Analysing connectivity with Granger causality and dynamic causal modelling

Analysing

connectivity

with

Granger

causality

and

dynamic

causal

modelling

Karl

Friston

,

Rosalyn

Moran

and

Anil

K

Seth

Thisreviewconsidersstate-of-the-artanalysesoffunctional integrationinneuronalmacrocircuits.Wefocusondetecting andestimatingdirectedconnectivityinneuronalnetworks usingGrangercausality(GC)anddynamiccausalmodelling (DCM).Theseapproachesareconsideredinthecontextof functionalsegregationandintegrationand—withinfunctional integration—thedistinctionbetweenfunctionalandeffective connectivity.Wereviewrecentdevelopmentsthathave enjoyedarapiduptakeinthediscoveryandquantificationof functionalbrainarchitectures.GCandDCMhavedistinctand complementaryambitionsthatareusefullyconsideredin relationtothedetectionoffunctionalconnectivityandthe identificationofmodelsofeffectiveconnectivity.Wehighlight thebasicideasuponwhichtheyaregrounded,providea comparativeevaluationandpointtosomeoutstandingissues. Addresses

1

TheWellcomeTrustCentreforNeuroimaging,UniversityCollege London,QueenSquare,LondonWC1N3BG,UK

2_{SacklerCentreforConsciousnessScienceandDepartmentof} Informatics,UniversityofSussex,BrightonBN19QJ,UK

Correspondingauthor:Friston,Karl([email protected])

CurrentOpinioninNeurobiology2013,23:172–178 ThisreviewcomesfromathemedissueonMacrocircuits

EditedbyStevePetersenandWolfSinger

ForacompleteoverviewseetheIssueandtheEditorial

Availableonline21stDecember2012 0959-4388#_{2012ElsevierLtd.}

http://dx.doi.org/10.1016/j.conb.2012.11.010

Introduction

Severaldichotomieshaveprovedusefulinthinkingabout analytic approaches to functional brain architectures. Perhapsthemostfundamentalisthedistinctionbetween functional segregation and integration. Functional segre-gationreferstotheanatomicalsegregationoffunctionally specialisedcortical andsubcortical systems, while func-tionalintegrationreferstothecoordinationandcoupling of functionally segregated systems [1]. Within func-tionalintegration,twomainclassesofconnectivityhave emerged—functional and effective connectivity. Func-tional connectivity refers to the statistical dependence or mutual information between two neuronal systems, while effective connectivity refers to the influence that one neuralsystem exerts over another [2,3].This distinction is particularly acute when considering the

differentanalysesonemightapplytoelectrophysiological orneuroimagingtimeseries.

Functionalandeffectiveconnectivity

Because functional connectivity is defined in terms of statisticaldependencies,itisanoperationalconceptthat underliesthedetectionof(inferenceabout)afunctional connection, without any commitmentto how that con-nectionwascaused.Inotherwords,onetestsfor depen-denciesbetweentwoormoretimeseries,torejectthenull hypothesisofstatisticalindependence.Thisisequivalent toassessingthemutualinformationandtestingfor signifi-cantdeparturesfrom zero. Atits simplest,this involves assessing (patterns of) correlations—of the sort that define intrinsic brain networks. An important distinc-tion—within functional connectivity—rests on whether dependencies are instantaneous or reflect an underlyingdynamical process, in which causesprecede consequences. This leads to the distinction between analysesofdirectedandundirectedfunctionalconnectivity thatdoanddonotappealtotemporalprecedence respect-ively. Common examples of techniques used to assess undirected functional connectivity (dependencies) in-cludeindependent componentsanalysis [4]and various measures of synchrony, correlation, or coherence [5]. However,wewillfocusonanalysesofdirectedfunctional connectivity—of which theprime example isGranger causality(GC)[6].Thisisbecausecouplinginthebrain isbothdirected andlargelyreciprocal(producingcyclic graphs or networks with loops that preclude structural causalmodelling).Aswewillseebelow,GCandrelated conceptssuchastransferentropy(TE)reston establish-ingastatisticaldependencebetweenalocalmeasurement of neuronalactivity and measurements of activity else-whereinthepast.

