please cite this article in press as: Stam CJ, vanj.clinph.2012.01.011

� The brain can be represented as a complex network with functionally connected units at several levelsthat changes in neurological and psychiatric disease.� Existing clinical neurophysiology techniques and network models to explain network properties arereviewed.� In addition to the already established network models, we suggest a heuristic model including hierar-chical 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 nor-mal conditions and how this architecture breaks down in neurological disease. It has become clear thatsimple activation studies are no longer sufficient. There is an urgent need to understand the brain as acomplex structural and functional network. Interest in brain network studies has increased strongly withthe advent of modern network theory and increasingly powerful investigative techniques such as ‘‘high-density EEG’’, MEG, functional and structural MRI. Modern network studies of the brain have demon-strated that healthy brains self-organize towards so-called ‘‘small-world networks’’ characterized by acombination of dense local connectivity and critical long-distance connections. In addition, normal brainnetworks display hierarchical modularity, and a connectivity backbone that consists of interconnectedhub nodes. This complex architecture is believed to arise under genetic control and to underlie cognitionand intelligence. Optimal brain network organization becomes disrupted in neurological disease in char-acteristic ways. This review gives an overview of modern network theory and its applications to healthybrain function and neurological disease, in particular using techniques from clinical neurophysiology,such as EEG and MEG.� 2012 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights

reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 002. The brain as a network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

2.1. Representing the brain as a network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 002.2. Anatomical and neurophysiological evidence for a network perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 002.3. Networks are not enough: the need for a theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 002.4. Networks in relation to complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 002.5. Relevance of brain networks for clinical neurophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

3. Functional interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
3.1. Time series and brain function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 003.2. Functional and effective connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
f Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.

linical Neurophysiology, VUmsterdam, The Netherlands.

Straaten).

Straaten ECW. The organization of physiological brain networks. Clin Neurophysiol (2012), doi:10.1016/

http://dx.doi.org/10.1016/j.clinph.2012.01.011

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Pleasej.clinp

3.3. Fragile binding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 003.4. Excessive connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 003.5. Disconnection syndromes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 003.6. Strengths and limitations of brain connectivity studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

4. Network studies in brain research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
4.1. Modern network theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
4.1.1. Graph theory, statistical physics and dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 004.1.2. Measures of complex networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 004.1.3. Representing structural and functional networks as graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

4.2. Healthy subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 004.3. Neuropsychiatric disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 004.4. Epilepsy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 004.5. Dementia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

5. Toward a theory of brain networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
5.1. From local activation to network organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 005.2. A heuristic model of complex brain networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 005.3. Development, aging and neurological disease in network space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 005.4. Future challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
1. Introduction

The brain can be seen as a complex anatomical and functionalnetwork. Theories that apply to complex networks in general, suchas modern network theory and complex systems theory, areincreasingly used in neuroscience to understand how normal brainfunction arises and what happens in disease. They provide a totallynew perspective leading to some very new insights. This reviewgives an overview of the fundamentals and applications of modernnetwork theory in the study of normal and developing brain func-tion. In addition, we aim to explain how it is possible that variousbrain diseases can be understood in terms of failing network char-acteristics. The focus will be mainly on functional aspects of brainnetworks, but structural connectivity, animal studies, and model-ing will also be discussed to provide a wider perspective. Since net-work studies are based upon a fundament of functionalconnectivity, the neurophysiological basis of normal and disruptedsynchronization is discussed first. Subsequently, the basic conceptsand tools of modern network theory are introduced. The applica-tion 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 partic-ular neurophysiological approaches. Finally we suggest a generalframework for the study of the organization of physiological brainnetworks.

2. The brain as a network

2.1. Representing the brain as a network

In 1906, Ramon Y Cajal and Camillo Golgi shared the NobelPrize 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, thebrain should be viewed as a large syncytium, or conglomerate, ofdirectly connected neurons. Instead, Cajal, using the silver nitratepreparation developed by Golgi, interpreted his findings in supportof the neuron theory that maintained that the brain consists of dis-crete neurons. In a sense, both turned out to be right: the brain isnow considered to be a large network of about 1010 neurons, witheach neuron communicating with about 104 other neurons, mostlyby synapses, but to a small extent also by gap junctions that di-rectly connect the cytoplasm of two adjacent cells and may play

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an important role in physiological and pathological synchroniza-tion processes. Alternative views of the brain, stressing its diffuseand holonomic nature, have been proposed by Lashley (1923), Pri-bram and Carlton (1986), and more recently by Nunez (1995). De-spite these suggestions, the concept of the brain as a large complexnetwork of interconnected elements, at multiple scales, has ob-tained a dominant place in modern neuroscience (Nunez, 2010).

2.2. Anatomical and neurophysiological evidence for a networkperspective

The concept of the brain as a complex network is increasinglysupported by a wealth of neuroanatomical and neurophysiologicaldata. Many different types of neurons have been recognized, andthey connect in characteristic ways, from the level of micro orminicolumns (the smallest functional unit of the brain containingabout 80–100 neurons) and macrocolumns (consisting of 60–80minicolumns; for a recent review see: Buxhoeveden and Casanova,2002) up to the level of local brain regions, lobes and functionalbrain systems. Izhikevich and Edelman (2008) included 22 differ-ent types of neurons in their model of the whole brain. Advancedtechniques are now being used to dissect the precise anatomy atthe smallest levels of the brain. Recently, using the micro-opticalsectioning tomography (MOST) system, a major breakthroughwas achieved in the detailed description of the connectivity of amouse brain (Li et al., 2010). Major efforts to describe the micro-scopic anatomy of the brain are the Blue Brain Project and the Hu-man Brain Project (Markram, 2006; Perin et al., 2011;www.humanbrainproject.eu). At higher levels of resolution, classicneuroanatomic techniques of tracing, successful in showing thenetwork blueprint of the brain of macaques, and cats, are now re-placed by modern imaging techniques, in particular high-fieldstrength MRI in combination with DTI (diffusion tensor imaging)and tractography. These methods have revealed the large-scalepatterns of brain connectivity from the level of voxels to the levelof Brodmann areas (Hagmann et al., 2007, 2008; Gong et al.,2009a).

Since the early nineties of the last century, invasive neurophysi-ological studies in animals have been able to demonstrate commu-nication, mostly by synchronization in the gamma band, betweenwidely separated but connected brain regions (Eckhorn et al.,1988; Engel et al., 1991; Gray et al., 1989; König et al., 1995). Manyof these findings have now been confirmed with EEG and MEG,

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demonstrating the existence of functional brain networks, even ina ‘‘resting-state’’ (Fell and Axmacher, 2011; Uhlhaas and Singer,2006). In a classic review of neurophysiological synchronizationstudies, 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 provideevidence for the existence of multiple resting-state networks, ofwhich the default mode network has become the most well known(Damoiseaux et al., 2006). Other resting-state networks or modulescorrespond 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 inparticular hierarchical network structure of the brain at multiplelevels (Bassett et al., 2008). From a network perspective, anatomi-cal and physiological studies present an increasingly interrelatedview upon the organization of the brain. Donald Hebb’s notion of‘‘synchronous cell assemblies’’ has developed from an abstract con-cept 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 functionalnetwork thus has a relatively long history, and a high face value gi-ven what we know about brain anatomy and neurophysiology.However, the notion of brain networks is too general to be opti-mally useful in understanding normal and disturbed brain physiol-ogy unless it is linked to a specific theory of networks. As indicatedby Carter Butts: ‘‘To represent an empirical phenomenon as a net-work is a theoretical act.’’ (Butts, 2009). A complex network ismore than the sum of the elements that constitute its buildingblocks; it is also more than the sum of all its pair-wise interactions.Complex systems have emergent properties that require exactmathematical theories for their proper understanding (Sporns,2011a). In this respect, the development of modern network theoryin the late nineties of the last century constitutes an important stepforward. The introduction of powerful new mathematical modelsof complex networks such as the small-world networks andscale-free networks gave rise to a widespread and rapidly growinginterest in network studies in many fields of science, ranging fromphysics, communication and transport systems to biological andsociological networks (Boccaletti et al., 2006). The developmentof modern network theory has greatly stimulated the interest inbrain networks, and has enabled a more exact, quantitative studyof the determinants of growth, learning, plasticity and failure inneural networks.

2.4. Networks in relation to complexity theory

While a network perspective is rapidly becoming a promisingapproach to understanding the complex nature of the brain, it is re-lated to other scientific fields that deal with complex systems.Cybernetics, information theory and general system theory consti-tute early examples of attempts to define general mathematicalframeworks for complex systems including neural networks (Shan-non and Weaver, 1949; Simon, 1962; von Bertalanffy, 1969; Wie-ner, 1948). An excellent introduction to modern complexityscience can be found in Melanie Mitchell’s book Complexity(Mitchell, 2009). The modern theory of dynamical systems, in par-ticular chaos theory, has had a considerable influence on ourunderstanding of complex behavior in deterministic but highlynonlinear systems. These ideas have influenced our concepts ofbrain function, in particular with respect to the predictability ofepileptic seizures (Stam, 2005). Perhaps the most fruitful contribu-tion of nonlinear dynamics has been the development of a generaltheory of synchronization processes, with considerable relevance

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for the study of communication between neurons and brain areas(Boccaletti et al., 2002).

Other important contributions have come from statistical phys-ics and the theory of phase transitions. According to the theory ofself-organized criticality, introduced by Per Bak, complex systemsevolve autonomously to a critical state characterized by powerlaws (Bak et al., 1987). There is now increasing evidence that suchphenomena may also be observed in the brain (Beggs and Plenz,2003; Werner, 2007). The plenitude of complexity theories mightsuggest the lack of a unifying framework. However, modern net-work theory is a promising candidate for integrating many of theemerging concepts in complexity studies. Herbert Simon stressedthat complex systems could be viewed in general as large collec-tions of interacting elements displaying a hierarchical organization(Simon, 1962). Modern network theory is based upon graph theoryas an exact mathematical formalism for the description of net-works, including their hierarchical organization. In addition, it usesideas from probability theory and statistical mechanics to dealwith stochastic aspects of large networks, and finally it involvesthe theory of dynamical systems and synchronization for the studyof processes taking place upon complex networks (Barrat et al.,2008). Therefore, modern network theory presents an attractiveand versatile framework for the study of the organization of com-plex brain networks.

