DCM: Advanced issues

1. DCM: Advanced issues Klaas Enno Stephan Centre for the Study of Social & Neural Systems Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London SPM Course 2008Zurich

2. Neural state equation intrinsic connectivity modulation of connectivity direct inputs modulatory input u2(t) driving input u1(t) t t y BOLD y y y   λ hemodynamic model  activity z2(t) activity z3(t) activity z1(t) z neuronal states integration Stephan & Friston (2007),Handbook of Connectivity

3. Overview • Bayesian model selection (BMS) • Timing errors & sampling accuracy • The hemodynamic model in DCM • Advanced DCM formulations for fMRI • two-state DCMs • nonlinear DCMs • An outlook to the future

4. Pitt & Miyung (2002) TICS Model comparison and selection Given competing hypotheses on structure & functional mechanisms of a system, which model is the best? Which model represents thebest balance between model fit and model complexity? For which model m does p(y|m) become maximal?

5. Bayesian model selection (BMS) Bayes’ rule: Model evidence: accounts for both accuracy and complexity of the model allows for inference about structure (generalisability) of the model integral usually not analytically solvable, approximations necessary (e.g. AIC or BIC)

6. Model evidence p(y|m) Balance between fit and complexity Generalisability of the model Gharamani, 2004 p(y|m) a specific y all possible datasets y Model evidence: probability of generating data y from parameters that are randomly sampled from the prior p(m). Maximum likelihood: probability of the data y for the specific parameter vector  that maximises p(y|,m).

7. Approximations to the model evidence in DCM Maximizing log model evidence = Maximizing model evidence Logarithm is a monotonic function Log model evidence = balance between fit and complexity No. of parameters At the moment, two approximations available in SPM interface: No. of data points Akaike Information Criterion: Bayesian Information Criterion: AIC favours more complex models, BIC favours simpler models. Penny et al. 2004, NeuroImage

8. Bayes factors To compare two models, we can just compare their log evidences. But: the log evidence is just some number – not very intuitive! A more intuitive interpretation of model comparisons is made possible by Bayes factors: positive value, [0;[ Raftery classification:

9. Two models with identical numbers of parameters AIC: BF = 3.3 BMS result: BF = 3.3 BIC: BF = 3.3

10. Two models with different numbers of parameters & compatible AIC/BIC based decisions about models AIC: BF = 0.1 BMS result: BF = 0.7 BIC: BF = 0.7

11. Two models with different numbers of parameters & incompatible AIC/BIC based decisions about models AIC: BF = 0.3 BMS result: “AIC and BIC disagree about which model is superior - no decision can be made.” BIC: BF = 2.2

12. Further reading on BMS of DCMs • Theoretical papers: • Penny et al. (2004) Comparing dynamic causal models. NeuroImage 22: 1157-1172. • Stephan et al. (2007) Comparing hemodynamic models with DCM. NeuroImage 38: 387-401. • Applications of BMS & DCM (selection): • Grol et al. (2007) Parieto-frontal connectivity during visually-guided grasping. J. Neurosci. 27: 11877-11887. • Kumar et al. (2007) Hierarchical processing of auditory objects in humans. PLoS Computat. Biol. 3: e100. • Smith et al. (2006) Task and content modulate amygdala-hippocampal connectivity in emotional retrieval. Neuron 49: 631-638. • Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27: 3512-3522.

13. Overview • Bayesian model selection (BMS) • Timing errors & sampling accuracy • The hemodynamic model in DCM • Advanced DCM formulations for fMRI • two-state DCMs • nonlinear DCMs • An outlook to the future

14. Timing problems at long TRs/TAs • Two potential timing problems in DCM: • wrong timing of inputs • temporal shift between regional time series because of multi-slice acquisition 2 slice acquisition 1 visualinput • DCM is robust against timing errors up to approx. ± 1 s • compensatory changes of σ and θh • Possible corrections: • slice-timing (not for long TAs) • restriction of the model to neighbouring regions • in both cases: adjust temporal reference bin in SPM defaults (defaults.stats.fmri.t0)

15. Slice timing in DCM: three-level model sampled BOLD response 3rd level 2nd level BOLD response neuronal response 1st level z = neuronal states u = inputs zh = hemodynamic states v = BOLD responses n, h = neuronal and hemodynamic parameters T = sampling time points Kiebel et al. 2007, NeuroImage

16. Slice timing in DCM: an example 3 TR 1 TR 2 TR 4 TR 5 TR t Original DCM 3 TR 1 TR 2 TR 4 TR 5 TR Present DCM t

17. Overview • Bayesian model selection (BMS) • Timing errors & sampling accuracy • The hemodynamic model in DCM • Advanced DCM formulations for fMRI • two-state DCMs • nonlinear DCMs • An outlook to the future