Functionalconnectivityconsidersdependenciesbetween measured neurophysiological responses. In contrast, effectiveconnectivityisbetweenhiddenneuronalstates generating measurements. Crucially, effective connec-tivity is always directed and rests on an explicit (para-meterised) model of causal influences—usually expressedintermsofdifference(discretetime)or differ-ential (continuous time) equations. The most popular approach to effective connectivity is dynamic causal modelling(DCM)[7–10,11,12].Inthiscontext, caus-alityisinherentintheformof themodel,where fluctu-ationsinhiddenneuronalstatescausechangesinothers: for example, changes in postsynaptic potentials in one area are caused by inputs from other areas. The Open access under CC BY license.

parametersofdynamiccausalmodelscorrespondto effec-tive connectivity—usually cast as synaptic density or coupling parameters—thatareoptimised byfittingthe modeltodata.Thenotionofeffectiveconnectivitystems fromthepioneeringworkofGersteinandPerkel[13]in earlyattemptstointerpretmultivariate electrophysiologi-cal recordings. At its inception, effective connectivity referredtomodels;inthesenseofthesimplestpossible circuitdiagramsthatexplainobservedresponses[14].In modern parlance, these correspond to dynamic causal modelswiththegreatestevidence:namely,modelswith theminimumcomplexitythatfurnishanaccurate expla-nation for data(seebelow). Inwhatfollows, wereview recentdevelopmentsintheanalysisofdirectedfunctional connectivity with GC and TE, the analysis of directed effectiveconnectivitywith DCMandthenconsiderthe approaches in light of eachother. Figure 1 provides an overview ofrecentdevelopmentsinthesetechniques.

Granger

causality

and

transfer

entropy

ThecoreideabehindGCisthatX‘Grangercauses’YifX contains information that helps predict the future of Y betterthaninformationalreadyinthepastofY(andinthe past ofother‘conditioning’variablesZ).Themost com-monimplementationofGCisvialinearvector autoregres-sive (VAR) modellingof timeseries data, enabling both statisticalsignificancetestingandestimationofGC mag-nitudes [6,15,16]. However, GC is not limited to this implementation; it can use nonlinear,time-varying, and non-parametric models [17,18]. In particular, TE [19] representsaninformation-theoreticgeneralisationofGC that does notrequire a parameterisedmodel (is model-free).Specifically,theTEfromXtoYiszeroif,andonlyif, YisconditionallyindependentofX’spast,givenitsown past.Importantly,forGaussiandata,TE isequivalentto GC[20],furnishingausefulinterpretationofGCinterms of information transfer in ‘bits’. Related approaches in-cludepartialdirectedcoherenceandthedirectedtransfer function;see[21]forareview.Herewefocusonthemost popularofthesetechniques,namelyGC:

Following itsintroduction withineconometrics [6,15], GChasbeenappliedinneurosciencepartlybecauseitis simple to estimate, given (stationary stochastic) time-series. Such data are generated by a wide range of neuroimaging and neurophysiologicalmethods. GC has some useful properties including a decomposition of causal influence by frequency [15] and formulation in an‘ensemble’form,allowingevaluationofGCbetween multivariate sets of responses [22]. GC has provided useful descriptions of directed functional connectivity in many electrophysiological studies[23–25]. Recently, Bosmanet al.[26]analysedelectrocorticographic data frommacaquemonkeystoshowthat‘bottom-up’signals across multiplecortical regionsweremostprominentin thegammaband,while‘topdown’influencesdominated atbetafrequencies—afindingthatisstrikingly

congru-ent with neural implementations of predictive coding [27].GC canalsobeappliedto standardEEGor MEG signals, either at the source or sensor level (following spatialfilteringtoreducetheimpactof volume conduc-tion).Forexample,Barrettetal.[28]used source-loca-lised EEG to show that gamma-band GC between posteriorandanteriorcingulatecorticesreliablyincreased during anaesthetic loss of consciousness, extending previous resultsobtainedusing (undirected)phase syn-chrony [29]. We will turn to this example later in the context ofDCM.

