Dynamic causal modls of nural systm dynamics 9 Dynamic causal modls of nural systm dynamics: currnt stat and futur xtnsions KLAAS E STEPHAN *, LEE M HARRISON, STEFAN J KIEBEL, OLIVIER DAVID, WILL D PENNY and KARL J FRISTON * Wllcom Dpartmnt of Imaging Nuroscinc, Institut of Nurology, Univrsity Collg London, Qun Squar, London WCN 3BG, UK INSERM U594 Nuroimagri Fonctionnll t Métaboliqu, Univrsité Josph Fourir, CHU Pavillon B BP 7, 3843 Grnobl Cdx 9, Franc *Corrsponding author (Fax, 44-7-834; Email, k.stphan@fi l.ion.ucl.ac.uk) Complx procsss rsulting from intraction of multipl lmnts can rarly b undrstood by analytical scintific approachs alon; additional, mathmatical modls of systm dynamics ar rquird. This insight, which disciplins lik physics hav mbracd for a long tim alrady, is gradually gaining importanc in th study of cognitiv procsss by functional nuroimaging. In this fild, causal mchanisms in nural systms ar dscribd in trms of ffctiv connctivity. Rcntly, dynamic causal modlling (DCM) was introducd as a gnric mthod to stimat ffctiv connctivity from nuroimaging data in a Baysian fashion. On of th ky advantags of DCM ovr prvious mthods is that it distinguishs btwn nural stat quations and modality-spcific forward modls that translat nural activity into a masurd signal. Anothr strngth is its natural rlation to Baysian modl slction (BMS) procdurs. In this articl, w rviw th concptual and mathmatical basis of DCM and its implmntation for functional magntic rsonanc imaging data and vnt-rlatd potntials. Aftr introducing th application of BMS in th contxt of DCM, w conclud with an outlook to futur xtnsions of DCM. Ths xtnsions ar guidd by th long-trm goal of using dynamic systm modls for pharmacological and clinical applications, particularly with rgard to synaptic plasticity. [Stphan K E, Harrison L M, Kibl S J, David O, Pnny W D and Friston K 6 Dynamic causal modls of nural systm dynamics: currnt stat and futur xtnsions; J. Biosci. 3 9 44]. Introduction Modrn cognitiv nuroscinc uss a varity of noninvasiv tchniqus for masuring brain activity. Ths tchniqus includ lctrophysiological mthods,.g. lctroncphalography (EEG) and magntoncphalograpy (MEG), and functional imaging mthods,.g. positron mission tomography (PET) and functional magntic rsonanc imaging (fmri). Two intrtwind concpts, functional spcialization and functional intgration, hav bn guiding nuroimaging applications ovr th last dcads (Friston a). Functional spcialization assums a local spcialization for crtain aspcts of information procssing, allowing for th possibility that this spcialization is anatomically sgrgatd across diffrnt cortical aras. Most currnt functional nuroimaging xprimnts us this prspctiv and intrprt th aras that ar activatd by a crtain task componnt as th lmnts of a distributd systm. Howvr, this xplanation is somwhat spculativ and clarly incomplt as long as on dos not charactriz Kywords. Dynamic causal modlling; EEG; ffctiv connctivity; vnt-rlatd potntials; fmri; nural systm Abbrviations usd: AIC, Akaik information critrion, BF, Bays factor; BIC, Baysian information critrion; BMS, Baysian modl slction; DCM, dynamic causal modlling; EEG, lctroncphalography; ERPs, vnt-rlatd potntials; fmri, functional magntic rsourc imaging; IFG, intrior frontal gyrus; MEG; magntoncphalography; SPC, suprior parital cortx. http://www.ias.ac.in/jbiosci J. Biosci. 3(), January 7, 9 44, Indian J. Acadmy Biosci. 3(), of Scincs January 7 9
3 Klaas E Stphan t al how th local computations ar bound togthr by contxtdpndnt intractions among ths aras. This binding is th functional intgration within th systm which can b charactrizd in two ways, namly in trms of functional connctivity and ffctiv connctivity. Whil functional connctivity dscribs statistical dpndncis btwn data, ffctiv connctivity rsts on a mchanistic modl of th causal ffcts that gnratd th data (Friston 994). This articl focuss xclusivly on a rcntly stablishd tchniqu for rmining th ffctiv connctivity in nural systms of intrst on th basis of masurd fmri and EEG/ MEG data: Dynamic causal modlling (DCM; Friston t al 3). W rviw th concptual and mathmatical basis of DCM and Baysian modl slction (BMS; Pnny t al 4a) and dmonstrat som applications, using mpirical and simulatd data. W also touch on som futur xtnsions of DCM that ar drivn by th long-trm goal of using DCM for pharmacological and clinical applications, particularly with rgard to qustions about synaptic plasticity.. Effctiv connctivity and a gnral stat quation for nural systms Th trm ffctiv connctivity has bn dfind by various authors in convrgnt ways. A gnral dfinition is that ffctiv connctivity dscribs th causal influncs that nural units xrt ovr anothr (Friston 994). Mor spcifically, othr authors hav proposd that ffctiv connctivity should b undrstood as th xprimnt- and tim-dpndnt, simplst possibl circuit diagram that would rplicat th obsrvd timing rlationships btwn th rcordd nurons (Artsn and Prißl 99). Both dfinitions mphasiz that rmining ffctiv connctivity rquirs a causal modl of th intractions btwn th lmnts of th nural systm of intrst. Bfor w dscrib th spcifics of th modl on which DCM rsts, lt us driv a gnral mathmatical form of modls of ffctiv connctivity. For this purpos, w choos rministic diffrntial quations with tim-invariant paramtrs as a mathmatical framwork. Not that ths ar not th only possibl mathmatical rprsntation of systms; in fact, many altrnativs xist,.g. stat spac modls or itrativ maps. Th undrlying concpt, howvr, is quit univrsal: a systm is dfind by a st of lmnts with n tim-variant proprtis that intract with ach othr. Each tim-variant proprty x i ( i n) is calld a stat variabl, and th n- vctor x(t) of all stat variabls in th systm is calld th stat vctor (or simply stat) of th systm at tim t: x () t xt () =.. () xn () t Taking an nsmbl of intracting nurons as an xampl, th systm lmnts would corrspond to th individual nurons, ach of which is rprsntd by on or svral stat variabls. Ths stat variabls could rfr to various nurophysiological proprtis,.g. postsynaptic potntials, status of ion channls, tc. Critically, th stat variabls intract with ach othr, i.. th volution of ach stat variabl dpnds on at last on othr stat variabl. For xampl, th postsynaptic mmbran potntial dpnds on which and how many ion channls ar opn; vic vrsa, th probability of voltag-dpndnt ion channls opning dpnds on th mmbran potntial. Such mutual functional dpndncis btwn th stat variabls of th systm can b xprssd quit naturally by a st of ordinary diffrntial quations that oprat on th