Exploring the network dynamics underlying brain activity during rest

Exploring

the

network

dynamics

underlying

brain

activity

during

rest

Joana

Cabral

a,b,

*

,

Morten

L.

Kringelbach

b,c

,

Gustavo

Deco

a,d

a

TheoreticalandComputationalNeuroscienceGroup,CenterofBrainandCognition,UniversitatPompeuFabra,Barcelona,Spain b

DepartmentofPsychiatry,UniversityofOxford,Oxford,UnitedKingdom c_{CFIN/MindLab,AarhusUniversity,Aarhus,Denmark}

d

Institucio´ CatalanadeRecercaiEstudisAvanc¸ats(ICREA),Barcelona,Spain

Contents

1. Brainactivityduringrest... 103

2. Signaturesofresting-stateactivity... 104

2.1. Resting-statehaemodynamicfluctuations... 104

2.2. Electrophysiologicalsignaturesofresting-stateactivity... 105

2.3. Detectionofresting-statepatternsusingMEG... 107

2.4. Electrophysiologicalcorrelatesofresting-stateBOLDfluctuations... 107

2.5. Alteredresting-stateactivityindisease... 108

3. Networksofthebrain... 109

3.1. Brainparcellation... 109

3.2. Anatomicalstructuralnetworks ... 110

ARTICLE INFO Articlehistory:

Received17December2012

Receivedinrevisedform4November2013 Accepted17December2013

Availableonline31December2013 Keywords: Resting-state Functionalnetworks Anatomicalnetworks Computationalmodelling ABSTRACT

Sincethemid1990s,theintriguingdynamicsofthebrainatresthasbeenattractingagrowingbodyof researchinneuroscience.Neuroimagingstudieshaverevealeddistinctfunctionalnetworksthatslowly activate and deactivate, pointing to the existence of anunderlying network dynamics emerging spontaneously during rest, with specific spatial, temporal and spectral characteristics. Several theoreticalscenarioshavebeenproposedandtestedwiththeuseoflarge-scalecomputationalmodels ofcoupled brainareas.However, amechanistic explanationthat encompassesallthephenomena observedinthebrainduringrestisstilltocome.

Inthisreview,weprovideanoverviewofthekeyfindingsofresting-stateactivitycoveringarangeof neuroimagingmodalitiesincludingfMRI,EEGandMEG.Wedescribehowtobestdefineandanalyze anatomicalandfunctionalbrainnetworksandhowunbalancingthesenetworksmayleadtoproblems withmentalhealth.Finally,wereviewexistinglarge-scalemodelsofresting-statedynamicsinhealth anddisease.

An important common feature of resting-state models is that the emergence of resting-state functionalnetworksisobtainedwhenthemodelparametersaresuchthatthesystemoperatesatthe edgeofabifurcation.Atthiscriticalworkingpoint,theglobalnetworkdynamicsrevealscorrelation patternsthatarespatiallyshapedbytheunderlyinganatomicalstructure,leadingtoanoptimalfitwith the empirical BOLD functional connectivity. However, new insights coming from recent studies, includingfasteroscillatorydynamicsandnon-stationaryfunctionalconnectivity,mustbetakeninto accountinfuturemodelstofullyunderstandthenetworkmechanismsleadingtotheresting-state activity.

ß2014TheAuthors.PublishedbyElsevierLtd.

Abbreviations:fMRI,functionalmagneticresonanceimaging;MEG,magnetoencephalography;EEG,electroencephalography;BOLD,bloodoxygenleveldependent;BLP, band-limitedpower;RSN,resting-statenetwork;DMN,defaultmodenetwork;ICA,independentcomponentanalysis;AAL,automatedanatomicallabeling;ROI,regionof interest;DTI/DSI,diffusiontensor/spectrumimaging.

* Correspondingauthorat:RocBoronat,138,08018Barcelona,Spain.Tel.:+351919289649. E-mailaddress:[email protected](J.Cabral).

0301-0082ß2014TheAuthors.PublishedbyElsevierLtd.

http://dx.doi.org/10.1016/j.pneurobio.2013.12.005

Open access under CC BY-NC-ND license.

3.3. Dynamicfunctionalnetworks ... 111

3.4. Characterizingbrainnetworksusinggraphtheory... 113

3.5. Modelsofbrainnetworkdevelopmentandorganization... 114

4. Large-scalemodelsofresting-statedynamics... 115

4.1. Conductance-basedbiophysicalmodel... 115

4.2. TheFitzHugh–Nagumomodel... 116

4.3. TheWilson–Cowanmodel... 117

4.4. TheKuramotomodel ... 118

4.5. Nodemodelinasynchronousstate... 118

4.6. Attractornetworkofspikingneurons... 119

4.7. TransformingneuronalactivityintoBOLDsignal... 121

4.8. Modellingtheimpactofstructurallesions... 122

5. Discussion ... 123

5.1. Experimentalevidenceofresting-stateactivity ... 123

5.2. Modelsofresting-stateactivity... 124

5.2.1. Theanatomicalconnectivity... 124

5.2.2. Thedynamicalregimeofbrainareas... 124

5.2.3. Timedelaysandnoise ... 124

5.2.4. Potentialsandpitfallsofthedifferentmodels... 125

5.3. Outlookandfuturedevelopments... 126

Acknowledgements... 126

References... 126

1. Brainactivityduringrest

Someone whois awake but notconsciously performing any task,physicalormental,issaidtoberesting.Inthisstate,unlike sleeping,thepersonisconsciousandreadytorespondpromptlyto anysort of external stimulation or cognitiverequirement.One couldsaythatthepersonissomehowonstand-by:althoughstill andquiet,sheisawake,readytosuddenlychaseaflythatlightly landson herarm,ortoimmediatelyturnherheadtowardsthe leastdisturbingsound.Notably,whilethepersonisrestingandthe body is static, thebrain instead seemsto beactively engaged, exhibitingslowspatiotemporallyorganizedfluctuationsof neuro-nalactivity.Thepatternsofbrainactivityobservedduringquiet wakefulrestaredistinguishablefromtheonesobservedduring goal-directedbehaviourorwhenthebrainfallsasleep(Decoetal., 2013a;Kalcheretal.,2013;Larson-Prioretal.,2011;Mennesetal., 2011).

Severalstudieshavespeculatedonthelinkbetweenthisresting brainactivityandunderlyinghigh-ordercognitiveprocessessuch asmoralreasoning,self-consciousness,rememberingpast experi-encesorplanningforthefuture(Buckneretal.,2008;Louetal., 1999; Morcom and Fletcher, 2007; Saxe and Kanwisher,2003; Wagneretal.,2005).However,findingsofrestingbrainpatternsin anesthetizedmonkeys(Vincentetal.,2007)and,morerecently,in rats(Luetal.,2012),pointstoamorefundamentaloriginofresting brainactivations(Fig.1)(evenifanimalsmayalsohaveaneedfor selfrepresentations).

Todate,theblood-oxygen-leveldependent(BOLD)signalused infunctionalmagneticresonanceimaging(fMRI)hasbeenthefirst andmostwidelyusedtechniqueinstudiesofbrainactivityduring rest (Biswal et al., 1995, 2010). But evidence of coordinated spontaneous activity has been detected in data collected with otherfunctionalimagingtechniquessuchasopticalimaging(Arieli etal.,1996),positron-emissiontomography(PET)(Raichleetal., 2001),electroencephalography(EEG)(Goldmanetal.,2002;Laufs etal.,2003;Moosmannetal.,2003),electrophysiology(Leopold etal.,2003)andmorerecentlymagnetoencephalography(MEG) (Brookesetal.,2011;dePasqualeetal.,2010).Allthesetechniques have their own specificities, sensitivities and spatio-temporal resolutionsand the data maybe affectedby differentkinds of physiologicalsignals andartefacts(see Section2).Assuch, itis crucialtotakeintoaccountthetypeofsignalbeingmeasuredby

each technique in order to understand the neurophysiological meaningof resting-stateactivityandhowit relates atmultiple spatialandtemporalscales.

Explorationsintotheorganizationofresting-stateactivityin the brain have revealed theexistence of temporally correlated activity–orfunctionalconnectivity–betweendifferentvoxelsin thebrain,somebelongingtobrainareasspecificfordifferenttasks, definingtheso-calledrestingstatenetworks(RSNs).Despitethe evidence fromtheempiricalside, thesignificance of functional connectivityinbrainactivityduringrestremainsunderdebate. Over the last years, a growing number of theoretical and experimentalstudieshaveaimedtoinvestigatetheoriginofthe correlationpatternsdefiningRSNsusingdifferentneuroimaging techniques. However, it is still not clearwhether RSNs are an epiphenomenon or not. Studies using diffusion-MRI to detect white matter pathwaysin the living brain have inspected the relationshipbetweenRSNsandtheunderlyingmapoflong-range axonalconnections(Hagmannetal.,2008;Spornsetal.,2000)(see Section 3). Importantly, a remarkable match has been found between the neuroanatomical network and resting-state func-tional connectivity, indicating that functional connections be-tweenbrainareasmaybemediatedthroughwhite-matterfibres. Bottom-up computationalmodels can be used tosimulate the interactions between brainareas inthestructural network and comparetheresultswithempiricalfunctionaldata.Althoughthe empiricaldataistypicallycontaminatedbyphysiologicalsignals (which varystronglyacrossthehumanbrainand thereforeare difficulttoremoveentirely),agoodfitwiththemodelresults(free fromanytypeofphysiologicalandbehaviouralartefacts)indicates that at least somepart of resting-state functional connectivity originatesfromneuralinteractionsinthewhite-matternetwork. InSection4wereviewtheexistingmodelsofresting-stateactivity obtainedthroughdifferentreductionlines.