TheapplicationGCtofMRIismorecontroversial,given theslowdynamicsandregionalvariabilityofthe haemo-dynamicresponsetounderlyingneuronalactivity[30,31]; andsee‘ProsandCons’below.Whilenaı¨veapplicationof GC to fMRIdata isunlikely to beinformative, careful considerationofthemethodologicalissueshaspermitted some useful applications that have produced testable hypotheses.Forexample,Wenetal.[32]analysedfMRI data obtainedfrom acued spatial visualattention task; finding that GC from dorsal to ventral frontoparietal regions predicted enhanced performance, while GC in the reciprocal direction was associated with degraded performance. These findings are consistent with the notion that dorsal attentional regions mediate goal-orientedtop-downdeploymentofattention,whileventral regions mediatestimulus-drivenbottom-up reorienting. Inasimilar paradigm,Bressleretal.[33]foundthatGC from parietal to occipital areaswas predictive of beha-vioural performance. In a final and unusual example, Schippersetal.[34]usedGCoffMRIsignalstoanalyse directed interactions between the brains of two subjects engaged in a social game (charades), providing novel evidencefor‘mirrorneuronsystem’formulationsofsocial interaction. Another promising application of GC is to intracranial local field potentials, which possess high temporalandspatialresolutionandwhichcomprise com-paratively fewvariables(ascompared tofMRIvoxelsor EEGsensors).Anearlyapplicationinthisarea,Gaillard et al. [35] examined directed functional connectivity during supraliminal as compared to subliminal visual word processing.

Dynamic

causal

modelling

The basic idea behind DCM is that neural activity propagates throughbrain networksas in an input-state-outputsystem,wherecausalinteractionsaremediatedby unobservable (hidden) neuronal dynamics. This multi-inputmulti-outputneuronalmodelisaugmentedwitha forward,orobservationmodelthatdescribesthemapping from neural activity to observed responses. Together neuronalandobservationmodelcomprise afull genera-tive model that takes a particular form depending on the data modality. The key outputs of DCM are the evidence for different models and the posterior parameter estimates of the (best) model, particularly

Figure1

Granger causality (GC) is first applied to fMRI time-series using the method of ‘Granger

causality mapping’ (GCM) from a seed voxel to other

voxels [43]

Important extension of the GCM approach, examining the influence of confounding

hemodynamic response functions [31]

GC combined with sparse regression techniques to allow estimation of high-dimensional dynamical models reflecting brain

networks [50]

GC applied to BOLD signals to reveal top-down influences during human visual

attention [33]

Theory and modelling showing invariance of GC to hemodynamic convolution given fast

sampling and low noise

GC applied within a state-space framework incorporating explicit observation equations for

modelling hemodynamic responses [49]

GC applied to local-field potentials (LFPs) recorded

from cat visual cortex; important early clarification

of statistical issues [23]

GC analysis of LFP data obtained from awake monkeys reveal

directional beta-frequency interactions in a large –scale

network during motor maintenance behaviour [25]

Nonparametric GC introduced (based on

Fourier and wavelet transforms) and validated on monkey

LFP data [18]

Equivalence shown between GC and transfer entropy for Gaussian data [20]

Adaptive multivariate autoregressive (AMVAR) modelling applied to multichannel event-related potentials, showing rapidly changing cortical dynamics during visuomotor integration [24]

GC validated on electrophysiological data from

rats given deconvolution of hemodynamic responses [30] Granger Causality fMRI fMRI 1999/ 2000 ₂₀₀₃ 2003 2004 2004 2006 2007 2005 2008 2008 2009 2009 2011 2011 2012 electrophysiology electrophysiology

Dynamic Causal Modelling

Dynamic causal modelling is introduced as the Bayesian inversion of dynamic (bilinear)

neurophysiological models of fMRI time-series [2]

Bayesian model comparison is described for selecting

among alternative DCMs [38]

Neural drivers in DCM for fMRI validated using concurrent electrophysiology in rodents [30]

Post-hoc Bayesian model selection allows rapid estimation of model

evidence for very large model spaces [51]