stat vctor: dx f( x,..., xn ) =. = F( x). () fn( x xn,..., ) Howvr, this dscription is not yt sufficint. First of all, th spcific form of th dpndncis f i nds to b spcifid, i.. th natur of th causal rlations btwn stat variabls. This rquirs a st of paramtrs θ which rmin th form and strngth of influncs btwn stat variabls. In nural systms, ths paramtrs usually corrspond to tim constants or synaptic strngths of th connctions btwn th systm lmnts. Th Boolan natur of θ, i.. th pattrn of absnt and prsnt connctions, and th mathmatical form of th dpndncis f i rprsnt th structur of th systm. And scond, in th cas of non-autonomous systms (i.. systms that xchang mattr, nrgy or information with thir nvironmnt) w nd to considr th inputs into th systm,.g. snsory information ntring th brain. W rprsnt th st of all m known inputs by th m-vctor function u(t). Extnding q. accordingly lads to a gnral stat quation for non-autonomous rministic systms: dx = F( x, u, θ ). (3) A modl whos form follows this gnral stat quation provids a causal dscription of how systm dynamics rsults from systm structur, bcaus it dscribs (i) whn and whr xtrnal inputs ntr th systm; and (ii) how th stat changs inducd by ths inputs volv in tim dpnding on th systm s structur. Givn a particular tmporal squnc of inputs u(t) and an initial stat x(), on obtains a complt dscription of how th dynamics of th systm (i.. th trajctory of its stat vctor x in tim) rsults from its structur by intgration of q. 3: x( ) x( ) F( x, u, θ). = + (4) J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 3 Equation 3 thrfor provids a gnral form for modls of ffctiv connctivity in nural systms. As dscribd lswhr (Friston t al 3; Stphan 4), all stablishd modls of ffctiv connctivity, including rgrssion-lik modls (.g. McIntosh and Gonzalz-Lima 994; Harrison t al 3), can b rlatd to this gnral quation. In th nxt sctions, w show how DCM modls nural population dynamics using a bilinar implmntation of this gnral form. This is combind with a forward modl that translats nural activity into a masurd signal. Bfor w procd to DCM, it is worth pointing out that w hav mad two main assumptions in this sction to simplify th xposition to th gnral stat quation. First, it is assumd that all procsss in th systm ar rministic and occur instantanously. Whthr or not this assumption is valid dpnds on th particular systm of intrst. If ncssary, random componnts (nois) and dlays could b accountd for by using stochastic diffrntial quations and dlay diffrntial quations, rspctivly. An xampl of th lattr is found in DCM for ERPs (s blow). Scond, w assum that w know th inputs that ntr th systm. This is a tnabl assumption in nuroimaging bcaus th inputs ar xprimntally controlld variabls,.g. changs in stimuli or instructions. It may also b hlpful to point out that using tim-invariant dpndncis f i and paramtrs θ is not a rstriction. Although th mathmatical form of f i pr s is static, th us of tim-varying inputs u allows for dynamic changs in what componnts of f i ar activatd. For xampl, input functions that can only tak valus of on or zro and that ar multiplid with th diffrnt trms of a polynomial function can b usd to induc tim-dpndnt changs from nonlinar to linar bhaviour (.g. by switching off all highr ordr trms in th polynomial) or vic vrsa. Also, thr is no principld distinction btwn stats and tim-invariant paramtrs. Thrfor, stimating tim-varying paramtrs can b tratd as a stat stimation problm. 3. Principls of DCM An important limitation of prvious mthods for rmining ffctiv connctivity from functional imaging data,.g. structural quation modlling (McIntosh and Gonzalz- Lima 994; Büchl and Friston 997) or multivariat autorgrssiv modls (Gobl t al 3; Harrison t al 3), is that thy oprat at th lvl of th masurd signals. This is a srious problm bcaus th causal architctur of th systm that w would lik to idntify is xprssd at th lvl of nuronal dynamics which is not dirctly obsrvd using non-invasiv tchniqus. In th cas of fmri data, for xampl, prvious modls of ffctiv connctivity wr fittd to th masurd tim sris which rsult from a hamodynamic convolution of th undrlying nural activity. Sinc classical modls do not includ th forward modl linking nuronal activity to th masurd hamodynamic data, analyss of intr-rgional connctivity prformd at th lvl of hamodynamic rsponss ar problmatic. For xampl, diffrnt brain rgions can xhibit markd diffrncs in nurovascular coupling, and ths diffrncs, xprssd in diffrnt latncis, undrshoots, tc. may lad to fals infrnc about connctivity. A similar situation is sn with EEG data whr thr is a big diffrnc btwn signals masurd at ach lctrod and th undrlying nuronal activity: changs in nural activity in diffrnt brain rgions lad to changs in lctric potntials that suprimpos linarly. Th scalp lctrods thrfor rcord a mixtur, with unknown wightings, of potntials gnratd by a numbr of diffrnt sourcs. Thrfor, to nabl infrncs about connctivity btwn nural units w nd modls that combin two things: (i) a parsimonious but nurobiologically plausibl modl of nural population dynamics, and (ii) a biophysically plausibl forward modl that dscribs th transformation from nural activity to th masurd signal. Such modls mak it possibl to fit jointly th paramtrs of th nural and of th forward modl such that th prdictd tim sris ar optimally similar to th obsrvd tim sris. This combination of a modl of nural dynamics with a biophysical forward modl is a cor fatur of DCM. Currntly, DCM implmntations xist both for fmri data and vnt-rlatd potntials (ERPs) as masurd by EEG/ MEG. Ths modality-spcific implmntations ar brifly summarizd in th nxt sctions. 4. DCM for fmri DCM for fmri uss a simpl modl of nural dynamics in a systm of n intracting brain rgions. It modls th chang of a nural stat vctor x in tim, with ach rgion in th systm bing rprsntd by a singl stat variabl, using th following bilinar diffrntial quation: dx ( n) = F( x, u, θ m ( j) = A+ ujb x+ Cu. j= (5) Not that this nural stat quation follows th gnral form for rministic systm modls introducd by q.3, i.. th modlld stat changs ar a function of th systm stat itslf, th inputs u and som paramtrs θ (n) that dfin th functional architctur and intractions among brain rgions at a nuronal lvl. Th nural stat variabls rprsnt a summary indx of nural population dynamics in th rspctiv rgions. Th nural dynamics ar drivn by xprimntally controlld xtrnal inputs that can ntr J. Biosci. 3(), January 7
