DCM: Advanced issues

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 Methods & models for fMRI data analysis, University of Zurich27 May 2009

2. Overview • Bayesian model selection (BMS) • Nonlinear DCM for fMRI • Timing errors & sampling accuracy • Integrating tractography and DCM • DCMs for electrophysiological data

3. 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?

4. 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

5. 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).

6. 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 In SPM2 & SPM5, interface offers 2 approximations: No. of data points Akaike Information Criterion: Bayesian Information Criterion: AIC favours more complex models, BIC favours simpler models. Penny et al. 2004, NeuroImage

7. 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;[ Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

8. 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:

9. 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 !

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

11. 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

12. Random effects BMS for group studies: a variational Bayesian approach Dirichlet parameters = “occurrences” of models in the population Dirichlet distribution of model probabilities Multinomial distribution of model labels Measured data Stephan et al. 2009, NeuroImage

13. • • • Task-driven lateralisation Does the word contain the letter A or not? letter decisions > spatial decisions group analysis (random effects),n=16, p<0.05 whole-brain corrected time Is the red letter left or right from the midline of the word? spatial decisions > letter decisions Stephan et al. 2003, Science

14. LG left MOG right MOG left FG right LG right FG left Inter-hemispheric connectivity in the visual ventral stream Right FG 38,-52,-20 Left MOG -38,-90,-4 Left FG -44,-52,-18 Right MOG -38,-94,0 LD|LVF 0.20  0.04 0.00  0.01 0.07  0.02 LD>SD, p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) p<0.01 uncorrected 0.27  0.06 0.11  0.03 LD LD 0.01  0.03 0.00  0.04 Left LG -12,-70,-6 Left LG -14,-68,-2 0.01  0.01 0.01  0.01 0.06  0.02 LD|RVF RVF stim. LVF stim. LD>SD masked incl. with RVF>LVF p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) LD>SD masked incl. with LVF>RVF p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) Stephan et al. 2007, J. Neurosci.

15. LD LD|LVF LD|RVF LD|LVF LD LD RVF stim. LD LVF stim. RVF stim. LD|RVF LVF stim. MOG MOG MOG MOG LG LG LG LG FG FG FG FG m2 m1 Stephan et al. 2009, NeuroImage

17. LD LD|LVF LD|RVF LD|LVF LD LD RVF stim. LD LVF stim. RVF stim. LD|RVF LVF stim. MOG MOG MOG MOG LG LG LG LG FG FG FG FG Simulation study: sampling subjects from a heterogenous population m1 • Population where 70% of all subjects' data are generated by model m1 and 30% by model m2 • Random sampling of subjects from this population and generating synthetic data with observation noise • Fitting both m1 and m2 to all data sets and performing BMS m2 Stephan et al. 2009, NeuroImage

18. A B true values: 1=220.7=15.4 2=220.3=6.6 mean estimates: 1=15.4, 2=6.6 true values: r1 = 0.7, r2=0.3 mean estimates: r1 = 0.7, r2=0.3  <r> m2 m2 m1 m1 C D true values: 1 = 1, 2=0 mean estimates: 1 = 0.89, 2=0.11  m2 log GBF12 m1

19. Overview • Bayesian model selection (BMS) • Nonlinear DCM for fMRI • Timing errors & sampling accuracy • Integrating tractography and DCM • DCMs for electrophysiological data

20. 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 x2(t) activity x3(t) activity x1(t) x neuronal states integration Stephan & Friston (2007),Handbook of Brain Connectivity

21. 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:

22. Neural population activity x3 fMRI signal change (%) x1 x2 u2 u1 Nonlinear dynamic causal model (DCM): Stephan et al. 2008, NeuroImage

23. SPC V1 IFG Attention V5 Photic .52 (98%) .37 (90%) .42 (100%) .56 (99%) .69 (100%) .47 (100%) .82 (100%) Motion .65 (100%) Nonlinear DCM: Attention to motion Stimuli + Task Previous bilinear DCM Büchel & Friston (1997) 250 radially moving dots (4.7 °/s) Friston et al. (2003) Conditions: F – fixation only A – motion + attention (“detect changes”) N – motion without attention S – stationary dots 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?

