1. Sensor & Source Space Statistics Sensor & Source Space Statistics Rik Henson (MRC CBU, Cambridge) With thanks to Jason Taylor, Vladimir Litvak, Guillaume Flandin, James Kilner & Karl Friston Sensor & Source Space Statistics Sensor & Source Space Statistics Rik Henson (MRC CBU, Cambridge) With thanks to Jason Taylor, Vladimir Litvak, Guillaume Flandin, James Kilner & Karl Friston

2. OverviewOverview A mass-univariate statistical approach to localising effects in space/time/frequency (using replications across trials/subjects)…

3. OverviewOverview Sensor Space:Sensor Space: 1.Random Field Theory (RFT) 2.2D Time-Freq (within-subject) 3.3D Scalp-Time (within-subject) 4.3D Scalp-Time (between-subjects) Source Space:Source Space: 1.3D contrast images 2.SPM vs SnPM vs PPM (vs FDR) 3.Other issues & Future directions 4.Multivariate Sensor Space:Sensor Space: 1.Random Field Theory (RFT) 2.2D Time-Freq (within-subject) 3.3D Scalp-Time (within-subject) 4.3D Scalp-Time (between-subjects) Source Space:Source Space: 1.3D contrast images 2.SPM vs SnPM vs PPM (vs FDR) 3.Other issues & Future directions 4.Multivariate

4. 1. Random Field Theory (RFT) RFT is a method for correcting for multiple statistical comparisons with N-dimensional spaces (for parametric statistics, eg Z-, T-, F- statistics)… 1. When is there an effect in time, eg GFP (1D)? 2. Where is there an effect in time-frequency space (2D)? 3. Where is there an effect in time-sensor space (3D)? 4. Where is there an effect in time-source space (4D)? RFT is a method for correcting for multiple statistical comparisons with N-dimensional spaces (for parametric statistics, eg Z-, T-, F- statistics)… 1. When is there an effect in time, eg GFP (1D)? 2. Where is there an effect in time-frequency space (2D)? 3. Where is there an effect in time-sensor space (3D)? 4. Where is there an effect in time-source space (4D)? Worsley Et Al (1996). Human Brain Mapping, 4:58-73

5. “Multimodal” Dataset in SPM8 manual (and website)“Multimodal” Dataset in SPM8 manual (and website) Single subject:Single subject: 128 EEG 275 MEG 3T fMRI (with nulls) 1mm 3 sMRI Two sessionsTwo sessions ~160 face trials and ~160 scrambled trials per session~160 face trials and ~160 scrambled trials per session (N=12 subjects soon, as in Henson et al, 2009 a, b, c)(N=12 subjects soon, as in Henson et al, 2009 a, b, c) “Multimodal” Dataset in SPM8 manual (and website)“Multimodal” Dataset in SPM8 manual (and website) Single subject:Single subject: 128 EEG 275 MEG 3T fMRI (with nulls) 1mm 3 sMRI Two sessionsTwo sessions ~160 face trials and ~160 scrambled trials per session~160 face trials and ~160 scrambled trials per session (N=12 subjects soon, as in Henson et al, 2009 a, b, c)(N=12 subjects soon, as in Henson et al, 2009 a, b, c) 2. Single-subject Example Chapter 33, SPM8 Manual

6. Faces Scrambled Faces > Scrambled 2. Where is an effect in time-frequency (2D)? Single MEG channelSingle MEG channel Mean over trials of Morlet Wavelet projection (i.e, induced + evoked)Mean over trials of Morlet Wavelet projection (i.e, induced + evoked) Write as t x f x 1 image per trialWrite as t x f x 1 image per trial SPM, correct on extent / heightSPM, correct on extent / height Single MEG channelSingle MEG channel Mean over trials of Morlet Wavelet projection (i.e, induced + evoked)Mean over trials of Morlet Wavelet projection (i.e, induced + evoked) Write as t x f x 1 image per trialWrite as t x f x 1 image per trial SPM, correct on extent / heightSPM, correct on extent / height Chapter 33, SPM8 Manual Kilner Et Al (2005) Neurosci. Letters