To understand the mechanisms leading to the dynamics observed in the brain at rest, one can look at the brain as a dynamicalsystem.Indeed,thecomplexspace–timestructure of thebrain’swiringdiagram,togetherwithamyriadofbiochemical processes, form a dynamical framework capable of holding an infinite number of mental states over which cognition unfolds (Kelso,2012;Tononietal.,1994).Theexistenceofdifferent input-dependentstablestatesinthebrainhasbeenevidentsincethefirst humanelectrophysiologicalrecordings,whichrevealedthatstrong

alpharhythms(8–13Hz)weresubstitutedbybetarhythms(13– 30Hz)whensubjectsopenedtheireyes(Berger,1929).Overthe years,electrophysiological studieshave identified characteristic brain rhythms ranging from <0.1 to 600Hz that appear and disappearaccording to the mental state in which the brain in engaged(Buzsa´ki,2006;NiedermeyerandLopesdaSilva,2005). Moreover,withtheimprovementofneuroimagingtechniques,it becamepossibletomapthesourcesofsuchrhythmsacrossthe brain,resultinginatemporalandaspatialpatternforeachbrain regime.Notably, consistency wasfoundin thespatio-temporal signatureofbrainstatesacrosshealthyhumans.However,current neuroimaging techniques detect only epiphenomena of brain activity, such as blood oxygenation levels or as electrical discharges,and thecomplex neuralmechanismsat thegenesis oftheseobservationsremainuntiltodayincompletelyunderstood. Among all brain regimes, the resting state is particularly interestingfromtheperspectiveofdynamicalsystemsbecauseit seemstoexhibitanexploratorydynamics,inwhichoneormore states (i.e. RSNs) may be visited (i.e. activated) over time, but subsequentlydeactivated,neversettinginafixedpoint,resulting inanon-stationaryregime.Howsuchtypeofbehaviouremerges from the brain architecture, in particular from the interaction betweenneurons,maybeinvestigatedbymeansofcomputational models, bringing together concepts of neurophysiology and theoreticalphysics.In moredetail, thecollective dynamicsof a group of neurons may be represented as a dynamical unit (governed by a set of dynamical equations) with a particular behaviourinthespontaneousstate.Several ofthese dynamical units may be coupled together according to realistic brain connectivity. The dynamical repertoire of the system (i.e. the different regimes it may display depending on the model parameters)maybeinvestigatedthroughsimulations(ananalytic solutionisrarelypossibleduetothecomplexityofthesystem)in orderto determine theconditions under which an exploratory behaviour as the one observed during rest could emerge (see Section4).

2. Signaturesofresting-stateactivity

In the following we review some of the most prominent findings in the vast resting-state literature, including both haemodynamicandelectrophysiologicalstudies.

2.1. Resting-statehaemodynamicfluctuations

Whenunexpectedresultsemergefromascientific experimen-tation,historytellsusthattheyshouldnotbeleftunexamined. Thishappenedto Biswaland colleagues(1995)while tryingto identifythebrainregionsthatco-activatedduringbilateralfinger tapping.Againstanyprediction,theynotedthattheareasinvolved in finger movement exhibited correlated activations not only duringthetask,butalsoduringrest.Evenafterensuringthatthe handswereimmobilized,theyfoundslow(<0.1Hz)fluctuationsin theBOLD signalof the sensorimotorcortex that were strongly correlatedacrosshemispheres.Theexistenceofslowcorrelated fluctuations during rest was particularly disturbing for studies aimingtodetectonlytask-related(evoked)neuronalactivations sincetheyrelyonthecomparisonwithabaselineactivity.Indeed, this ongoing activity was foundto contribute for the variance observedinevokedcorticalresponses(Arielietal.,1996)(more recentobservationsindicatethatthisvarianceisalsointrinsically related tospontaneouspower fluctuations of ongoingneuronal oscillations(Becker etal., 2011)).In ordertotake into account thesetask-independentfunctionalactivationsinstudiesofevoked corticalresponses,itbecamenecessarytodefineabaseline–or defaultmode – ofbrain activity(Gusnardet al.,2001; Mazoyer et al.,2001; Shulmanet al., 1997). Later,Greiciuset al. (2003)

performedafunctionalconnectivityanalysisofthedefaultmode andprovidedevidencefortheexistenceofacohesivedefaultmode network(DMN).RegionswithintheDMNwerefoundtobemore functionallyconnectedduringpassiveresting-statethanduringa task.Forthisreason,theDMNiscalledatask-negativeRSN.Deeper explorationsintoresting-statedynamicsrevealedothergroupsof brainregionsexhibitingcorrelatedactivations,withBOLDsignal changes comparable to the ones found in task-related experi-ments. The mapping of such functional networks uncovered corticalsystems usually involved in active cognitive processes, suchasvision,language,movement,executiveprocessing,aswell asinthebasalganglia(Beckmannetal.,2005;Damoiseauxetal., 2006;DeLucaetal.,2006;Foxetal.,2005;Robinsonetal.,2009; Smithet al.,2009). UnliketheDMN,thesefunctional networks exhibitstrongerfunctionalconnectivitywhenengagedinatask andaredenominatedtask-positiveRSNs(seeFig.2).

Although fMRI has been widely used to study resting-state activity,itisimportanttonotethattheBOLDsignalisnotadirect

Fig.1.Comparisonofthedefaultmodenetwork(DMN)inrats,monkeysandhumans.TheregionscomposingtheDMNexhibitcorrelatedneuronalactivityduringrest. AdaptedfromLuetal.(2012).

measureofneuralactivity.FluctuationsintheBOLDsignalaredue tovariationsinthemagneticsusceptibilityofagivenvoxelinthe brain,whichinturnmayreflectaregionalchangeincerebralblood flowand/ordeoxyhemoglobinconcentration(Frahmetal.,1994; Logothetisetal.,2001;Ogawaetal.,1990).Raichleetal.(2001)

verified the neurophysiological basis of resting-state BOLD fluctuations bymeasuring theoxygen extraction fractionusing PET. However, the exact neural mechanisms at the genesis of resting-statedynamics(occurringatfastertime-scales)cannotbe accuratelyassessedusingthistechnique.Furthermore,theBOLD signal is sensitive to physiological noise (such as heart and respiratory signals, among others). The full removal of these signalsisnotyetachievable,whichresultsinalowsignaltonoise ratio(SNR)ofBOLDtime-series(Boubelaetal.,2013;Changetal., 2013;KimandOgawa,2012;Longetal.,2005;Triantafyllouetal., 2005).Nevertheless,theveryhighspatialresolutionofthefMRI technique(1mm)allowslocatingpreciselytheactivationsitesin thebrain,evenindeepsources.Inaddition,theanalysisof resting-state data recorded over long time periods (i.e. 20min) compensatesforthelowSNRoftheBOLDtime-series.

Overthelastdecade,therobustnessofresting-statedynamics hasbeen validated by consistency across healthy subjects and high-reproducibility across research groups (Cole et al., 2010). Recently, Biswal and colleagues assembled over 1400 healthy resting-statefMRIdatacollectedindependentlyat35international centresandperformedthebiggestfMRIstudyofthehealthybrain todate(www.nitrc.org/projects/fcon_1000)(Biswaletal.,2010). High reproducibility was found across studies and individuals revealing a universal architecture of positive and negative functionalconnectionsdespitethevariabilityinducedby physio-logicalsignalstotheBOLDtime-series.Inparticular,Kalcheretal. (2012)identified16networksconsistentoverthewholegroupof

1000subjects,someofwhichcloselymatchedtheRSNsmostoften reportedintheliterature.Inaddition,sexandagewerefoundto contributetointer-individualvariability. Sex-relateddifferences (alsoreportedinLiuetal.(2009)with300subjects,43%men)are supposedly related to phenotypic variations due to the sexual dimorphism in genomic expression. On the other hand, lower functional connectivity wasobserved in advanced ageing (free from Alzheimer’s disease) suggesting that the disruption of resting-statefunctionalconnectivitymayberelatedtocognitive decline(Andrews-Hannaetal.,2007;Kringelbachetal.,2011).

2.2. Electrophysiologicalsignaturesofresting-stateactivity

TherelationshipbetweentheBOLDsignalandtheunderlying neural activity was deeply investigated by Logothetis and colleagues(2001),whoperformedsimultaneousfMRIand intra-corticalelectrical recordings in anesthetized monkeys,showing thatspontaneousfluctuationsinlocal-fieldpotentialscorrelated positivelywiththelocalBOLDsignalaroundtheelectrode. Intra-corticalelectrophysiologicalrecordings,aswellasEEGandMEG, aretechniquesthatregisterthevoltagefluctuations(witheither electric ormagneticsensors) resultingfromionic currentflows that occur when a large number of neighbouring neurons dischargesimultaneouslyandthereforearemoredirectlyrelated toneuralactivitythantheBOLDsignal(Hansenetal.,2010).

IfthespontaneousBOLDfluctuationsareindeedareflectionof underlyingneural activity,we canexpect somecomponentsof electrophysiologicalsignalstoexhibitspontaneouslow-frequency fluctuationswithlarge-scalecorrelationpatternssimilartothose observedwithresting-statefMRI.Asthesemeasureshaveabetter temporalresolution(intherangeof1kHz)theyarevaluabletools toperformtemporalandspectralanalysisofresting-stateactivity.