2.5. Relevance of brain networks for clinical neurophysiology

Normal brain functioning and its breakdown in neurologicaland neuropsychiatric disease cannot be properly understood with-out taking the network perspective into account. However, a gen-eral, informal notion of brain networks is no longer sufficient.Modern network theory is beginning to have an impact on ourideas about the way brain networks develop, how this develop-ment is organized and constrained by genetic and geometric fac-tors, how the architecture of brain networks comes to displayuniversal features such as hierarchical modularity and how this re-lates to cognitive function and intelligence. In addition, networktheory influences notions of localization of brain function and glo-bal effects of local lesions. The idea that such different conditionsas Alzheimer’s disease and epilepsy may both be explained interms of ‘‘hub failure’’ clearly illustrates how network science isbeginning to have an impact upon clinical concepts.

Clinical neurophysiology is the medical specialism that dealswith diagnosis, monitoring, and prognosis of clinical neurologicalconditions using a variety of neurophysiological techniques. Mod-ern network theory may be particularly relevant for clinical neuro-physiology since two of its main techniques to study centralnervous system function, the EEG and MEG, present an enormousand hardly explored potential to investigate how brain areas com-municate, and how this communication fails in neurological dis-ease. The fact that both EEG and MEG measure neural activitydirectly, and not indirectly such as is the case with fMRI BOLD, isa major advantage. Combining functional connectivity with mod-ern network theory presents a unique opportunity for clinical neu-rophysiology. EEG and MEG are not only the most direct tools tomeasure brain function; they also can provide a window on thebrain 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 func-tional network, several elements are required: (i) we need to havea reliable measurement of the level of activity of the network ele-





ments; (ii) communication between network elements needs to becharacterized and quantified; (iii) the full pattern of pair wiseinteractions between network elements needs to be integratedand analyzed within the framework of network theory; (iv) theinterdependencies between functional and structural network lev-els need to be assessed. Here, we focus upon the first element. Theother three aspects will be addressed in subsequent sections.

Neurons are the obvious structural and functional buildingblocks of the nervous system. Neuronal function is reflected intwo different types of activity: (i) small fluctuations of the restingmembrane potential of the dendrites induced by excitatory andinhibitory synapses; (ii) action potentials along the axon. A typicalcortical pyramidal neuron may be covered by up to 1000–10,000synapses, with the excitatory synapses predominantly located atthe dendritic tree far away from the soma and the inhibitory syn-apses located closer to the nerve cell body. Activation of an excit-atory synapse induces a brief depolarization of the membranepotential (excitatory post synaptic potential or EPSP) and activa-tion of an inhibitory synapse causes a somewhat larger and longerlasting hyper polarization of the membrane potential (inhibitorypost synaptic potential or IPSP). If the joint effect of all EPSPs andIPSPs on the dendritic tree, determined by spatial and temporalsummation, causes the membrane potential near the axon hillockto exceed the threshold, an action potential is generated that willbe propagated along the axon, but also backwards to the dendritictree. 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 contro-versy in neuroscience concerns the question whether the relevantinformation is carried only by the firing rate, or also by the exacttiming of action potentials. The fact that the exact timing of firingof 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 ac-count (Abbott and Nelson, 2000).

While neuronal spiking is fundamental for inter neuronal com-munication, the most commonly used techniques to assess humanbrain function such as EEG, MEG, and fMRI do not measure actionpotentials directly. Both EEG and MEG reflect large-scale sum-mated field potentials that are ultimately caused by EPSPs andIPSPs of cortical, mainly pyramidal neurons. Characteristically,the field potentials recorded with EEG and MEG show oscillationsin a wide range of frequency bands, ranging from (sub) delta togamma, high-frequency oscillations (HFO) and ripples (Buzsaki

Fig. 1. Schematic illustration of time series analysis. Measures of activation can be detercorrelation reflect statistical interdependencies between two time series. These correconnectivity). Both types of connectivity can be studied with linear techniques (corgeneralized synchronization). Shown are two EEG signals recorded at O2 and O1 positio


and Draguhn, 2004). The temporal resolution of EEG and MEG isessentially unlimited, which is of interest since both very low aswell as very high frequency oscillations have been shown to haveneurophysiological significance (Bragin et al., 2010; Van Somerenet al., 2011). The functional significance of oscillatory activity indifferent frequency bands has long remained a mystery, and ithas even been suggested that these oscillations might be mereepiphenomena (Niedermeyer and Schomer, 2011). There is nowincreasing evidence, however, that a phase relation exists betweenoscillatory field potentials and neuronal actions potentials(Eckhorn and Obermueller, 1993; Lee et al., 2005). This observationstresses that oscillatory brain activity contains information that isrelevant for understanding local and interregional communication.

The BOLD signal reflects the presence of excess oxyhemoglobinin the smallest blood vessels near activated brain regions; it can bethought of as a low pass, integrated measure of neuronal spikingwith a typical time scale of seconds. Studies combining EEG withfMRI are now beginning to reveal the complex relations betweenoscillatory band power and BOLD signals (Rosenkranz and Lemi-eux, 2010). In particular, a relation between the envelope of gam-ma band oscillations and BOLD time series has been suggested(Logothetis, 2002).

Time series techniques can be used to analyze the recordings oflocal brain activity by EEG, MEG and BOLD. The most commontechnique is spectral analysis, that represents signal power as afunction of frequency. This approach is especially helpful to deter-mine oscillatory components of the signal, such as a dominantpeak due to the alpha rhythm, and power changes in certain fre-quency bands. Activation of specific brain regions is often associ-ated with a decrease or increase of power in a particularfrequency band; these changes can be captured with a techniquecalled event-related desynchronization (ERD) or event-related syn-chronization (ERS) (Pfurtscheller and Lopes da Silva, 1999). Short-lasting, non-stationary changes in oscillatory brain activity can alsobe characterized with evoked or event-related potentials. In con-trast to ERD and ERS, that measure induced power changes, evokedand event-related potentials only measure oscillatory changes thatare exactly phase-locked to some event. Another effective tech-nique to assess non-stationary aspects of oscillatory power iswavelet analysis. Oscillatory brain activity may show correlationson very long time scales. These correlations can be characterizedby spectral analysis, the Hurst exponent, or detrended fluctuationanalysis (Benayoun et al., 2010; Linkenkaer-Hansen et al., 2007;Stam and de Bruin, 2004). Finally, the non-random structure of

mined from individual time series using for instance spectral analysis. Measures oflations can be bidirectional (functional connectivity) or unidirectional (effectiverelation coefficient, coherence) or nonlinear techniques (phase synchronization,ns; vertical bars are second markers.





neurophysiological time-series analysis can also be characterizedwith techniques derived from nonlinear dynamics. Here, the timeseries is assumed to reflect a trajectory in a high-dimensional statespace of the underlying dynamical system, and this trajectory canbe 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 neu-rons or local brain regions can be measured and characterized withneurophysiological techniques and time series analysis. The nextstep is to consider how this information can be used to infer howcommunication between neurons and brain regions takes place(Fig. 1). An important concept for understanding communicationin brain networks is synchronization. Unfortunately, synchroniza-tion is often used in neurophysiology in a rather informal way,referring to a vague notion of the level of cooperation betweenneurons. It is often assumed that band power of field potentialsoscillations reflect this cooperation between neurons; however,band power of an EEG or MEG signal recorded at a particular elec-trode or sensor may depend upon many factors, and its interpreta-tion in terms of (local) levels of neuronal synchronization is notstraightforward (Daffertshofer and van Wijk, 2011). In physics,and in particular in dynamical systems theory, the notion of syn-chronization has been given a far more specific and quantitativeinterpretation, that goes back all the way to the groundbreakingwork of the Dutch physicist Christiaan Huygens. Much of the workon synchronization has been based upon the notion of phase cou-pling between pairs of harmonic oscillators (Boccaletti et al., 2002;Rosenblum and Pikovsky, 2003). With the advent of nonlineardynamics and chaos theory, the concept of synchronization hasbeen broadened, and now includes any type of statistical interde-pendency between dynamical systems, even if they do not displayregular oscillations and even if strict phase coupling is absent: socalled generalized synchronization (Rulkov et al., 1995). Theseadvances in the physics of synchronization influenced the possibil-ities for detecting and quantifying synchronization in neurophysi-ological data.

While synchronization in its various manifestations should beconsidered the fundamental physical concept for understandingcorrelations between neurophysiological time series, different no-tions 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 betweenneurophysiological time series. Originally, this concept referred torecordings of neuronal spiking, but it is now also used for EEG,MEG and in particular fMRI (Lowe, 2010; Stevens, 2009). Whilefunctional connectivity is a model free concept, the notion of effec-tive connectivity stresses asymmetric causal interactions betweenneural systems (Friston, 2002). The notion of effective connectivityis especially clear in the approach called dynamic causal modeling(DCM) introduced by Karl Friston (Penny et al., 2004). However, weshould stress that both effective and functional connectivity couldbe defined in terms of synchronization theory, that is: in the lan-guage of interacting dynamical systems. As we will see, this is ofparticular importance if we want to obtain a formal understandingof communication processes taking place on complex networks(Arenas et al., 2008).