24. 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. 2008, NeuroImage

25. 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. 2008, NeuroImage

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

27. rivalry non-rivalry 0.02 -0.03 MFG 1.05 0.08 2.43 2.41 -0.31 0.51 0.30 PPA FFA -0.80 0.04 -0.03 0.02 0.06 faces houses faces houses Nonlinear DCM: Binocular rivalry Stephan et al. 2008, NeuroImage

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

29. Overview • Bayesian model selection (BMS) • Nonlinear DCM for fMRI • Timing errors & sampling accuracy • Integrating tractography and DCM • DCMs for electrophysiological data

30. 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 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

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

32. Slice timing in DCM: an example 3 TR 1 TR 2 TR 4 TR 5 TR Default sampling t 3 TR 1 TR 2 TR 4 TR 5 TR Slice-specific sampling t Kiebel et al. 2007, NeuroImage

33. Overview • Bayesian model selection (BMS) • Nonlinear DCM for fMRI • Timing errors & sampling accuracy • Integrating tractography and DCM • DCMs for electrophysiological data

34. Diffusion-weighted imaging Parker & Alexander, 2005, Phil. Trans. B

35. Probabilistic tractography: Kaden et al. 2007, NeuroImage • computes local fibre orientation density by spherical deconvolution of the diffusion-weighted signal • estimates the spatial probability distribution of connectivity from given seed regions • anatomical connectivity = proportion of fibre pathways originating in a specific source region that intersect a target region • If the area or volume of the source region approaches a point, this measure reduces to method by Behrens et al. (2003)

36. Integration of tractography and DCM R1 R2 low probability of anatomical connection  small prior variance of effective connectivity parameter R1 R2 high probability of anatomical connection  large prior variance of effective connectivity parameter Stephan, Tittgemeyer, Knoesche, Moran, Friston, in revision

37. probabilistic tractography FG right LG right anatomical connectivity connection-specific priors for coupling parameters LG left LG (x1) FG (x4) LG (x2) FG (x3) FG left LD|LVF LD LD DCM structure LD|RVF BVF stim. RVF stim. LVF stim.

41. Overview • Bayesian model selection (BMS) • Nonlinear DCM for fMRI • Timing errors & sampling accuracy • Integrating tractography and DCM • DCMs for electrophysiological data

42. 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

43. DCMs for M/EEG and LFPs • can be fitted both to frequency spectra and ERPs • models different neuronal cell types, different synaptic types (and their plasticity) and spike-frequency adaptation (SFA) • ongoing model validation by LFP recordings in rats, combined with pharmacological manipulations standards deviants A1 A2 Example of single-neuron SFA Tombaugh et al. 2005, J.Neurosci.

44. 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. (2003) NeuroImage

45. g 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Synaptic ‘alpha’ kernel Inhibitory cells in agranular layers Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers Exogenous input u Sigmoid function Excitatory pyramidal cells in agranular layers Extrinsic Connections: Forward Backward Lateral David et al. 2006, NeuroImage Kiebel et al. 2007, NeuroImage Moran et al. 2009, NeuroImage

46. Electromagnetic forward model for M/EEG Forward model: lead field & gain matrix Depolarisation of pyramidal cells Scalp data Forward model Kiebel et al. 2006, NeuroImage

47. DCM for steady-state responses • models the cross-spectral density of recorded data • feature extraction by means of p-order VAR model • spectral form of neuronal innovations (i.e. baseline cortical activity) are estimated using a mixture of white and pink (1/f) components • assumes quasi-stationary responses (i.e. changes in neuronal states are approximated by small perturbations around some fixed point) 10 Frequency (Hz) 20 30 Time (s) 0 10 Moran et al. 2009, NeuroImage

48. Validation study using microdialysis (in collaboration with Conway Inst., UC Dublin) • two groups of rats with different rearing conditions • LFP recordings and microdialysis measurements (Glu & GABA) from mPFC Moran et al. 2008, NeuroImage

49. Experimental data FFT 10 mins time series: one area (mPFC) blue: control animals red: isolated animals * p<0.05, Bonferroni-corrected Moran et al. 2008, NeuroImage

50. Predictions about expected parameter estimates from the microdialysis measurements upregulation of AMPA receptors amplitude of synaptic kernels ( He) • SFA (2) chronic reduction in extracellular glutamate levels  EPSPs  activation of voltage-sensitive Ca2+ channels → intracellular Ca2+→ Ca-dependent K+ currents → IAHP sensitisation of postsynaptic mechanisms Van den Pool et al. 1996, Neuroscience Sanchez-Vives et al. 2000, J. Neurosci.