Nonlinear DCM for fMRI is described, allowing

for (neuronal) state-dependent changes in connectivity [48] DCM for fMRI parameterises inhibitory and excitatory neuronal processes [52] Stochastic DCM developed in generalised coordinates of motion to provide estimates of hidden neuronal states [47] DCM for evoked electrophysiological responses

is introduced, using neural mass models with multiple neuronal populations [7] Reciprocal connections are shown to be necessary for generating late responses in EEG [8] DCM for steady state responses applied to intracranial EEG from rodents: synaptic measures validated using microdialysis [40] Conductance based neuronal models allow for the testing of connectivity through specific ion channels [39] Validation of receptor-specific contributions using pharmacological challenge in humans [11]

Current Opinion in Neurobiology

AtimelineofrecentadvancesinGrangercausality(toppanel)anddynamiccausalmodelling(bottompanel).Entriesabovethetimelinespertainto functionalmagneticresonanceimaging(MRI)andthosebelowthelinesreportspecificdevelopmentsforelectrophysiology.

thosedescribingthecouplingamongbrainregions.These allow formodelandsystemidentification,respectively. DCMwasintroducedforfMRItimeseries[36],wherethe neuronal modelcomprises one or twohidden (lumped) neuronalstatesforeachregion.Theneuronaldynamicsof eachregiondependonthestrengthofconnectionswithin thatregion (parameterisedbyaself-connection),onthe strength of external inputs (experimental input parameters) and on inputs from other regions in the network (thecouplingparameters).Neuronal activityis thentransformedthroughahaemodynamicmodel(with region-specificparameters)tomodelmeasuredresponses [37]. The coupling between brain regions can then be estimated foraparticularmodelarchitectureusing stan-dardvariationalBayesiantechniques[36].Inpractice,itis usualtospecifydifferentarchitecturesorhypothesesand formallycomparetheevidence forthesemodels,before examining parameter estimates [38]. DCM necessarily accounts for directed connections among brain regions and disambiguates the neuronal drivers of a particular eventandsubsequentsignalpropagation. Electrophysio-logicalmeasurementssupportrichermodelsofneuronal dynamics in DCM that comprise sources with laminar specific mixtures of neuronal populations. These have evolvedfromkernel-based models[7]thatuse postsyn-apticconvolutionoperatorstodescriberesponsesat excit-atory and inhibitory synapses to conductance-based models, where particular ionchannelscan bemodelled andidentified[39].Theseneuralmassmodelsare accom-paniedbylinearelectromagneticforwardmodelsto gen-erateresponsesatEEGscalpelectrodes,atMEGsensors or atintracranialrecordingsites.ApplicationofDCMto animallocal-fieldpotentialdatahasfacilitatedvalidation studies,whereindependent,invasivemeasurements(e.g. microdialysis or pharmacological perturbations) suggest that DCM can be used to estimate the physiological mechanismsresponsibleformediatingeffective connec-tivity [40].

Pros

and

cons

Clearly, GC and DCM have complementary aims and strengths:GCcanbeapplieddirectlytoanygiven time-seriestodetectthecouplingamongempiricallysampled neuronal systems. Thiscanprovide usefulinsights into thesystem’sdynamicalbehaviourindifferentconditions orinspontaneouslyactive‘resting’states.Onemightthen proceedtoamoremechanistic(modelorhypothesis— driven)characterisationusingDCM.However,thiscalls for bespokemodelsofthesystemin question[41].In other words,GC isagenericinferentialprocedure char-acterisingdirectedfunctionalconnectivity,whileDCMis aframeworkthatenforces(orenables)specificmodelsor hypotheses to be tested. Crucially, both rest on model selection:InDCMthisinvolvescomparingtheevidence for differentmodelsdirectly[38],whilemodelselection inGCisimplicitinthetestforthepresenceofGC—and

also arises in the selection of VAR model order, using standardapproximationstomodelevidence,suchasthe Akaikeor Bayesianinformationcriteria [42].