3 Klaas E Stphan t al th modl in two diffrnt ways: thy can licit rsponss through dirct influncs on spcific rgions (.g. vokd rsponss in arly snsory cortics; th C matrix) or thy can modulat th coupling among rgions (.g. during larning or attntion; th B matrics). Not that q. 5 dos not account for conduction dlays in ithr inputs or intr-rgional influncs. This is not ncssary bcaus, du to th larg rgional variability in hmodynamic rspons latncis, fmri data do not posss nough tmporal information to nabl stimation of intr-rgional axonal conduction dlays which ar typically in th ordr of - ms (not that th diffrntial latncis of th hmodynamic rspons ar accommodatd by rgion-spcific biophysical paramtrs in th hmodynamic modl dscribd blow). This was vrifid by Friston t al (3) who showd in simulations that DCM paramtr stimats wr not affctd by introducing artificial dlays of up to ± s. In contrast, conduction dlays ar an important part of DCM for ERPs (s blow). Givn th bilinar stat quation (q. 5), th nural paramtrs θ (n) = {A, B, C} can b xprssd as partial drivativs of F: F A = x u= j F B = ( ) xuj F C =. u x= (6) As can b sn from ths quations, th matrix A rprsnts th fixd connctivity among th rgions in th absnc of input, th matrics B (j) ncod th chang in connctivity inducd by th jth input u j, and C mbodis th strngth of dirct influncs of inputs on nuronal activity. Figur summariss this bilinar stat quation and shows a spcific xampl modl. A Gnral bilinar stat quation dx m j = A + ( ) ( u j B ) x + j= Cu B y dx = ax + ax + cu dx = ax + ax + a3x3 + b dx3 = a3x + a33x3 y y 3 () u x + b (3) 3 u x 3 3 Figur. (A) Th bilinar stat quation of DCM for fmri. (B) An xampl of a DCM dscribing th dynamics in a hirarchical systm of visual aras. This systm consists of aras V and V5 and th suprior parital cortx (SPC). Each ara is rprsntd by a singl stat variabl (x...x 3 ). Black arrows rprsnt connctions, gry arrows rprsnt xtrnal inputs into th systm and thin dottd arrows indicat th transformation from nural stats into hamodynamic obsrvations (thin boxs; s figur for th hamodynamic forward modl). In this xampl, visual stimuli (photic) driv activity in V which is propagatd to V5 and SPC through th connctions btwn th aras. Th V V5 connction is allowd to chang whnvr th visual stimuli ar moving, and th SPC V5 connction is modulatd whnvr attntion is dirctd to motion. Th stat quation for this particular xampl is shown on th right.. J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 33 nuronal input x stat variabls z = { x, s, f, v, q} f activity-dpndnt signal s= x κs γ ( f ) s flow induc tion f = s f changs in volum v changs in dhb / v f v α = q = f E( f, ρ) qρ v v q / α q/v y = λ( v,q) hamodynamic rspons Figur. Summary of th hamodynamic modl usd by DCM for fmri. Nuronal activity inducs a vasodilatory and activity-dpndnt signal s that incrass blood flow f. Blood flow causs changs in volum and doxyhamoglobin (v and q). Ths two hamodynamic stats ntr th output nonlinarity which rsults in a prdictd BOLD rspons y. Th modl has 5 hmodynamic paramtrs: th rat constant of th vasodilatory signal dcay (κ), th rat constant for auto-rgulatory fdback by blood flow (γ), transit tim (), Grubb s vssl stiffnss xponnt (α), and capillary rsting nt oxygn xtraction (ρ). E is th oxygn xtraction function. Adaptd, with prmission by Elsvir Ltd., from Friston t al (3). DCM for fmri combins this modl of nural dynamics with an xprimntally validatd hamodynamic modl that dscribs th transformation of nuronal activity into a BOLD rspons. This so-calld Balloon modl was initially formulatd by Buxton t al (998) and latr xtndd by Friston t al (). Brifly, it consists of a st of diffrntial quations that dscrib th rlations btwn four hamodynamic stat variabls, using fiv paramtrs θ (h). Mor spcifically, changs in nural activity licit a vasodilatory signal that lads to incrass in blood flow and subsquntly to changs in blood volum and doxyhmoglobin contnt. Th prdictd BOLD signal is a non-linar function of blood volum and doxyhamoglobin contnt. This hamodynamic modl is summarisd by figur and dscribd in ail by Friston t al (). Th combind nural and hamodynamic paramtr st θ = {θ (n), θ (h) } is stimatd from th masurd BOLD data, using a fully Baysian approach with mpirical priors for th hamodynamic paramtrs and consrvativ shrinkag priors for th coupling paramtrs. Dtails of th paramtr stimation schm, which rsts on an xpctation maximization (EM; Dmpstr t al 977) algorithm and uss a Laplac (i.. Gaussian) approximation to th tru postrior, can b found in Friston (b). J. Biosci. 3(), January 7
34 Klaas E Stphan t al Onc th paramtrs of a DCM hav bn stimatd from masurd BOLD data, th postrior distributions of th paramtr stimats can b usd to tst hypothss about connction strngths. Du to th Laplac approximation, th postrior distributions ar dfind by thir postrior mod or maximum a postriori (MAP) stimat and thir postrior covarianc. Usually, th hypothss to b tstd concrn contxt-dpndnt changs in coupling. A classical xampl is givn by figur 3. Hr, DCM was applid to fmri data from a singl subjct, tsting th hypothsis that in a hirarchical systm of visual aras (c.f. figur ) attntion to motion nhancd th backward connctions from th infrior frontal gyrus (IFG) onto suprior parital cortx (SPC) and from SPC onto V5, rspctivly. Othr xampls of singl-subjct analyss can b found in Mchlli t al (3), Pnny t al (4b) and Stphan t al (5). For statistical infrnc at th group lvl, various options xist. Th simplst approach is to ntr th conditional stimats of intrst into a classical scond-lvl analysis; for xampls s Bitan t al (5) and Smith t al (6). A mor cohrnt approach may b to us Baysian analyss at th group lvl as wll (M Garrido, J M Kilnr, S J Kibl, K E Stphan and K J Friston, unpublishd rsults). Fittd to rgional fmri tim sris, a givn DCM xplains how local brain rsponss wr gnratd from th intrplay of th thr mchanisms dscribd by th stat quation (q. 5): intr-rgional connctions, thir contxtual modulation and driving inputs. Figur 4 provids a simpl fictitious xampl that is basd on simulatd data. In this xampl w fix th paramtrs and us DCM as a modl to gnrat synthtic data, as opposd to its usual us, i.. stimating paramtr valus from mpirical data. Lt us imagin w ar daling with a x factorial xprimnt (figur 4A) whr on xprimntal factor controls snsory stimulation (stimulus S vs. stimulus S ) and a scond factor controls task rquirmnts (task T vs. task T ). Lt us furthr imagin that, using convntional statistical paramtric mapping, w had found a main ffct of snsory stimulation in a particular brain ara x (with obsrvd tim sris y ; s figur 4B, uppr panl) and a stimulus-by-task