7. 3. Where is an effect in scalp-time space (3D)? 2D sensor positions specified or projected from 3D digitised positions2D sensor positions specified or projected from 3D digitised positions Each sample projected to a 32x32 grid using linear interpolationEach sample projected to a 32x32 grid using linear interpolation Samples tiled to created a 3D volumeSamples tiled to created a 3D volume 2D sensor positions specified or projected from 3D digitised positions2D sensor positions specified or projected from 3D digitised positions Each sample projected to a 32x32 grid using linear interpolationEach sample projected to a 32x32 grid using linear interpolation Samples tiled to created a 3D volumeSamples tiled to created a 3D volume Chapter 33, SPM8 Manual F-test of means of ~150 EEG trials of each type (since polarity not of interest)F-test of means of ~150 EEG trials of each type (since polarity not of interest) (Note that clusters depend on reference)(Note that clusters depend on reference) F-test of means of ~150 EEG trials of each type (since polarity not of interest)F-test of means of ~150 EEG trials of each type (since polarity not of interest) (Note that clusters depend on reference)(Note that clusters depend on reference) y x t

8. More sophisticated 1 st -level design matrices, e.g, to remove trial-by-trial confounds within each subject, and create mean adjusted ERP for 2 nd –level analysis across subjects Each trial Each trial-type (6) Henson Et Al (2008) Neuroimage 3. Where is an effect in scalp-time space (3D)? beta_00* images reflect mean (adjusted) 3D scalp-time volume for each condition Within-subject (1 st -level) Within-subject Across-subjects (2 nd -level) Across-subjects Confounds (4)

9. Taylor & Henson (2008) Biomag 4. Where is an effect in scalp-time space (3D)? Mean ERP/ERF images can also be tested between-subjects. Note however for MEG, some alignment of sensors may be necessary (e.g, SSS, Taulu et al, 2005) Without transformation to Device Space Stats over 18 subjects on RMS of 102 planar gradiometers With transformation to Device Space

10. OverviewOverview Sensor Space:Sensor Space: 1.Random Field Theory (RFT) 2.2D Time-Freq (within-subject) 3.3D Scalp-Time (within-subject) 4.3D Scalp-Time (between-subjects) Source Space:Source Space: 1.3D contrast images 2.SPM vs SnPM vs PPM (vs FDR) 3.Other issues & Future directions 4.Multivariate Sensor Space:Sensor Space: 1.Random Field Theory (RFT) 2.2D Time-Freq (within-subject) 3.3D Scalp-Time (within-subject) 4.3D Scalp-Time (between-subjects) Source Space:Source Space: 1.3D contrast images 2.SPM vs SnPM vs PPM (vs FDR) 3.Other issues & Future directions 4.Multivariate

11. Henson Et Al (2007) Neuroimage 1. Estimate evoked/induced energy (RMS) at each dipole for a certain time-frequency contrast (e.g, from sensor stats, e.g 0-20Hz, 150- 200ms), for each condition (e.g, faces & scrambled) and subject 2. Smooth along the 2D surface 3. Write these data into a 3D image in MNI space (if canonical / template mesh used) 4. Smooth by 8-12mm in 3D (to allow for normalisation errors) 1. Estimate evoked/induced energy (RMS) at each dipole for a certain time-frequency contrast (e.g, from sensor stats, e.g 0-20Hz, 150- 200ms), for each condition (e.g, faces & scrambled) and subject 2. Smooth along the 2D surface 3. Write these data into a 3D image in MNI space (if canonical / template mesh used) 4. Smooth by 8-12mm in 3D (to allow for normalisation errors) Where is an effect in source space (3D)? Source analysis of N=12 subjects; 102 magnetometers; MSP; evoked; RMS; smooth 12mm Note sparseness of MSP inversions…. Analysis Mask

12. Where is an effect in source space (3D)? Source analysis of N=12 subjects; 102 magnetometers; MSP; evoked; RMS; smooth 12mm 1. Classical SPM approach Caveats: Inverse operator induces long-range error correlations (e.g, similar gain vectors from non-adjacent dipoles with similar orientation), making RFT conservativeInverse operator induces long-range error correlations (e.g, similar gain vectors from non-adjacent dipoles with similar orientation), making RFT conservative Need a cortical mask, else activity “smoothed” outsideNeed a cortical mask, else activity “smoothed” outside Distributions over subjects may not be Gaussian…Distributions over subjects may not be Gaussian… 1. Classical SPM approach Caveats: Inverse operator induces long-range error correlations (e.g, similar gain vectors from non-adjacent dipoles with similar orientation), making RFT conservativeInverse operator induces long-range error correlations (e.g, similar gain vectors from non-adjacent dipoles with similar orientation), making RFT conservative Need a cortical mask, else activity “smoothed” outsideNeed a cortical mask, else activity “smoothed” outside Distributions over subjects may not be Gaussian…Distributions over subjects may not be Gaussian… SPM p<.05 FWE