Many time series analysis techniques have been developed tocharacterize statistical interdependencies between two or moretime series of neurophysiological activity. An excellent overviewis the review by Pereda et al. (2005). A useful distinction is that be-tween linear and nonlinear techniques. The most important linearmeasure of correlation between time series is coherence. It can be


considered a generalization of the (time-lagged) correlation be-tween time series. Coherence describes the strength of the correla-tion between two time series as a function of frequency; it dependsupon the consistency of the phase difference as well as the powerof the two time series (Nunez et al., 1997). To track correlations innon-stationary signals, coherence can also be based upon waveletanalysis (Lachaux et al., 2002). At least three categories of nonlin-ear synchronization measures can be distinguished. The first groupconsists of measures that quantify the consistency of the phase dif-ference between two signals, discarding the influence of signalamplitude (Rosenblum et al., 1996). Phase synchronization analy-sis can be based upon a Hilbert transform of the data (Mormannet al., 2000) as well as wavelet analysis (Lachaux et al., 1999); bothapproaches are equivalent (Burns, 2004). A second group of mea-sures is based upon nonlinear dynamics and quantifies generalizedsynchronization between attractors reconstructed from the timeseries (Arnhold et al., 1999; Hu and Nenov, 2004; Pecora and Car-roll, 2000; Schiff et al., 1996; Schmitz, 2000). The synchronizationlikelihood is a relatively simple unbiased estimator of generalizedsynchronization (Stam and van Dijk, 2002; Montez et al., 2006). Fi-nally, there is a group of nonlinear coupling measures such asevent-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 aboveare generally symmetric, and do not provide information on cau-sality or the direction of the influence. One approach to determineeffective connectivity is based upon Granger causality, originallydeveloped in economics (Granger, 1969). In the case of Grangercausality, the future of time series x can be predicted better, ifnot only its own past, but also the past of another time series yis taken into account. If including the information of y improvesthe prediction of x, y is said to have a causal effect on x. An impor-tant technique to apply Granger causality in a frequency depen-dent way is the directed transfer function (DTF) (Blinowska,2011). The direction of coupling can also be determined with tech-niques based upon phase synchronization. An important exampleis the phase slope index (Nolte et al., 2008). Measures of causalinteractions have also been derived in a nonlinear dynamical sys-tems context (Arnhold et al., 1999; Hu and Nenov, 2004; Pecoraand Carroll, 2000; Schiff et al., 1996; Schmitz, 2000). Finally, thenonlinear correlation coefficient h2 has also been used to derive anonlinear directed coupling measure (Wendling et al., 2005).

In contrast to fMRI, neurophysiological techniques for deter-mining the coupling between different brain regions suffer fromthe common source bias. If the activity recorded at two differentelectrodes or sensors is influenced by a single underlying source,the estimated correlations do not reflect true interactions betweendifferent neural systems. In the case of EEG, the activity recorded atthe reference electrode can also influence the estimated correla-tions between different EEG channels (Nunez et al., 1997). Theseare serious methodological problems that cannot simply be solvedby performing the connectivity analysis in source space instead ofsignal space. While source or Laplacian derivations may diminishthese influences, they do not solve the problem completely, andthey may obscure true long-distance interactions. Nunez et al.(1997) proposed a corrected version of the coherence to accountfor the influence of volume conduction in EEG. Nolte et al. (2004)suggested to use the imaginary component of coherence as a mea-sure of coupling, that is not influenced by volume conduction. Adisadvantage of the imaginary coherence is the fact that its magni-tude is influenced by signal power as well as the magnitude ofphase lag between the channels; therefore, increases or decreasescannot be interpreted easily. The only conclusion that is certain,is that the existence of a non-zero imaginary coherence cannotbe explained by volume conduction. The phase lag index is basedupon the idea of the imaginary coherence but it is not influenced





by the signal amplitude nor by the magnitude of the phase lag(Stam et al., 2007b). The recently proposed weighted phase lag in-dex WPLI is claimed to be less sensitive to noise, but re-introducesthe dependency upon the magnitude of the phase difference (Vincket al., 2011). Interestingly, coherence, imaginary coherence, phasecoherence, WPLI and PLI are all intimately related to each othersince they can all be computed from the Hilbert transform of thedata (Stam et al., 2007b).

The availability of so many different measures of synchroniza-tion raises the question whether a rational choice can be made be-tween them in terms of performance on real data. A few studieshave attempted to compare a subset of measures with respect totheir ability to correctly detect the presence of synchronizationin a multivariate data set (Ansari-Asl et al., 2006; David et al.,2004; Quian Quiroga et al., 2002a). Unfortunately the results arenot unequivocal. Performance of the measures seems to dependupon characteristics of the data set as well as characteristics ofthe method itself. In the case of complex synchronization measuresbased upon nonlinear dynamical systems theory, many parametershave to be taken into account (Montez et al., 2006). Performing theanalysis in source space does not simplify matters, since eachsource reconstruction algorithm has its own assumptions thatmay influence correlations between source-based signals in un-known ways. At present, the most rational approach may be touse more than one technique, and to search for results that areas consistent as possible, irrespective of the technique that wasused.

3.3. Fragile binding

Interregional synchronization, or functional/effective connec-tivity, conveys important information about healthy brain func-tioning. Here we briefly indicate what connectivity studies haveshown even without a formal network perspective; below we willcontinue this discussion in the context of modern network theory.In healthy subjects the strength of interregional synchronizationdepends upon age. Long distance synchronization is relativelylow at birth, and increases during development, possibly due tomaturation and myelinization of long distance association path-ways (Barry et al., 2004; Gmehlin et al., 2011; González et al.,2011; Thatcher et al., 2008). The strength of synchronization de-pends not only upon maturation and age, but is also strongly influ-enced by genetic factors (Chorlian et al., 2007; Posthuma et al.,2005; Van Beijsterveldt et al., 1998). How genetic factors can deter-mine the strength of functional interactions between brain regionsis not yet understood.

The level of interregional functional connectivity is different fordifferent frequency bands and seems to be related to local bandpower as well as distance. According to some authors, long dis-tance communication is mainly supported by synchronization inlow frequency bands, while short distance local communicationdepends upon synchronization in beta and gamma frequencybands (Von Stein and Sarnthein, 2000). The strength of synchroni-zation between different brain regions shows characteristic fluctu-ations that may not simply reflect noise effects, but could also bedue to the rapid formation and dissolution of functional connec-tions. This dynamic nature of synchronization, even during a rest-ing-state, has been referred to as ‘‘fragile binding’’ (Stam and deBruin, 2004). Two different models of synchronization dynamicshave been suggested. According to the idea of microstates the brainmanifests a succession of relatively stable states of a few hundredmilliseconds duration connected by abrupt transitions (Lehmannet al., 2006). However, empirical studies show that the distributionof synchronization and desynchronization times is rather complex,pointing in the direction of a process with (self-organized) critical-ity (Beggs and Plenz, 2003; Breakspear et al., 2004; Stam and de


Bruin, 2004; Gong et al., 2007). Interregional synchronization isalso clearly influenced by behavioral state and cognition. Duringsleep, characteristic changes in synchronization occur in differentsleep stages (Ferri et al., 2005). Tasks that involve working memoryare associated with increased synchronization, especially in thetheta band, and possibly also in the alpha band (Jensen Axmacheret al., 2008; Fell and Axmacher, 2011; Jensen et al., 2002; Klimesch,1997; Sarnthein et al., 1998; Tesche and Karhu, 2000). Gammaband synchronization is now believed to be an important neuro-physiological mechanism underlying binding, attention and evenconsciousness (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 char-acterized by the rapid formation and disappearance of functionalconnections between different brain regions. This ‘‘fragile binding’’can become disturbed in two different ways: excessive connectiv-ity or disconnection. The most obvious example of a syndromecharacterized by excessive synchronization would seem to be epi-lepsy (Lehnertz et al., 2009). At the neuronal level, epileptic phe-nomena are associated with paroxysmal depolarization shifts(Gorji and Speckmann, 2009). Recently, the significance of gapjunctions as an important cause of hypersynchronization in epi-lepsy has also been pointed out (Volman et al., 2011). The classicinterictal and ictal EEG patterns of epilepsy, such as spikes andgeneralized spike-wave discharges, are usually assumed to reflectexcessive synchronization at the neuronal as well as the macro-scopic level (Gorji and Speckmann, 2009). However, recent obser-vations of the activity of individual neurons during seizures inhumans suggest that a hypersynchronization-only concept of epi-leptic activity may be too simple (Truccolo et al., 2011).

Although some types of seizures, notably absence seizures char-acterized by generalized 3-Hz spike-wave discharges, may indeedreflect excessive synchronization compared to pre- and postictalstates, this pattern may not hold for all seizures, and in particularnot for partial seizures. Partial seizures may represent a complexsequence of increases as well as decreases of synchronization indifferent frequency bands during different stages of the seizure.In fact, the highest level of synchronization during partial seizuresmay be reached only at the very end of the seizure (Schindler et al.,2007). Pre-ictal changes in EEG synchronization have also been thetopic of intense research, especially since papers by German andFrench groups suggested the possibility of seizure prediction(Lehnertz and Elger, 1998; Le van Quyen et al., 2001a; Martinerieet al., 1998). Of all EEG measures that have been tested for theirperformance in predicting seizures, measures of interregional syn-chronization turn out to be the most successful (Mormann et al.,2005). Surprisingly, the pre-ictal state may be characterized by adecrease rather than an increase in synchronization (Mormannet al., 2003). It has been suggested that this pre-ictal drop in syn-chronization might reflect a release of the ictal focus from theinhibitory influence of other brain regions (Le van Quyen et al.,2001b). In this respect, the interictal state should be distinguishedfrom the pre-ictal state. There are indications that synchronizationlevels during the inter ictal state are abnormally high, not onlycompared to the pre-ictal state, but also to synchronization levelsin healthy subjects. Increased synchronization in the theta bandhas been described in patients with different types of epilepsy,and may predict the risk of epilepsy even independently of theoccurrence 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 basalganglia disorder, increased interregional synchronization in the al-pha band has been found in early, untreated Parkinson’s disease





(Stoffers et al., 2008a). In later phases of the disorder, this hypersynchronization extends to lower as well as higher frequencybands (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 childrenwith obesitas, excessive synchronization during a no-task resting-state has been shown in very low and very high frequency bands(Olde Dubbelink et al., 2008). Of interest, MRI has shown excessivewhite matter volume in subjects with obesitas. In patients withdiabetes, excessive synchronization has also been demonstrated(van Duinkerken et al., 2009) but a decrease of connectivity hasalso been observed (Cooray et al., 2011). In subjects with mild cog-nitive impairment (MCI), a precursor of dementia in a substantialproportion of cases, increased levels of synchronization have alsobeen reported, either at rest, or during the performance of cogni-tive tasks (Buldú et al., 2011; Jiang, 2005; Pijnenburg et al., 2004).

