Dynamic Causal Modelling: Theory and Methods for Dummies

1. 18th February 2009 Stephanie Burnett Christian Lambert Methods for Dummies 2009 Dynamic Causal Modelling Part I: Theory

2. Psychophysiological interactions (PPI) and structural equation modelling (SEM) Functional vs. effective connectivity Functional connectivity: temporal correlation between spatially remote neurophysiological events Effective connectivity: the influence that the elements of a neuronal system exert over each other Standard fMRI analysis PPIs, SEM, DCM Last time, in MfD…

3. 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 BOLD signal Task

4. PPIs Represent how the (experimental) context modulates connectivity between a brain region of interest, and anywhere else E.g. (Whatever gives rise to the) signal in one brain region (V1) will lead to a signal in V5, and the strength of this signal in V5 depends on attention DCM models how neuronal activity causes the BOLD signal (forward model) V1 V5 V5 V1 attention attention DCM models bidirectional and modulatory interactions, between multiple brain regions V1 V1 Introduction: DCM and its place in the methods family tree That is, your conclusions are about neural events

5. 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 signals across the brain) “This sounds more like something I’d enjoy writing up!” Introduction: DCM and its place in the methods family tree

6. DCM models interactions between neuronal populations fMRI, MEG, EEG The aim is to estimate, and make inferences about: The coupling among brain areas How that coupling is influenced by changes in experimental context DCM basics

7. DCM starts with a realistic model of how brain regions interact and where the inputs can come in 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 Neural and hemodynamic models (more on this in a few minutes) DCM basics

8. Inputs State variables Outputs DCM basics

9. 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

10. 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

11. 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

12. Output The BOLD signal (for example) that you’ve measured in the brain regions specified in your model DCM basics

13. 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

14. Linear Dynamic Model X1= A11X1 + A21X2 + C11U1 X2= A22X2 + A12X1 + C22U2 The Linear Approximation fL(x,u)=Ax + Cu Intrinsic Connectivity Extrinsic (input) Connectivity

15. 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

16. 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

19. Bi-Linear Dynamic Model (DCM) X1= A11X1 + (A21+ B212U1(t))X+ C11U1 X2= A22X2 + A12X1 + C22U2 The Bilinear Approximation fB(x,u)=(A+jUjBj)x + Cu Intrinsic Connectivity Extrinsic (input) Connectivity INDUCED CONNECTIVITY

20. Bilinearstateequation in DCM modulation of connectivity systemstate direct inputs state changes intrinsic connectivity m externalinputs

21. Bilinearstateequation in DCM state changes intrinsic connectivity modulation of connectivity systemstate direct inputs m externalinputs

22. z4 z3 z1 z2 CONTEXT RVF LVF u2 u3 u1 LG left FG right LG right FG left

23. Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ). z λ y DCM for fMRI: the basic idea 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.

24. Priors on biophysical parameters

25. The hemodynamic “Balloon” model • 5 hemodynamic parameters: Vasodilatory signal • Empirically determineda priori distributions. • Computed separately for each area (like the neural parameters).

26. z λ y Conceptual overview Neural state equation The bilinear model effective connectivity modulation of connectivity Input u(t) direct inputs c1 integration neuronal states b23 activity z2(t) a12 activity z3(t) activity z1(t) hemodynamic model y y BOLD y Friston et al. 2003,NeuroImage

27. Estimating model parameters DCMs are biologically plausible (i.e. complicated) - they have lots of free parameters A Bayesian framework is a good way to embody the constraints on these parameters Bayes Theorem posterior  likelihood ∙ prior q µ q × q p ( | y ) p ( y | ) p ( )

28. Use Bayes’ theorem to estimate model parameters Priors – empirical (haemodynamic parameters) and non-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 Bayes Theorem posterior  likelihood ∙ prior q µ q × q p ( | y ) p ( y | ) p ( ) Inferences about the strength (= speed) of connections between the brain regions in your model

29. EM algorithm – works out the parameters in a model Bayesian model selection to test between alternative models Single subject analysis Use the cumulative normal distribution to test the probability with which a certain parameter is above a chosen threshold γ:  ηθ|y Interpretation of parameters

30. 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 Model comparison and selection

31. Bayes’ theorem: Model evidence: The log model evidence can be represented as: Bayes factor: Bayesian Model Selection Penny et al. 2004, NeuroImage

32. - Group analysis: One sample t-test: Parameter > 0? Paired t-test: Parameter 1 > parameter 2? rm ANOVA: For multiple sessions per subject Interpretation of parameters • 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

33. 1. DCM now accounts for the slice timing problem New stuff in DCM

34. Extension I: Slice timing model Potential timing problem in DCM: Temporal shift between regional time series because of multi-slice acquisition 2 slice acquisition 1 visualinput 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)

35. 1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.) New stuff in DCM

36. Single-state DCM Two-state DCM input Extrinsic (between-region) coupling Intrinsic (within-region) coupling Extension II: Two-state model

37. 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) New stuff in DCM

38. Dynamic Causal Modelling of fMRI Model comparison Network dynamics Priors State space Model Model inversion using Expectation-maximization Posterior distribution of parameters Haemodynamic response fMRI data (y)

39. Practical steps of a DCM study - I • 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? • Defining criteria for inference: • single-subject analysis: stat. threshold? contrast? • group analysis: which 2nd-level model? • 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

40. Practical steps of a DCM study - II • Extraction of time series, e.g. via VOI tool in SPM • caveat: anatomical & functional standardisation important for group analyses • 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. • Definition of model • via DCM-GUI or directlyin MATLAB

41. Practical steps of a DCM study - III • DCM parameter estimation • caveat: models with many regions & scans can crash MATLAB! • Model comparison and selection: • Which of all models considered is the optimal one? Bayesian model selection • Testing the hypothesisStatistical test onthe relevant parametersof the optimal model

42. DCM button ‘specify’ NB: in order!

43. 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………

44. 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 REFERENCES

45. ANY QUESTIONS???