intraction in ara x (with obsrvd tim sris y ). This intraction mans that th diffrnc btwn stimulus S and stimulus S is largr during task T than during task T (s figur 4B, lowr panl). W can gnrat th (nois-fr) data shown in figur 4B using th DCM displayd by figur 4C. Th stimulus main ffct in ara x rsults from th driving inputs to x bing much strongr for stimulus S than for stimulus S. This diffrntial ffct is thn convyd onto ara x by th connction from x to x. Critically, th strngth of this connction is strongly nhancd during task T, but only marginally influncd during task T. This diffrnc in modulation causs th intraction in ara x (not that this modl would hav producd an intraction in ara x as wll if w had chosn a strongr back-connction from x to x ). Usually, of cours, DCM is applid in th rvrs fashion, i.. to stimat th paramtrs θ (n) = {A, B, C} from masurd fmri data as in figur 4B. Th goal is to infr th nuronal mchanisms that hav shapd local brain rsponss,.g. th prsnc of main ffcts or intractions. Simulations lik th on dscribd abov can also b usd to xplor th robustnss of paramtr stimation in DCM. For xampl, on can gnrat data multipl tims, adding obsrvation nois (s figur 4D), and thn trying to r-stimat th paramtrs from th noisy data. W ar currntly working on various xtnsions to DCM for fmri. Concrning th forward modl, S J Kibl, S K löppl, N Wiskopt and K J Friston (unpublishd rsults) hav augmntd th obsrvation quation by taking into account th slic-spcific sampling tims in multi-slic MRI acquisitions. This nabls DCM to b applid to fmri data from any acquisition schm (compar Friston t al 3 for rstrictions of th original DCM formulation in this rgard) and provids for mor vridical rsults. With rgard to th nural stat quation, on currnt xtnsion is to rprsnt ach rgion in th modl by multipl stat variabls,.g. populations of xcitatory and inhibitory nurons (Marriros t al in prparation; s also Harrison t al 5). A similar approach has alrady bn implmntd in DCM for ERPs which is dscribd in th following sction. Finally, w ar currntly augmnting th stat quation of DCM for fmri by including additional non-linar trms (Stphan t al in prparation). An xampl is th following xtnsion: T () xd x dx m j = A+ u B ( ) j x+ Cu+ j. = T ( n) xd x. (7) This xtnsion nhancs th kind of dynamics that DCM can captur and nabls th usr to implmnt additional typs of modls. For xampl, byond modlling how connction strngths ar modulatd by xtrnal inputs, on can now modl how connction strngths chang as a function of th output from aras. This ability is critical for various applications,.g. for marrying rinforcmnt larning modls with DCM (c.f. Stphan 4). In a nural systm modl of dscriptiv larning thoris lik tmporal diffrnc larning, th prdiction rror, ncodd by th activity of a particular nural unit, rmins th chang of connction strngth btwn othr nural units that ncod proprtis of conditional and unconditional stimuli (s Schultz and Dickinson ). Figur 5A shows a simulation xampl whr th connction from an ara x to anothr ara x is nhancd multiplicativly by th output from a third rgion x 3, i.. dx ( ) = ax+ ax+ d 3 xx3. J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 35 (A) Attntion Photic.5 (98%).37 (9%) SPC.4 (%) Motion V.8 (%).65 (%).56 (99%) V5.47 (%).69 (%) IFG (B) Figur 3. (A) DCM applid to data from a study on attntion to visual motion by Büchl and Friston (997). Th modl is similar to th on shown in figur, xcpt for th addition of anothr ara, th infrior frontal gyrus (IFG). Th most intrsting aspct of this modl concrns th rol of motion and attntion in xrting bilinar ffcts on connctions in th modl. Th prsnc of motion in th visual stimulation nhancs th connction from ara V to th motion snsitiv ara V5. Th influnc of attntion is to nabl backward connctions from th IFG to th suprior parital cortx (SPC) from SPC to V5. Dottd arrows conncting rgions rprsnt significant bilinar affcts in th absnc of a significant intrinsic coupling. Inhibitory slf-connctions ar not displayd for clarity. (B) Fittd rsponss basd upon th conditional stimats and th adjustd data. Th insrt shows th approximat location of th rgions. Adaptd, with prmission by Elsvir Ltd., from Friston t al (3). J. Biosci. 3(), January 7
36 Klaas E Stphan t al (B) y (A) Task factor Task Task Stimulus factor Stim Stim T /S T /S T /S T /S y (C) (D) y y Figur 4. (A) Summary of a fictitious x factorial xprimntal dsign, comprising task and stimulus factors. (B) Simulatd BOLD rsponss of two aras, y and y (without obsrvation nois). Th first ara shows a main ffct of stimulus and th scond ara additionally shows a stimulus-by-task intraction. Th rd and grn bars dnot whn task and task ar prformd, rspctivly. (C) Th DCM which was usd to gnrat th nois-fr rsponss shown in (B). As shown schmatically, all inputs wr box-car functions. +++ dnots strongly positiv and + dnots wakly positiv inputs and connction strngths, dnots ngativ connction strngths. Th diffrnt strngths of th driving inputs induc a main ffct of stimulus in th first ara, x. This ffct is convyd onto th scond ara, x, by mans of th x x connction. Critically, th strngth of this connction varis as a function of which task is prformd. This bilinar modulation inducs a stimulus-by-task intraction in x (c.f. B). (D). Data gnratd from th modl shown in (C) but with additional obsrvation nois (signalto-nois ratio of unity). J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 37 Critically, x 3 is not only drivn by xtrnal inputs, but also rcivs an input from x. This mans that for an xcitatory connction from x to x 3, a positiv rinforcmnt ffct rsults: th highr activity in x 3, th mor strongly inputs from x to x will b nhancd, lading to highr activity in x, which, in turn, drivs x 3 vn furthr. Figur 5B shows an xampl of this ffct, using simulatd data. Such a modl, of cours, livs on th brink of stability and is pron to runaway xcitation, which rquirs rgularisation with suitabl priors on th paramtrs. In contrast, an inhibitory connction from x to x 3 maks th modl xtrmly stabl bcaus th highr th activity in x 3, th highr th rspons in x to x inputs and thus th strongr th inhibitory fdback onto x 3 (not shown hr). 