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Methods for Dummies 2009 Dynamic Causal Modelling Part I: Theory 18th February 2009 Stephanie Burnett Christian Lambert Last time, in MfD… Psychophysiological interactions (PPI) and structural equation modelling (SEM) Functional vs. effective connectivity Functional connectivity: temporal correlation between spatially remote Standard fMRI analysis neurophysiological events Effective connectivity: the influence that the elements of PPIs, SEM, DCM a neuronal system exert over each other Introduction: DCM and its place in the methods family tree Task BOLD signal Standard fMRI analysis: The BOLD signal (related to brain activity in some implicit way) in some set of brain is correlated, and is also correlated with your task “This is a fronto-parietal network collection of brain regions involved in activated while processing coffee” Introduction: DCM and its place in the methods family tree attention PPIs Represent how the (experimental) context modulates connectivity between a brain region of V1 V5 interest, and anywhere else DCM models how neuronal activity causes the BOLD signal (forward model) E.g. (Whatever gives rise to the) signal in one brain attention That is, your conclusions are about region (V1) will lead to a neural events signal in V5, and the strength of this signal in V5 DCM models bidirectional and modulatory depends on attention interactions, between multiple brain regions V1 V5 Introduction: DCM and its place in the methods family tree DCM Your experimental task causes neuronal activity in an input brain region, and this generates a BOLD signal. The neuronal activity in this input region, due to your task, then causes or modulates neuronal activity in other brain regions (with resultant patterns of BOLD “This sounds more signals across the brain) like something I’d enjoy writing up!” DCM basics DCM models interactions between neuronal populations fMRI, MEG, EEG The aim is to estimate, and make inferences about: 1. The coupling among brain areas 2. How that coupling is influenced by changes in experimental context DCM basics DCM starts with a realistic model of how brain regions interact and where the inputs Neural and hemodynamic models can come in (more on this in a few minutes) Adds a forward model of how neuronal activity causes the signals you observe (e.g. BOLD) …and estimates the parameters in your model (effective connectivity), given your observed data DCM basics Inputs State variables Outputs DCM basics Inputs In functional connectivity models (e.g. standard fMRI analysis), conceptually your input could have entered anywhere In effective connectivity models (e.g. DCM), input only enters at certain places DCM basics Inputs can exert their influence in two ways: 1. Direct influence e.g. visual input to V1 2. Vicarious (indirect) influence e.g. attentional modulation of the coupling between V1 and V5 DCM basics State variables Neuronal activities, and other neuro- or bio-physical variables needed to form the outputs Neuronal priors Haemodynamic priors What you’re modelling is how the inputs modulate the coupling among these state variables DCM basics Output The BOLD signal (for example) that you’ve measured in the brain regions specified in your model Dynamic Modelling (i) Generate equations to model the dynamics of physical systems. These will be LINEAR or NON-LINEAR Linear models provide good approximation However neuronal dynamics are non-linear in nature Linear Dynamic Model INPUT U1 INPUT U2 C11 C22 A21 A11 X1 A12 X2 A22 X1= A11X1 + A21X2 + C11U1 X2= A22X2 + A12X1 + C22U2 The Linear Approximation fL(x,u)=Ax + Cu x1 A11 A12 x1 C11 0 U 1 x 2 0 C 22 U 2 x 2 A21 A22 Intrinsic Connectivity Extrinsic (input) Connectivity Dynamic Modelling (ii) In DCM we are modelling the brain as a: “Deterministic non-linear dynamic system” Effective connectivity is parameterised in terms of coupling between unobserved brain states Bilinear approximation is useful: Reduces the parameters of the model to three sets 1) Direct/extrinsic 