3.5. Disconnection syndromes

While abnormally increased levels of synchronization in neuro-logical disease may seem a bit counterintuitive, except in the con-text of epilepsy, the notion that destruction or loss of anatomicalconnections between brain regions would be reflected in dimin-ished functional connectivity would seem more natural. A niceillustration of this idea is the loss of inter hemispheric EEG coher-ence in children with agenesis of the corpus callosum (Koeda et al.,1995). The most obvious example of this idea is Alzheimer’s dis-ease, a degenerative type of dementia characterized by a progres-sive and severe loss of neurons and connections between brainregions. An important concept is the idea that at least some typesof cognitive dysfunction should be explained directly in terms oflost connectivity. Delbeuck et al. (2003) suggested referring to Alz-heimer’s disease as a disconnection syndrome. Evidence for a lossof interregional synchronization comes from a large number ofEEG, MEG, and fMRI studies (Babiloni et al., 2004; Dickerson andSperling, 2009; Rombouts et al., 2005; Sorg et al., 2009). Dimin-ished connectivity has been reported mostly during no-task rest-ing-states, but may also be present during the performance ofcognitive tasks (Sperling et al., 2010). Loss of functional connectiv-ity in Alzheimer’s dementia correlates both with structuralchanges as well as cognitive impairments, and might improvethe prediction of progression from MCI to AD (Rossini et al.,2006; Smits et al., 2011; Stam et al., 2006). A particularly nice illus-tration of functional, structural, and cognitive correlations isshown in the study of Pogarell et al. (2005), where decreased inter

Fig. 2. Graph and adjacency matrix. Networks can be represented mathematically as gradots correspond to nodes and lines between dots correspond to connections. This is anrepresented as an adjacency matrix. This is a square matrix with the same number ofinformation on the presence (value 1; indicated by color red) or absence (value 0; indicaan unweighted, binary graph it is symmetric.


hemispheric EEG coherence correlates with atrophy of the corre-sponding part of the corpus callosum on MRI. Of interest, dementiain Parkinson’s disease may also result in a loss of connectivity, inparticular between frontal and temporal areas (Bosboom et al.,2009). In non-demented Parkinson patients, EEG changes mayaccurately predict the later development of dementia (Klassenet al., 2011).

Although the concept of a disconnection syndrome in dementiadisorders seems appealing, there are some important caveats. First,even in diffuse dementia disorders increases of synchronization incertain frequency bands can be observed in the same patients whoalso display decreases of connectivity in other bands. In particular,loss of connectivity in the alpha band may be accompanied by in-creased connectivity in theta and beta/gamma bands (Stam et al.,2006; Bosboom et al., 2009). Furthermore, as discussed in the pre-vious section, MCI and other diffuse neurological disorders withmild cognitive dysfunction are sometimes characterized by in-creased rather than decreased synchronization. Such paradoxicalfindings are often explained in terms of putative compensationmechanisms, but really present a serious challenge to any simpleinterpretation of connectivity changes in neurological disease.

Another challenge comes from findings in local brain disease.Bartolomei et al. (2006a) showed that brain tumors may causewidespread changes in resting-state functional connectivity, alsoin the unaffected hemisphere. To complicate matters even further,connectivity in different frequency bands may be increased as wellas decreased following local lesions (Bartolomei et al., 2006b).These findings have been replicated in several other studies (Gug-gisberg et al., 2008; Martino et al., 2011; Zaveri et al., 2009). Con-nectivity changes are also related to cognitive changes in thesepatients (Bosma et al., 2008). The most consistent finding seemsto be an increase in theta band connectivity; this increase is partlyreversed after surgery and appears to be related to the severity ofthe accompanying epilepsy (Douw et al., 2008, 2010b). During theWada procedure, where the activity of one hemisphere is tempo-rarily depressed by amobarbital, similar patterns have also beenobserved: increased low frequency connectivity ipsilaterally anddecreased in the opposite hemisphere, and widespread increaseof 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 regionscan add important information to observations derived from localchanges in brain activity only. As an obvious example, levodopa

phs that consist of nodes (vertices) and connections (edges). In the left panel blackexample of an undirected, unweighted graph. In the right panel the same graph is

rows and columns as the number of nodes in the graph. Each matrix cell providested by color blue) of an edge between two vertices. Since this matrix corresponds to




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) eachvertex 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 fewrandom 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 bandpower in Parkinson’s disease (Stoffers et al., 2008a,b). Neurophysi-ological assessment of functional connectivity has the advantage ofa high time resolution and a direct measurement of neural activity.Although functional interactions are constrained by anatomicalconnectivity, they are not identical to it (Adachi et al., 2011; Honeyet al., 2009). Synchronization studies show fragile binding inhealthy subjects and patterns of hyperconnectivity or disconnec-tion in various neurological disorders. However, the patterns ofsynchronization changes in neurological disease are difficult tointerpret. In particular the occurrence of increased as well as de-creased synchronization (in different frequency bands and differ-ent brain areas) challenges our understanding of the underlyingmechanisms, and demonstrates the need for a more consistent the-oretical 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 systemsWhile 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 anaverage degree of 7.72. Vertices are displayed closer to the middle if their degree ishigher. A few vertices (the hubs) have very high degrees, whereas most nodes aremore in the periphery of the graph. The high diversity of node degrees ischaracteristic of scale-free graphs and is absent in lattice, small-world and randomgraphs. However, scale-free graphs have a vanishingly small clustering coefficient.


that is hard to face without a proper theoretical framework. Oneattempt to develop such a framework is modern network theory,also referred to as graph theory (Newman, 2010). While networktheory and graph theory are sometimes referred to as if they weresynonyms, they are not, and understanding how they are relatedmay help to obtain a better idea what network theory is about,and how it might be helpful in dealing with connectivity data inneuroscience. Modern network theory is a highly interdisciplinaryfield of science, devoted to the study of complex networks. It isrooted in at least three different but connected branches of math-ematics and physics: (i) graph theory; (ii) statistical mechanics ofnetworks; (iii) dynamical systems theory.

Graph theory is a relatively old branch of discrete mathematicsthat deals with networks (Gross and Yellen, 2006). It originatedwith the work of Euler, in particular with his solution of the prob-lem of the seven bridges of Konigsberg. Typically, graph theorydeals with relatively small networks that are represented as setsof network nodes (vertices) and network connections (edges)(Fig. 2). A network represented as a set of vertices and edges iscalled a graph. Different categories of graphs can be distinguished.Graphs where edges are either present or absent are called binaryor unweighted; when a weight representing for instance thestrength, length or importance of a connection is assigned to eachedge the resulting graph is called weighted. When edges only indi-cate that a connection or relation exists between two vertices thegraph is undirected. When edges indicate the direction of influencethe graph is directed or a digraph. Many important theorems havebeen proven for graphs, in particular in relation to special casessuch as fully connected graphs, Harary graphs, wheel graphs ortrees, but this mathematical knowledge is mainly of interest inspecific fields of science such as computer and communication sci-ence (Gross and Yellen, 2006). Classic graph theory could not dealwith large networks with stochastic features.

The scope of graph theory was extended to realistic complexnetworks when models were introduced that aided the descriptionand analysis of large networks with random components. The firstexample of such a model was the random graph model proposedby Erdös and Rényi (1959) (ER). In this model, an edge exists be-tween any randomly chosen pair of vertices with a fixed probabil-ity p. While this may seem a very simple model, it turns out to havemany surprising and important mathematical properties, such asthe sudden emergence of a giant connected component for increas-ing values of p and short path length. The ER random graph is not arealistic model of many real complex networks, but is often used asa null model for comparison. The breakthrough in the statisticalapproach to networks occurred with the publication of two papers




Table 1Explanation of basic network terminology and definitions of most important network concepts and measures.

Symbol Concept Explanation

G Graph Mathematical representation of a networkV Vertex A vertex is a node of the network. The capital V refers to the set of all vertices in a graphE Edge An edge is a connection or relation between two vertices. The capital E refers to the set of all edges in a graphAij Adjacency matrix Binary square matrix. If two vertices i and j are connected the matrix cell aij will have value 1; otherwise it will have value 0Wij Weight Edges in a network can be assigned a weight w that may contain information on the relative importance of the edgesk Degree 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)Neighbor Two vertices that are connected by a common edge are neighborsCost Number of edges in a graph expressed as fraction of all possible edges

P(k) Degree distribution The probability distribution P(k) is the probability that a randomly chosen vertex of the graph will have degree kr Degree correlation Correlation coefficient of degrees of pairs of neighbors. Can be defined for unweighted as well as weighted graphsj Degree divergence Measure of the broadness of the degree distribution. Related to the spectral radius and the threshold for the onset of synchronizationC Clustering

coefficientThe clustering coefficient C is a measure of the local connectedness of a graph. It can be defined as the number of edges presentbetween the neighbors of a vertex divided by the total possible number of edges between the neighbors

L Path length Number of edges in the shortest path between two vertices. In weighted graph: path with lowest sum of edge weights between twovertices (note that this is not necessarily the path with the smallest number of edges)

D Diameter Longest shortest path of a graphEglobal Global efficiency Average of the inverse of the shortest paths by setting the value to zero for disconnected verticesElocal Local efficiency Efficiency computed from the set of neighbors of a vertex

Motif Small characteristic subgraph. A triangle (three connected vertices) is an example of a motif. The prevalence of different types ofmotifs is indicated by a motif count

Q Modularity Modules are subgraphs (typically larger than motifs) such that vertices within a module are more strongly connected than vertices ofdifferent modules. For a given partitioning of a graph Newmans modularity index Q indicates the quality of the division into modules

Hierarchy A graph with a hierarchical structure is characterized by a power law relation between C(k) and k with an exponent about �1b Betweenness

centralityFraction of all shortest paths that pass through a particular vertex or edge

Closeness centrality Average shortest path length from a reference vertex to the rest of the graphEccentricity Longest shortest path from reference vertex to any other vertex of the graphHub Vertex with high centrality. This centrality can be defined in terms of degree, or (combinations of) any of the other centrality measures

k1 Spectral radius Largest eigenvalue of adjacency matrix. Inverse is measure of synchronization threshold. Spectral radius is also related to shortestpath, overall connectedness and degree diversity of a graph

