1. DCM: Advanced issues Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London SPM Course Zurich 2009

2. intrinsic connectivity direct inputs modulation of connectivity Neural state equation hemodynamic model λ x y integration BOLD yy y activity x 1 (t) activity x 2 (t) activity x 3 (t) neuronal states t driving input u 1 (t) modulatory input u 2 (t) t Stephan & Friston (2007), Handbook of Brain Connectivity   

3. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI The hemodynamic model in DCM Timing errors & sampling accuracy DCMs for electrophysiological data

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

5. Model evidence: Bayesian model selection (BMS) Bayes’ rule: 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

6. Model evidence p(y|m) Gharamani, 2004 p(y|m) all possible datasets y a specific y Balance between fit and complexity Generalisability of the model 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. Logarithm is a monotonic function Maximizing log model evidence = Maximizing model evidence In SPM2 & SPM5, interface offers 2 approximations: Akaike Information Criterion: Bayesian Information Criterion: Log model evidence = balance between fit and complexity Penny et al. 2004, NeuroImage Approximations to the model evidence in DCM No. of parameters No. of data points AIC favours more complex models, BIC favours simpler models.

8. Bayes factors positive value, [0;  [ But: the log evidence is just some number – not very intuitive! A more intuitive interpretation of model comparisons is made possible by Bayes factors: To compare two models, we can just compare their log evidences. B 12 p(m 1 |y)Evidence 1 to 350-75%weak 3 to 2075-95%positive 20 to 15095-99%strong  150  99% Very strong Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

9. AIC: BF = 3.3 BIC: BF = 3.3 BMS result: BF = 3.3 SPM2/SPM5: Two models with identical numbers of parameters

10. AIC: BF = 0.1 BIC: BF = 0.7 BMS result: BF = 0.7 SPM2/SPM5: Two models with different numbers of parameters & compatible AIC/BIC based decisions about models

11. AIC: BF = 0.3 BIC: BF = 2.2 BMS result: “AIC and BIC disagree about which model is superior - no decision can be made.” SPM2/SPM5: Two models with different numbers of parameters & incompatible AIC/BIC based decisions about models

12. The negative free energy approximation Under Gaussian assumptions about the posterior (Laplace approximation), the negative free energy F is a lower bound on the log model evidence:

13. The complexity term in F In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies. The complexity term of F is higher –the more independent the prior parameters (  effective DFs) –the more dependent the posterior parameters –the more the posterior mean deviates from the prior mean NB: SPM8 only uses F for model selection !

14. V1 V5 stim PPC M2 attention V1 V5 stim PPC M1 attention V1 V5 stim PPC M3 attention V1 V5 stim PPC M4 attention BF  2966  F = 7.995 M2 better than M1 BF  12  F = 2.450 M3 better than M2 BF  23  F = 3.144 M4 better than M3 M1 M2 M3 M4 BMS in SPM8: an example

15. Fixed effects BMS at group level Group Bayes factor (GBF) for 1...K subjects: Average Bayes factor (ABF): Problems: -blind with regard to group heterogeneity -sensitive to outliers

16. A suboptimal solution... Positive Evidence Ratio (PER): i.e. number of subjects in which there is positive evidence for model i divided by number of subjects in which there is positive evidence for model j

17. Random effects BMS for group studies Dirichlet parameters = “occurrences” of models in the populations Dirichlet distribution of model probabilities Multinomial distribution of subject-specific models Measured data Stephan et al. 2009, in revision

18. x1x1 x2x2 u1u1 x3x3 u2u2 x1x1 x2x2 u1u1 x3x3 u2u2 incorrect model (m 2 )correct model (m 1 ) m2m2 m1m1

19. Exceedance probability Estimates of Dirichlet parameters Post. expectations of model probabilities Stephan et al. 2009, in revision

20. MOG LG RVF stim. LVF stim. FG LD|RVF LD|LVF LD MOG LG RVF stim. LVF stim. FG LD LD|RVFLD|LVF MOG m2m2 m1m1 Stephan et al. 2009, in revision

22. Interface in SPM8 Post. expectations of model probabilities Exceedance probability

23. 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. –Stephan et al. (2009) Bayesian model selection for group studies. NeuroImage, in revision. Examples of application: –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. –Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27: 3512-3522.

24. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI The hemodynamic model in DCM Timing errors & sampling accuracy DCMs for electrophysiological data

25. bilinear DCM Bilinear state equation: driving input modulation nonlinear DCM driving input modulation Two-dimensional Taylor series (around x 0 =0, u 0 =0): Nonlinear state equation:

26. Neural population activity fMRI signal change (%) x1x1 x2x2 x3x3 Nonlinear dynamic causal model (DCM): Stephan et al. 2008, NeuroImage u1u1 u2u2

27. Nonlinear DCM: Attention to motion V1IFG V5 SPC Motion Photic Attention.82 (100%).42 (100%).37 (90%).69 (100%).47 (100%).65 (100%).52 (98%).56 (99%) Stimuli + Task 250 radially moving dots (4.7 °/s) Conditions: F – fixation only A – motion + attention (“detect changes”) N – motion without attention S – stationary dots Previous bilinear DCM Friston et al. (2003) Friston et al. (2003): attention modulates backward connections IFG→SPC and SPC→V5. Q: Is a nonlinear mechanism (gain control) a better explanation of the data? Büchel & Friston (1997)

28. modulation of back- ward or forward connection? additional driving effect of attention on PPC? bilinear or nonlinear modulation of forward connection? V1 V5 stim PPC M2 attention V1 V5 stim PPC M1 attention V1 V5 stim PPC M3 attention V1 V5 stim PPC M4 attention BF = 2966 M2 better than M1 M3 better than M2 BF = 12 M4 better than M3 BF = 23    Stephan et al. 2008, NeuroImage

29. V1 V5 stim PPC attention motion 1.25 0.13 0.46 0.39 0.26 0.50 0.26 0.10 MAP = 1.25 Stephan et al. 2008, NeuroImage

30. V1 V5 PPC observed fitted motion & attention motion & no attention static dots Stephan et al. 2008, NeuroImage

31. FFA PPA MFG -0.80 -0.31 faceshouses faceshouses rivalrynon-rivalry 1.050.08 0.30 0.51 2.43 2.41 0.04-0.030.020.06 0.02 -0.03 Nonlinear DCM: Binocular rivalry Stephan et al. 2008, NeuroImage

32. BR nBR FFA PPA MFG time (s) Stephan et al. 2008, NeuroImage

33. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI The hemodynamic model in DCM Timing errors & sampling accuracy DCMs for electrophysiological data

34. stimulus functions u t neural state equation hemodynamic state equations Balloon model BOLD signal change equation important for model fitting, but of no interest for statistical inference 6 hemodynamic parameters: Empirically determined a priori distributions. Area-specific estimates (like neural parameters)  region-specific HRFs! The hemodynamic model in DCM Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage

35. LG left LG right RVFLVF FG right FG left black: measured BOLD signalred: predicted BOLD signal Region-specific HRFs E 0 =0.1 E 0 =0.5 E 0 =0.9

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

37. A B C hh ε How interdependent are our neural and hemodynamic parameter estimates? Stephan et al. 2007, NeuroImage

38. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI The hemodynamic model in DCM Timing errors & sampling accuracy DCMs for electrophysiological data

39. Timing problems at long TRs/TAs Two potential timing problems in DCM: 1.wrong timing of inputs 2.temporal shift between regional time series because of multi-slice acquisition DCM is robust against timing errors up to approx. ± 1 s –compensatory changes of σ and θ h Possible corrections: –slice-timing in SPM (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) Best solution: Slice-specific sampling within DCM 1 2 slice acquisition visual input

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

41. Slice timing in DCM: an example t 1 TR2 TR 3 TR 4 TR5 TR t 1 TR2 TR 3 TR 4 TR5 TR Default sampling Slice-specific sampling

42. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI The hemodynamic model in DCM Timing errors & sampling accuracy DCMs for electrophysiological data

43. Neural state equation: Electric/magnetic forward model: neural activity  EEG MEG LFP (linear) DCM: generative model for fMRI and 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 fMRI ERPs inputs Hemodynamic forward model: neural activity  BOLD (nonlinear)

44. DCMs for M/EEG and LFPs can be fitted both to frequency spectra and ERPs models synaptic plasticity and of spike-frequency adaptation (SFA) ongoing model validation by LFP recordings in rats, combined with pharmacological manipulations Moran et al. 2008, NeuroImage standardsdeviants A1 A2 Tombaugh et al. 2005, J.Neurosci. Example of single-neuron SFA

45. Neural mass model of a cortical macrocolumn Excitatory Interneurons H e,  e Pyramidal Cells H e,  e Inhibitory Interneurons H i,  e Extrinsic inputsExtrinsic inputs Excitatory connection Inhibitory connection   e,  i : synaptic time constant (excitatory and inhibitory)  H e, H i : synaptic efficacy (excitatory and inhibitory)   1,…,   : intrinsic connection strengths  propagation delays 22 11 44 33 MEG/EEG signal MEG/EEG signal Parameters: Jansen & Rit (1995) Biol. Cybern. David et al. (2006) NeuroImage mean firing rate  mean postsynaptic potential (PSP) mean PSP  mean firing rate

46. 4  3  1  2  1 2 4914 41 2))((xxuaxsHx xx eeee     Excitatory spiny cells in granular layers Exogenous input u 4  3  1  2  Intrinsic connections 5  Excitatory spiny cells in granular layers Excitatory pyramidal cells in agranular layers Inhibitory cells in agranular layers Synaptic ‘alpha’ kernel Sigmoid function Extrinsic Connections: Forward Backward Lateral Moran et al. 2008, NeuroImage

47. Electromagnetic forward model for M/EEG Depolarisation of pyramidal cells Forward model: lead field & gain matrix Scalp data Forward model

48. Thank you