13. Where is an effect in source space (3D)? Source analysis of N=12 subjects; 102 magnetometers; MSP; evoked; RMS; smooth 12mm 2. Nonparametric, SnPM Robust to non-Gaussian distributionsRobust to non-Gaussian distributions Less conservative than RFT when dfs<20Less conservative than RFT when dfs<20Caveats: No idea of effect size (e.g, for future experiments)No idea of effect size (e.g, for future experiments) Exchangeability difficult for more complex designsExchangeability difficult for more complex designs 2. Nonparametric, SnPM Robust to non-Gaussian distributionsRobust to non-Gaussian distributions Less conservative than RFT when dfs<20Less conservative than RFT when dfs<20Caveats: No idea of effect size (e.g, for future experiments)No idea of effect size (e.g, for future experiments) Exchangeability difficult for more complex designsExchangeability difficult for more complex designs SnPM p<.05 FWE

14. Where is an effect in source space (3D)? Source analysis of N=12 subjects; 102 magnetometers; MSP; evoked; RMS; smooth 12mm 3. PPMs No need for RFT (no MCP!)No need for RFT (no MCP!) Threshold on posterior probability of an effect (greater than some size)Threshold on posterior probability of an effect (greater than some size) Can show effect size after thresholding…Can show effect size after thresholding…Caveats: Assume normal distributions (e.g, of mean over voxels); sometimes not met for MSP (though usually fine for IID)Assume normal distributions (e.g, of mean over voxels); sometimes not met for MSP (though usually fine for IID) 3. PPMs No need for RFT (no MCP!)No need for RFT (no MCP!) Threshold on posterior probability of an effect (greater than some size)Threshold on posterior probability of an effect (greater than some size) Can show effect size after thresholding…Can show effect size after thresholding…Caveats: Assume normal distributions (e.g, of mean over voxels); sometimes not met for MSP (though usually fine for IID)Assume normal distributions (e.g, of mean over voxels); sometimes not met for MSP (though usually fine for IID) PPM p>.95 (γ>1SD) Grayscale= Effect Size

15. Where is an effect in source space (3D)? Source analysis of N=12 subjects; 102 magnetometers; MSP; evoked; RMS; smooth 12mm 4. FDR? Topological issues…?Topological issues…? 4. FDR? Topological issues…?Topological issues…? SPM p<.05 FWE

16. Some further thoughts: Since data live in sensor space, why not perform stats there, and just report some mean localisation (e.g, across subjects)?Since data live in sensor space, why not perform stats there, and just report some mean localisation (e.g, across subjects)? True but: What if sensor data not aligned (e.g, MEG)? (Taylor & Henson, 2008)? What if want to fuse modalities (e.g, MEG+EEG) (Henson et al, 2009)? What if want to use source priors (e.g, fMRI) (Henson et al, submitted)? Contrast localisations of conditions, or localise contrast of conditions?Contrast localisations of conditions, or localise contrast of conditions? “DoL” or “LoD” (Henson et al, 2007, Neuroimage) LoD has higher SNR (though difference only lives in trial-average, i.e evoked)? But how test localised energy of a difference (versus baseline?) Construct inverse operator (MAP) from a difference, but then apply that operator to individual conditions (Taylor & Henson, in prep) Some further thoughts: Since data live in sensor space, why not perform stats there, and just report some mean localisation (e.g, across subjects)?Since data live in sensor space, why not perform stats there, and just report some mean localisation (e.g, across subjects)? True but: What if sensor data not aligned (e.g, MEG)? (Taylor & Henson, 2008)? What if want to fuse modalities (e.g, MEG+EEG) (Henson et al, 2009)? What if want to use source priors (e.g, fMRI) (Henson et al, submitted)? Contrast localisations of conditions, or localise contrast of conditions?Contrast localisations of conditions, or localise contrast of conditions? “DoL” or “LoD” (Henson et al, 2007, Neuroimage) LoD has higher SNR (though difference only lives in trial-average, i.e evoked)? But how test localised energy of a difference (versus baseline?) Construct inverse operator (MAP) from a difference, but then apply that operator to individual conditions (Taylor & Henson, in prep) Where is an effect in source space (3D)?