EC Eigenvectorcentrality

Value of the ith coordinate of the eigenvector corresponding to the largest eigenvector the adjacency matrix. Measure of theconnectedness/centrality of the ith vertex

Spectral gap Difference between largest and second largest eigenvalue of adjacency matrix. Index how rapidly dynamics on the graph will convergeto the synchronized state

lN�1 Algebraicconnectivity

Second smallest eigenvalue of Laplacian matrix

S Synchronizability Ratio of largest to second smallest eigenvalue of Laplacian matrix. Its inverse is a measure of the stability of the synchronized state of adynamic process on the graph

R Effective GraphResistance

Measure of the resistance to flow on a graph analogous to Kirchoffs law for electronic circuits (Van Mieghem et al., 2010)

GIC Graph indexcomplexity

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 aregular graph on a ring where each vertex is connected to its kneighbors (Fig. 3). With a rewiring probability p, an edge is chosenand re connected to another randomly chosen vertex. For p = 0 thismodel corresponds to a regular graph or lattice characterized byhigh clustering (each vertex being connected to a few other verti-ces most of which are also connected to each other). In such a reg-ular graph, it takes on average many steps to travel from one vertexto any other vertex on the graph, that is: a regular graph has a longpath length. When p = 1 the graph is in fact a random graph, likethe ER random graph. Such a random graph has a low clusteringcoefficient, but the average path length is now also very short.The really interesting behavior is seen for small but non-zero val-ues of p: rewiring even a very small number of edges does not af-fect the high clustering, but results in a dramatic decrease ofaverage path length. Such intermediate networks that combinehigh clustering with short path lengths are called small-worldnetworks.

The strength of the small-world model is that it explains thecombination of high clustering and short paths encountered inmany real complex networks. However, it does not explain anotherproperty of realistic networks. In ER and WS graphs the vertices allhave about the same number of edges, that is: they have a narrowdegree 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 distributioncorresponds to a so-called power law: P(k) � k�c, with c a scalingexponent between 2 and 3. Barabasi and Albert (1999) were thefirst to suggest a simple algorithm for growing networks, wherenewly added vertices connect preferentially to existing edges withhigher degrees (Fig. 4). The scale-free network of Barabasi and Al-bert (BA) is now considered one of the most important models toexplain the presence of highly skewed power law degree distribu-tions in many complex networks, such as the Internet and theWorld Wide Web (Barabasi, 2009). An important implication ofsuch a degree distribution is the presence of so-called hubs, verti-ces with an exceptionally high number of connections.

Graph theory and statistical physics present two of the majorpillars of modern network theory; the third is dynamical systemstheory. Networks in general, and complex networks in particular,are mainly important because we are interested in the processesthat take place on these networks (Barrat et al., 2008). For instance,we could assume that each vertex corresponds to a dynamical sys-tem, perhaps an oscillator, and each edge indicates which oscilla-tors interact and how strong this interaction is. More generally,we can imagine that some transport of traffic, goods, or informa-tion is taking place on the graph. The important questions hereare: how does the topology of the graph influence the dynamicalprocesses on the graph? In particular, do topological properties




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 isthe 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 arealso 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 thered 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. Betweennesscentrality 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 betweennesscentrality 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. Inthe 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 degreecorrelation, the left with a higher degree correlation. Modularity refers to the existence of subgraphs that consist of vertices with more connections to other vertices withinthe 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 informa-tion transport in the network? Of interest, Watts and Strogatz wereinvestigating such questions about oscillator synchronizationwhen they discovered their small-world model. Finally, perhapsthe most challenging questions of all relate to mutual interactionsbetween dynamics and topology: how do dynamical processeschange the architecture of the networks, and how does altered net-work structure constrain dynamics?

4.1.2. Measures of complex networksModern network science, with its roots in graph theory, statis-

tical physics, and nonlinear dynamics, has developed a large andstill increasing number of measures that can be used to character-ize large and complex networks (Table 1 and Fig. 5). We have al-ready encountered a few of these measures, such as theclustering coefficient, the average shortest path, and the degreedistribution. Often, new measures are developed when new mod-els or new observations on real networks suggest that a particularfeature of networks may be of interest and deserves exact quanti-fication. An overview of currently used network measures can befound in several review papers and textbooks (Kaiser, 2011; New-man, 2010; Rubinov and Sporns, 2010). Here we briefly mentionthe most important categories.

The first category involves measures that characterize the pres-ence and nature of structural building blocks and sub networks.The simplest example is in fact the clustering coefficient, which re-flects the presence of triangles (complete subgraphs of three verti-ces) in networks. Triangles are important since they are directlyrelated to the robustness and error tolerance of the network (Boc-caletti et al., 2006). Triangles are the simplest example of motifs.Motifs are small subgraphs consisting of a few vertices connectedin a particular way and possibly associated with a specific function.Motif analysis involves counting the prevalence of different motifsin a network and comparing the results with those for random net-works (Milo et al., 2002). Exponential random graph analysis is a


sophisticated statistical technique popular in the social networkresearch, that also involves reconstructing complex networks froma 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 eachother than to vertices outside their own module (Newman, 2006).Finally, modules can sometimes combine to form higher orderstructures, giving rise to a true hierarchical modularity (Ravasz,2009).

A second category involves measures sensitive to the level ofintegration in a network. Global integration can be assessed withthe average shortest path, the diameter (the longest shortest path),global efficiency (Latora and Marchiori, 2001, 2003) and the spec-tral radius (the largest eigenvalue of the adjacency matrix of thenetwork) (Van Mieghem, 2011).

A third category involves measures of relative vertex impor-tance. The most basic measure is the degree of vertex (the numberof edges connected to a vertex), and the degree distribution P(k). Aswe have seen, different types of complex networks may be distin-guished on the basis of their degree distribution. Degree is thus ameasure of the importance of a vertex in a network. This conceptis referred to as centrality. The degree correlation indicateswhether high degree vertices tend to connect preferentially toother high degree vertices (assortative mixing) or to low degreevertices (disassortative mixing) (Newman, 2003). Centrality isnot always the same as degree, and other more sophisticated cen-trality measures have been developed as well. Betweenness cen-trality of a vertex is the number of shortest paths through thatvertex divided by the total number of shortest paths (in a similarway it is also possible to define edge betweenness) (Wang et al.,2008). The eccentricity of a vertex is the length of its longest short-est path to any other vertex; the lower the eccentricity, the morecentral a vertex is in a network. Eigenvalue centrality determinesthe importance of a vertex on the basis of its connections to othervertices, but also with respect to how those other vertices are




Fig. 6. Representing multi channel time series as a graph. First, all pair-wisecorrelations (using any of the synchronization measures shown in Fig. 1) betweenall time series are computed. The results constitute a matrix, with the cell valuesindicating the strength of the correlation between two time series. Next, all cellvalues below a threshold are set to zero, and all cell values above the threshold areset to 1. This converts the weighted matrix to an unweighted binary matrix (thebinary matrix and unweighted graph representations are equivalent). From thismatrix all network measures of interest (see Fig. 3) can be computed. As a controlprocedure, a large ensemble (indicated by the open arrow) of random networks isgenerated from the unweighted matrix, preserving the number of nodes, edges andusually also the degree distribution. The network measures are also computed forall the random graphs. Combining the network measures from the original networkand the average values obtained from the ensemble of random graphs normalizedmeasures can be computed. As an alternative procedure (indicated by interruptedlines), the computation of network measures can also be done directly on theweighted matrix skipping the threshold and binarization steps.


connected (and so on); this is also the idea behind the PageRankalgorithm used by Google to determine the importance of webpag-es (Lohmann et al., 2010).

The final and fourth category we want to mention consists ofmeasures that relate network topology to the dynamics of pro-cesses on the network, in particular the synchronization thresholdand the stability of the synchronous state (Arenas et al., 2008).Topological features that are associated with a lower synchroniza-tion threshold are the inverse of the spectral radius (the largesteigenvalue of the adjacency matrix of a graph) and a measure ofdegree diversity called kappa (hk2i=hki). Kappa is especially highin networks with a scale-free degree distribution and high-degreehubs. These concepts are based upon the assumption that thedynamics can be described by non-identical Kuramoto oscillators(Restrepo et al., 2005). If we assume identical oscillators, synchro-nization can be analyzed with the so-called master stability equa-tion (Barahona and Pecora, 2002; Chen and Duann, 2008;Nishikawa et al., 2003). This suggests that the stability (not thethreshold) of the synchronous state depends upon the ratio ofthe second smallest to the largest eigenvalue of the Laplacian ofthe adjacency matrix of the graph (Arenas et al., 2008). The secondsmallest Laplacian eigenvalue, referred to as the algebraic connec-tivity, 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 magni-tude of the algebraic connectivity is not only important for syn-chronization, but also reflects network robustness (resilienceagainst damage).

4.1.3. Representing structural and functional networks as graphsTo be able to use the concepts and measures of network theory

for a better understanding of the organization of brain networks, itis important to have a strategy to represent neuroanatomical and


neurophysiological data in the form of graphs. For structural data,this is relatively easier than for physiological data. Since the fullinformation on the connectivity of the neuronal network of Caeno-rhabditis elegans is known, Watts and Strogatz were able to repre-sent this nervous system as a graph by taking the neurons asvertices and connections between neurons as edges (Watts andStrogatz, 1998). For several animals, maps are available of thestructural connections between macroscopic brain areas; this al-lows a representation as graphs by taking brain areas instead ofneurons as vertices (Sporns et al., 2000; Stephan et al., 2000). Mod-ern MRI studies in humans present two possibilities to models ana-tomical connectivity as graphs. Matrices of correlations in corticalthickness can be taken as surrogate measures of anatomical con-nectivity (He et al., 2007). More sophisticated techniques allowthe reconstruction of anatomical networks in individuals (Tijmset al., 2011). Alternatively, DTI and tractography can be used toconstruct connectivity matrices at various levels of anatomical res-olution (Hagmann et al., 2007, 2008; Iturria-Medina et al., 2007).