2) Intrinsic/Latent 3) Changes in intrinsic coupling induced by inputs The idea behind DCM is not limited to bilinear forms AIM: Estimate the parameters by perturbing the system and observing the response. Important in experimental design: 1) One factor controls sensory perturbation 2)One factor manipulates the context of sensory evoked responses INPUT U1 INPUT U2 C11 C22 A21 A11 X1 A12 X2 A22 INPUT U1 INPUT U2 C11 C22 B221 A21 A11 X1 A12 X2 A22 Bi-Linear Dynamic Model (DCM) INPUT U1 INPUT U2 C11 C22 B221 A21 A11 X1 A12 X2 A22 2 X1= A11X1 + (A21+ B 12U1(t))X + C11U1 X2= A22X2 + A12X1 + C22U2 The Bilinear Approximation j fB(x,u)=(A+jUjB )x + Cu x1 A11 A12 0 B 2 12 x1 C11 0 U 1 U 2 x2 A21 A22 0 0 x 2 0 C 22 U 2 Intrinsic Extrinsic (input) INDUCED CONNECTIVITY Connectivity Connectivity Bilinear state equation in DCM state intrinsic modulation of system direct m external changes connectivity connectivity state inputs inputs z1 a11 a1n m b11 b1n z1 c11 c1m u1 j j u j zn an1 ann j 1 bnj1 bnn zn cn1 cnm um j m z ( A u j B ) z Cu j j 1 Bilinear state equation in DCM state intrinsic modulation of system direct m external changes connectivity connectivity state inputs inputs z1 a11 a1n m b11 b1n z1 c11 c1m u1 j j u j zn an1 ann j 1 bnj1 bnn zn cn1 cnm um j m ( A u j B j ) z Cu z {A, B ...B , C} n 1 m j 1 FG FG z3 left right z4 m z ( A u j B ) z Cu j j 1 LG LG z1 left right z2 RVF CONTEXT LVF u2 u3 u1 z1 a11 a12 a13 0 0 b12 3 0 0 z1 0 c12 0 z a a 2 21 22 0 a24 u1 u 0 0 0 0 z2 c21 0 0 3 u2 z3 a31 0 a33 a34 3 0 0 0 b34 z3 0 0 0 u3 z4 0 a42 a43 a44 0 0 0 0 z4 0 0 0 DCM for fMRI: the basic idea Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). z The modelled neuronal dynamics (z) is λ 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 BOLD signals are maximally similar to the experimentally measured BOLD signals. Priors on biophysical parameters The hemodynamic “Balloon” model • 5 hemodynamic Neural activity parameters: z (t ) { , , , , } h vasodilatory signal Vasodilatory signal s z s γ( f 1) f s • Empirically flow induction f s determined f a priori distributions. changes in volume v changes in dHb τv f v 1 /α τq f E ( f, ) v1/α q/v q v q • Computed separately for each area (like the BOLD signal neural parameters). y (t ) v, q Conceptual Neural state equation z F ( z, u, n ) overview The bilinear model z ( A u j B j ) z Cu F z effective connectivity A z z 2F z modulation of B j connectivity zu j u j z Input F z u(t) direct inputs C u u c1 integration neuronal z activity b23 a12 z2(t) states activity activity hemodynamic z1(t) z3(t) λ model y y y BOLD y Friston et al. 2003, NeuroImage Estimating model parameters Bayes Theorem DCMs are biologically plausible (i.e. complicated) - posterior likelihood ∙ prior p( | y ) p( y | ) p( ) they have lots of free parameters A Bayesian framework is a good way to embody the constraints on these parameters Use Bayes’ theorem to estimate model parameters Priors – empirical Bayes Theorem (haemodynamic parameters) and non- posterior likelihood ∙ prior p( | y ) p( y | ) p( ) empirical (eg. shrinkage priors, temporal scaling) Likelihood derived from error and confounds (eg. drift) Calculate the Posterior probability for each effect, and the probability that it exceeds a set threshold Inferences about the strength (= speed) of connections between the brain regions in your model Interpretation of parameters Single subject analysis - EM algorithm – works out Use the cumulative normal distribution to test the probability the parameters in a model with which a certain parameter is above a chosen threshold γ: - Bayesian model selection to ηθ|y test between alternative models Model comparison and selection A good model of your data will balance model fit with complexity (overfitting models noise) You find this by taking evidence ratios (the “Bayes factor”) The “Bayes factor” is a summary of the evidence in favour of one model as opposed to another Bayesian Model Selection Bayes’ theorem: p( y | , m) p( | m) p( | y, m) p ( y | m) Model evidence: p( y | m) p( y | , m) p( | m) d