Understanding Dynamic Causal Modeling in Brain Connectivity Analysis

1. DCM Theory 5th March 2008 Andreina Mendez Stephanie Burnett

2. Recap • Last session: PPIs (psycho-physiological interactions) • 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

3. “This is a frontoparietal network collection of brain regions involved in activated while processing coffee” Introduction: DCM and its place in the methods family tree • 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 BOLD signal Task

4. V1 V5 attention DCM models bidirectional and modulatory interactions, between multiple brain regions V1 DCM models how neuronal activity causes the BOLD signal (forward model) i.e. your conclusions are about neural processes Introduction: DCM and its place in the methods family tree • 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

5. 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/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!”

6. 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

7. DCM starts with a realistic model of how brain regions interact 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 Neural and hemodynamic models (more later on)

8. DCM treats the brain as a nonlinear dynamic system The system has inputs, state variables, and outputs Your experiment is a designed perturbation of the system DCM basics

9. Inputs In functional connectivity models (e.g. standard fMRI analysis), conceptually your input can enter 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 influence e.g. attentional modulation of the coupling between SPC and V5 DCM basics

11. State variables neuronal activities, and other neuro/biophysical variables needed to form the outputs (later on) Neuronal priors Hemodynamic priors DCM basics

12. Output the BOLD signal you’ve measured in the brain regions specified in your model DCM basics

13. Models of effective connectivity = system models. • System = set of elements which interact in a spatially and temporally specific fashion. • System dynamics = change of state vector in time • Causal effects in the system: • interactions between elements • external inputs u • System parameters  :specify the nature of the interactions • general state equation for non-autonomous systems overall system staterepresented by state variables change ofstate vectorin time

14. LG left FG right LG right FG left Example: linear dynamic system z4 z3 LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF)visual field. z1 z2 RVF LVF u2 u1 systemstate input parameters state changes effective connectivity externalinputs

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

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

17. z4 z3 z1 z2 CONTEXT RVF LVF u2 u3 u1 LG left FG right LG right FG left Extension: bilinear dynamic system

18. z λ y 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). • The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ). 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.

19. The hemodynamic “Balloon” model • 5 hemodynamic parameters: • Empirically determineda priori distributions. • Computed separately for each area (like the neural parameters). • Buxton et al. 1998

20. Priorsonbiophysicalparameters

21. 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 a12 activity z2(t) activity z3(t) activity z1(t) hemodynamic model y y BOLD y Friston et al. 2003,NeuroImage

22. 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 ( ) Estimating model parameters

23. 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

24. 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

25. 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

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

27. - 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

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

29. 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)

30. 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.)

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

32. 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)

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

34. DCM good because Causal – models effective connectivity, not functional connectivity Neuronally plausible Forward model of how neuronal activity causes BOLD signal Inputs only enter at certain places; can model vicarious (modulatory) input; can have reciprocal connections and loops Next week: practical issues in DCM SUMMARY

35. 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

36. THANK YOU Special thanks to: - Andre Marreiros - Maria Joao