5. DCM for ERPs ERPs as masurd with EEG or MEG hav bn usd for dcads to study lctrophysiological corrlats of cognitiv oprations. Nvrthlss, th nurobiological mchanisms that undrli thir gnration ar still largly unknown. DCM for ERPs was dvlopd as a biologically plausibl modl to undrstand how vnt-rlatd rsponss rsult from th dynamics in coupld nural nsmbls. It rsts on a nural mass modl which uss stablishd connctivity ruls in hirarchical snsory systms to assmbl a ntwork of coupld cortical sourcs (Jansn and Rit 995; David and Friston 3; David t al 5). Ths ruls charactris connctions with rspct to thir laminar pattrns of origin and trmination and distinguish btwn (i) forward (or bottom-up) connctions originating in agranular layrs and trminating in layr 4, (ii) backward (or top-down) connctions originating and trminating in agranular layrs, and (iii) latral connctions originating in agranular layrs and targting all layrs. Ths intraral cortico-cortical connctions ar xcitatory, using glutamat as nurotransmittr, and aris from pyramidal clls (figur 6). Each rgion or sourc is modlld as a microcircuit in which thr nuronal subpopulations ar combind and assignd to granular and supra-/infragranular layrs. A population of xcitatory pyramidal (output) clls rcivs inputs from inhibitory and xcitatory populations of intrnurons via intrinsic (intra-aral) connctions. Within this modl, xcitatory intrnurons can b rgardd as spiny stllat clls which ar found in layr 4 and rciv forward connctions. Although xcitatory pyramidal clls and inhibitory intrnurons ar found in both infra- and supragranular layrs in cortx, on dos not nd to rprsnt both cll typs in both layrs in th modl. To modl th clltyp spcific targts of backward and latral connctions, it is sufficint to rprsnt, for xampl, pyramidal clls in infragranular layrs and intrnurons in supragranular layrs and constrain th origins and targts of backward and latral connctions as shown in figur. 6. Th nural stat quations ar summarizd in figur 7. To prturb th systm and modl vnt-rlatd rsponss, th ntwork rcivs inputs via input connctions. Ths connctions ar xactly th sam as forward connctions and dlivr input u to th spiny stllat clls in layr 4. Input u rprsnts affrnt activity rlayd by subcortical structurs and ar modlld as two paramtrizd componnts, a gamma dnsity function (rprsnting an vnt-rlatd burst of input that is dlayd and disprsd by subcortical synapss and axonal conduction) and a discrt cosin st (rprsnting fluctuations in input ovr pristimulus tim). Th influnc of this input on ach sourc is controlld by a paramtr vctor C. Ovrall, th DCM is spcifid in trms of th stat quations shown in figur 7 and a linar forward modl dx = f( x, u, θ) y = Lx + ε, (8) whr x rprsnts th transmmbran potntial of pyramidal clls, y is th masurd data at th snsor lvl, L is a lad fild matrix coupling lctrical sourcs to th EEG channls, and ε is obsrvation rror. In comparison to DCM for fmri, th forward modl is a simpl linarity as opposd to th nonlinar hamodynamic modl in DCM for fmri. In contrast, as vidnt from th dscriptions abov and a comparison of figurs and 7, th stat quations of DCM for ERPs ar much mor aild and ralistic. On could rgard th bilinar approximation for fmri as a bilinar approximation to th stat quations for EEG. Howvr, th DCMs for fmri ar furthr simplifid bcaus thr is only on nuronal stat for ach rgion or sourc. As an xampl for th addd complxity in DCM for ERPs, considr th stat quation for th inhibitory subpopulation: x 7 = x8 H x x x C B C L I S x 8 7 = + + (( γ ) ( )). 8 3 (9) Hr, th paramtr matrics C F, C B, C L ncod forward, backward and latral connctions rspctivly. Within ach subpopulation, th dynamics of nural stats ar rmind by two oprators. Th first transforms th avrag dnsity of prsynaptic inputs into th avrag postsynaptic mmbran potntial. This is modlld by a linar transformation with xcitatory () and inhibitory (i) krnls paramtrizd by H,i and,i. H,i control th maximum postsynaptic potntial and,i rprsnt lumpd rat constants (i.. lumpd across dndritic spins and th dndritic tr). Th scond oprator S transforms th avrag potntial of ach subpopulation into an avrag firing rat. This is assumd to b instantanous and is a sigmoid function. Intra-aral intractions among th subpopulations dpnd on constants γ 4 which control th J. Biosci. 3(), January 7
38 Klaas E Stphan t al (A) mod mod +++ + x 3 stim + +++ stim + +++ + +++ x x (B) stim stim.4. -..4. -..4. -..5.5.5 3 x 4 3 4 5 6 7 8 9.5.5.5 3 x 4 3 4 5 6 7 8 9 mod mod mod mod.5.5.5 3 x 4 3 4 5 6 7 8 9 Figur 5. (A) Exampl of a DCM with scond-ordr trms in th stat quation. In this xampl, th third ara modulats th connction from th first to th scond ara. Th first ara is drivn by two diffrnt stimuli (stim, stim; randomly mixd vnts, rprsntd as dlta functions, 4 s apart) and th third ara is drivn by som inputs rprsnting cognitiv st (mod, mod; altrnating blocks of 5 s duration, shown as gry boxs in (B). Not that th third ara is not only drivn by xtrnal input but also rcivs an input from th scond ara. +++ dnots strongly positiv and + dnots wakly positiv inputs and connction strngths, dnots ngativ connction strngths. (B) Simulatd rsponss of this systm (not that all inputs and connctions wr givn positiv wights in this simulation). From top to bottom, th plots show th nural (x) and hamodynamic (y) rsponss in altrnating fashion. Th x-axis dnots tim (for hamodynamic rsponss in sconds, for nural rsponss in tim bins of 4 ms), th y-axis dnots arbitrary units. It can b sn asily that vokd activity in th first ara only causs a significant rspons in th scond ara if th third ara shows a high lvl of activity and thus nabls th x x connction. Furthrmor, du to th xcitatory x x 3 connction, a positiv rinforcmnt ffct rsults. Both mchanisms lad to obvious nonlinaritis in th gnratd data (s thick arrows for an xampl). Not that this modl, similar th on in figur 4, also gnrats a stim mod intraction in th scond ara. This is hardr to s by y than in figur 4 bcaus hr th driving inputs ar randomly mixd vnts and additionally, strong non-linar ffcts occur. J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 39 inhibitory intrnurons spiny stllat clls pyramidal clls Intrinsic Forward Backward Latral Input u 3 that on dos not ncssarily hav to assum known lad fild paramtrs (L in q. 8) for th forward modl. Instad, it is possibl to stimat lad-fild and coupling paramtrs simultanously and thus us DCM for ERPs as a sourc rconstruction approach with physiologically informd constraints. Futur fforts will concntrat on furthr nhancing th biological ralism of th modl. On approach may b to introduc a modulation of coupling paramtrs btwn th nuronal populations, within rgions. This nabls on to modl within-rgion adaptation, as opposd to changs in coupling btwn rgions (Kibl t al in prparation). Anothr and mor long-trm goal will b to includ mchanisms rlatd to particular nurotransmittrs in