Representing neurophysiological data as a graph is less straight-forward. We first describe the basic approach (Fig. 6). Then wemention some alternative possibilities and point out methodolog-ical caveats. Assume we have a recording of n channels of ‘‘high-density EEG’’ or MEG. After preprocessing, artefact removal and fil-tering we can compute the strength of synchronization between allpossible pairs of channels or sensors, using one of the techniquesdescribed in Section 3.2. The resulting information can be repre-sented in the form of an n by n matrix, with each ni,j matrix cellcontaining a real number reflecting the strength of synchronizationbetween channel i and j. In graph theory, such a matrix is referredto as the adjacency matrix A. To represent this information in theform 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 be-tween vertex i and j. If ni,j < T, no edge is present between vertices iand j. If we indicate the presence of an edge by 1, and the absenceby 0, we have converted the n channel EEG or MEG data into a bin-ary graph. Once the data are represented as a graph (with its equiv-alent adjacency matrix), in principle all measures available innetwork theory can be applied to the network. Clearly, this ap-proach opens up new and potentially interesting ways to analyzefunctional as well as structural network data.

There are a number of variations to this basic scheme, and a fewmethodological problems. First, a strong point of the approach isthat it can be applied to all sorts of data, structural, functional(EEG, MEG, fMRI BOLD time series, correlations in spike trains orcalcium fluctuations). However, it is not always clear what shouldbe considered a vertex, and what is an edge, especially in the caseof functional data (Kaiser, 2011). EEG and MEG recorded in signalspace do not have a simple relation to underlying sources, andtherefore according to some authors might not qualify as bona fidevertices (Antiqueira et al., 2010). The alternative would be to donetwork analysis in source space rather than signal space; how-ever, even this will not solve problems related to volume conduc-tion and spatial sampling that might interfere with properties ofthe reconstructed networks (Bialonski et al., 2010). A further prob-lem concerns the choice of a threshold. This can be based upon thesignificance of the synchronization, or a requirement that theresulting graph is connected. The same threshold can be used forall networks to be compared, or it can be adjusted individually toobtain networks with the same degree for all subjects and condi-tions (Stam et al., 2007a). In many cases, a range of thresholds isstudied to avoid the rather arbitrary choice of a single threshold.Clustering coefficients, motif prevalence, and path lengths are of-ten normalized by comparing the results with those of randomnetworks with the same number of nodes, edges, and degree distri-bution (Achard and Bullmore, 2007). Unfortunately, comparison ofnetworks with different numbers of vertices and edges remains





problematic, even after normalization by random network data(van Wijk et al., 2010).

4.2. Healthy subjects

Network studies of structural and functional brain networkshave revealed a number of consistent characteristics. First, thereis considerable evidence that brain networks are characterized bya combination of high clustering (the fraction of neighbors of anode that are connected to each other) and short average pathlengths (the minimum number of edges to be travelled to get fromone vertex to another). This has first been shown for functionalnetworks with MEG and fMRI (Eguíluz et al., 2005; Stam, 2004;Stephan et al., 2000). Similar results have been obtained withstructural MRI techniques, in particular cortical thickness correla-tions and DTI tractography (He et al., 2007; Hagmann et al.,2007,2008; Iturria-Medina et al., 2007; Gong et al., 2009a). Humanbrain networks should thus be classified as small-world networks.This is important since the strong clustering of small-world net-works might support the segregation, and short path length mightsupport the integration that are both assumed to be necessary foroptimal information processing in networks (Sporns, 2011b; Stam,2010). In addition to clustering, human brain networks also displaystructure at the larger scales of network motifs, and in particularnetwork modules (Meunier et al., 2010; Sporns and Kötter,2004). Structural and functional MRI studies have shown that hu-man brain networks consist of around five to six modules, eachcorresponding with known functional subsystems (Gong et al.,2009a; He and Evans, 2010). These large-scale modules may becomposed of smaller sub modules, suggesting a hierarchical orga-nization of brain networks (Ferrarini et al., 2009; Gleiser andSpoormaker, 2010; Meunier et al., 2010). It is interesting to see thatthe hierarchical organization of human brain networks is exactlythe kind of organization that Herbert Simon predicted as optimalfor any complex system (Simon, 1962).

Not all nodes or vertices in brain networks have the same signif-icance. There is accumulating evidence that the degree distributionof human brain networks is highly skewed, with a relatively highprevalence of high-degree nodes or hubs (Eguíluz et al., 2005;Van den Heuvel et al., 2008). There is still some controversywhether human brain networks are truly scale-free with apower-law degree distribution. Some studies suggest that the de-gree distribution might be more properly modeled by an exponen-tially truncated power-law (Achard et al., 2006). This controversyis probably due, at least to some extent, to methodological issues,such as the as the level of resolution (ROIs versus voxels) used forthe network analysis (Fornito et al., 2010). There is little doubt thatthe so-called default mode network, consisting of medial frontal,anterior and posterior cingulate cortex, precuneus and parietal cor-tex, constitutes a connectivity backbone that links the hubs withthe highest degrees (Hagmann et al., 2008). The highly connectedhubs of the default network are strongly correlated with intelli-gence, and may also be of central importance in consciousness(Douw et al., 2011; Li et al., 2009; Van den Heuvel et al., 2009; Sod-du et al., 2011).

Network changes have also been described in relation to thephysiological changes of consciousness related to sleep (Ferriet al., 2007, 2008). As we will discuss below, brain network hubsmay not only play a central role in normal brain function, but alsoin some types of brain disease. The fact that high degree nodes inbrain networks preferentially connect to each other suggests a po-sitive degree correlation or assortative mixing (De Haan et al.,2009; Hagmann et al., 2008). A similar type of degree correlationis also found in social networks, whereas most other networks innature tend to be disassortative (Newman, 2003). However, atthe neuronal level, brain networks are disassortative rather than


assortative (Bettencourt et al., 2007). This implies that the humanbrain is the only type of complex network known that displaysopposite patterns of mixing at different spatial scales. At the cellu-lar level, the brain is organized as a communication network, andat the macroscopic level the brain is organized as a social network.The origin and significance of this type of organization of brain net-works are currently unknown.

The complex architecture of human brain networks arises dur-ing development and seems to be under strong genetic control. Alarge EEG study of children investigated at age 5 and 7 showedan evolution of a relatively more random to a more small-worldlike topology of functional brain networks (Boersma et al., 2011;Micheloyannis et al., 2009). In a follow up study, it was shown thatthe optimal small-world pattern of adult age is gradually replacedby a more random topology at higher ages (Smit et al., 2010). Thus,the evolution of brain network topology with respect to clusteringand path length seems to follow an inverted u-curve. In addition,clustering coefficients and path lengths were shown to have a highheritability in twin studies (Smit et al., 2008). The evolution fromrandom to small-world topology has been replicated in modelstudies (Siri et al., 2007). Another interesting finding is the differ-ence in brain network topology in males and females, with higherconnectivity 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, char-acterized by cognitive and psychiatric symptoms and a suspectedbut generally unknown neurobiological substrate. The need forobjective diagnostic findings and the failure of structural imagingmethods, in particular MRI, to show consistent patterns of abnor-malities in neuropsychiatric disease presents a challenge for amore physiological and network based approach. One of the mostprevalent and disabling psychiatric conditions is schizophrenia.Schizophrenia is not a simple Mendelian disorder, and complex ge-netic and developmental factors are assumed to play an importantrole (Kim et al., 2011). In particular, there is a suspicion that thenormal development of long-range fronto-temporal connectionsis disrupted. Network studies provide some support for this thesis.In an early EEG study, functional brain networks of schizophrenicpatients were shown to have lower normalized clustering and pathlength compared to healthy controls (Micheloyannis et al., 2006).An EEG study based upon network analysis of the level of nonlinearcoupling also showed more random network topology in schizo-phrenic patients (Rubinov et al., 2009a). A few MRI studies showedrather subtle network changes in schizophrenia. Alexander-Blochet al. (2010) reported a disruption of modular network structurein schizophrenia, with a preservation of global clustering and pathlength. Abnormal modular structure in schizophrenia has also beendetected with EEG (Jalili and Knyazeva, 2011). Van den Heuvelet al. (2010) demonstrated very local network changes in frontaland temporal areas, with a preservation of global network proper-ties. An overview of modern network research in schizophrenia isgiven in the review by van den Heuvel and Hulshoff Pol (2010).As we have discussed above, brain networks undergo systematicchanges during development and such changes depend uponstrong genetic influences as well as experience. If schizophreniais indeed – at least to some extent – a genetic disorder of braindevelopment subtle abnormalities of brain network topology inpatients should be expected. These may be characterized by a rel-atively local fronto-temporal disturbance of normal networkarchitecture.

Other neuropsychiatric conditions have not been studied exten-sively. In acutely depressed patients, global connectivity assessedwith synchronization likelihood was lower and EEG functional





network topology during sleep was more random compared tohealthy controls (Leistedt et al., 2009). In a large fMRI study, de-pressed patients had lower normalized path length and increasednode centrality of their functional brain networks compared tocontrols (Zhang et al., 2011). These changes are also consistentwith a randomization of network topology in neuropsychiatric dis-ease. Of interest, some of the altered node centrality measures cor-related with disease duration and severity.

Structural and functional network changes, possibly under ge-netic control, have also been implicated in attention deficit hyper-activity disorder or ADHD (Konrad and Eickhoff, 2010). In a resting-state fMRI study, an increase of local and a decrease of global effi-ciency, suggesting abnormally regular networks, was reported inADHD patients (Wang et al., 2009). Local changes in nodal effi-ciency mainly involved frontal, temporal and occipital areas. Com-munity analysis of EEG networks, based upon a fuzzysynchronization likelihood, has been shown to be able to distin-guish between ADHD patients and healthy controls (Ahmadlouand Adeli, 2011).

