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 Zurich 27 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. 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

4. 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

5. 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).

6. 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.

7. 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.

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. 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

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

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

15. 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, NeuroImage

17. Simulation study: sampling subjects from a heterogenous population Population where 70% of all subjects' data are generated by model m 1 and 30% by model m 2 Random sampling of subjects from this population and generating synthetic data with observation noise Fitting both m 1 and m 2 to all data sets and performing BMS MOG LG RVF stim. LVF stim. FG LD|RVF LD|LVF LD MOG LG RVF stim. LVF stim. FG LD LD|RVFLD|LVF MOG m1m1 m2m2 Stephan et al. 2009, NeuroImage

18. B A m1m1 m2m2 m1m1 m2m2 m1m1 m2m2 log GBF 12 C D  <r><r>  true values:  1 =22  0.7=15.4  2 =22  0.3=6.6 mean estimates:  1 =15.4,  2 =6.6 true values: r 1 = 0.7, r 2 =0.3 mean estimates: r 1 = 0.7, r 2 =0.3 true values:  1 = 1,  2 =0 mean estimates:  1 = 0.89,  2 =0.11

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

20. 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   

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

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

23. 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)

24. 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

25. 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

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

27. 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

28. BR nBR FFA PPA MFG 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: 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

31. 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

32. 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 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. R2R2 R1R1 R2R2 R1R1 low probability of anatomical connection  small prior variance of effective connectivity parameter high probability of anatomical connection  large prior variance of effective connectivity parameter Integration of tractography and DCM Stephan, Tittgemeyer, Knoesche, Moran, Friston, in revision

37. LG ( x 1 ) LG ( x 2 ) RVF stim. LVF stim. FG ( x 4 ) FG ( x 3 ) LD|LVF LD BVF stim. LD|RVF  DCM structure LG left LG right FG right FG left  anatomical connectivity probabilistic tractography  connection- specific priors for coupling parameters

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

42. 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)

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 standardsdeviants A1 A2 Tombaugh et al. 2005, J.Neurosci. Example of single-neuron SFA

44. 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. (2003) NeuroImage mean firing rate  mean postsynaptic potential (PSP) mean PSP  mean firing rate

45. 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 David et al. 2006, NeuroImage Kiebel et al. 2007, NeuroImage Moran et al. 2009, NeuroImage

46. Electromagnetic forward model for M/EEG Depolarisation of pyramidal cells Forward model: lead field & gain matrix 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 20 30 Frequency (Hz) 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 chronic reduction in extracellular glutamate levels upregulation of AMPA receptors sensitisation of postsynaptic mechanisms  EPSPs  amplitude of synaptic kernels (  H e )  activation of voltage-sensitive Ca 2+ channels →  intracellular Ca 2+ →  Ca-dependent K + currents →  IAHP  SFA (  2 ) Van den Pool et al. 1996, Neuroscience Sanchez-Vives et al. 2000, J. Neurosci.

51. Extrinsic forward connections 4  1  2  u 5  Excitatory spiny cells in granular layers Excitatory pyramidal cells in infragranular layers Extrinsic forward connections 4  3  u 5  Excitatory spiny cells in granular layers Inhibitory cells in supragranular layers [161, 210] [29,37] [195, 233] (0.4) (0.37)(0. 13) [3.8 6.3] [0.76,1.34] (0.0003) (0.04) Control group estimates in blue, isolated animals in red, p values in parentheses. sensitization of post- synaptic mechanisms Increased neuronal adaption: decreased firing rate Moran et al. 2008, NeuroImage

52. Take-home messages Bayesian model selection (BMS): generic approach to selecting an optimal model from an arbitrarily large number of competing models random effects BMS for group studies: posterior model probabilities and exceedance probabilities nonlinear DCM: enables one to investigate synaptic gating processes via activity-dependent changes in connection strengths DCM & tractography: probabilities of anatomical connections can be used to inform the prior variance of DCM coupling parameters DCMs for electrophysiology: based on neurophysiologically fairly detailed neural mass models

53. Thank you