The Where, How and What of brain function. Effective Connectivity & Dynamic Causal Modelling (DCM) Structure of this talk. Concepts of brain function

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1 Effective Connectivit & Dnic Cusl odelling (DC) Bsed on slides fro: K. Stephn SP course t CRC, ULg, 9 The Where, How nd Wht of brin function Where in the brin is certin cognitive process ipleented? GL nlses (e.g. SP) How does this ipleenttion work (in ters of functionl principles)? odels of effective connectivit Wht does this process en (in coputtionl ters)? odels of neurl coding Structure of this tlk Connectivit: concepts & definitions Wring up: Pscho-phsiologicl interctions (PPI) Structurl Eqution odelling (SE) Dnic Cusl odelling (DC): Conceptul bsis The biliner odel t the neurl level The heodnic odel Priors & preters Plnning DC-coptible fri stud Prcticl steps in SP5 Exple: Attention to otion in the visul sste Concepts of brin function Functionl specilistion nlses of of regionll specific effects: which res constitute neuronl sste? Functionl connectivit the teporl correltion between sptill reote neurophsiologicl events ODEL-FREE Functionl integrtion nlses of of inter-regionl regionl effects: wht re re the the interctions between the the coponents of of given neuronl sste? Effective connectivit the influence one one neuronl sste exerts over over nother ODEL-DEPENDENT

2 Effective vs. functionl connectivit odel: A V fri tie-series B.5 * A e C.3 * A e A.3.49 Correct odel C B -. Correltions: A B C Pscho-phsiologicl interctions (PPI) biliner odel of the chnge in coupling between regions A nd B, depending on the pschologicl context C: A x C B C cn be contrst of two conditions (C, C -, else) or pretric vrible. A PPI corresponds to context-dependent difference in the slope of the regression between two regionl tie series. PPI exple: ttentionl odultion of V Attention ctivit SP{Z} PPI: the sttisticl odel nd its interprettion [ V C] β V β C β 3 G β G e V V x Att. ctivit tie ttention no ttention V ttention Two possible interprettions: ttention V Friston et l. 997, NeuroIge 6:8-9 Büchel & Friston 997, Cereb. Cortex 7: V ctivit odultion of V b ttention odultion of the ipct of ttention on b V.

3 PPI: proble nd solution PPI en of identifing regions whose responses cn be explined in ters of n interction between : - Activit in specified re (x n, phsiologicl fctor) - Soe experientl effect (C, pschologicl fctor) Proble: esured signl x (BOLD signl) is the neuronl ctivit convolved with the hrf! Conv (x n,hrf) x One cnnot sipl convolve the pschologicl vrible C with the hrf nd ultipl the signl x. Conv (C,hrf) * x conv ((C * x n ),hrf) PPI: proble nd solution PPI en of identifing regions whose responses cn be explined in ters of n interction between : - Activit in specified re (x n, phsiologicl fctor) - Soe experientl effect (C, pschologicl fctor) Prcticll :. V bold tie series x hs to be decorelted to estite the neuronl tie series x n.. ultipl x n nd C, then convolve with hrf ppi 3. Convolve C with hrf Ch 4. Enter in design trix: ppi s covrite of interest, x nd Ch s covrite of no interest. Points to 3 re done with sp_peb_ppi, clled b the PPIs button. Structurl Eqution odeling (SE) SE tests hpothesis how severl vribles interct with ech other cusll in the context of fri: vribles tie series of res interctions ntoicl connections u strength of interctions between vribles is quntified b pth coeffcients odultor vribles llow to ssess the influence of pschologicl fctor on the strength of specific connections P 4 3 u 43 u 4 u 3 theticl exple of structurl odel u u 4 u u A u ( I A) Σ T 3 ( I A) ( I A) u(( I A) T uu ( I A) u) T SE estites pth coefficients in A such tht the difference between odelled covrince Σ nd observed covrince S becoes inil 3 4 u 43 structurl odel odelled covrince Σ u u 3 u3 4 u4 genertive odel T

4 Liittions of PPIs nd SE PPIs: ver siple odel: onl llows for contributions fro single re SE: coplex odels esil becoe unidentifible both: not esil used with event-relted dt operte t the level of BOLD tie series liited cusl interpretbilit in neurl ters! DC conceptul overview DC llows to odel cognitive sste t the neuronl level (which is not directl ccessible for fri). The odelled neuronl dnics (z) is trnsfored into re-specific BOLD signls () b heodnic forwrd odel (λ). The i of DC is to estite preters t the neuronl level such tht the odelled BOLD signls re xill siilr to the experientll esured BOLD signls. z λ DC: the neuronl level Wht does DC odel t the neuronl level? For ech re, DC odels the chnge of n bstrct neuronl stte in tie. This neuronl stte is represented b single stte vrible (z). NB: z hs no direct biophsicl correlte. DC trets the brin s non-liner, deterinistic sste whose stte chnges in tie entirel depend on: the current stte (z), externl inputs into the sste (u) n perturbtion, F( z, θ ) intrinsic sste structure & properties (preters θ n ). Which preters does θ n contin nd which echniss do the concern? Conceptul overview: Neurl stte equtions ctivit z (t) Input u(t) c b 3 ctivit z (t) ctivit z 3 (t) neuronl sttes BOLD Friston et l. 3