The log model log p ( y | m) accuracy(m) evidence can be complexity(m) represented as: p( y | m i) Bayes factor: Bij p( y | m j ) Penny et al. 2004, NeuroImage Interpretation of parameters - Group analysis: • Like “random effects” analysis in SPM, 2nd level analysis 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: rm ANOVA: Parameter > 0? Parameter 1 > parameter 2? For multiple sessions per subject New stuff in DCM 1. DCM now accounts for the slice timing problem Extension I: Slice timing model slice acquisition Potential timing problem in DCM: Temporal shift between regional time 2 series because of multi-slice 1 acquisition visual input Solution: Modelling of (known) slice timing of each area. Slice timing extension now allows for any slice timing differences. Long TRs (> 2 sec) no longer a limitation. (Kiebel et al., 2007) New stuff in DCM 1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.) Extension II: Two-state model Single-state DCM Two-state DCM input x1E u (t ) x1 x1E , I exp( A11 uB11 ) IE IE x1I Aij uBij exp( Aij uBij ) x ( ABu ) x Cu x t ( A uB) x Cu t e A11 EE e A11 EI e A1 N 0 x1E A11 A1N x1 A11 I e A11 IE II e 0 0 x1 A x(t ) A x(t ) AN 1 ANN xN A EI E EE e N1 0 e ANN e ANN xN 0 IE e ANN e ANN II xI 0 N Extrinsic (between- Intrinsic (within- z Az u B z Cu j j region) coupling region) coupling New stuff in DCM 1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.) 3. Biological plausibility: more complex balloon model (SPM5) 4. Non-linear version of DCM as well as bilinear (SPM8) Dynamic Causal Modelling of fMRI Network Model dynamics comparison Model inversion Priors State space using Model Expectation-maximization Haemodynamic Posterior distribution response fMRI of parameters data (y) Practical steps of a DCM study - I 1. Definition of the hypothesis & the model (on paper) • Structure: which areas, connections and inputs? • Which parameters in the model concern my hypothesis? • How can I demonstrate the specificity of my results? • What are the alternative models to test? 2. Defining criteria for inference: • single-subject analysis: stat. threshold? contrast? • group analysis: which 2nd-level model? 3. Conventional SPM analysis (subject-specific) • DCMs are fitted separately for each session (subject) → for multi-session experiments, consider concatenation of sessions or adequate 2nd level analysis Practical steps of a DCM study - II 4. Extraction of time series, e.g. via VOI tool in SPM • caveat: anatomical & functional standardisation important for group analyses 5. Possibly definition of a new design matrix, if the “normal” design matrix does not represent the inputs appropriately. • NB: DCM only reads timing information of each input from the design matrix, no parameter estimation necessary. 6. Definition of model • via DCM-GUI or directly in MATLAB Practical steps of a DCM study - III 7. DCM parameter estimation • caveat: models with many regions & scans can crash MATLAB! 8. Model comparison and selection: • Which of all models considered is the optimal one? Bayesian model selection 9. Testing the hypothesis Statistical test on the relevant parameters of the optimal model DCM button ‘specify’ NB: in order! Summary DCM is NOT EXPLORATORY Used to test the hypothesis that motivated the experimental design BUILD A MODEL TO EXPRESS HYPOTHESIS IN TERMS OF NEURAL CONNECTIVITY The GLM used in typical fMRI data analysis uses the same architecture as DCM but embodies more assumptions Note: In DCM a “Strong Connection” means an influence that is expressed quickly or with a small time constant. When constructing experiments, consider whether you want to use DCM early When in doubt, ask the experts……… REFERENCES Karl J. Friston. Dynamic Causal Modelling. Human brain function. Chapter 22. Second Edition. http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/ K.J Friston, L. Harrison and W. Penny. Dynamic Causal Modelling. Neuroimage 2003; 19:1273-1302. SPM Manual Last year’s presentation ANY QUESTIONS???

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