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Models of effective connectivity & Dynamic Causal Modelling (DCM) Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London Methods & Models for fMRI data analysis 03 December 2008 Overview • Brain connectivity: types & definitions – anatomical connectivity – functional connectivity – effective connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – DCM for fMRI: Neural and hemodynamic levels – Parameter estimation & inference • Applications of DCM to fMRI data – Design of experiments and models – Some empirical examples and simulations Connectivity A central property of any system Communication systems Social networks (internet) (Canberra, Australia) FIgs. by Stephen Eick and A. Klovdahl; see http://www.nd.edu/~networks/gallery.htm Structural, functional & effective connectivity • anatomical/structural connectivity Sporns 2007, Scholarpedia = presence of axonal connections • functional connectivity = statistical dependencies between regional time series • effective connectivity = causal (directed) influences between neurons or neuronal populations Anatomical connectivity • neuronal communication via synaptic contacts • visualisation by tracing techniques • long-range axons “association fibres” Diffusion-weighted imaging Parker & Alexander, 2005, Phil. Trans. B Diffusion-weighted imaging of the cortico- spinal tract Parker, Stephan et al. 2002, NeuroImage However, knowing anatomical connectivity is not enough... 1. Connections are recruited in a context-dependent fashion 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 Local functions are context-sensitive: They depend on network activity. 2. Connections show plasticity • synaptic plasticity = change in the structure and transmission properties of a chemical synapse NMDA receptor • critical for learning • can occur both rapidly and slowly • NMDA receptors play a critical role • NMDA receptors are regulated by modulatory neurotransmitters like dopamine, serotonine, acetylcholine Gu 2002, Neuroscience Different approaches to analysing functional connectivity • Seed voxel correlation analysis • Eigen-decomposition (PCA, SVD) • Independent component analysis (ICA) • any other technique describing statistical dependencies amongst regional time series Seed-voxel correlation analyses • Very simple idea: seed voxel – hypothesis-driven choice of a seed voxel → extract reference time series – voxel-wise correlation with time series from all other voxels in the brain Drug-induced changes in functional connectivity Finger-tapping task in first-episode schizophrenic patients: voxels that showed changes in functional connectivity (p<0.005) with the left ant. cerebellum after medication with olanzapine Stephan et al. 2001, Psychol. Med. Does functional connectivity not simply correspond to co-activation in SPMs? No, it does not - see regional task T regional response A2 the fictitious example response A1 on the right: Here both areas A1 and A2 are correlated identically to task T, yet they have zero correlation among themselves: r(A1,T) = r(A2,T) = 0.71 but r(A1,A2) = 0 ! Stephan 2004, J. Anat. Pros & Cons of functional connectivity analyses • Pros: – useful when we have no experimental control over the system of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.) • Cons: – interpretation of resulting patterns is difficult / arbitrary – no mechanistic insight into the neural system of interest – usually suboptimal for situations where we have a priori knowledge and experimental control about the system of interest models of effective connectivity necessary For understanding brain function mechanistically, we need models of effective connectivity, i.e. models of causal interactions among neuronal populations. Some models for computing effective connectivity from fMRI data • Structural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000 • regression models (e.g. psycho-physiological interactions, PPIs) Friston et al. 1997 • Volterra kernels Friston & Büchel 2000 • Time series models (e.g. MAR, Granger causality) Harrison et al. 2003, Goebel et al. 2003 • Dynamic Causal Modelling (DCM) bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008 Overview • Brain connectivity: types & definitions – anatomical connectivity – functional connectivity – effective connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – DCM for fMRI: Neural and hemodynamic levels – Parameter estimation & inference • Applications of DCM to fMRI data – Design of experiments and models – Some empirical examples and simulations Psycho-physiological interaction (PPI) Task factor GLM of a 2x2 factorial design: Task A Task B y (TA TB ) 1 main effect of task Stim 1 Stimulus factor TA/S1 TB/S1 ( S1 S 2 ) β 2 main effect of stim. type (TA TB ) ( S1 S 2 ) β 3 interaction Stim 2 TA/S2 TB/S2 e We can replace one main effect in the GLM by the time series of an y (TA TB ) 1 main effect of task V1 time series area that shows this main effect. V 1β 2 main effect of stim. type E.g. let's replace the main effect of stimulus type by the time series of (TA TB ) V 1β 3 psycho- physiological area V1: interaction e Friston et al. 1997, NeuroImage Attentional modulation of V1→V5 SPM{Z} Attention V5 activity V1 V5 time = attention V5 activity V1 x Att. V5 no attention Friston et al. 1997, NeuroImage Büchel & Friston 1997, Cereb. Cortex V1 activity PPI: interpretation y (TA TB ) 1 V 1β 2 (TA TB ) V 1β 3 e Two possible interpretations of the PPI term: attention attention V1 V5 V1 V5 Modulation of V1V5 by Modulation of the impact of attention on V5 attention by V1 Two PPI variants • "Classical" PPI: – Friston et al. 1997, NeuroImage – depends on factorial design – in the GLM, physiological time series replaces one experimental factor – physio-physiological interactions: two experimental factors are replaced by physiological time series • Alternative PPI: – Macaluso et al. 2000, Science – interaction term is added to an existing GLM – can be used with any design Task-driven lateralisation Does the word contain the letter A or not? letter decisions > spatial decisions • • group analysis (random effects), • n=16, p<0.05 corrected analysis with SPM2 Is the red letter left or right from the midline of the spatial decisions > letter decisions word? Stephan et al. 2003, Science Bilateral ACC activation in both tasks – but asymmetric connectivity ! group analysis random effects (n=15) p<0.05, corrected (SVC) IFG left ACC (-6, 16, 42) letter vs spatial Left ACC left inf. frontal gyrus (IFG): decisions increase during letter decisions. IPS spatial vs letter right ACC (8, 16, 48) decisions Right ACC right IPS: Stephan et al. 2003, Science increase during spatial decisions. PPI single-subject example letter decisions spatial Signal in right ant. IPS decisions Signal in left IFG bVS= -0.16 bL= -0.19 spatial letter decisions decisions bVS=0.50 bL=0.63 Signal in left ACC Signal in right ACC Left ACC signal plotted against left IFG Right ACC signal plotted against right IPS Stephan et al. 2003, Science Pros & Cons of PPIs • Pros: – given a single source region, we can test for its context-dependent connectivity across the entire brain – easy to implement • Cons: – very simplistic model: only allows to model contributions from a single area – ignores time-series properties of data – application to event-related data relies deconvolution procedure (Gitelman et al. 2003, NeuroImage) – operates at the level of BOLD time series sometimes very useful, but limited causal interpretability; in most cases, we need more powerful models Overview • Brain connectivity: types & definitions – anatomical connectivity – functional connectivity – effective connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – DCM for fMRI: Neural and hemodynamic levels – Parameter estimation & inference • Applications of DCM to fMRI data – Design of experiments and models – Some empirical examples and simulations Example: FG FG LG = lingual gyrus x3 x4 a linear system left right FG = fusiform gyrus of dynamics in Visual input in the visual cortex - left (LVF) - right (RVF) LG LG x1 left right x2 visual field. RVF LVF u2 u1 x1 a11 x1 a12 x2 a13 x3 c12u2 x2 a21 x1 a22 x2 a24 x4 c21u1 x3 a31 x1 a33 x3 a34 x4 x4 a42 x2 a43 x3 a44 x4 Example: FG FG LG = lingual gyrus x3 x4 a linear system left right FG = fusiform gyrus of dynamics in Visual input in the visual cortex - left (LVF) - right (RVF) LG LG