Fig.2.Consistentresting-statenetworksacrosshealthysubjectsdetectedwithfMRI.10distinctpatternswithpotentialfunctionalrelevanceweredetectedusingtensor probabilisticindependentcomponentanalysis,consistingofregionsknowntobeinvolvedinmotorfunction,visualprocessing,executivefunctioning,auditoryprocessing, memory,andthedefault-modenetwork,eachwithBOLDsignalchangesupto3%.Allfunctionalimageshavebeenco-registeredintoastandardstructuralMRItemplate (MNI).

counterpartofBOLDsignalfluctuationsusingsimultaneous record-ingsof fMRIandEEGsignals(EEG–fMRI) inrestinghumans.Goldman etal.(2002)mappedtheregionswhoseBOLDsignalchangedreliably withmodulationinposterioralphaactivity,findingthatincreased alphapowerwascorrelatedwithdecreasedMRIsignalinmultiple regionsofoccipital,superiortemporal,inferiorfrontal,andcingulate cortex, and with increased signal in the thalamus and insula. Subsequentstudieshavecorroboratedthesefindings,pointingto an inverse relationship between occipital alpha activity and the BOLD signalinthecortex,suggestingthatalphaactivityintheoccipital cortexmaybeassociatedwithmetabolicdeactivation(deMunck et al., 2007; Difrancesco et al., 2008; Feige et al., 2005; Goncalves et al., 2006;Laufsetal.,2003;Moosmannetal.,2003).Consideringthe pericentral(rolandic) EEGrhythms,Ritterand colleagues(2009)

foundthat the rolandicalpha and beta spectralpowers wereinversely correlated with the BOLD signal in the postcentral cortex and precentral cortex, correspondingly. These findings point to the directionthatEEG‘‘idlerhythms’’maybeassociatedwith lower metabolicactivity.Exploringawiderrangeoffrequencies,Laufsand colleagues(2003)filteredtheEEGsignalsintodistinctfrequency bands andcompared themodulations of thepowerin each frequency band(orband-limitedpower,BLP, fluctuations,seeFig.3foran illustration)withtheBOLDtimecourses.Theyfoundthatthepowerof 17–23Hzoscillations(inthebeta-band)waspositivelycorrelated with the haemodynamic fluctuations found in the posterior cingulate,the precuneusandthe leftandrighttemporo-parietal and dorsomedial prefrontal, which are similar to the regions identifiedbyGreiciusetal.astheDMN(Greiciusetal.,2003).These findings suggest that fluctuations in BOLD signal at rest may at least in partreflectband-limitedpowerfluctuations ofneuronalactivity happeningat fasterfrequencies.However, EEGsensorsfailto capture signalsfromsubcorticalregionswhichinturnaremoreproneto physiological fluctuations. As such, the correspondence between EEG power and BOLD signal may vary strongly across the brain and should berestrainedtothecorticalsurface.

For a deeper exploration of power fluctuations during rest, Leopoldandcolleaguesusedmultipleelectrodestorecordneural

frequency range. They proposed that such power fluctuations might make a significant contribution to the high amplitude fluctuationsobservedinthetimecourseofrestingstatesignals obtainedwithfMRI.

In2005,developmentsinresting-statefMRIstudiesrevealed theexistenceofrobustRSNscharacterizedbyparticulartemporal signalsusingindependent componentanalysis(ICA)(Beckmann etal., 2005). Toinvestigateifthetemporal signalsof eachRSN could be related to EEG power fluctuations in a particular frequencyband,Mantinietal.(2007)recordedsimultaneousfMRI andEEGand,applyingICAtotheBOLDsignalsidentifiedsixrobust RSNs over a groupof 15 healthy subjects.The temporal signal associated witheach RSN (i.e. thecorresponding temporal ICs) werethencorrelatedwiththeEEGpowervariationsofdelta(

d

), theta(

u

),alpha(

a

),beta(

b

),andgamma(

g

)rhythms.Theyfound thateachRSNwascharacterizedbyaspecificelectrophysiological signature that involved the combination of different brain rhythms.Forexample,inagreementwiththeresultsfromLaufs etal.(2003)theirfindingsindicatethatthefunctionalactivationof theDMNcorrelatesbetterwithbeta-frequencyEEGpower(13– 30Hz). However, due to the low spatial resolution of EEG, investigations were limited to temporal signals and did not explore the correspondence between the cortical maps of EEG rhythmsandthespatialmapsoftheRSNsdetectedwithfMRI.

Later,ShmuelandLeopoldsimultaneouslyrecordedfMRI time-seriesandintra-corticalelectrophysiologicalsignalsfrom1single recording site in the visual cortex of anesthetized monkeys (Shmuel and Leopold, 2008). They demonstrated correlations betweenslowfluctuationsinBOLDsignalsinwidespreadareasin visual cortex of both hemispheres and concurrent neuronal activityfromtherecordingsite,wheretheneuralsignalconsisted of either the spiking rate of a small group of neurons or LFP fluctuations (especially in thegammaband). Using intracranial electrophysiologicalrecordingsinhumans,Niretal.(2008)found slow (<0.1Hz, following 1/f-like profiles) spontaneous fluctua-tions of neuronal activity with significant inter-hemispheric correlations.Thesefluctuationswereevidentmainlyinneuronal firingratesandingamma(40–100Hz)LFPpowermodulationsin agreement with the results from Shmuel and Leopold (2008). Furthermore, multiple intracranial recordings revealed clear selectivity for RSNs in the spontaneous gamma LFP power modulations(Heet al., 2008;Miller et al.,2009). Theseresults pointtoslowspontaneousmodulationsinfiringrateandgamma LFPasthelikelycorrelatesofspontaneousfMRIfluctuationsinthe humansensorycortex.

Finally, a recent study investigated how theBOLD signalin humans performing a cognitive task is related to neuronal synchronization across different frequency bands (Scheeringa etal.,2011).TheauthorssimultaneouslyrecordedEEGandBOLD whilesubjectsengagedinavisualattentiontaskknowntoinduce sustainedchangesinneuronalsynchronizationacrossawiderange of frequencies. Trial-by-trial BOLD fluctuations correlated posi-tivelywithtrial-by-trialfluctuationsinhigh-EEG gammapower (60–80Hz)and negatively withalphaand betapower. Gamma powerontheonehand,andalphaandbetapowerontheother hand,independentlycontributed(inananti-correlatedmanner)to explaining BOLD variance. These resultsindicate that neuronal dynamics underlying high- and low-frequency synchronization contributedifferentlytotheBOLDsignal.

Fig.3.Amplitudefluctuationsofband-passedoscillationsatdifferentfrequency bands.Theamplitudeenvelope(red)ofoscillationsfilteredinarestrictedfrequency range(black)areproportionaltothesquaredrootofthepower.RecentEEGand MEGstudiespointtoarelationshipbetweentheseenvelopefluctuationsandthe BOLDsignalfluctuationsobservedduringrest.However,thefrequencyrange(or carrierfrequency)thatbestcapturesBOLDfluctuationsremainsunderdebate.

2.3. Detectionofresting-statepatternsusingMEG

Perhapsthefirststudydedicatedtoinvestigatetheneuronal correlates ofrestingBOLDsignalusing MEG wasperformed by Nikouline and colleagues in 2001 (Nikouline et al., 2001). The authors investigated inter-hemispheric phase synchrony and amplitudecorrelationofbetaoscillationsinarestingcondition. Betaoscillationsintheleftandrighthemispherewerefoundto exhibit transientsynchronizedactivity. Importantly,the ampli-tudeoftheseoscillations(whichisproportionaltothesquaredroot of thepower) waslarger when thesynchronization index was strongest, and correlated across hemispheres over long time intervals (>1s). The authors suggested that the low-frequency amplitudemodulationofspontaneousrhythmicactivitymaybe the source of correlations of low-frequency haemodynamic responses,generally interpretedasfunctional connectivity.This workappearstobethefirstdemonstrationofband-limitedpower correlationsbetweenthetwohemispheresduringrest.

Almostadecadelater,andfollowingtheelectrophysiological studiesindicating thatspontaneousBOLDfluctuations couldbe drivenbyslowamplitudemodulationsofneuraloscillations,Liu etal.(2010)characterizedthepowermodulationsofspontaneous MEGrhythmsrecordedfromhumansubjectsduringwakefulrest (witheyesopenandeyesclosed)andlightsleep.Throughspectral, correlationandcoherenceanalyses,theyfoundthatresting-state MEG rhythms demonstrated ultraslow (<0.1Hz) spontaneous powermodulationsthatsynchronizedoveralargespatialdistance, especially between bilaterally homologous regions in opposite hemispheres. Their observations suggest that coherent power modulationsofspontaneousrhythmicactivity(especiallyinthe

b

-band)reflecttheelectrophysiologicalsignatureofthelarge-scale functionalnetworks.A coupleof monthslater dePasquale and colleaguesused a seed-basedmethod to characterizethe MEG signaturesof twowell-characterized RSNs:thedorsal attention andthedefaultmodenetworks(dePasqualeetal.,2010).Taking intoaccountthenon-stationarityofMEGactivity,theyfoundthat theband-limitedpoweroftheRSNseedswereonlysynchronized forrestrictedperiodsintime,resultinginatransientformationof RSNs. Their results indicate that RSNs manifest in MEG as synchronousmodulationsofband-limitedpowerprimarilywithin thetheta,alpha,and betabands, whichcorrespond torhythms slowerthanthe

g

-frequencyoscillationsgenerallyassociatedwith the electro-physiological correlates of event-related BOLD responses.