th modl,.g. modulation of NMDA-dpndnt synaptic plasticity by dopamin or actylcholin (Stphan t al 6). This will b particularly important for potntial clinical applications of DCM (s blow). Howvr, prior to any clinical applications, this approach will rquir carful validation using pharmacological paradigms in humans and animals. In particular, on will nd to dmonstrat a clos rlationship btwn rcptor status (that is systmatically changd by pharmacological manipulation) and th corrsponding paramtr stimats in th DCM. Figur 6. A schma of th nural populations which ar modlld sparatly for ach rgion in DCM for ERPs. Diffrnt rgions ar coupld by forward, backward and latral connctions, all of which originat from xcitatory pyramidal clls but targt spcific populations. Th figur shows a typical hirarchical ntwork composd of thr cortical aras. Extrinsic inputs vok transint prturbations around th rsting stat by acting on a subst of sourcs, usually th lowst in th hirarchy. Rproducd with prmission by Elsvir Ltd. from David t al (6). strngth of intrinsic connctions and rflct th total numbr of synapss xprssd by ach subpopulation. In q. 9, th top lin xprsss th rat of chang of voltag as a function of currnt (assuming constant capacitanc of th cll mmbran). Th scond lin spcifis how currnt changs as a function of voltag and currnt. For simplification, our dscription hr has omittd th fact that in DCM for ERPs all intra- and intr-aral connctions hav conduction dlays. This is implmntd by dlay diffrntial quations. Just as with DCM for fmri, th DCM for ERPs is usually usd to invstigat whthr coupling strngths chang as a function of xprimntal contxt. Figur 8 shows an xampl of a DCM applid to EEG data from a singl subjct prforming an auditory oddball task (David t al 6): forward and backward connctions btwn primary auditory and orbitofrontal cortx ar strongr during procssing of oddball stimuli compard to standard stimuli. Similar to DCM for fmri, svral xtnsions of DCMs for lctrophysiological masurs ar plannd or alrady undr way. For xampl, Kibl t al (6) dmonstratd 6. Baysian modl slction A gnric problm ncountrd by any kind of modlling approach is th qustion of modl slction: givn som obsrvd data, which of svral altrnativ modls is th optimal on? This problm is not trivial bcaus th dcision cannot b mad solly by comparing th rlativ fit of th compting modls. On also nds to tak into account th rlativ complxity of th modls as xprssd, for xampl, by th numbr of fr paramtrs in ach modl. Modl complxity is important to considr bcaus thr is a tradoff btwn modl fit and gnralisability (i.. how wll th modl xplains diffrnt data sts that wr all gnratd from th sam undrlying procss). As th numbr of fr paramtrs is incrasd, modl fit incrass monotonically whras byond a crtain point modl gnralisability dcrass. Th rason for this is ovrfitting : an incrasingly complx modl will, at som point, start to fit nois that is spcific to on data st and thus bcom lss gnralisabl across multipl ralizations of th sam undrlying gnrativ procss. [Gnrally, in addition to th numbr of fr paramtrs, th complxity of a modl also dpnds on its functional form; s Pitt and Myung (). This is not an issu for DCM, howvr, bcaus hr compting modls usually hav th sam functional form.] Thrfor, th qustion Which is th optimal modl among svral altrnativs? can b rformulatd mor prcisly as Givn svral altrnativs, which modl rprsnts th bst balanc btwn fit and complxity? J. Biosci. 3(), January 7
4 Klaas E Stphan t al inhibitory intrnurons spiny stllat clls pyramidal clls Extrinsic forward connctions C F S( x ) 7 γ γ 4 3 4 5 3 6 = x 8 H 8 = (( C 4 5 = x 5 6 = x H = H = = x i (( C = x x 6 (( C B F B + C + C H i x6 = γ 4S( x7 ) L L x + γ 3I) S( x )) + γ I) S( x ) + C i x 3 i U 8 x 7 x u) γ γ Intrinsic connctions L x5 + C ) S( x ) + γ S( x )) 4 x x Extrinsic latral connctions Extrinsic backward connctions C L S( x ) C B S( x ) Figur 7. Schmatic of th nural modl in DCM for ERPs. This schma shows th stat quations dscribing th dynamics of a microcircuit rprsnting an individual rgion (sourc). Each rgion contains thr subpopulations (pyramidal, spiny stllat and inhibitory intrnurons) that ar linkd by intrinsic connctions and hav bn assignd to supragranular, granular and infragranular cortical layrs. Diffrnt rgions ar coupld through xtrinsic (long-rang) xcitatory connctions. Rproducd with prmission by Elsvir Ltd. from David t al (6). In a Baysian contxt, th lattr qustion can b addrssd by comparing th vidnc, P(y m), of diffrnt modls. According to Bays thorm py ( θ, mp ) ( θ m) p( θ ym, ) =, py ( m) () th modl vidnc can b considrd as a normalization constant for th product of th liklihood of th data and th prior probability of th paramtrs, thrfor p( y m) = p( y θ, m) p( θ m) dθ. () Hr, th numbr of fr paramtrs (as wll as th functional form) ar considrd by th intgration. Unfortunatly, this intgral cannot usually b solvd analytically, thrfor an approximation to th modl vidnc is ndd. In th contxt of DCM, on potntial solution could b to mak us of th Laplac approximation, i.. to approximat th modl vidnc by a Gaussian that is cntrd on its mod. As shown by Pnny t al (4a), this yilds th following xprssion for th natural logarithm (ln) of th modl vidnc (η θ y dnots th MAP stimat, C θ y is th postrior covarianc of th paramtrs, C ε is th rror covarianc, θ P is th prior man of th paramtrs, and C P is th prior covarianc): ln p( y m) = accuracy( m) complxity( m) ( ) ( ) T = ln C y h( u, η y C θ ε y h u hθ y (, ε T ln Cp ln Cθ y + ( ηθ y θp) Cp ( ηθ y p). θ () This xprssion proprly rflcts th rquirmnt, as discussd abov, that th optimal modl should rprsnt th bst compromis btwn modl fit (accuracy) and modl complxity. W us it routinly in th contxt of DCM for ERPs (compar David t al 6). In th cas of DCM for fmri, a complication ariss. This is du to th complxity trm which dpnds on th prior dnsity, for xampl, th prior covarianc of th intrinsic connctions. This is problmatic in th contxt of DCM for fmri bcaus th prior covarianc is dfind in a modl-spcific fashion to nsur that th probability of J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 4 3 mod PC - - 4 OF OF 3 - mod.74 (98%) 3.58 (%).7 (95%) STG - 4 3.3 (97%) 3 mod 3 A A.93 (%) - - 4 6 tim (ms) standard (prdictd) standard (obsrvd) oddball (prdictd) oddball (obsrvd) input 4 Figur 8. DCM for ERPs masurd during an auditory oddball paradigm. Lft: Prdictd (thick) and obsrvd (thin) rsponss in masurmnt spac. Ths ar a projction of th scalp or channl data onto th first thr spatial mods or ignvctors of