Although GC is generic, its naive application is not alwaysjustified.Forexample,applicationtofMRImust recognise the indirect relation between neuronal activity and haemodynamic responses. In particular, regionalvariationsinhaemodynamiclatencycould con-found the temporal precedence assumptions of GC [30].While thesevariations canbe partiallycontrolled forbycontrastingGCbetweenexperimentalconditions [17,43,31] false inferences remain possible. Interest-ingly, recent theory and modelling suggests that GC may be robust to haemodynamic variations but not whencombinedwithdown-samplingandmeasurement noise [44]. In contrast, DCM models haemodynamic variationsexplicitlyandtriestoexplainthedataatthe level of hidden neuronal states—in other words, it triestogetbeneaththesurfacestructureofthedatato explain how they were generated: see [45,46] for further discussion.

Inanalysisofelectrophysiologicaltimeseries,GCismore widely accepted because there is no temporal lag be-tween the responses recorded and their underlying (neuronal)causesand becausethedatacanbesampled at fast timescales. The advantages of GC in furnishing frequency-dependent and multivariate measures have beenclearlydemonstrated[22,26,28].However,there is an unresolved issue in this setting—the random fluctuations assumed by GC are serially independent (shownotemporalcorrelationsandfluctuateatveryfast timescales). This is an issue because neuronal fluctu-ations in the brain are produced by neuronal systems thathavethesametimeconstantsasthesystemstudied. Whileserialindependencecanbecheckedfor,thenature of neuronalfluctuations maydeserve more attention in thefuture.

AkeyfeatureofDCMisthatitcanincludevariablesthat describedynamicsthatarehiddenfromobservation.For example,theGCanalysisofanaestheticlossof conscious-ness by Barrettetal. [28] mentionedabove, was com-plemented by a mechanistic study by Boly et al. [47] using DCM. She found that a DCM that included a hidden thalamic source performed better than DCMs based solelyon observedcortical timeseries,and estab-lished a dissociationbetween theeffects of (measured) corticaland(inferred)subcortical structuresonlevelsof consciousness. In contrast to GC, being able to model hiddensourcesmeansthemodel(hypothesis)spacecan be very large and calls for a principled approach to Bayesian modelcomparisonofmodelsthatare(a priori) considered equally plausible. The specification and interrogationof themodel spaceis anoutstanding con-ceptualissuefor DCM.

DCMpositsandidentifiesneuronalmechanisms respon-sibleforfunctionalintegrationinthebrain.Connectivity in this setting necessitates biologically plausible expla-nations. In DCM for fMRI, new developments[48,49] enabletheincorporationofbackgroundorongoing spon-taneouscorticalfluctuations,nonlinearitiesandinhibitory neuronalpopulations[53].Theaddition ofspontaneous orstochasticfluctuationsenhancestheplausibilityofthe generativemodelattheneuronallevel,where non-Mar-kovian noise processes sit atop experimentally induced brain activations [48]. In DCM for electrophysiological data, the models will potentially allow the characteris-ationof receptor-specificcontributionsto brain connec-tivity,whichmaybeimportantinapharmacologicaland clinicalsetting[11].

Conclusion

In conclusion, GC and DCM are complementary:both model neuralinteractions and both are concerned with directed causal interactions. GC models dependency amongobservedresponses,whileDCMmodelscoupling among the hidden states generating observations. Despitethisfundamentaldifference,thetwoapproaches maybeconverging.OntheonehandDCMforstochastic systems[48] cannow accommodate the random fluctu-ationsassumedbyGC.Ontheotherhand,state-spaceGC approachescanincorporatemodalityspecificobservation equations [50]. The ability to handle large numbers of sourcesforregionsisfacilitated bymultivariate (ensem-ble) GC [22] and sparse regression techniques [51], as wellasrecentdevelopmentsin posthocmodel optimis-ationfor networkdiscoverywithDCM [52].One might hope that both approaches—perhaps GC disclosing candidate models for DCM—will counter the claims that modern brain mapping is a neo-phrenology and providecharacterisationsof brain circuitsthatmayhold promiseforthetreatmentofneurologicalandpsychiatric disorders.

Acknowledgements

KJFandRMarefundedbytheWellcomeTrust.AKSisfundedbythe EPSRC (EP/G007543/1) and the Dr. Mortimer and Theresa Sackler Foundation.

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