ThefirstworktoshowthatMEGcanindependentlydetectthe spatial patterns of RSNs, in the same manner that has been demonstratedin fMRI, wasrecentlyperformed by Brookes and colleagues(Brookesetal.,2011)Asafirststep,theMEGdatawas frequencyfilteredinto bandsof interest(

d

,

u

,

a

,

b

,and

g

)and projectedintosourcespaceusingabeamformerspatialfilter.Then, ICA was applied to the band-limited amplitude fluctuations, resultingin8RSNswithsignificantsimilaritytotheRSNsderived independentlyusingfMRI(seeFig.4).Importantly,most resting-statenetworkswerelinkedto

b

-bandamplitudefluctuations.This outcomeconfirms the neuralbasisof haemodynamic networks anddemonstratesthepotentialofMEGasatoolforunderstanding themechanismsthatunderlieRSNs.

In a recent MEG study, Hipp and colleagues found that spontaneous oscillatory neuronal activity exhibits frequency-specific spatial correlationstructure in thehuman brain (Hipp et al., 2012). To discount spurious correlationof signal power caused by linear signal leakage, a new analysis approach was developedinwhichthesourceestimatesofspontaneousneuronal activityreconstructedfromMEGareorthogonalized,discardingin this way in-phase relationships. They found power envelope correlationsbetweenhomologousearlysensoryareas(seeFig.5).

Overall,correlationofpoweracrosscorticalregionswasstrongest in thealphatobetafrequency range(8–32Hz)and correlation patterns depended on the underlying oscillation frequency. Furthermore, theyidentifiedfrequency-specific hubsresiding in themedialtemporallobeforthethetafrequencyrange(4–6Hz),in lateralparietal areasfor thealphatobetafrequency range (8– 23Hz) and in sensorimotor areas for higher frequencies (32– 45Hz).Theseresultsreinforcetheideathatinteractionsinvarious large-scale cortical networks may be reflected in frequency-specificpowercorrelations.

MEGhasahigherspatialresolutionthanEEGbecauseofthe highersensitivityofMEGsensorsandbecausemagneticfieldsare lessdistortedbytheskullandscalpthanelectricfields(Hansen et al.,2010). Makingtheappropriatecorrectionstoaccount for linearsignalleakage(Brookesetal.,2012;Hippetal.,2012)the spatialresolutionofMEGcanbeaspreciseas2mmforsourcesin the corticalsurface. However, MEG detects onlythe tangential components of cortical activity (from the sulci) and the local magneticfieldsoriginatingindeep(subcortical)sourcesmaybe distorted,limitingspatialaccuracy.Assuch,theanalysisof resting-stateMEGdatahasoftenbeenrestrictedtocorticalareas.

2.4. Electrophysiologicalcorrelatesofresting-stateBOLDfluctuations

All electrophysiological studies presented in the previous sections have aimed to investigate the neurophysiological counterpart of resting-state BOLD fluctuations. Overall, results indicatethatresting-stateBOLDfluctuationsaredrivenbyslow modulationsofneuralactivity,whicharemanifestedbytransient increasesintheamplitude(orinthepower)offastoscillationsina certainfrequencyrange.

However, regarding thequestionwhether thereis a specific carrier frequency whose power modulations serve as the best candidatefortheneuronalcorrelatesofspontaneousBOLDsignal fluctuations,thereseemstobeadebate.Ononeside,the intra-corticalrecordings performedbyLeopoldetal.(2003),Niretal. (2008)andShmuelandLeopold(2008)proposeadirect relation-shipwiththepower ofgamma-frequencyoscillations.Although restrictedtotherecordingsite,theirstudiesindicatethattheBOLD signalispositivelycorrelatedwiththelocalgammaLFP.Tothe extentthattheirfindingscanbegeneralizedtoothercorticalareas, thesefindingsindicatethatBOLDfunctionalconnectivitybetween remote regions during rest can be linked to slow increases in neuronalactivationlevelshappeninginthegamma-band.Indeed, the elevated energy requirements of high-frequency neural oscillationsrepresentamechanisticlinkbetweenthefunctional connectivity of brain regions and their respective metabolic demands(detectedintheBOLDsignal)(Lordetal.,2013).

On the other hand, most of the studies using external recordings,suchasEEGandMEG(Brookesetal.,2011;deMunck etal.,2007;dePasqualeetal.,2010;Difrancescoetal.,2008;Feige etal.,2005;Goncalvesetal.,2006;Hippetal.,2012;Laufsetal., 2003; Liuet al.,2010; Moosmannet al.,2003;Nikoulineet al., 2001)haveidentifiedcorrelatedfluctuationsinthepower(orthe amplitude)ofoscillationsinthealphaandbetafrequencyranges, revealing functional networks that closely match the ones obtainedinfMRIstudies.Thisrelationshipwasinvestigatedusing simultaneous recordings of EEG–fMRI and results point to an inversecorrelationbetweentheBOLDsignalandthepowerinthe alphaband,indicatingthatthese‘‘idlerythms’’maybeassociated withdeactivationanddecreasedmetabolicrate.

Sinceeachtechniquehasintrinsiclimitationsandtheresults arenotexclusive,bothscenarioscouldcoexist.Indeed,according totheresultsfromScheeringaetal.(2011),anegativecorrelation existsbetweenthepowersathighandlowfrequencies,indicating thattheBOLDsignalmightbeassociatedtointeractionsbetween

gamma-band activity(related to increasedmetabolic rate) and alpha/beta-frequency oscillations (decreased metabolic rate). Furthermore, rhythms from different frequency bands lead to different correlation patterns, suggesting a complex interplay between multiple frequency bands (de Pasquale et al., 2010; Mantinietal.,2007)whichinturnmayreflectdifferentaspectsof neuronalprocessing (Palva and Palva, 2012;Womelsdorfet al., 2007). However, an accurate answer requires simultaneous recordingsofboth fMRIandelectro-physiologicalsignals across the whole brain, which is unavailable in the present days. Alternatively,onecanexplore themechanismat thegenesisof these slow electrophysiological power fluctuations and their relationship with the BOLD signal by means of computational models,whichservetotesttheoreticalpredictions(inSection4we explorethesemodelsindetail).

2.5. Alteredresting-stateactivityindisease

Overthelastdecade,alargenumberofstudieshavereported alteredrestingbrainactivityinawiderangeofmentalillnesses.

Theseresultsnotonlyillustratetheimportanceofbalancing resting-statedynamicsforanoptimalcognitivefunction,butalsoprovide insights tounderstand the intrinsicmechanisms leadingtoand potentiallytreatingthediseased brain(Kringelbachetal.,2011). However,itisimportanttonotethatsubstantialmethodological limitationsarisewhenrecordingdatafrompatientswith psycho-logicaldisordersduetohighermotionandphysiologicalartefacts. ThecorrectionoftheseartefactslikelyreducesanysignificantBOLD signalandthereforethealterationsreportedinresting-stateactivity indiseasemustbediscussedwithcaution.

InAlzheimer’sdisease(AD),forexample,functional connectiv-ityduringrestwasfoundtodeterioratewithinallsystemswiththe progressionofthedisease(Binnewijzendetal.,2012;Damoiseaux etal.,2012;Greiciusetal.,2004;Zhouetal.,2010).Furthermore,

Supekaret al.(2008)characterized resting-statefunctional net-works using graph theory and found that functional brain networksinADshowedlossofsmall-worldproperties, character-izedbyasignificantlylowerclusteringcoefficient.

Regarding schizophrenia, several studies have reported a widespread decrease in the functional connectivity of patients

Fig.4.ComparisonofbrainnetworksobtainedusingICAindependentlyonMEGandfMRIdata.(A)DMN;(B)leftlateralfrontoparietalnetwork;(C)rightlateralfrontoparietal network;(D)sensorimotornetwork;(E)medialparietalregions;(F)visualnetwork;(G)frontallobesincludinganteriorcingulatecortex;(H)cerebellum.Allnetworkswere identifiedusingb-bandpowerfluctuations,excepttheDMN,whichwasdetectedusinga-bandpowerfluctuations.

AdaptedfromBrookesetal.(2011).

Fig.5.Powerenvelopecorrelationsbetweenorthogonalizedspontaneoussignalsfromhomologousearlysensoryareas.(a)Inter-hemisphericcorrelationsbetweenauditory (b),somatosensory(c)andvisual(d)cortices,asafunctionofthecarrierfrequency.Correlationsareincreasedinthealphaandbetafrequencyranges(8–30Hz). AdaptedfromHippetal.(2012).

duringrest(Bassettetal.,2012; Liangetal.,2006;Lynalletal., 2010; Skudlarski et al., 2010), supporting the hypothesis that schizophreniamayarisefromthedisruptedfunctionalintegration ofsegregatedbrainareas.Furtheranalysisofresting-stateactivity inschizophreniausinggraphtheoryindicateasubtle randomiza-tionoffunctionalnetworks,withdecreasedsmall-world proper-ties,lowerclusteringcoefficientsandlesshigh-degreehubs(Liu etal.,2008;Lynalletal.,2010;Bassettetal.,2012).

Resting-statealterationshavebeenfoundinmanyothermental illnessesincludingdementia(Buckneretal.,2000;Romboutsetal., 2009), autism (Cornew et al., 2012; Cherkassky et al., 2006; Kennedyet al., 2006;Lai et al.,2010; Wenget al.,2010), mild cognitiveimpairment(Romboutsetal.,2005),multiplesclerosis (Bonavitaetal.,2011;Faivreetal.,2012;Schoonheimetal.,2013) andmajordepression(Greiciusetal.,2007;Shelineetal.,2009; Veeretal.,2010).Despitethestrongevidenceofdisrupted resting-stateactivityindisease,theuseofresting-statedataforclinical purposesisstill inprogresssince it requiresconsistencyacross studiesbeforeitcanbeusedinameaningfulwayatthe single-patientlevel(Greicius,2008;FoxandGreicius,2010).