Functional network analysis of delta band EEG in autism spec-trum disorder (ASD) revealed increased short-range fronto-lateralconnectivity and a loss of long-range fronto-occipital connections(Barttfeld et al., 2011). These changes were accompanied by a low-er clustering coefficient, longer path length, and increased modu-larity in ASD patients. An earlier EEG study also reported loss ofglobal (alpha band) coherence and an increase in local (theta band)connectivity in ASD (Murias et al., 2007). In this study, no formalnetwork analysis was applied. Graph theoretical analysis of struc-tural brain networks in subjects with grapheme-color synesthesiashowed a shift from global to more local processing (Hänggiet al., 2011).

4.4. Epilepsy

Epilepsy refers to a group of neurological conditions, character-ized by the unexpected and unpredictable emergence of an abnor-mal dynamic state of the brain with excessive neuronal firing andsynchronization. Epilepsy is one of the most prevalent and impor-tant neurological conditions. Because of its highly dynamic charac-ter, proper investigation of epilepsy requires tools with the highestpossible temporal resolution such as EEG and MEG. How and whyepileptic seizures arise in the brain is still essentially an enigma.Network theory can be of help here by studying changes in theorganization of brain networks, that might lower the thresholdfor pathological synchronization and facilitate the spread of abnor-mal activity throughout the brain.

The first indications that network analysis might be helpful inunderstanding epilepsy, came from model studies. One studyshowed that changing the topology of simulated hippocampal neu-ronal networks from regular to small-world and then to randomlowers the threshold for synchronization and gives rise to sei-zure-like activity and ultimately to epileptic bursting (Netoffet al., 2004). Percha et al. (2005) also suggested a relation betweensmall-world topology and epileptic seizures. Another topologicalfeature, that may be relevant for understanding seizures, is the de-gree-distribution, in particular the presence of hubs. Morgan andSoltesz (2008) showed that scale-free networks with hub nodeshad the lowest threshold for the emergence and spreading of sei-zures. Healthy brain networks are probably characterized by ahierarchical modular structure, that could arise naturally duringdevelopment and might subserve functional specialization (Rubi-nov et al., 2009b; Stam et al., 2010; Wang et al., 2011). Interest-ingly, modular structure might actually present a threshold forsystem-wide synchronization. Epileptic seizures could arise whenthe normal modular structure breaks down in brain disease (Kaiserand Hilgetag, 2010).


There is growing empirical support for the idea that changes inbrain network topology might play a crucial role in epilepsy, assuggested by the model studies. Analysis of EEG depth recordingsin patients suffering from temporal lobe seizures showed a transi-tion from a more random to a more regular topology of functionalbrain networks during seizures (Ponten et al., 2007). The observa-tion of abnormal regularization of functional brain networks dur-ing seizures has been confirmed by other studies, based uponelectro corticographic recordings (Kramer et al., 2008; Schindleret al., 2008). A similar pattern has been demonstrated with surfaceEEG and MEG during absence seizures (Gupta et al., 2011; Pontenet al., 2009). Several studies have shown that interictal functionalnetworks in epilepsy patients may be characterized by increasedconnectivity, especially in the theta band, a shift to a more regulartopology, changes in modular structure and prominent hub-like re-gions (Chavez et al., 2010; Douw et al., 2010b; Horstmann et al.,2010; Wilke et al., 2011). Inter-ictal network changes as well ascognitive impairment may also be related to the duration of theepileptic condition (van Dellen et al., 2009; Vlooswijk et al.,2011). MRI studies have also demonstrated abnormal brain net-work organization in epilepsy (Bernhardt et al., 2011; Liao et al.,2010; Vlooswijk et al., 2011; Zhang et al., 2011). These observa-tions are clinically relevant since it has been shown, for instance,that removal of hub regions, identified with corticography duringepilepsy surgery, is associated with a better outcome (Ortegaet al., 2008; Wilke et al., 2011). Of interest, in the study of Ortegaet al., the minimum spanning tree, based upon the corticographictime series, was used to identify the hub nodes. Characterizationof interictal changes in connectivity could also improve the diag-nostic yield of routine EEG (Douw et al., 2010a).

4.5. Dementia

Alzheimer’s disease is one of the most common causes ofdementia in the aging population. It was also one of the first neu-rological conditions to be studied with the tools of modern net-work theory (Pievani et al., 2011). Functional network analysis ofEEG filtered in the beta band showed changes in the path lengthin Alzheimer patients (Stam et al., 2007a). These changes corre-lated with a lower mini mental state score in the patient group.This study also showed the methodological problems associatedwith different ways for reconstructing networks from multi-chan-nel EEG data, in particular using a fixed threshold or a fixed degreefor all networks. In a later MEG study, resting-state functionalbrain networks in Alzheimer patients were characterized by a low-er normalized clustering coefficient and path-length, especially inthe lower alpha band (Stam et al., 2009). The network changes inAlzheimer patients could be replicated with a simple damage mod-el that assumed preferential damage of hub-like network connec-tions. A decrease of the normalized clustering coefficient andpath-length in Alzheimer’s disease could be confirmed in a morerecent EEG study (De Haan et al., 2009). This study revealed anopposite pattern of network changes in frontal lobe dementia(Fig. 7). All patients as well as healthy control subjects displayedpositive degree correlations, but the degree correlations were sig-nificantly lower in Alzheimer’s disease and frontal lobe dementia.Functional connectivity of MEG during a working memory studywas investigated in patients with mild cognitive impairment (Bul-dú et al., 2011). The average level of connectivity and a so-calledoutreach parameter were increased in the MCI group. Thesechanges were interpreted in terms of higher energy expenditureand a tendency toward random structure in MCI patients. The in-creased synchronization likelihood during a working memory taskin MCI is in line with the EEG study of Pijnenburg et al. (2004).

Network changes in dementia have also been shown with struc-tural and functional MRI studies (Bokde et al., 2009; Filippi and




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 referencemontage. For the Alzheimer’s disease (AD), frontotemporal dementia (FTLD) and control groups, the functional connectivity (synchronization likelihood, SL) based graphs areshown 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-groupdifferences. From: De Haan W et al. Functional neural network analysis in frontotemporal dementia and Alzheimer’s disease using EEG and graph theory. BMC Neurosci2009;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 t-test, 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 synchronizationbetween 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 ofFMRI resting-state functional connectivity. PLoS One 2010;5:e13788. (Copyright by the authors under the license agreement that the work can be freely distributed whenfully cited.)


Agosta, 2011). The first resting-state fMRI network study in Alzhei-mer’s disease reported a significant decrease of the clustering coef-ficient, interpreted as a signature of local connection loss (Supekaret al., 2008). In contrast, a more recent study showed normalclustering coefficients but decreased normalized path length(Sanz-Arigita et al., 2010) (Fig. 8). In this study, the average levelof connectivity, quantified as the synchronization likelihood ofthe resting-state BOLD time series, was the same in both groups,but Alzheimer patients had a relatively high connectivity level infrontal areas, and a relatively lower connectivity level in parietaland occipital areas. This anterior to posterior disconnection wasalso observed in another fMRI study (Wang et al., 2007).


There is increasing evidence from these studies, that differenttypes of dementia might be characterized by specific patterns ofnetwork changes (Zhou et al., 2010). This observation suggests thatnetwork analysis could be of clinical use in the differential diagno-sis of dementia (Pievani et al., 2011; Zamrini et al., 2011).

One of the most challenging questions at this moment iswhether the network changes in Alzheimer’s disease can be relatedto specific underlying mechanisms. One of the classic neuropatho-logical hallmarks of Alzheimer’s disease is the deposition of amy-loid beta. Deposition of amyloid beta in the brain is not randombut follows a specific pattern with the highest concentrations inmultimodal association areas. There is a considerable overlap be-




Fig. 9. Local activation, pair-wise interactions, networks organization. Brain function can be studied from different viewpoints depending on the way in which interactionsare 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 brainareas); 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 functionalconnectivity explicitly or implicitly are based on this ‘‘first order’’ model of interaction. Finally (right panel) all higher order interactions (interactions of interactions) can alsobe taken into account. This ‘‘second and higher order’’ or network model allows the definition and quantification of concepts such as clustering, path length, modularity thatcannot be handled by lower order models.

Fig. 10. A heuristic model of complex brain networks. The model consists of threeaxes 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 theright. Intermediate networks on this axis are small-world networks. The verticalaxis indicating the level of degree diversity connects the random network below tothe scale-free network in the upper right corner. Finally, the diagonal axis thatreflects 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 cornercorresponds 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 rest-ing-state fMRI studies (Buckner et al., 2008; Damoiseaux and Grei-cius, 2009; Greicius et al., 2009; Van den Heuvel and Hulshoff Pol,2010). These areas show the highest levels of activity in the rest-ing-state, and also have the highest level of connectivity (Hagmannet al., 2007,2008; van den Heuvel et al., 2008). In a combined PETand fMRI study, an intriguing correlation was demonstrated be-tween brain network node degree and amyloid deposition: areaswith high degree hubs were also areas with the largest concentra-tion of amyloid (Buckner et al., 2009). Amyloid deposition and hubconnectivity may even be correlated in non-demented subjects(Drzezga et al., 2011). Simulations on the basis of MEG in Alzhei-mer’s disease also pointed to a specific vulnerability of hub-likebrain regions (Stam et al., 2009). Network theory predicts thatthe distribution of traffic in communication networks dependsupon the degree distribution, with high-degree hubs receivingmost of the traffic. If the traffic load becomes excessive, hubs willbe the most likely network nodes to fail. If network traffic can besomehow related to neuronal firing, the missing link betweenamyloid deposition and high degree hubs might be excessive syn-aptic activation (Bero et al., 2011).


5. Toward a theory of brain networks

5.1. From local activation to network organization

In this review, we have argued for the need and potential use-fulness of studying the brain from the perspective of a complex or-ganized network (Fig. 9). The underlying question is simple anddaunting at the same time: why is our brain so complex? If we as-sume that our brain is the result of a long evolutionary process wecan expect that its complex organization is not random but ratherreflects some kind of optimal solution to multiple, possibly con-flicting, constraints. The question then becomes whether a net-work perspective can provide an explanation, even if onlytentatively, of the origins and functional significance of the organi-zation of brain networks.