17. Future Directions Extend RFT to 2D cortical surfaces (“surfstat”)Extend RFT to 2D cortical surfaces (“surfstat”) Go multivariate…Go multivariate… – To localise (linear combinations) of spatial (sensor or source) effects in time, using Hotelling-T 2 and RFT – To detect spatiotemporal patterns in 3D images (MLM / PLS) Extend RFT to 2D cortical surfaces (“surfstat”)Extend RFT to 2D cortical surfaces (“surfstat”) Go multivariate…Go multivariate… – To localise (linear combinations) of spatial (sensor or source) effects in time, using Hotelling-T 2 and RFT – To detect spatiotemporal patterns in 3D images (MLM / PLS) Pantazis Et Al (2005) NeuroImage Carbonell Et Al (2004) NeuroImage Duzel Et Al (2003) Neuroimage Kherif Et Al (2004) NeuroImage Duzel Et Al (2003) Neuroimage Kherif Et Al (2004) NeuroImage

18. Multivariate Model (MM) toolbox Famous Novel Scrambled “M170”? Multivariate Linear Model (MLM) across subjects on MEG Scalp-Time volumes (now with 3 conditions) X Famous Novel Scrambled Sensitive (and suggestive of spatiotemporal dynamic networks), but “imprecise”

19. The End

21. 2. Where is an effect in time-frequency (2D)? Kilner Et Al (2005) Neurosci. Letters

22. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

23. (Tikhonov) Linear system to be inverted: Since n<p, need to regularise, eg “weighted minimum (L2) norm” (WMN): W = Weighting matrix W = I minimum norm W = DD T coherent W = diag(L T L) -1 depth-weighted W p = (L p T C y -1 L p ) -1 SAM W = … …. Y = Data, n sensors x t=1 time-samples J = Sources, p sources x t time-samples L = Forward model, n sensors x p sources E = Multivariate Gaussian noise, n x t C e = error covariance over sensors “L-curve” method ||Y – LJ|| 2 ||WJ|| 2 = regularisation (hyperparameter) Weighted Minimum Norm, Regularisation Phillips Et Al (2002) Neuroimage, 17, 287–301

24. Equivalent “Parametric Empirical Bayes” formulation: Maximal A Posteriori (MAP) estimate is: Y = Data, n sensors x t=1 time-samples J = Sources, p sources x t time-samples L = Forward model, n sensors x p sources C (e) = covariance over sensors C (j) = covariance over sources Equivalent Bayesian Formulation Phillips Et Al (2005) Neuroimage, 997-1011 (Contrasting with Tikhonov): Posterior is product of likelihood and prior: W = Weighting matrix W = I minimum norm W = DD T coherent W = diag(L T L) -1 depth-weighted W p = (L p T C y -1 L p ) -1 SAM W = … ….

25. Covariance Constraints (Priors) How parameterise C (e) and C (j) ? Q = (co)variance components (Priors) λ = estimated hyperparameters “IID” constraint on sensors (Q (e) =I(n)) # sensors # sources “IID” constraint on sources (Q (j) =I(p))Sparse priors on sources (Q 1 (j), Q 2 (j), …) … # sources

26. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

27. How estimate λ? …. Use EM algorithm: Expectation-Maximisation (EM) Once estimated hyperparameters (iterated M-steps), get MAP for parameters (single E-step): (Note estimation in nxn sensor space) Phillips et al (2005) Neuroimage …to maximise the (negative) “free energy” (F): (Can also estimate conditional covariance of parameters, allowing inference:)

28. Multiple Constraints (Priors) Multiple constraints: Smooth sources (Q s ), plus valid (Q v ) or invalid (Q i ) focal prior QsQs QvQv QiQi Mattout Et Al (2006) Neuroimage, 753-767 QsQs Q s,Q v 500 simulations Q s,Q i Q s,Q i,Q v 500 simulations

29. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

30. The “model log-evidence” is bounded by the free energy: Model Evidence Friston Et Al (2007) Neuroimage, 34, 220-34 Also useful when comparing different forward models, ie L’s, Henson et al (submitted-b) A (generative) model, M, is defined by the set of {Q (e), Q (j), L}: (F can also be viewed the difference of an “accuracy” term and a “complexity” term): Two models can be compared using the “Bayes factor”:

31. Model Comparison (Bayes Factors) Mattout Et Al (2006) Neuroimage, 753-767 Log- Evidence Bayes Factor QsQsQsQs205.2 7047 7047 1.8 1.8 (1/9899) (1/9899) Q s,Q v 214.1 Q s,Q v,Q i 214.7 (Q s,Q i ) 204.9 QsQs QvQv QiQi Multiple constraints: Smooth sources (Q s ), plus valid (Q v ) or invalid (Q i ) focal prior

32. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

33. To handle temporally-extended solutions, first assume temporal-spatial factorisation: V typically Gaussian autocorrelations… Temporal Correlations Friston Et Al (2006) Human Brain Mapping, 27:722–735 In general, temporal correlation of signal (sources) and noise (sensors) will differ, but can project onto a temporal subspace (via S) such that: S typically an SVD into N r temporal modes… Then turns out that EM can simply operate on prewhitened data (covariance), where Y size n x t: Y = vectorised data, nt x 1 C (e) = spatial error covariance over sensors V (e) = temporal error covariance over sensors C (j) = spatial error covariance over sources V (j) = temporal error covariance over sources ~

34. Localising Power (eg induced) Friston Et Al (2006) Human Brain Mapping, 27:722–735

35. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

36. Automatic Relevance Detection (ARD) When have many constraints (Q’s), pairwise model comparison becomes arduous Moreover, when Q’s are correlated, F-maximisation can be difficult (eg local maxima), and hyperparameters can become negative (improper for covariances) Note: Even though Qs may be uncorrelated in source space, they can become correlated when projected through L to sensor space (where F is optimised) Prestim Baseline Anti-Averaging Smoothness Depth-Weighting Sensor-level Source-level Henson Et Al (2007) Neuroimage, 38, 422-38

37. Automatic Relevance Detection (ARD) When have many constraints (Q’s), pairwise model comparison becomes arduous Moreover, when Q’s are correlated, F-maximisation can be difficult (eg local maxima), and hyperparameters can become negative (improper for covariances) To overcome this, one can: 2) impose (sparse) hyperpriors on the (log-normal) hyperparameters: Uninformative priors are then “turned-off” as(“ARD”) 1) impose positivity constraint on hyperparameters: (…where η and Σ λ are the posterior mean and covariance of hyperparameters) Complexity

38. Automatic Relevance Detection (ARD) When have many constraints (Q’s), pairwise model comparison becomes arduous Moreover, when Q’s are correlated, F-maximisation can be difficult (eg local maxima), and hyperparameters can become negative (improper for covariances) Prestim Baseline Anti-Averaging Smoothness Depth-Weighting Sensor-level Source-level Henson Et Al (2007) Neuroimage, 38, 422-38

39. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

40. Multiple Sparse Priors (MSP) Friston Et Al (2008) Neuroimage So why not use ARD to select from a large number of sparse source priors….!? Q (2) 1 Q (2) N … Q (2) j Left patch … … Q (2) j+1 Right patch … Q (2) j+2 Bilateral patches …

41. Multiple Sparse Priors (MSP) So why not use ARD to select from a large number of sparse source priors….! Friston Et Al (2008) Neuroimage No depth bias!

42. 2. Parametric Empirical Bayes (PEB) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion) Weighted Minimum Norm & Bayesian equivalent Weighted Minimum Norm & Bayesian equivalent EM estimation of hyperparameters (regularisation) EM estimation of hyperparameters (regularisation) Model evidence and Model Comparison Model evidence and Model Comparison Spatiotemporal factorisation and Induced Power Spatiotemporal factorisation and Induced Power Automatic Relevance Detection (hyperpriors) Automatic Relevance Detection (hyperpriors) Multiple Sparse Priors Multiple Sparse Priors MEG and EEG fusion (simultaneous inversion) MEG and EEG fusion (simultaneous inversion)

43. Fusion of MEG/EEG Remember, EM returns conditional precisions (Σ) of sources (J), which can be used to compare separate vs fused inversions… Henson Et Al (2009b) Neuroimage Separate Error Covariance components for each of i=1..M modalities (C i (e) ): Data and leadfields scaled (with m i spatial modes):