5 Use differentil equtions to represent neuronl sste Stte vector Chnges with tie Rte of chnge of stte vector Interctions between eleents Externl inputs, u Sste preters θ z( t) z( t) zn( t) sste represented b stte vribles f( z... zn, θ) n fn( z... zn, θn) f(, z θ ) dz dt sz DC preters rte constnts Generic solution to the ODEs in DC: Dec function: z ( t) z ()ex t) Hlf-life τ: z ( τ ).5 z () z ()ex sτ ) z ( t) z ()ex st), z () z ().4. s ln/ τ -... τ z Liner dnics: nodes s s 4; Neurodnics: nodes with input u z u u s sz s( z z) z () z () s 4; s 8; z z z z z () t ex st) z () t s tex st) > z s t z z c s > u z ctivit in z is coupled to z vi coefficient

6 Neurodnics: positive odultion Neurodnics: reciprocl connections u u u u u z u u z u z z z z index, not squred z z c s > u u b z b z odultor input u ctivit through the coupling z z reciprocl connection disclosed b u z z c s,, > u u b z b z Heodnics: reciprocl connections Heodnics: reciprocl connections u u z h BOLD (without noise) 4 u u z h BOLD with Noise dded z h BOLD (without noise) 4 z h BOLD with Noise dded seconds h(θ) represents the BOLD response (blloon odel) to input blue: red: neuronl ctivit bold response 4 6 seconds h(θ) represents the BOLD response (blloon odel) to input blue: red: neuronl ctivit bold response

7 Biliner stte eqution in DC for fri stte chnges z n n connectivit n nn b u bn stte vector ( A u B ) z Cu b n z c b nn zn cn direct inputs externl inputs & L L L n regions O L od inputs odultion of connectivit O L O L c u cn u drv inputs In DC, the neurl dnics of the odelled sste depends on 4 preters θ n {A,B,C,σ}: intrinsic connectivit deterines, which res cn influence ech other contextul inputs chnge connection strengths direct (e.g. sensor) inputs inect ctivit into the odel re-intrinsic inhibition dec of induced ctivit Activit in the sste is onl induced b direct inputs (C) no spontneous ctivit of the res θ n is deterined b Besin estition schee (see below). direct inputs - u (e.g. visul stiuli) V contextul inputs - u (e.g. ttention) A B C σ SPC θ n The heodnic Blloon odel 5 heodnic preters: h θ { κ, γ, τ, α, ρ} iportnt for odel fitting, but of no interest for sttisticl inference Epiricll deterined priori distributions. Coputed seprtel for ech re (like the neurl preters). f chnges in volue /α τv& f v v ctivit z(t) vsodiltor signl s& z κs γ( f ) s flow induction f& s v f BOLD signl ( t) λ( v, q) chnges in dhb /α τq& f E( f, ρ) qρ v q/v q Neurl stte equtions ctivit z (t) Input u(t) c b 3 ctivit z (t) ctivit z 3 (t) Neurl stte eqution z F( z, θ ) neuronl stte chnges & z n n BOLD intrinsic connectivit L n O L nn The biliner odel neuronl sttes context-dependent connectivit b u bn & direct inputs L b n z c L c u O O L b nn zn cn L cn u ( A u B ) z Cu integrtion z λ heodnic odel Friston et l. 3

8 DC rodp Priors in DC Priors fri dt Neuronl dnics Stte spce odel Heodnics odel inversion using Expecttion-xiiztion Posterior densities of preters odel coprison needed for Besin estition, ebod constrints on preter estition express our prior knowledge or belief bout preters of the odel heodnic preters: epiricl priors teporl scling: principled prior coupling preters: shrinkge priors Bes Theore θ ) θ ) θ ) posterior likelihood prior Gussin Bivrite Gussin Likelihood nd Prior p p ( ) ( ) ( θ ) N ( θ, λ ( ) ) ( ) ( ) ( ) ( θ N θ, λ ) Posterior ( ) Posterior Likelihood p ( ) ( θ ) N (, p ) p λ λ ( ) ( ) p λ θ ( ) ( ) λ p ( ) θ ( ) Prior Reltive Precision Weighting θ ( ) θ ()