18. RVF LVF LG left LG right FG right FG left Example: BOLD signal modelled with DCM black: measured BOLD signal red: predicted BOLD signal

19. t The hemodynamic model in DCM u stimulus functions neural state equation • 6 hemodynamic parameters: important for model fitting, but of no interest for statistical inference hemodynamic state equations Balloon model • Empirically determineda priori distributions. • Computed separately for each area (like the neural parameters) region-specific HRFs! BOLD signal change equation Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage

20. Recent changes in the hemodynamic model(Stephan et al. 2007, NeuroImage) • new output non-linearity, based on new exp. data and mathematical derivations BMS indicates that new model performs better than original Buxton model • field-dependency of output coefficients is handled better, e.g. by estimating intra-/extravascular BOLD signal ratio less problematic to apply DCM to high-field fMRI data

21. How independent are our neural and hemodynamic parameter estimates? A B C h ε r,A r,B r,C Stephan et al. 2007, NeuroImage

22. Overview • Bayesian model selection (BMS) • Timing errors & sampling accuracy • The hemodynamic model in DCM • Advanced DCM formulations for fMRI • two-state DCMs • nonlinear DCMs • An outlook to the future

23. Single-state DCM Two-state DCM input Extrinsic (between-region) coupling Intrinsic (within-region) coupling Marreiros et al. 2008, NeuroImage

24. non-linear DCM modulation driving input bilinear DCM driving input modulation Two-dimensional Taylor series (around x0=0, u0=0): Nonlinear state equation: Bilinear state equation:

25. Neural population activity u2 +++ x3 – +++ + + BOLD signal change (%) + x1 x2 u1 +++ + – – Neuronal state equation: Stephan et al., submitted

26. M3 attention M2 better than M1 PPC BF= 2966 stim V1 V5 M4 BF= 12 attention PPC M3 better than M2 stim V1 V5 BF= 23 M4 better than M3 attention M1 M2  modulation of back- ward or forward connection? PPC PPC attention stim V1 V5 stim V1 V5  additional driving effect of attention on PPC?  bilinear or nonlinear modulation of forward connection? Stephan et al., submitted

27. B A attention MAP = 1.25 0.10 PPC 0.26 0.39 1.25 0.26 V1 stim 0.13 V5 0.46 0.50 motion Stephan et al., submitted

28. motion & attention static dots motion & no attention V1 V5 PPC observed fitted Stephan et al., submitted

29. Overview • Bayesian model selection (BMS) • Timing errors & sampling accuracy • The hemodynamic model in DCM • Advanced DCM formulations for fMRI • two-state DCMs • nonlinear DCMs • An outlook to the future

30. DCM: generative model for fMRI and ERPs Hemodynamicforward model:neural activityBOLD (nonlinear) Electric/magnetic forward model:neural activityEEGMEG LFP (linear) Neural state equation: fMRI ERPs Neural model: 1 state variable per region bilinear state equation no propagation delays Neural model: 8 state variables per region nonlinear state equation propagation delays inputs

31. Neural mass model of a cortical macrocolumn E x t r i n s i c i n p u t s Excitatory Interneurons He, e mean firing rate  mean postsynaptic potential (PSP) 1 2 Pyramidal Cells He, e MEG/EEG signal 3 4 mean PSP mean firing rate Inhibitory Interneurons Hi, e Excitatory connection Inhibitory connection • te, ti : synaptic time constant (excitatory and inhibitory) • He, Hi: synaptic efficacy (excitatory and inhibitory) • g1,…,g4: intrinsic connection strengths • propagation delays Parameters: Jansen & Rit (1995) Biol. Cybern. David et al. (2006) NeuroImage

32. DCM for ERPs: neural state equations Extrinsic lateral connections Inhibitory cells in supra/infragranular layers activity inhibitory interneurons Extrinsic forward connections Excitatory spiny cells in granular layers spiny stellate cells Intrinsic connections pyramidal cells Excitatory pyramidal cellsin supra/infragranular layers MEG/EEG signal mV Extrinsic backward connections State equations David et al. (2006) NeuroImage neuronal (source) model

33. DCM for LFPs • extended neural mass models that can be fitted to LFP data (both frequency spectra and ERPs) • explicit model of spike-frequency adaptation (SFA) • current validation work to establish the sensitivity of various parameters wrt. specific neurotransmitter effects • validation of this model by LFP recordings in rats, combined with pharmacological manipulations standards deviants A1 A2 Moran et al. (2007, 2008) NeuroImage