th channl data. Th prdictd rsponss ar basd on th conditional xpctations of th DCM paramtrs and show vry good agrmnt with th masurd data. Right: Graph dpicting th sourcs and connctions of a DCM in which both forward and backward connctions wr allowd to chang btwn oddball and standard trials. Th rlativ strngth of coupling strngths for oddball rlativ to standard stimuli ar shown alongsid th connctions. Th prcnt conditional confidnc that this diffrnc is gratr than zro is shown in brackts. Only changs with 9% confidnc or mor (solid lins) ar shown numrically. In all connctions th coupling was strongr during oddball procssing, rlativ to standards. A, primary auditory cortx; OF, orbitofrontal cortx; PC, postrior cingulat cortx; STG, suprior tmporal gyrus. Rproducd with prmission by Elsvir Ltd. from David t al (6). obtaining an unstabl systm is vry small. (Spcifically, this is achivd by choosing th prior covarianc of th intrinsic coupling matrix A such that th probability of obtaining a positiv Lyapunov xponnt of A is P <.; s Friston t al 3 for ails.) Consquntly, in this particular contxt, usag of th Laplacian approximation complicats comparison of modls with diffrnt numbrs of connctions. In DCM for fmri, mor suitabl approximations, which do not dpnd on th prior dnsity, ar affordd by th Baysian information critrion (BIC) and Akaik Information Critrion (AIC), rspctivly. As shown by Pnny t al (4a), for DCM ths approximations ar givn by BIC accuracy m d θ = ( ) lnn AIC = accuracy( m) d, θ (3) whr d θ is th numbr of paramtrs and N is th numbr of data points (scans). If on compars th complxity trms of BIC and AIC, it bcoms obvious that BIC pays a havir pnalty than AIC as soon as on dals with 8 or mor scans (which is virtually always th cas for fmri data). Thrfor, J. Biosci. 3(), January 7
4 Klaas E Stphan t al BIC will b biasd towards simplr modls whras AIC will b biasd towards mor complx modls. This can lad to disagrmnt btwn th two approximations about which modl should b favourd. In DCM for fmri, w hav thrfor adoptd th convntion that, for any pairs of modls m i and m j to b compard, a dcision is only mad if AIC and BIC concur (s blow); th dcision is thn basd on that approximation which givs th smallr Bays factor (BF): py mi BFij = ( ) py ( m). (4) j Just as convntions hav dvlopd for using P-valus in frquntist statistics, thr ar convntions for th us of BFs. For xampl, Raftry (995) suggsts intrprtation of BFs as providing wak (BF < 3), positiv (3 BF < ), strong ( BF < 5) or vry strong (BF 5) vidnc for prfrring on modl ovr anothr. BMS plays a cntral rol in th application of DCM. Th sarch for th bst modl, amongst svral compting ons, prcds (and is oftn qually important to) th qustion which paramtrs of th modl rprsnt significant ffcts. Svral studis hav usd BMS succssfully to addrss complx qustions about th architctur of nural systms. For xampl, Pnny t al (4a) invstigatd which connctions in a systm of hirarchically connctd visual aras wr most likly to undrli th modulatory ffcts of attntion to motion that wr obsrvd in th BOLD rsponss of ara V5. Thy found, using data from a singl subjct, that th bst modl was on in which attntion nhancd V5 rsponss to V inputs. In anothr singlsubjct study, Stphan t al (5) systmatically drivd 6 diffrnt modls that could hav xplaind BOLD activity in visual aras during latralizd prsntation of visual word stimuli. Thy found vidnc that, in this subjct, intrhmisphric connctions srvd task-dpndnt information transfr from th non-dominant to th dominant hmisphr but only whn th stimulus was initially rcivd by th non-dominant hmisphr. Finally, M Garrido, J M Kilnr, S J Kibl, K E Stphn and K J Friston (unpublishd rsults) xtndd th prvious work by David t al (6) and applid BMS in th contxt of an auditory oddball study, masurd with EEG, to find th most likly xplanation, in trms of coupling changs, for th wll-known mismatch ngativity potntial. Thy found that thir group of halthy controls was dividd into two subgroups charactrizd by diffrnt optimal modls. R-xamining th ERPs of ths subgroups sparatly rvald a significant diffrnc in th xprssion of mismatch-rlatd rsponss that would hav bn missd in convntional ERP analyss. This xampl highlights that BMS may also b of considrabl intrst for dfining clinical populations for whom biological markrs ar prsntly lacking. This issu is takn up in th nxt and final sction. 7. Outlook to futur applications of DCM DCM is currntly th most advancd framwork for infrring th ffctiv connctivity in nural systms from masurd functional nuroimaging data. Our hop is that ovr th nxt yars, th gnric framwork of DCM and th ongoing dvlopmnts, som of which wr brifly dscribd in this articl, will contribut to a mor mchanistic undrstanding of brain function. Of particular intrst will b th us of nural systm modls lik DCM (i) to undrstand th mchanisms of drugs and (ii) to dvlop modls that can srv as diagnostic tools for disass linkd to abnormalitis of connctivity and synaptic plasticity,.g. schizophrnia. Concrning pharmacology, many drugs usd in psychiatry and nurology chang synaptic transmission and thus functional coupling btwn nurons. Thrfor, thir thraputic ffcts cannot b fully undrstood without modls of drug-inducd connctivity changs in particular nural systms. So far, only rlativly fw studis hav studid pharmacologically inducd changs in connctivity (.g. Hony t al 3). As highlightd in a rcnt rviw by Hony and Bullmor (4), an xciting possibility for th futur is to us systm modls at th arly stag of drug dvlopmnt to scrn for substancs that induc dsird changs of connctivity in nural systms of intrst with a rasonably wll undrstood physiology. Th succss of this approach will partially dpnd on dvloping modls that includ additional lvls of biological ail (.g. ffcts of diffrnt nurotransmittrs, s abov) whil bing parsimonious nough to nsur mathmatical idntifiability and physiological intrprtability; s Brakspar t al (3), Harrison t al (5), Jirsa (4) and Robinson t al () for xampls that mov in this dirction. Anothr important goal is to xplor th utility of modls of ffctiv connctivity as diagnostic tools (Stphan 4). This sms particularly attractiv for psychiatric disass whos phnotyps ar oftn vry htrognous and whr a lack of focal brain pathologis points to abnormal connctivity (dysconnctivity) as th caus of th illnss. Givn a pathophysiological thory of a spcific disas, connctivity modls might allow on to dfin an ndophnotyp of that disas, i.. a biological markr at intrmdiat lvls btwn gnom and bhaviour, which nabls a mor prcis and physiologically motivatd catgorization of patints (Gottsman and Gould 3). Such an approach has rcivd particular attntion in th fild of schizophrnia rsarch whr a rcnt focus has bn on abnormal synaptic plasticity lading to dysconnctivity in nural systms concrnd with motional and prcptual larning (Friston 998; Stphan t al 6). A major challng will b to stablish nural systms modls which ar snsitiv nough that thir connctivity paramtrs can b usd rliably for diagnostic classification and tratmnt rspons prdiction J. Biosci. 3(), January 7