Insomecases,resting-statefunctionalconnectivitywasfound tocorrelatewithcognitiveperformance(i.e.Lynalletal.,2010), which indicates that resting-state correlations may be closely relatedtothebindingmechanismsthatsupporttheintegrationof informationinthebrain.

Overall, more detailed investigations of the highly coherent functional and structural brain networksin health and disease have the potential not only to increase our understanding of fundamental brain function but of how best to modulate the balance.Inparticular,someoftheproblemsfoundin neuropsy-chiatricdisorderscouldinformfuturetreatment,e.g.potentially withdeepbrainstimulation(Kringelbachetal.,2011).

3. Networksofthebrain

Mappingthebrainiscomplexbutishelpedbythefactthatthe brain,inthesameway asmanybiological, socialand chemical systems, is a network composed by a large number of inter-connecteddynamicalunits.Themapofneuralconnectionsinthe brainhasbeencalledtheConnectome(Hagmann,2005;Sporns etal.,2005),aterminspiredbytheeffortinmappingthehuman geneticcode,orGenome.

Connectionsinthebrainmayrefertoanatomicallinks,suchas synapsesorfibrepathways(anatomical/structuralconnectivity), statisticaldependencies suchascorrelationorcoherence (func-tional connectivity) orto directed causalinteractions (effective connectivity)betweendistinctunitswithinanervoussystem.Such unitsmaycorrespondtoindividualneurons,neuronalpopulations, oranatomicallysegregatedbrainregions.

Exploringthewhole connectomeata cellularscaleishighly complexand hardly tractable,since thehumancerebral cortex containsatleast1010_{neuronslinkedby10}14_{synapses(Sporns,}

2009).Theonly organismfromwhich thefullmap ofneuronal connectionsis known inits entirety istheone millimetre-long wormCaenorhabditiselegans,butittookoveradecadetocomplete theidentificationofits300neuronsand7000connections(White etal.,1986).

Inthecontextofresting-stateactivity,itis convenienttogo beyondthecellularscaleandfocusessentiallyonthelong-range whitematterpathwaysconnectingsegregatedbrainareas.These long-rangeconnectionsare particularlyimportantbecausethey stronglyconstrainhowbrainregionscommunicate,definingthe wiring diagram over which spatiallysegregated information is integratedinthenervoussystem(Jirsaand Kelso,2000;Sporns et al., 2000; Tononi et al., 1992, 1994). Spatially distributed and functionally specialized brain areas are continuously

communicating and co-operating to respond to cognitive demands, perceive sensory stimuli and generate coordinated movement. Although the neuronal mechanisms under which corticalregionscommunicateremainunclear,the neuroanatomi-calnetworkisbelievedtoserveasthestructuralsubstrateupon which coordinated functional integration occurs (Bressler and Tognoli,2006;Ghoshetal.,2008a;JirsaandMcIntosh,2007;Knock etal.,2009;Spornsetal.,2000).

Combiningfunctional and structural imaging modalities has revealed thatresting-state functionalnetworksreflect, tosome extent, theunderlyingstructural connectivity(Damoiseauxand Greicius,2009;Greiciusetal.,2009;vandenHeuvelandHulshoff Pol, 2010; van den Heuvel et al., 2009a). However, the bond betweenstructuralandfunctionalconnectivityisnot straightfor-ward: although structural connectivity is a good predictor of functional connectivity – i.e. if there is a direct anatomical connection,thereislikelyafunctionalconnection–theoppositeis not necessarily true. Indeed,robust functional connectivityhas beenobservedin theabsenceof adirect anatomicallink(Koch etal.,2002).Recentstudiessuggestthatresting-statefunctional connectivity also reflects polysynaptic anatomical pathways relatedtocerebro-cerebellarcircuits(Buckneretal.,2011;Habas et al., 2009; Krienenand Buckner,2009; O’Reilly et al., 2010). Indeed,itwasfoundthatspontaneousfluctuationsinthemotor cortexarefunctionallycoupledtotheirtopographically appropri-atecontra-lateralregionsintheanteriorlobeofthecerebellum (Buckneretal., 2011). Furthermore, patientswithfocal pontine strokes, were found to display selectively reduced functional connectivitybetweenthemotorcortexaffectedand the contra-lateral cerebellum, providing additional evidence that resting-state functional connectivity reflects polysynaptic anatomical pathways(Luetal.,2011).

Consideringthatcoupledbrainregionshaveacausal relation-ship and the connectivity is directed, the functional network architecture may be inferred using Granger Causality (GC) or Dynamic Causal Modelling (DCM) (Friston et al., 2003, 2013; Kasessetal.,2010;Liaoetal.,2010;Pennyetal.,2004;Stephan etal.,2008).Thesemethodsareprincipallyusedtotestorpredict dynamical responses to causal events (Marreiros et al., 2010). Recently,Fristonetal.(2011)proposedanextensionto conven-tional DCM toinclude endogenous or random fluctuations and allowinthis waytheapplicationofDCMtoresting-stateBOLD timeseries.However,itsapplicationtoresting-statedataisstill limitedandsofarrestrictedtotheDMN(DiandBiswal,2013;Li etal.,2012).

While the origin of resting-state functional connectivity remains unclear, computational models of neural network dynamics(whichwedescribeindetailinSection4)arevaluable toolstoinvestigatetheroleoftheanatomicalnetworkinshaping functionalconnectivityduringrest(Fig.6)(Ritteretal.,2013).

3.1. Brainparcellation

Thefirststeptodefinebrainnetworksistodefinethenodesin thenetwork.Todoso,itisnecessarytodividegreymatteratthe desired scale according tospecific strategies for anatomical or functional partitioning. Severalbrain parcellationtemplates are availableintheliterature,rangingfromthecanonicalclassification intolobes(i.e.frontal,parietal,temporalandoccipitallobes)toas muchasseveralthousandregionsofinterest.Oncetheparcellation schemeisdefined,differentstrategiescanbeusedtodefinethe linksbetweennodes.

Oneofthefirstandmostwidelyknownparcellationschemesof the human cortex was performed by the German anatomist

Brodmann (1909) based on the cytoarchitectural organization of neurons. In 2002, a parcellation scheme was proposed by

Tzourio-Mazoyer et al. with theintention of standardizing the anatomical labelling of brain regions in neuroimaging studies (Tzourio-Mazoyer et al.,2002). Automated anatomicallabelling (AAL)wasperformedon abraintemplate consistingona high-resolutionstructuralMRIscanfromahealthymalesuppliedbythe MontrealNeurologicalInstitute(MNI)(Collinsetal., 1998).The corticalandsubcorticalgreymatterofthestandardMNIbrainwas dividedinto90brainregions(45ineach hemisphere).TheAAL parcellation template is included in the Statistical Parametric Mapping(SPM)package(Fristonetal.,1994)andisfreelyavailable totheneuroimagingcommunity.Todate,morethan2000research articleshaveusedtheAALparcellationtemplate(basedoncitation index),includingstudiesofresting-statefunctionalconnectivity (Achardand Bullmore,2007; Achardet al.,2006; Bassettetal., 2012; Braun et al., 2012; Liu et al., 2008; Lynall et al., 2010; Salvadoret al., 2005b; Sanz-Arigita etal., 2010; Supekaret al., 2008).

Anotherparcellationschemeusedinstudiesoflarge-scalebrain connectivityistheonecreatedbyHagmannandcolleagues(2007). Grey matter was divided into 66 cortical regions and then individually subdivided into small regions of interest (ROIs) resultingin998ROIs,eachcovering1.5cm2_{ofthecorticalsurface.}

Twolabelledmeshes(onemeshwiththe66regionsandanother withthe998ROIs)werecreatedontheaveragebrainandwere registeredonto thebrain of individual participantsusing Free-surfer (surfer.nmr.mgh.harvard.edu). Recently, the same group proposedarobustmethodforconstructingnormalized parcella-tionsatdifferentscales,iterativelyregroupingthe998ROIsinto biggerROIs,resultingin5scalesofcortexparcellation,e.g.66,133, 241,483and998regions(Cammounetal.,2012).

Theanalysisofbrainnetworksmayalsobeperformedatthe voxellevel,i.e.thesmallestcubicelementinavolumetricimage. Theexactsizeofa voxelwillvarydependingonthetechnology used.Typically,fMRIvoxelsrepresentavolumeof27mm3(acube with3mmlengthsides),whichcontainabout1millionneurons.In

vandenHeuveletal.(2009b),functionalnetworkpropertieswere explored at thevoxel scale (64mm3_{), resulting in graphs with}

9500nodes.Inanotherworkfromthesamegroup,RSNswere

defined by applyinga voxel-based clustering method(vanden Heuvel et al., 2009a). In large-scale models, however, the simulationofsystems withseveralthousandsofinterconnected nodesishighlyexpensivefromthecomputationalperspectiveand low-dimensionparcellationschemeshavebeentypicallyusedin modelsoflarge-scaleresting-stateactivity.