44. Fusion of MEG/EEG Magnetometers (MEG)Gradiometers (MEG)Electrodes (EEG) + Fused… Henson Et Al (2009b) Neuroimage

45. OverviewOverview 1.Random Field Theory for Space-Time images 2.Empirical Bayesian approach to the Inverse Problem 3.A Canonical Cortical mesh and Group Analyses 4.[ Dynamic Causal Modelling (DCM) ] 1.Random Field Theory for Space-Time images 2.Empirical Bayesian approach to the Inverse Problem 3.A Canonical Cortical mesh and Group Analyses 4.[ Dynamic Causal Modelling (DCM) ]

46. 3. Canonical Mesh & Group Analyses A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions

47. A “Canonical” Cortical Mesh Given the difficulty in (automatically) creating accurate cortical meshes from MRIs, how about inverse-normalising a (quality) template mesh in MNI space? Original MRI Template MRI (in “MNI” space) Spatial Normalisation Normalised MRI Warps… Ashburner & Friston (2005) Neuroimage

48. A “Canonical” Cortical Mesh Mattout Et Al (2007) Comp. Intelligence & Neuroscience Apply inverse of warps from spatial normalisation of whole MRI to a template cortical mesh… Individual CanonicalTemplate Individual CanonicalTemplate “Canonical” N=1

49. A “Canonical” Cortical Mesh Henson Et Al (2009a) Neuroimage Statistical tests of model evidence over N=9 MEG subjects show: 1.MSP > MMN 2.BEMs > Spheres (for CanInd) 3.(7000 > 3000 dipoles) 4.(Normal > Free for MSP) Free Energy/10 4 But warps from cortex not appropriate to skull/scalp, so use individually (and easily) defined skull/scalp meshes… Canonical Cortex Individual Skull Individual Scalp CanInd N=9

50. 3. Canonical Mesh & Group Analyses A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions

51. Group Analyses in 3D Once have a 1-to-1 mapping from M/EEG source to MNI space, can create 3D normalised images (like fMRI) and use SPM machinery to perform group-level classical inference… Smoothed, Interpolated J Taylor & Henson (2008), Biomag N=19, MNI space, Pseudowords>Words 300-400ms with >95% probability

52. 3. Canonical Mesh & Group Analyses A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions

53. fMRI spatial priors Henson Et Al (submitted) Group fMRI results in MNI space can be used as spatial priors on individual source space......importantly each fMRI cluster is separate prior, so is “weighted” independently … Thresholding and connected component labelling Project onto cortical surface using Voronoï diagram … … Prior covariance components

54. 3. Canonical Mesh & Group Analyses A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions A “canonical” (Inverse-normalised) cortical mesh A “canonical” (Inverse-normalised) cortical mesh Group analyses in 3D Group analyses in 3D Use of fMRI spatial priors (in MNI space) Use of fMRI spatial priors (in MNI space) Group-based inversions Group-based inversions

55. Group-based source priors Litvak & Friston (2008), Neuroimage Concatenate data and leadfields over i=1..N subjects… …projecting data and leadfields to a reference subject (0): Common source-level priors: Subject-specific sensor-level priors:

56. Group-based source priors Taylor & Henson (in prep) N=19, MNI space, Pseudowords>Words, 300-400ms with >95% probability Individual Inversions Group Inversion

57. SummarySummary SPM also implements Random Field Theory for principled correction of multiple comparisons over space/time/freq SPM implements a variant of the L2-distributed norm that: 1.effectively automatically “regularises” in principled fashion 2.allows for multiple constraints (priors), valid & invalid 3.allows model comparison, or automatic relevance detection… 4.…to the extent that multiple (100’s) of sparse priors possible 5.also offers a framework for MEG+EEG fusion SPM can also inverse-normalise a template cortical mesh that: 1.obviates manual cortex meshing 2.allows use of fMRI priors in MNI space 3.allows using group constraints on individual inversions SPM also implements Random Field Theory for principled correction of multiple comparisons over space/time/freq SPM implements a variant of the L2-distributed norm that: 1.effectively automatically “regularises” in principled fashion 2.allows for multiple constraints (priors), valid & invalid 3.allows model comparison, or automatic relevance detection… 4.…to the extent that multiple (100’s) of sparse priors possible 5.also offers a framework for MEG+EEG fusion SPM can also inverse-normalise a template cortical mesh that: 1.obviates manual cortex meshing 2.allows use of fMRI priors in MNI space 3.allows using group constraints on individual inversions