The study of the function of elementary units of the brain, fromthe level of neurons and microcolumns all the way up to macro-scopic brain regions, is a necessary first step for understandingbrain function (Section 3.1). Much of the progress in neurosciencein the past few decades is related to a better understanding of theconditions underlying local activation of functional brain units.However, it is also obvious that the activity of functional units ofthe brain makes little sense (literally as well as metaphorically) ifit cannot be communicated to other units of the brain. The functionof 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 offunctional units interact and exchange information. This realiza-tion underlies the rapidly increasing interest in studies of ‘‘func-tional connectivity’’ and related concepts (Section 3.2). A majorfocus in this kind of studies is the development of reliable mea-sures of pair-wise interactions. However, when this approach is ap-plied to the study of interactions in systems with large numbers ofelements, its limitations become apparent. Often, it is very difficultto make sense out of the multitude of increases and decreases offunctional connectivity levels that are observed in different behav-ioral states and neurological disorders. The point is, that the func-tional architecture of the brain cannot be understood as the sum ofa large number of pair-wise interactions (Section 3.6).

The point of modern network theory is that a kind of systemsperspective is needed to understand how communication is orga-nized in a brain, consisting of a large number of interacting ele-ments (von Bertalanffy, 1969). A network perspective introducesconcepts such as clustering, modularity, path length and degreedistribution, that do not make sense, and in fact do not even exist,at the level of pair-wise interacting brain regions or neurons (Sec-tion 4.1). Such high level concepts and measures, based upon theseconcepts, can help to identify organizational themes in brain net-





works, and common patterns in the way these networks are dis-rupted in disease.

5.2. A heuristic model of complex brain networks

The concept of a graph provides a general framework for deal-ing with all sorts of networks and allows studying the organizationof the brain using novel concepts and measures (Fig. 10). The con-cepts and measures derive their meaning from the fact that theyare defined in the context of specific structural models of complexnetworks. The introduction of random graphs bridged the gap be-tween small, deterministic graphs, and large networks that requirea statistical description (Erdös and Rényi, 1959). The ER model ex-plains some important aspects, but fails to explain the phenome-non of clustering. The small-world model solves this problem butcannot explain the diversity of node properties, observed in realnetworks. The scale-free model of Barabasi and Albert does explainnode diversity and the existence of hubs or highly connectednodes, and has a very short path length, but unfortunately also avanishing clustering coefficient.

The problems we are faced with can be described as follows:currently, we have two powerful models, the WS model (that con-tains 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 firstproblem is that of the relationship between the two models. Nei-ther model can be reduced to the other; some properties can onlybe explained by one model, and not by the other. In other words:we need them both, but they do not go well together. The secondproblem is that a number of network properties have been de-tected that are clearly present in structural and anatomical net-works, but cannot be explained by either model. The two mostobvious examples are (hierarchical) modularity and degree corre-lations. These two problems present an important challenge for anetwork theory of the brain.

To overcome these problems, and to guide future studies, wepresent a heuristic model of complex brain networks (Fig. 10).The heuristic model assumes an abstract ‘‘network space’’ definedby three corners and three axes. The first, horizontal axis, is a rep-resentation of the WS model. The left corner is occupied by the reg-ular lattice network, the right corner by the random ER network,and intermediate networks between the lattice and random net-work correspond to small-world configurations. The horizontalaxis indicates the balance between order and randomness. The ver-tical axis with the SF network in the upper right corner indicatesthe level of node diversity. While interpolation between ER andSF networks is not commonly used, it is easy to see how this couldbe done, for instance by randomly rewiring the edges of the SF net-work with a probability p. The degree diversity kappa (hk2i=hki)could be a useful measure to locate a network between the ex-tremes of ER and SF. The third diagonal axis connects the latticein the lower left corner to the SF network in the upper right corner.According to Ravasz and Barabási (2003), networks that are inter-mediate between a lattice and a scale-free network display hierar-chical modularity (hierarchical modular network, HMN). This canbe quantified by the exponent of the power law relation betweenC(k) and k (with C the clustering coefficient and k the degree). Innetworks with hierarchical modularity, the clustering coefficientis inversely correlated with the degree.

The heuristic model of ‘‘network space’’ suggests that the HMNpresents a very special case. This type of network is not only inter-mediate between the lattice and the SF network, but, if we projectits position on the X axis, also between the lattice and the ER mod-el, and if we project it on the Y axis, between the ER and the SF net-work. In fact, the HMN combines properties of a lattice, a randomand a scale-free network as a kind of compromise. It combines high


clustering with short path lengths and high degree diversity. Inaddition, it displays hierarchical modularity and could be linkedto positive degree correlations (Newman, 2003, 2006).

The concept of HMN is very attractive, since it suggests how thefundamental types of network models might be related. Further-more, HMNs are the only network model that explain all majorproperties (clustering, short paths, degree diversity, modularityand assortativity) observed in real structural and functional net-works. There are indications that modular structure can evolvenaturally in networks that use optimization algorithms such assimulated annealing (Kirkpatrick et al., 1983) to deal with multipleconflicting constraints (Pan and Sinha, 2007; Rubinov et al., 2009b;Stam et al., 2010; Yuan and Zhou, 2011). Furthermore, hierarchicalmodularity has been shown to explain critical dynamics as well asoscillations of neural activity, both of which are assumed to play akey role in neuronal information processing (Kaiser et al., 2010;Rubinov et al., 2011; Wang et al., 2011). Since HMNs present acompromise between many network properties and may evolvenaturally in networks dealing with multiple constraints, it isseductive to conceive of HMNs as optimal configurations function-ing as attractors in ‘‘network space’’. However, it should be stressedthat at present, in contrast to the ER, WS and SF models, we do nothave a simple, universal mathematical model of HMNs. The discov-ery of such a model could offer substantial support for the heuristicmodel presented here.

5.3. Development, aging and neurological disease in network space

Can we use our heuristic model of complex brain networks tomake sense out of the many network studies described in this re-view? We will address this question with respect to development,aging and various types of neurological disease. The central idea ofthe heuristic model is that hierarchical modular networks presentan optimal state characterized by high clustering, shorts paths, de-gree diversity, hierarchical modularity and assortativity. In thisview, network changes are either towards or away from this opti-mal state. Furthermore, the three corners in ‘‘network space’’ cancharacterize distinct patterns of network changes.

In general, network studies of brain development are in agree-ment with the notion of a movement toward an optimal HMN orga-nization (Boersma et al., 2011; Fair et al., 2009; Micheloyannis et al.,2009; Yap et al., 2011). Here we should note that changes fromeither a lattice or a random network to a small-world networkcould be translated to changes in the direction of a HMN network;in fact, the heuristic model predicts that node diversity should in-crease at the same time. Studies differ with respect to the startingpoint of the network trajectory. EEG studies suggest that networkdevelopment may start in the lower right corner, whereas MRI-based studies indicate that networks may be more lattice-like earlyin development (Boersma et al., 2011; Fair et al., 2009; Yap et al.,2011). Evolution toward a hierarchical modular structure, irrespec-tive of the initial state, is supported by model studies (Siri et al.,2007). Network changes in aging may be explained by a movementaway from the HMN pattern toward either a more random or amore lattice like structure (Smit et al., 2010). Loss of modular struc-ture has also been observed in aging (Meunier et al., 2009).

The heuristic model suggests that brain disease at the networklevel always implies a movement away from the optimal HMNstate. Depending upon the trajectory through ‘‘network space’’, dif-ferent patterns may be discerned. First, network disease could re-sult in a straightforward movement to the lower right corner.This network randomization is a pattern that has been observedin many different types of brain disease, ranging from Alzheimer’sdisease, brain tumors, depression to schizophrenia (De Haan et al.,2009; Leistedt et al., 2009; Micheloyannis et al., 2006; Rubinovet al., 2009a; Stam et al., 2009). Network randomization seems to





be a characteristic of relatively severe and advanced brain disease.Second, in some other conditions brain networks seem to moveaway from the HMN toward the lower left corner. This type ofchange would be characterized by a shift from global to local con-nectivity, and a pathological increase in ‘‘network regularity’’. Thispattern of brain network changes has been observed in develop-mental disorders (Barttfeld et al., 2011; Hänggi et al., 2011; Wanget al., 2009) as well as in the early stages of various neuropsychiat-ric conditions (Buldú et al., 2011; Pijnenburg et al., 2004; Schoon-heim et al., 2011b). Third, in some situations brain disease may becharacterized by the abnormal activation of hubs, or the emer-gence of new pathological hub-like brain areas. This would corre-spond to a movement to the upper right corner. Excessiveactivation of hub nodes has been associated with epileptic brain re-gions (Chavez et al., 2010; Douw et al., 2010b; Horstmann et al.,2010; Wilke et al., 2011; Zhang et al., 2011). In addition, excessivehub activation could be a temporary state in other neurologicaldisorders, followed by a more general network randomization.

While different trajectories in network space might characterizedifferent stages or patterns of disease, if the disease is long lastingand severe, the natural endpoint seems to be the random networkin the lower right corner. This suggests that the random networkcould 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 al-lows us to go beyond local activation and pair-wise interactionstudies, and define higher order concepts relevant for normal andabnormal brain function. One challenge is to develop better mea-sures of brain network characteristics, and use these measures tointegrate information from multiple imaging modalities such asEEG, MEG, structural and functional MRI. However, even more ur-gent is the need for new and better models that can fill the gap be-tween the existing small-world and scale-free models. A simplemathematical model of hierarchical networks could be an impor-tant step forward. A deeper understanding of complex brain net-works also requires the identification of the most relevantconstraints that confront developing brain networks, as well asthe algorithms that are used to deal with them. In particular, therole of space and geometric constraints may be important factors(Henderson and Robinson, 2011; Kaiser and Hilgetag, 2004). Itmay be helpful to think of the complex architecture of brain net-works in terms of solutions; the main task then becomes to identifythe problems these networks are trying to solve. Finally, the ulti-mate challenge for network studies in a clinical context, and in par-ticular in clinical neurophysiology is whether this approach willhave an impact not only on our understanding of pathophysiolog-ical mechanisms, but also on clinical diagnosis and treatment.

Acknowledgements

We thank Alexandra Linger for her secretary support.

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