Dynamic causal modls of nural systm dynamics 43 of individual patints. Idally, such modls should b usd in conjunction with paradigms that ar minimally dpndnt on patint complianc and ar not confoundd by factors lik attntion or prformanc. Givn stablishd validity and sufficint snsitivity and spcificity of such a modl, on could us it in analogy to biochmical tsts in intrnal mdicin, i.. to compar a particular modl paramtr (or combinations throf) against a rfrnc distribution drivd from a halthy population (Stphan 4). Anothr possibility is to us DCM paramtr sts as inputs to statistical classification mthods in ordr to dfin distinct patint subpopulations. Altrnativly, if diffrnt clinical subgroups xhibit diffrnt fingrprints of dysconnctivity, ach rprsntd by a particular DCM, modl slction could provid a powrful approach to classify patints. Such procdurs could hlp to dcompos currnt psychiatric ntitis lik schizophrnia into mor wll-dfind subgroups charactrizd by common pathophysiological mchanisms and may facilitat th sarch for gntic undrpinnings. Acknowldgmnts This work was supportd by th Wllcom Trust. Rfrncs Artsn A and Prißl H 99 Dynamics of activity and connctivity in physiological nuronal Ntworks; in Non linar dynamics and nuronal ntworks (d.) H G Schustr (Nw York: VCH Publishrs) pp 8 3 Bitan T, Booth J R, Choy J, Burman D D, Gitlman D R and Msulam M M 5 Shifts of ffctiv connctivity within a languag ntwork during rhyming and splling; J. Nurosci. 5 5397 543 Brakspar M, Trry J R and Friston K J 3 Modulation of xcitatory synaptic coupling facilitats synchronization and complx dynamics in a biophysical modl of nuronal dynamics; Ntwork: Comput. Nural Sys. 4 73 73 Büchl C and Friston K J 997 Modulation of connctivity in visual pathways by attntion: cortical intractions valuatd with structural quation modlling and fmri; Crbral Cortx 7 768 778 Buxton R B, Wong E C and Frank L R 998 Dynamics of blood flow and oxygnation changs during brain activation: th balloon modl; Magn. Rson. Md. 39 855 864 David O and Friston K J 3 A nural mass modl for MEG/EEG: coupling and nuronal dynamics; NuroImag 743 755 David O, Harrison L M and Friston K J 5 Modlling vntrlatd rsponss in th brain; NuroImag 5 756 77 David O, Kibl S J, Harrison L M, Mattout J, Kilnr J M and Friston K J 6 Dynamic causal modling of vokd rsponss in EEG and MEG; NuroImag 3 55 7 Dmpstr A P, Laird N M and Rubin D B 977 Maximum liklihood from incomplt data via th EM algorithm; J. R. Stat. Soc. Sris B: Stat. Mthodol. 39 38 Friston K J 994 Functional and ffctiv connctivity in nuroimaging: a synthsis; Hum. Brain Mapping 56 78 Friston K J 998 Th disconnction hypothsis; Schizophrnia Rs. 3 5 5 Friston K J a Byond phrnology: What can nuroimaging tll us about distributd circuitry; Annu. Rv. Nurosci. 5 5 Friston K J b Baysian stimation of dynamical systms: An application to fmri; NuroImag 6 53 53 Friston K J, Harrison L and Pnny W 3 Dynamic causal modlling; NuroImag 9 73 3 Friston K J, Mchlli A, Turnr R and Pric C J Nonlinar rsponss in fmri: th Balloon modl, Voltrra krnls, and othr hmodynamics; NuroImag 466 477 Gobl R, Robrock A, Kim D S and Formisano E 3 Invstigating dirctd cortical intractions in tim-rsolvd fmri data using vctor autorgrssiv modling and Grangr causality mapping; Magn. Rson. Imaging 5 6 Gottsman I I and Gould T D 3 Th ndophnotyp concpt in psychiatry: tymology and stratgic intntions; Am. J. Psychiatry 6 636 645 Harrison L M, Pnny W and Friston K J 3 Multivariat autorgrssiv modling of fmri tim sris; NuroImag 9 477 49 Harrison L M, David O and Friston K J 5 Stochastic modls of nuronal dynamics; Philos. Trans. R. Soc. London B Biol. Sci. 36 75 9 Hony G D, Suckling J, Zlaya F, Long C, Routldg C, Jackson S, Ng V, Fltchr P C, Williams S C R and Brown J and Bullmor E T 3 Dopaminrgic drug ffcts on physiological connctivity in a human cortico-striato-thalamic systm; Brain 6 767 8 Hony G and Bullmor E 4 Human pharmacological MRI; Trnds Pharmacol. Sci. 5 366 374 Jansn B H and Rit V G 995 Elctroncphalogram and visual vokd potntial gnration in a mathmatical modl of coupld cortical columns; Biol. Cybrntics 73 357 366 Jirsa V K 4 Connctivity and dynamics of nural information procssing; Nuroinformatics 83 4 Kibl S J, David O and Friston K J 6 Dynamic causal modlling of vokd rsponss in EEG/MEG with lad-fild paramtrization; NuroImag 3 73 84 Mchlli A, Pric C J, Noppny U and Friston K J 3 A dynamic causal modling study on catgory ffcts: bottom up or top down mdiation?; J. Cognitiv Nurosci. 5 95 934 McIntosh A R and Gonzalz-Lima F 994 Structural quation modling and its application to ntwork analysis in functional brain imaging; Hum. Brain Mapping Pnny W D, Stphan K E, Mchlli A and Friston K J 4a Comparing dynamic causal modls; NuroImag 57 7 Pnny W D, Stphan K E, Mchlli A and Friston K J 4b Modlling functional intgration: a comparison of structural quation and dynamic causal modls; NuroImag 3 S64 S74 Pitt M A and Myung I J Whn a good fit can b bad; Trnds Cognitiv Nurosci. 6 4 45 Raftry A E 995 Baysian modl slction in social rsarch; in Sociological mthodology (d.) P V Marsdn (Cambridg, MA : Cambridg Univrsity Prss) pp 96 J. Biosci. 3(), January 7
44 Klaas E Stphan t al Robinson P A, Rnni C J, Wright J J, Bahramali H, Gordon E and Row D L Prdiction of lctroncphalographic spctra from nurophysiology; Phys. Rv. E63 93 Schultz W and Dickinson A Nuronal coding of prdiction rrors; Annu. Rv. Nurosci, 3 473 5 Smith A P R, Stphan K E, Rugg M D and Dolan R J 6 Task and contnt modulat amygdala-hippocampal connctivity in motional rtrival; Nuron 49 63 638 Stphan K E 4 On th rol of gnral systm thory for functional nuroimaging; J. Anat. 5 443 47 Stphan K E, Baldwg T and Friston K J 6 Synaptic plasticity and dysconnction in schizophrnia; Biol. Psychiatry 59 99 939 Stphan K E, Pnny W D, Marshall J C, Fink G R and Friston K J 5 Invstigating th functional rol of callosal connctions with dynamic causal modls; Ann. N. Y. Acad. Sci. 64 6 36 Publication: 8 Sptmbr 6 J. Biosci. 3(), January 7