x1 left right x2 visual field. RVF LVF u2 u1 state effective system input external changes connectivity state parameters inputs x1 a11 a12 a13 0 x1 0 c12 x Az Cu x a a 2 0 a24 x2 c21 0 u1 21 22 x3 a31 0 a33 a34 x3 0 0 u2 { A, C} x4 0 a42 a43 a44 x4 0 0 Extension: FG FG x3 x4 bilinear left right dynamic m x ( A u j B ( j ) ) x Cu system j 1 LG LG x1 left right x2 RVF CONTEXT LVF u2 u3 u1 x1 a11 a12 a13 0 0 b12 (3) 0 0 x1 0 c12 0 x a a 2 21 22 0 a24 x c u1 u3 0 0 0 0 2 21 0 0 u2 x3 a31 0 a33 a34 0 0 0 b34 (3) x3 0 0 0 u3 x4 0 a42 a43 a44 0 0 0 0 x4 0 0 0 Dynamic Causal Modelling (DCM) Hemodynamic Electromagnetic forward model: forward model: neural activityBOLD neural activityEEG MEG LFP Neural state equation: dx F ( x , u, ) fMRI dt EEG/MEG simple neuronal model complicated neuronal model complicated forward model simple forward model Stephan & Friston 2007, inputs Handbook of Brain Connectivity Basic idea of DCM for fMRI (Friston et al. 2003, NeuroImage) • Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). x • The modelled neuronal dynamics (x) is transformed into area-specific BOLD signals (y) λ by a hemodynamic forward model (λ). y The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals. y y y BOLD y λ hemodynamic activity model x2(t) activity activity x3(t) x1(t) neuronal x states modulatory integration input u2(t) driving t Neural state equation x ( A u j B( j ) ) x Cu input u1(t) x intrinsic connectivity A x modulation of x t B( j) connectivity u j x x direct inputs C Stephan & Friston (2007), u Handbook of Brain Connectivity Bilinear DCM driving input modulation Two-dimensional Taylor series (around x0=0, u0=0): dx f f 2 f f ( x, u ) f ( x0 ,0) x u ux ... dt x u xu Bilinear state equation: dx m A ui B( i ) x Cu dt i 1 Example: u1 context-dependent decay u1 u2 stimuli context u1 u2 u2 - Z1 + - x x1 Z2 1 + x2 + x2 - x Ax u2 B (2) x Cu1 - x1 a12 b 2 0 c1 0 u1 x a 21 x u 2 11 x 2 0 0 0 u2 b 22 2 Penny, Stephan, Mechelli, Friston NeuroImage (2004) DCM parameters = rate constants Integration of a first-order linear differential equation gives an exponential function: dx x (t ) x0 exp( at ) ax dt Coupling parameter a is inversely The coupling parameter a proportional to the half life of z(t): thus describes the speed of the exponential change in x(t) 0.5x0 x( ) 0.5x0 x0 exp(a ) a ln 2 / ln 2 / a The hemodynamic u stimulus functions model in DCM t dx m A u j B( j ) x Cu dt neural state equation • 6 hemodynamic parameters: j 1 vasodilatory signal h { , , , , , } f s x s γ ( f 1) s s flow induction (rCBF) f s important for model fitting, hemodynamic f but of no interest for state equations statistical inference Balloon model changes in volume v changes in dHb τv f v1 /α τq f E ( f,E 0 ) q 0 v1 /α q/v E • Computed separately for each v q area (like the neural parameters) region-specific HRFs! S q ( q, v ) V0 k1 1 q k2 1 k3 1 v S0 v k1 4.30 E0TE Friston et al. 2000, NeuroImage k2 r0 E0TE BOLD signal Stephan et al. 2007, NeuroImage k3 1 change equation u stimulus functions The hemodynamic model in DCM t dx m neural state A u j B( j ) x Cu equation dt j 1 0.4 0.2 vasodilatory signal 0 s x s γ ( f 1) 0 2 4 6 8 10 12 14 s RBM N, = 0.5 f s CBM , = 0.5 N 1 RBM N, = 1 flow induction (rCBF) CBM , = 1 N 0.5 RBM , = 2 hemodynamic f s N CBM N, = 2 state 0 f 0 2 4 6 8 10 12 14 equations Balloon model 0.2 0 changes in volume v changes in dHb -0.2 τv f v1 /α τq f E ( f,E 0 ) q 0 v1 /α q/v E -0.4 v q -0.6 0 2 4 6 8 10 12 14 S q ( q, v ) V0 k1 1 q k2 1 k3 1 v S0 v k1 4.30 E0TE k2 r0 E0TE BOLD signal k3 1 change equation Stephan et al. 2007, NeuroImage How interdependent are our neural and hemodynamic parameter estimates? 