3.2. Anatomicalstructuralnetworks

Historically, the mapping of white matter connections was performedpost-mortembyhistologicaldissectionandstaining,by degeneration methods or by axonal tracing. These classical techniques,however,weretime-consumingandgenerallyapplied torestrictedareasinthebrain.Inanattempttoassembledatafrom differentwhitemattertracingstudiesofthemacaquebrain,Kotter (2004)createdtheonlinedatabaseforthe‘‘Collationof Connec-tivity on the Macaque brain’’ (CoCoMac www.cocomac.org), allowing for continuous updating and refinement of such anatomical connection maps. From this database, Ko¨tter and Wankederivedarealisticconnectivitymapofonehemisphereof theprimatebrain,proposingacoarseparcellationoftheprimate cerebralcortexinto38regions,whichreflectedbroadandrather uncontroversialdivisions(Fig.7)(KotterandWanke,2005).This connectivitymaphasbeenusedinmodelsoflarge-scale resting-statebrainactivity(Decoetal.,2009;Ghoshetal.,2008a;Honey etal.,2007)revealingthatthelarge-scaleconnectivitytopologyof cerebralcortex,togetherwithtimedelaysandinthepresenceof noise, defines a dynamic framework from which spontaneous patternssimilartotheonesobservedintherestingbrainemerge. Forhumanbrains,however,themanualtracingofwhitematter pathwayshasnotbeenperformed.Someworkshavetriedtoinfer human anatomical networks from cortical thickness/volume measurementsobtainedwithstructuralMRI.Thisapproachrelies on thefact that corticalvolumeis strongly correlatedbetween regionsthatareaxonallyconnected(Lerchetal.,2006).Inthisway, awhole-brainnetworkcanbeinferredfromhumanMRIdataby estimating,foreachbrainarea,thecorticalvolume(orthickness) foralargesampleofsubjectsandthencomputingthecorrelation

Fig.6.Exploringthestructure–functionrelationshipusingcomputationalmodels.Diagramillustratinghowcomputationalmodelscanservetoexploretherelationship betweenanatomicalstructuralnetworksandrestingfunctionalnetworks.Thedashedline(--)indicatesthefeedbackofthemodel’sperformance,thatisnecessarytotune theparametersofthemodel.

Fig.7.TheMacaqueconnectivity.ThecouplingweightsCnpindicatethestrengthofconnections(classifiedasweak(1),medium(2)orstrong(3))fromregionpton(directed). TheCoCoMacdatabase,createdbyRolfKo¨tter,representsoneofthefirstattemptstoobtainalarge-scaleanatomicalconnectomeofthemammalianbrain.

matrix(Bassettet al.,2008; Chenetal., 2008;Heet al.,2007). However, thereis still no direct proofthat correlations ofgrey mattervolumeoverthewholebrainacrosssubjectsareindicative ofaxonalconnectivity.

Inrecentyears, arevolution inconnectomics(i.e.thescience concernedwithassemblingandanalyzingconnectomedatasets,

Hagmann, 2005) has happened helped by the technological advancementsin Diffusion MRI. This methodpermits the non-invasivedetectionofwhitematterfibrepathwaysusingaspecific MRI sequence that is sensitive to water anisotropy, i.e. the directionofwaterdiffusioninabody.Thedetectionoffibretracts withdiffusionMRIreliesonthefactthatwaterpropagatesalong theorientationofthefibresbecausethemyelinsheathprovidesa barrierperpendiculartothefibres.Diffusiontensorimaging(DTI) (Wedeenetal.,1995)estimatesthemaindirectionandstrengthof anisotropic diffusion in each voxel while diffusion spectrum imaging(DSI)exploresthestrengthofanisotropyinalldirections, allowingthecrossingofmultiplefibresinasinglevoxel(Wedeen etal.,2005).

Oncediffusionimagesareobtained,thedetectionoffibretracts, or tractography, is obtained by constructing three-dimensional curves of maximal diffusion coherence using computational algorithms. Together withefficient computational tractography algorithms,thistechniquepermitstherapidandalmostautomatic constructionof comprehensive maps ofbrain connectivity.The Human Connectome Project ( www.humanconnectomeprojec-t.org) aims to provide an unparalleled compilation of neural connectivitydatabasedonDiffusionMRIstudies(Fig.8).

Todefinealow-resolutionstructuralnetworkusingDiffusion MRI,theconnectionstrengthCnpbetweentwobrainareasnandp

isscaledbythenumberofwhitemattertractsdetectedbetween voxelsineacharea,whichcanrangefrom0toasmuchasseveral thousandtractsperconnection.

Thefirstmappingsofhumanwhole-brainanatomical connec-tivity usingDiffusion MRItractography wereperformedalmost simultaneouslybyHagmannand colleagues(2007)and Iturria-Medina and colleagues (2007). In Hagmann et al. (2007), anatomical networks were composed by 500–4000 nodes and werederivedfromthebrainsof2healthysubjects.Subsequently, low-resolutionparcellationwasproposed,thistimedividingthe cortex into 66 regions (Fig. 9) (Hagmann et al., 2008). This anatomicalnetwork was used in the first large-scalemodel of humanresting-statefunctionalconnectivity(Honeyetal.,2009). Iturria-Medinaetal. analyzed anatomicalconnectivityat lower resolution, first focusing on the connectivity between 71 grey matterstructures,across5healthysubjects(Iturria-Medinaetal., 2007)extendedlaterto20healthyparticipantsand90corticaland subcortical regions according to the AAL parcellation scheme (Iturria-Medinaetal.,2008).Todate,thelargestsampleofhealthy

structuralbrainnetworksusingDTIandtheAALparcellationwas from80youngadults(Gongetal.,2009).

3.3. Dynamicfunctionalnetworks

Thesimultaneousactivationofspatiallysegregated functional-lyspecializedbrainregionsdefinesafunctionalnetwork,andthese regions aresaid tobefunctionally connected.Unlike structural networks,functionalnetworksareonlyactiveforaperiodintime, representinga transientbrainstatewhere differentbrainareas activatesimultaneously,supposedlyintegratingsegregated infor-mation(Tononietal.,1998,1994).

Evokedfunctionalnetworkscanbeeasilycapturedby compar-ing a measure of brain activity during a particular task with baseline data sets. During rest, however, the definition of an activationparadigmbecomesunfeasible.Still,evenwhenno task-relatedprocessistriggeringtheactivationofacertainfunctional network,severalstudieshavereportedthetransientactivationof distinctfunctionalnetworksduringrest.Functionalconnectivity can be inferred from resting-state recordings using different approaches, namely correlation measures, ICA, PCA, mutual information,covarianceandcoherenceanalysis.

Theclassicandmostwidelyusedmethodtoinferthestrength of network interactions (or functional connections) consists in estimating the linear (Pearson) correlation coefficient between temporalsignals(Bandettinietal.,1993;Biswaletal.,1995).Iftwo regionsactivateanddeactivateatthesametime,thereislikelya functionalconnection.ThePearsoncorrelationcoefficientbetween twoseriesXandYofsizeNisgivenbythefollowingequation:

r

X;Y¼ PN n¼1ðXn ¯XÞðYn ¯YÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN n¼1ðXn ¯XÞ 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_P N n¼1ðYn ¯YÞ 2 q

Using correlation measures to investigate resting-state pat-terns, two levels of analysis are possible: (1) looking at the correlationbetweenonespecificregionandtherestofthebrain, i.e.seed-basedcorrelation,or(2)exploringallpossiblefunctional connections by studying thecorrelation matrix. To performan analysisattheseedlevel,informationaboutthemainactivation sitesofacertainRSNmustbeprovided.Afteridentifyingtheseeds belongingtoaRSN,thenco-activationsseedmapscanbebuiltby overlappingthecorrelationmapsofeachseed(Foxetal.,2005). ThebrainregionsthatcorrelatewithalltheseedsfromaRSNare identifiedaspartofthatRSN.

Performingthecorrelationmatrixbetweensignals(estimated either at the voxel or regional level) provides information regardingallpaircorrelationsinthebrain.Eachlineinthematrix correspondstoaseed,andtheentriesinthatlinecorrespondtothe

correlationcoefficientbetweentheactivationlevelinthatseedand alltheremainingregions(columns).Oneadvantageofcomputing correlationmatricesisthattheycanbestudiedusinggraphtheory, not only to evaluate the topological properties of functional networks,but alsoforthedetection offunctional modules and hubs.Investigatingcorrelationmatricesatthevoxellevelcanbe computationallycostlysoitiscommontoaveragethesignalsfrom voxelsfallinginthesamecorticalregion(definedaccordingtoa parcellationscheme),andthenanalyzethecorrelationmatrixata much lower resolution (as in Honey et al., 2009). Moreover, waveletanalysiscanbeappliedtothetemporalsignalstocompute frequency-dependent correlation matrices. For example, this approach allowssearching for the frequency range over which maximaldifferencesareobservedbetweenhealthycontrolsand subjects(Bassettetal.,2012; Lynalletal.,2010; Supekaretal., 2008). Despite its usefulness for detecting linear statistical dependencies, the correlation analysis has certain limitations. ThemostimportantreliesonthefactthatRSNsarenotspatially independentandcanoverlap.In otherwords, thesamecortical regioncanbelongtomorethanoneRSNatatimeandtherefore activateswheneveroneoranotherRSNisengaged.Inthisway,the activationpattern of that region turns out to be a sumof the activationpatternsof each RSN it belongs to, which cannotbe capturedusingcorrelation-basedapproaches(Smithetal.,2012). RecentstudiesproposetheuseofICAtoextract RSNspatial maps from coordinated BOLD fluctuations (Beckmann, 2012; Beckmannet al.,2005;Damoiseauxetal., 2006;DeLuca etal., 2006;Mantinietal.,2007).ICAisacomputationaltechniquefor

identifyinghiddenstatisticallyindependentsourcesfrom multi-variatedata.Importantly,ICAcanbespatial(optimizingforspatial independencebetweencomponents)ortemporal(optimizingfor temporal independence between components). In other words, spatialICAallowsmappingtheareasinthebrainthatconsistently activatetogether,whiletemporalICAmapstheregionsinthebrain thatmostlycontributetoagiventemporalsignal.Todate,nearly allapplicationsofICAtofMRI(includingresting-statefMRI)have usedspatialICA,becauseICArequiresalargenumberofsamplesto functionwell, and infMRI there areorders ofmagnitude more voxels than time points. However, for finding temporally independent, potentially spatially overlapping, functional net-works,temporalICAprovidesa betterapproach (Boubelaet al., 2013; Smith etal., 2012). ApplyingICA toresting-statesignals reliesontheassumptionthatbrainactivityduringrestresultsfrom theadditivecombinationofindependentspatialsignals.ICAisa particularcaseofblindsourceseparationandthesignalineach point xn(t), n=1,...,Ncan be written as a composition of the

differentindependent componentsICk(t),k=1,...,Kwith

coeffi-cientsan,kinthefollowingway:

xnðtÞ¼an;1ICðtÞ1þan;2IC2ðtÞþþan;kICðtÞk

Animportant notetoconsiderin ICAis that thenumber of independentcomponentsKis,intheory,equaltothenumberof sourcesNbecauseICAcannotsort(norscale)thesourcesignals in correct order. However, reducing the number of sources