9 Priors in DC sste stbilit: in the bsence of input, the neuronl sttes return to stble ode lrgest rel eigenvlue of the intrinsic coupling trix (principl Lpunov exponent) ust be negtive constrints on prior vrince of intrinsic connections (A) shrinkge priors for coupling preters (η) conservtive estites! σ.5 i C k θ b i, ηθ, Cθ cik h θ h η θ A C B C C C h self-inhibition: ensured b priors on σ (η σ, C σ.5) these llow for neurl trnsients with hlf life in the rnge of 3 s to seconds probbilit of negtive Lpunov exponent. ησ σ σ ) ln /ησ τ z ) σ ) z τ z σ τ Cobining the neurl nd heodnic sttes gives the coplete forwrd odel: x { z, s, f, v, q} n h θ θ θ x& f ( x, θ ) λ( x) h( θ ) The observtion odel includes esureent error ε nd confounds X (e.g. drift): h( θ ) Xβ ε Preter estition Besin preter estition under Gussin ssuptions b ens of the E lgorith (expecttion xiistion). h( η ) in Result: Gussin posteriori preter distributions, chrcterised b en η θ nd covrince C θ. η θ θ Inference bout DC preters: single-subect nlsis Besin preter estition in DC: Gussin ssuptions bout the posteriori distributions of the preters Use of the cuultive norl distribution to test the probbilit b which certin preter (or contrst of preters c T η θ ) is bove chosen threshold γ: Besin odel selection Bes theore in slightl extended fshion: odel evidence is coputed b θ, ) θ ) θ, ) ) p ( ) θ, ) θ ) dθ T c η θ γ p φ N T c C c θ γ cn be chosen s function of the expected hlf life of the neurl process, e.g. γ ln / τ γ η θ Log of the odel evidence cn be expressed s Bes fctors: log ) ccurc ( ) coplexit ( ) i) B i )

10 odel coprison nd selection Given copeting hpotheses, which odel is the best? log ) ccurc ( ) coplexit ( ) i) B i ) B to 3 3 to to 5 5 Y) Evidence Wek Positive Strong Ver strong Pitt & iung (), TICS Plnning DC-coptible stud Suitble experientl design: preferbl ulti-fctoril (e.g. x ) t lest one fctor tht vries the sensor input t lest one fctor tht vries the contextul input TR: s short s possible (optil: < s) Hpothesis nd odel: define specific priori hpothesis which preters re relevnt? ensure tht intended odel is suitble to test this hpothesis initil siultion define criteri for inference Attention to otion in the visul sste Coprison of two siple odels We used this odel to ssess the site of ttention odultion during visul otion processing in n fri prdig reported b Büchel & Friston. Photic Attention Tie [s]? SPC odel : ttentionl odultion of V Photic V.36 otion.3 Attention -..7 SPC.84 odel : ttentionl odultion of SPC Attention Photic SPC V otion Friston et l. 3, NeuroIge - fixtion onl - observe sttic dots photic V - observe oving dots otion - tsk on oving dots ttention prietl cortex V otion Besin odel selection: odel better thn odel Decision for odel : log ) >> log ) in this experient, ttention priril odultes V

11 Extension : Nonliner DC for fri Extension III: Nonliner DC for fri u biliner DC u nonliner DC Cn ctivit during ttention to otion be explined b llowing ctivit in SPC to odulte the V-to- connection? u u ttention.9 (%). Biliner stte eqution dx ( i) A ui B x Cu dt i dx dt Nonliner stte eqution A n ( i) ( ) ui B x D x Cu i visul stiultion.65 (%).3 (%) V SPC. (97.4%) ( SPC) D V The posterior densit of 5, V indictes tht this gting existed with 97.4% confidence. (The D trix encodes which of the n neurl units gte which connections in the sste) Here DC cn odel ctivit-dependent chnges in connectivit; how connections re enbled or gted b ctivit in one or ore res..4 (%) otion Attentionl odel coprison Stephn et l., NeuroIge, 8 ColBlck Go for ANY odels! A. Low Attention B. High Attention ColBlck Low Attention High Attention A. B. V V4 P V V4 P ColBlck Low Attention High Attention A.3 B.3 V V4 P ColBlck Low Attention High Attention A.4 B.4 ColBlck Low Attention High Attention A.5 B.5 V V4 P V V4 P All odels fitted on dt fro 4 subects 4 odel fitting!

12 Group odel selection: FFX or RFX? odel spce prtitioning Tpe A or B, i.e with or without the low ttention input, s odel? Conclusions Dnic Cusl odelling (DC) of of fri is isechnistic odel tht thtisis infored b bntoicl nd nd phsiologicl principles. DC uses uses deterinistic differentil eqution to to odel neuro-dnics (represented b b trices A,B A,B nd nd C) C) DC uses uses Besin frework to to estite odel preters DC provides n n observtion odel for forneuroiging dt, dt, e.g. e.g. fri, /EEG Express hpothesis s s concurrent odels which cn cn be be copred, t t the the individul nd/or group level level