1 A 5 0.8 0.6 10 B 0.4 15 C 0.2 20 0 25 -0.2 h 30 -0.4 35 -0.6 -0.8 40 ε -1 5 10 15 20 25 30 35 40 r,A r,B r,C Stephan et al. 2007, NeuroImage Bayesian statistics new data prior knowledge p( y | ) p( ) p( | y) p( y | ) p( ) posterior likelihood ∙ prior Bayes theorem allows one to formally incorporate prior knowledge into The “posterior” probability of the computing statistical probabilities. parameters given the data is an optimal combination of prior knowledge In DCM: empirical, principled & and new data, weighted by their shrinkage priors. relative precision. Shrinkage priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect stimulus function u Overview: parameter estimation x ( A u j B j ) x Cu neural state equation • Combining the neural and hemodynamic states gives the complete forward model. activity- dependent vasodilatory signal s z s γ( f 1) • An observation model s s includes measurement f error e and confounds X flow - induction (rCBF) parameters (e.g. drift). hidden states f s h { , , , , } z {x, s, f , v, q} f n { A, B1... B m , C} • Bayesian parameter state equation { h , n } estimation by means of a z F ( x , u, ) Levenberg-Marquardt gradient ascent, embedded changes in volume v changes in dHb into an EM algorithm. τv f v1/α τq f E ( f, ) q v1/α q/v v q • Result: Gaussian a posteriori parameter distributions, characterised by ηθ|y mean ηθ|y and y (x ) y h(u, ) X e covariance Cθ|y. observation model modelled BOLD response Inference about DCM parameters: Bayesian single-subject analysis • Gaussian assumptions about the posterior distributions of the parameters • Use of the cumulative normal distribution to test the probability that a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ: cT ηθ|y y p N cT C y c • By default, γ is chosen as zero ("does the effect exist?"). Bayesian single subject inference LD|LVF 0.13 0.34 p(cT>0|y) 0.19 0.14 = 98.7% FG FG left right LD LD 0.44 0.29 0.14 0.14 LG LG left right 0.01 -0.08 0.17 0.16 RVF LD|RVF LVF stim. stim. Contrast: Modulation LG right LG links by LD|LVF vs. Stephan et al. 2005, modulation LG left LG right by LD|RVF Ann. N.Y. Acad. Sci. Inference about DCM parameters: Bayesian fixed-effects group analysis Because the likelihood distributions Under Gaussian assumptions this is from different subjects are independent, easy to compute: one can combine their posterior densities by using the posterior of one group individual subject as the prior for the next: posterior posterior covariance covariances p( | y1 ) p( y1 | ) p( ) N p( | y1 , y2 ) p( y2 | ) p( y1 | ) p( ) C|1y1 ,..., y N C|1yi i 1 p( y2 | ) p( | y1 ) N 1 1 ... | y ,..., y C | yi | yi C | y1 ,..., y N p( | y1 ,..., y N ) p( y N | ) p( | y N 1 )...p( | y1 ) i 1 1 N group individual posterior posterior covariances and means mean Inference about DCM parameters: group analysis (classical) • In analogy to “random effects” analyses in SPM, 2nd level analyses can be applied to DCM parameters: Separate fitting of identical models for each subject Selection of bilinear parameters of interest one-sample t-test: paired t-test: rmANOVA: parameter > 0 ? parameter 1 > e.g. in case of multiple parameter 2 ? sessions per subject Overview • Brain connectivity: types & definitions – anatomical connectivity – functional connectivity – effective connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – DCM for fMRI: Neural and hemodynamic levels – Parameter estimation & inference • Applications of DCM to fMRI data – Design of experiments and models – Some empirical examples and simulations What type of design is good for DCM? Any design that is good for a GLM of fMRI data. GLM vs. DCM DCM tries to model the same phenomena as a GLM, just in a different way: It is a model, based on connectivity and its modulation, for explaining experimentally controlled variance in local responses. No activation detected by a GLM → inclusion of this region in a DCM is useless! Stephan 2004, J. Anat. Multifactorial design: explaining interactions with DCM Task factor Stim1/ Stim2/ Task A Task A Task A Task B Stim 1 Stimulus factor TA/S1 TB/S1 X1 X2 GLM Stim 2 Stim 1/ Stim 2/ TA/S2 TB/S2 Task B Task B Let’s assume that an SPM analysis Stim1 shows a main effect of stimulus in X1 and a stimulus task interaction in X2. X1 X2 DCM How do we model this using DCM? Stim2 Task A Task B Simulated data X1 +++ – – Stimulus 1 + X1 +++ X2 Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B Stimulus 2 + +++ + Task A Task B X2 Stephan et al. 2007, J. Biosci. X1 Stim 1 Stim 2 Stim 1 Stim 2 Task A Task A Task B Task B X2 plus added noise (SNR=1) Recent fMRI studies with DCM • DCM now an established tool for fMRI analysis • >50 studies published, incl. high- profile journals • combinations of DCM with computational models Thank you