Fig.9.Anatomicalnetworkwith66corticalregions.AnatomicalconnectomederivedbyHagmannetal.(2008)usingDSIaveragedover5healthysubjects.Top-left: Parcellationschemedividingeachhemisphereinto33anatomicallysegregatedregions(adaptedfromHagmannetal.(2008)).Top-middle:Whitemattertractsdetected usingDSIandtractography.Top-right:Schematicrepresentationoftheanatomicalnetwork,whereregionsarerepresentedbyredspheresplacedattheircentreofgravity andthelink’sthicknessisproportionaltothenumberoffibretractsdetectedineachconnection.Bottom-left:Thecouplingweightsareproportionaltothenumberoftracts detected.Whitecolourmeansthatnofibreconnectingthetwocorrespondingregionswasdetected.Weightswerenormalizedsothat0Cnp1.White=noconnection. Bottom-right:Distancebetweenregionsgivenastheaveragelengthofthefibresconnectingapairofregions.

(or dimensions) is possible by previously performing principal componentanalysis(PCA).PCAperformsalinearmappingofthe datatoalowerdimensionalspaceinsuchawaythatthevarianceof thedatainthelow-dimensionalrepresentationismaximized.In practice,itisobtainedbycomputingtheeigenvaluedecomposition ofthecovariancematrixofthesignals.Then,theeigenvaluesare sortedaccordingtothe proportionofvariancethey accountfor. Dimensionreductionispossiblebyselectingtheeigenvalues(i.e. principalcomponents)thatrepresentmostofthevariance.

Beckmannetal.(2005)usedprobabilistic-ICA(PICA)todetect RSNs that exhibit high spatial consistency across subjects and closely resemble discrete corticalfunctional networks(such as visualcorticalareasorsensory-motorcortex).Later,Damoiseaux etal.(2006)usedthesametechniqueandidentified10robustRSNs across healthy subjects with potential functional relevance, consisting of regions known to be involved in motor function, visual processing, executive functioning, auditory processing, memory,andtheDMN(seeFig.2).Moreover,theyreportthese networksexhibitsignificantbaselinedynamics,withpercentage BOLDsignalchangesupto3%(comparablewiththesignalchanges foundintask-relatedexperiments).TheextractionofRSNsfrom whole-brainresting-stateBOLDsignalsusing ICAis particularly important because it does not require any prior information regardingtheexpectedlocationofRSNs(seeFig.10).Brookesetal. (2011) appliedICAto theamplitude envelopes of band-passed source-projectedMEGsignalsanddetected8ICspatialmapsthat closely matched the RSNs found with fMRI. These ICs were estimated from the envelopes of alpha- and beta-frequency oscillations (see Fig. 4). Other methods for characterizing resting-statefMRInetworksincludepartialcorrelation(Fransson andMarrelec,2008),coherence andpartialcoherence(Salvador etal.,2005b),andphaserelationships(Sunetal.,2005).

BothcorrelationandICanalysisrelyonthelinearrelationship between regions. Other nonlinear functional connectivity

measuresfrominformation theory,suchasmutual information, have been applied toresting-state data (Hartman et al., 2011; Hlinka et al., 2011). Hlinka and colleagues defined functional connectivitymatricesusingbothlinear(Pearsoncorrelation)and non-linear functional connectivity measures and evaluated the resultingnetworksusinggraphtheory(Hlinkaetal.,2011).Their resultsshowthat,atleastfromagraph-theoreticalperspective,the nonlinearities in resting-state activityare practically negligible when compared to the inter-subject variability of the graph measures.Onthegroup-averagelevel,thenonlinearityeffectsare unnoticeable as they vary individually. In addition Lynall and colleaguesanalyzed resting-statedata usingboth mutual infor-mationandcorrelationmeasuresatdifferentwaveletscales,and found that correlation measures allowed a better distinction betweenhealthycontrolsandschizophreniapatients(Lynalletal., 2010).Thelineardependencyofresting-stateBOLDtime-seriesis probablylinkedtothelowtemporalresolutiontypicallyusedin BOLD-fMRIstudies(i.e.intheorderofacoupleofseconds),which doesnotallowthedetectionoffastnon-linearinteractions.Recent advancesinultra-fastspatialencodingandinversereconstruction have increasedthe temporal resolutionof whole-brain fMRI to timescalesontheorderof100msandfaster.Thisunprecedented temporalresolutionhasbeenshowntosignificantlyincreaseBOLD sensitivity resulting in improved detection of single trial task activationandrestingstatenetworks(Posseetal.,2012,2013).

3.4. Characterizingbrainnetworksusinggraphtheory

Thefirstapproachtocapturetheorganizationalpropertiesof brainnetworksistomodelthemasgraphswhosenodesrepresent dynamical units and whose links stand for the connections (structural or functional) between them. A large number of measures have been defined in the field of graph theory to characterize the topology of graphs (Boccaletti et al., 2006).

Fig.10.IdentificationofanRSNusingtemporalICA.AnindependenttemporalcomponentICk(t)(green)isextractedfromtheBOLDsignalsusingICAandthenmappedover thebrain(top)byselectingthesetofvoxelsnwhoseBOLDsignals,xn(t)havestrongercontributionsfromICk(t)asindicatedbythecoefficientsan,k.Thespectralpropertiesof theICwaveformrevealapeak<0.1Hz.

A brain connectome,be it anatomical or functional, withN corticalregionscanberepresentedintheformofamatrixC,withN columnsandNrows,whereeachentryinthematrixC(n,p)orCnp

encodestheconnectionstrengthbetweentwobrainareasnandp (withn,p=1,...,N).Connectionscanbebinary,simplydenoting whetheralinkispresentorabsent.However,inbrainnetworks, theconnectionis usuallyweighted:instructuralnetworks,it is generallyscaledbythenumber,density, orcoherence ofwhite matterfibretracts;infunctionalnetworksitmaycorrespondsto magnitudesofcorrelationalorcausalinteractionsbetweenthetwo brain areas. Studying weighted graphs, however, imposes an increaseddegreeofcomplexityandmostgraphtheoreticalstudies of resting-state functional networks have used binary graphs instead. To transform the connectivity matrix C into a binary adjacencymatrix A, oneneeds todefinea threshold,th, where Anp=1 if Cnp>th, or Anp=0 otherwise. The definition of this

thresholdhasdirect impacts on thedensity of connections(or sparsity)ofthenetwork,whichmayhavenon-negligibleeffectson thegraphpropertiesofthenetwork(Bassettetal.,2012;vanWijk et al., 2010). Therefore, to compare graph measures between networksof differentsubjects,it is usualtodefine equi-sparse graphs (instead of equi-threshold graphs), to ensure the same numberofedgesinallgraphs(Bassettetal.,2012;Lynalletal., 2010)(seeFig.11).

Anothercrucialsteponthedefinitionofbraingraphsisthesize ofthenetwork,whichistakenasthenumberofnodes.Sincethe braincanbeparcellatedatdifferentscales(suchasneurons,voxels or cortical regions) different graph properties may arise, and thereforethepropertiesofbrainnetworksmustbetakeninthe lightoftheparcellationandthethresholdingtechniqueemployed. Evenso,tworeviewstudiesonhumanbraingraphs(Bassettand Bullmore,2009;HeandEvans,2010)reportedthatsomecomplex networkpropertiesareconsistentoverarangeofspatialandtime scales,and across modalities of neuroimaging data. Conserved principles include small worldness, high efficiency/low wiring cost, modularity and hubs. These properties were consistently found in brain networks obtained with both structural MRI (Bassettetal.,2008;Chenetal.,2008;Heetal.,2007),diffusion MRI (Hagmann et al., 2008; Iturria-Medina et al., 2008), fMRI (Ferrarini etal.,2009;vandenHeuvel etal.,2008; Wangetal., 2009a),EEG(Micheloyannisetal.,2009;Rubinovetal.,2009),and MEG(Stametal.,2009;Valenciaetal.,2008).

