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Models of Effective Connectivity & Dynamic Causal Modelling Hanneke den Ouden Wellcome Trust Centre for Neuroimaging, University College London, UK Donders Institute for Brain, Cognition and Behaviour, Nijmegen, the Netherlands SPM course Thanks to Klaas Stephan and Meike Grol for slides Zurich, February 2009 Systems analysis in functional neuroimaging Functional specialisation: Functional integration: What regions respond to a particular How do regions influence each other? experimental input? Brain Connectivity ? ? Overview • Brain connectivity: types & definitions – anatomical connectivity – functional connectivity – effective connectivity • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) • Applications of DCM to fMRI data Structural, functional & effective connectivity Sporns 2007, Scholarpedia • anatomical/structural connectivity = presence of axonal connections • functional connectivity = statistical dependencies between regional time series • effective connectivity = causal (directed) influences between neurons or neuronal populations Anatomical connectivity • presence of axonal connections • neuronal communication via synaptic contacts • visualisation by – tracing techniques – diffusion tensor imaging However, knowing anatomical connectivity is not enough... • Connections are recruited in a context-dependent fashion: – Local functions depend on network activity 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 However, knowing anatomical connectivity is not enough... • Connections are recruited in a context-dependent fashion: – Local functions depend on network activity • Connections show plasticity – Synaptic plasticity = change in the structure and transmission properties of a synapse – Critical for learning – Can occur both rapidly and slowly Need to look at functional and effective connectivity Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) • Applications of DCM to fMRI data Different approaches to analysing functional connectivity Definition: statistical dependencies between regional time series • 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 SVCA example: Task-induced changes in functional connectivity 2 bimanual finger-tapping tasks: During task that required more bimanual coordination, SMA, PPC, M1 and PM showed increased functional connectivity (p<0.001) with left M1 No difference in SPMs! Sun et al. 2003, Neuroimage Does functional connectivity not simply correspond to co-activation in SPMs? regional task T regional response A2 response A1 No, it does not - see the fictitious example 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 For understanding brain function mechanistically, we need models of effective connectivity, i.e. models of causal interactions among neuronal populations to explain regional effects in terms of interregional connectivity 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 • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) • Applications of DCM to fMRI data Psycho-physiological interaction (PPI) • bilinear model of how the influence of area A on area B changes by the psychological context C: AxCB • a PPI corresponds to differences in regression slopes for different contexts. Psycho-physiological interaction (PPI) Task factor GLM of a 2x2 factorial design: Task A Task B y (TA TB ) 1 main effect of task Stimulus factor Stim 1 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 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 Example PPI: 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 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 – operates at the level of BOLD time series sometimes very useful, but limited causal interpretability; in most cases, we need more powerful models DCM! Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – Basic idea – Neural level – Hemodynamic level – Priors & Parameter estimation • Applications of DCM to fMRI data Basic idea of DCM for fMRI (Friston et al. 2003, NeuroImage) • Investigate functional integration & modulation of specific cortical pathways • Using a bilinear state equation, a cognitive system is modelled at its x underlying neuronal level (which is not directly accessible for fMRI). λ • The modelled neuronal dynamics (x) is transformed into area- specific BOLD signals (y) by a hemodynamic forward model (λ). y The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are maximally similar. Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – Basic idea – Neural level – Hemodynamic level – Priors & Parameter estimation • Applications of DCM to fMRI data 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 u1 u2 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 Az x Ax Cu x a a 0 a24 x2 c21 0 u1 2 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 0 a24 x c u1 2 21 22 0 0 u3 0 0 0 0 2 21 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 y y y BOLD y hemodynamic activity λ 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) endogenous x 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 Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – Basic idea – Neural level – Hemodynamic level – Priors & Parameter estimation • Applications of DCM to fMRI data The hemodynamic model in DCM • 6 hemodynamic u stimulus functions parameters: t activity neural state equation h { , , , , , } x (t ) vasodilatory signal important for model fitting, but s x s γ ( f 1) s of no interest for statistical f s inference flow induction (rCBF) hemodynamic state f s equations f • Computed separately for Balloon model each area (like the neural changes in volume v changes in dHb parameters) τv f v1 /α τq f E ( f,E 0 ) q 0 v1 /α q/v q E region-specific HRFs! v BOLD signal Estimated BOLD y (t ) v, q Friston et al. 2000, NeuroImage response Stephan et al. 2007, NeuroImage Example: modelled BOLD signal FG FG left right LG LG left right RVF LVF black: observed BOLD signal red: modelled BOLD signal Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) – Basic idea – Neural level – Hemodynamic level – Priors & Parameter estimation • Applications of DCM to fMRI data Bayesian statistics new data prior knowledge p( y | ) p( ) p( | y) p( y | ) p( ) posterior likelihood ∙ prior Bayes theorem allows us to express The posterior probability of the our prior knowledge or “belief” about parameters given the data is an parameters of the model optimal combination of prior knowledge and new data, weighted by their relative precision. Priors in DCM • embody constraints on parameter estimation – hemodynamic parameters: empirical priors – coupling parameters of self-connections: principled priors – coupling parameters other connections: shrinkage priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect DCM parameters = rate constants Integration of a first-order linear differential equation gives an exponential function: dx ax x (t ) x0 exp( at ) dt Coupling parameter a is inversely The coupling parameter a proportional to the half life of x(t): thus describes the speed of x( ) 0.5x0 the exponential change in x(t) 0.5x0 x0 exp(a ) a ln 2 / ln 2 / a If AB is 0.10 s-1 this means that, per unit time, the increase in activity in B corresponds to 10% of the activity in A 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 0 0 0 u2 b 22 2 2 Penny, Stephan, Mechelli, Friston NeuroImage (2004) DCM Summary Select areas you want to model • Extract timeseries of these areas Modulatory input Driving input (x(t)) (e.g. context/learning/drugs) (e.g. sensory stim) • Specify at neuronal level b12 – what drives areas (c) c1 – how areas interact (a) c2 neuronal – what modulates interactions (b) states activity • State-space model with 2 levels: x1(t) a12 activity x2(t) – Hidden neural dynamics – Predicted BOLD response y BOLD y • Estimate model parameters: Gaussian a posteriori parameter ηθ|y distributions, characterised by mean ηθ|y and covariance Cθ|y. 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?"). 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 • Functional connectivity • Psycho-physiological interactions (PPI) • Dynamic causal models (DCMs) • Applications of DCM to fMRI data – Design of experiments and models – Some empirical examples and simulations Planning a DCM-compatible study • Suitable experimental design: – any design that is suitable for a GLM – preferably multi-factorial (e.g. 2 x 2) • e.g. one factor that varies the driving (sensory) input • and one factor that varies the contextual input • Hypothesis and model: – Define specific a priori hypothesis – Which parameters are relevant to test this hypothesis? – If you want to verify that intended model is suitable to test this hypothesis, then use simulations – Define criteria for inference – What are the alternative models to test? 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 A1 +++ Stim1 + Stim 1 Stim 2 Stim 1 Stim 2 A1 A2 Task A Task A Task B Task B +++ Stim2 +++ + + Task A Task B A2 X1 Stim 1 Stim 2 Stim 1 Stim 2 Task A Task A Task B Task B X2 plus added noise (SNR=1) Final point: 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. If there is no evidence for an experimental effect (no activation detected by a GLM) → inclusion of this region in a DCM is not meaningful. Thank you Stay tuned to find out how to … select the best model comparing various DCMs … test whether one region influences the connection between other regions … do DCM on your M/EEG & LFP data … and lots more!

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fMRI data, causal models, main effect, Functional connectivity, physiological interactions, time series

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posted: | 2/28/2011 |

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