The small-world architecture is particularly important in functionalnetworksbecauseitsupportsbothsegregatedmodular specializationanddistributedfunctionalintegration(Spornsand Honey, 2006). In addition, it maximizes the efficiency of

disruptedsmall-worldpropertiesinresting-statefunctional net-worksofpeoplewithschizophrenia(Bassettetal.,2012;Liuetal., 2008;Lynalletal.,2010),Alzheimer’sdisease(Supekaretal.,2008) andattention-deficit/hyperactivitydisorder(Wangetal.,2009b). Another important graph measure is the modularity,which identifiesmodulesoflinkednodesthatworktogethertoachieve distinctivefunctions(Newman,2006;LeichtandNewman,2008). Connections are usually denser within modules than between them.Detectingandcharacterizingmodulesofthebraincanallow us to identify groups of anatomically and/or functionally associated componentsthat may subserve specific behavioural functions. For example, the modularity of resting-state fMRI functionalnetworkshasbeenfoundtochangewithnormalageing (Meunieretal.,2009).Notehoweverthatthesechangesmaynotbe directly related to neuronal activity but to heart function and vesselcompliancechanges,naturallyoccurringinnormalageing. In amodel ofresting-statedynamics,Decoet al.(2009)useda modularityalgorithmtodivideonehemisphereofthemacaque anatomicalnetworkinto2communities(seemoredetailslater). Exploringthedynamicsofthetwomodulesindependently,they foundthateachmoduleexhibitedaspecifictemporalpatternof synchronization,andnotably,thesynchronizationlevelbetween thetwomoduleswasanti-correlated,whichcouldbeexplainedby stochasticresonanceinasystemwithmultistability.Theseresults reinforcetheideathatthetopologicalorganizationofstructural brainnetworksconstrainandmouldbraindynamicsatmultiple levels.

3.5. Modelsofbrainnetworkdevelopmentandorganization

In therapidly growing field of brainnetwork science,some modelshavebeenproposedtoexplaintheevolutionary organiza-tionprinciplesofbrainnetworksandtheirimplicationsinbrain function.Forexample,studiesofanatomicalandfunctionalbrain networkshave shown that theconnection probability between twobrainregionsdecreaseswiththephysicaldistancebetween them (Chklovskii and Koulakov, 2004; Salvador et al., 2005a). Based on this observation, generative models can be built by definingtheprobabilityofafunctionalconnection(edge)between twocorticalregions(nodes)separatedbysomeEuclideandistance inanatomicalspace(Kaiserand Hilgetag,2004a,b;Vertes etal., 2012).Suchmodelsmaybeoptimizedtoreflectgraph-theoretical aspectsofbrainnetworks,suchashighefficiencyorhighclustering coefficient,leadingtogenerativemodelsofnetworkswith small-worldorganization,likethebrain.Theeconomicnetworkmodel from Bullmore and Sporns (2012) addresses how the brain

Fig.11.Thresholdingweightedmatricesintobinarygraphs.Aresting-statefunctionalconnectivity(FC)matrix,whereentriesindicatethedegreeoftemporalcorrelation betweenBOLDsignals,canbetransformedintoabinarygraphwithdifferentconnectiondensities,dependingonthethreshold(1ifthecorrelationcoefficientisabovea certainthreshold,0otherwise).Importantly,distinctconnectiondensitiesleadtodistinctgraphproperties.

optimizesaneconomictrade-offbetweenthecost(i.e.minimizing connectiondensity) andthe efficiencyof networkfunction (i.e. minimizing the characteristic path length). Another theoretical model(Vertesetal.,2012)proposesthatthetopologyofhuman brainfunctionalnetworksemergesfromtwocompetingfactors: lowerprobabilityoflong-rangeconnectionsandhigherconnection probabilitybetweenregionssharingsimilarinput.Althoughsuch models allow investigating certain organizational features of resting-statefunctionalconnectivity,theydonotactuallymodel theneuralactivity(i.e.dynamics)ofinteractingbrainareasnor investigatetheneurophysiologicaland/orbiophysicalmechanisms coupling brain areas together. As such, the network dynamics underlyingbrainactivityduringrestmaynot befullyexplored usingthesemodels.

4. Large-scalemodelsofresting-statedynamics

The dynamics emerging spontaneously from the interplay betweenbrainareaswhentheseareembeddedinthe neuroana-tomicalnetwork,hasbeenattractingagrowingbodyofresearchin computationalneuroscience(Decoetal.,2011,2013b;Jirsaetal., 2010; Nakagawa et al., 2013; Ritter et al., 2013). By meansof whole-brainnetwork models,constrainedby realisticstructural connectivity,onecanexplorehowlarge-scaleinteractionsgiverise to the spontaneous emergence of resting-state fluctuations. Importantly,these modelscan be used totest predictions and evaluatetheimpactoflesionsinthebrain.Therecentlylaunched Virtual Brain (www.thevirtualbrain.org) is a neuroinformatics platformwithauser-friendlybrainsimulatorwhichallowsusers to perform customized simulations, analyze the results and compare them with macroscopic neuroimaging signals (Ritter etal.,2013).

Toinvestigatethespontaneousdynamicsofinteractingbrain areas at the whole-brain scale, it is useful to go beyond the microscopicactivityofindividualneuronsandconsiderinsteadthe mesoscopicbehaviouroflargeensemblesofneurons,orneuronal populations(Decoetal.,2008).Althoughsimulationsofdetailed modelsatthecellular(andevensub-cellular)levelarebecoming computationallyfeasible (Izhikevichand Edelman, 2008; Mark-ram,2006),reducedneural-massmodels,despitetheirlowspatial resolution, allow a comprehensive study of the large-scale interactivedynamicswithrelativelylowparametriccomplexity. Thisapproachismotivatedbyneuroimagingobservationsshowing thatneuronswithinadenselyconnectedneuralensembletendto sharethesamephysiologicalproperties,exhibitdensereciprocal interconnectivityandshowstrongdynamicalcorrelations. Neural-massmodelscanbeextendedtoneural-fieldmodels,wherethe expectedstateofaneuronalpopulationbecomesafunctionofboth timeandpositiononthebrain’sspatiallycontinuouscorticalsheet. To date, however, all the models of whole-brain resting-state dynamicshaverepresentedbrainareasasisolatedpointsinspace (orpoint-masses).

Followingdifferentreductionlines,anumberofstudieshave contributedfortheunderstandingofresting-statedynamicsusing neural-massmodelscoupledaccordingtothebrain’sanatomical architecture(Cabraletal.,2011;Decoetal.,2009;DecoandJirsa, 2012; Ghosh etal., 2008c; Honey et al.,2007, 2009). All these modelsrelyontheassumptionthat resting-stateactivityarises solely from synaptic interactions between brain areas and disregardthe role of physiological signals (such as blood flow, vessel structure and cerebrospinal fluid) which vary strongly across thehuman brain and even display very slow (<0.1Hz) fluctuations (Chang et al., 2013; Kim and Ogawa, 2012; Moser etal.,1999;Striketal.,2002).Intheapproachfollowedbythese models,theBOLDsignalisobtained fromthesimulated neural activityusingasimplemodel(i.e.theWindkessel-Balloonmodel

applied uniformly to all brain areas (see Section 4.7). In the following,wereviewexistingmodelsofresting-stateactivity.

4.1. Conductance-basedbiophysicalmodel

The first work to investigate the cooperative behaviour of neural systems coupled through a realistic anatomical wiring schemewasachievedbyHoneyandcolleagues(2007).Theyuseda biophysical neural-mass model introducedby Breakspear et al. (2003)togetherwiththeanatomicalconnectivityofthemacaque cortex (Kotter, 2004). Later, the same model was extended to incorporatehumanneuroanatomicalconnectivityandresultswere compared with resting-state fMRI functional connectivity from healthyhumans(Honeyetal.,2009).Finally,thesamegroupused thismodeltostudythedynamicalimpactoflesionsinthebrain (Alstottetal.,2009;HoneyandSporns,2008).

Theneural-massdynamicsintheseworkswasderivedfroma conductance-based model of neuronal dynamics (Morris and Lecar,1981)extendedforneuralpopulationactivity(Larteretal., 1999).Thecouplingbetweenneuralmasses(orcorticalregions) was introduced via weak long-range excitatory-to-excitatory connections, mimicking glutamate-induced synaptic currents (Breakspear,2004;Breakspearetal.,2003).

The main dynamical variable in that model is the mean membranepotentialofpyramidalcellsV,whichisgovernedbythe conductanceofsodium(gNa),potassium(gk)andcalcium(gCa)ions

throughvoltage-gatedchannels,plusthepassiveconductanceof ‘leaky’ ions (gL). The total current flow across pyramidal cell

membranesisgivenby:

CdV

dt ¼gCamCaðVVCaÞgNamNaðVVCNaÞgKWðVVKÞ

gLðVVLÞ

wheregionisthemaximumconductanceofeachpopulationofions,

mion isthe fractionofopen ion channels(W for potassium ion

channels),andVionistheNernstpotentialforthationspecies.All

equationsandparametersarenon-dimensionalandnormalizedto neuralcapacitanceC=1.Eachvoltage-gatedchannelopenswhen themembranepotentialovercomesagiventhreshold,Tion.Fora

large population of ion channels, Tion assumes a Gaussian

distribution(withvariance

d

ion)and hence,thefractionof open

ionchannelsisgivenbythefollowing(sigmoid-shaped)function:

mion¼0:5 1þtanh VTion

d

ion     :

The fraction of open potassium channels W is defined differently because these channels ‘relax’ from one state to anotheratanexponentialrate.Therefore,Wisgovernedby

dW dt ¼

f

ðmKWÞ

t

where

f

isa temperaturescalingfactorandis

t

the‘relaxation’ timeconstant.

Tointroducesynapticinteractionsbetweenneuronswithinthe sameneuralensemble,theaveragefiring-ratesofexcitatoryðQVÞ

and inhibitory neurons ðQZÞ is calculated and introduced as a

feedbacktermsubsequenttocellfiringtorepresent neurotrans-mitterrelease.Atthecellsoma,themembranepotentialtriggers anactionpotentialifitexceedsathreshold.Averagingthisoverthe ensemble of neurons and assuming once again a Gaussian distribution,thecellfiringratescanbeobtainedbythefollowing equations: