M/EEG Analysis in SPM Rik Henson (MRC CBU, Cambridge) Jeremie Mattout, Christophe Phillips, Stefan Kiebel, Olivier David, Vladimir Litvak, ... & Karl Friston (UCL, London)

Overview As in the spirit of this talk series about Statistical Parametric Mapping, this talk is only about localising experimental M/EEG effects in space (sensors or sources) and/or time... [plus a few words about testing effective connectivity] ...it does not review all the other types of analyses and information one can obtain from M/EEG... ...and nor does it cover M/EEG preprocessing (epoching, filtering, averaging, time-freq transformations, etc), since these procedures are fairly standard nowadays

Overview Random Field Theory for Space-Time images Empirical Bayesian approach to the Inverse Problem A Canonical Cortical mesh and Group Analyses [ Dynamic Causal Modelling (DCM) ]

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

Where is an effect in time-sensor space (3D)? F-statistic for ANOVA across EEG subjects (Henson et al, 2008, Neuroimage) MEG data first requires sensor-level realignment, using e.g, SSS... SPM{F} Taylor & Henson (2008) Biomag

1. Localising in Space/Freq/Time Extended to 2D cortical mesh surface RFT generally requires Gaussian smoothing, but exerts exact FWE control for sufficient smoothing Nonparametric (permutation) methods of FWE control make fewer distributional assumptions (do not require smoothing), but do require exchangeability Pantazis Et Al (2005) NeuroImage, 25:383-394

Overview Random Field Theory for Space-Time images Empirical Bayesian approach to the Inverse Problem A Canonical Cortical mesh and Group Analyses [ Dynamic Causal Modelling (DCM) ]

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

Weighted Minimum Norm, Regularisation Linear system to be inverted: 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 Ce= error covariance over sensors Since n<p, need to regularise, eg “weighted minimum (L2) norm” (WMN): (Tikhonov) W = Weighting matrix W = I minimum norm W = DDT coherent W = diag(LTL)-1 depth-weighted Wp = (LpTCy-1Lp)-1 SAM W = … …. “L-curve” method ||Y – LJ||2 ||WJ||2 = regularisation (hyperparameter) Phillips Et Al (2002) Neuroimage, 17, 287–301

Equivalent Bayesian Formulation Equivalent “Parametric Empirical Bayes” formulation: 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 Posterior is product of likelihood and prior: W = Weighting matrix W = I minimum norm W = DDT coherent W = diag(LTL)-1 depth-weighted Wp = (LpTCy-1Lp)-1 SAM W = … …. Maximal A Posteriori (MAP) estimate is: (Contrasting with Tikhonov): Phillips Et Al (2005) Neuroimage, 997-1011

Covariance Constraints (Priors) How parameterise C(e) and C(j)? “IID” constraint on sensors (Q(e)=I(n)) # sensors Q = (co)variance components (Priors) λ = estimated hyperparameters Sparse priors on sources (Q1(j), Q2(j), …) … # sources # sources “IID” constraint on sources (Q(j)=I(p))

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

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

Multiple Constraints (Priors) Multiple constraints: Smooth sources (Qs), plus valid (Qv) or invalid (Qi) focal prior Qs Qs Qs,Qv 500 simulations Qs,Qi Qs,Qi,Qv 500 simulations Qv Qi Mattout Et Al (2006) Neuroimage, 753-767

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

Model Evidence Friston Et Al (2007) Neuroimage, 34, 220-34 A (generative) model, M, is defined by the set of {Q(e), Q(j), L}: The “model log-evidence” is bounded by the free energy: Friston Et Al (2007) Neuroimage, 34, 220-34 (F can also be viewed the difference of an “accuracy” term and a “complexity” term): Two models can be compared using the “Bayes factor”: Also useful when comparing different forward models, ie L’s, Henson et al (2009b), NI

Model Comparison (Bayes Factors) Multiple constraints: Smooth sources (Qs), plus valid (Qv) or invalid (Qi) focal prior Log-Evidence Bayes Factor Qs 205.2 7047 1.8 (1/9899) Qs,Qv 214.1 Qs,Qv,Qi 214.7 (Qs,Qi) 204.9 Qs Qv Qi Mattout Et Al (2006) Neuroimage, 753-767

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

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

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

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

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

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: 1) impose positivity constraint on hyperparameters: 2) impose (sparse) hyperpriors on the (log-normal) hyperparameters: Uninformative priors are then “turned-off” as (“ARD”) Complexity (…where η and Σλ are the posterior mean and covariance of hyperparameters)

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

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

Multiple Sparse Priors (MSP) 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 … … Friston Et Al (2008) Neuroimage

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!

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

Fusion of MEG/EEG Henson Et Al (2009a), Neuroimage Separate Error Covariance components for each of i=1..M modalities (Ci(e)): Data and leadfields scaled (with mi spatial modes): Henson Et Al (2009a), Neuroimage

Fusion of MEG/EEG + Fused Henson Et Al (2009a), Neuroimage Magnetometers (MEG) Gradiometers (MEG) Electrodes (EEG) + Fused Henson Et Al (2009a), Neuroimage

Overview Random Field Theory for Space-Time images Empirical Bayesian approach to the Inverse Problem A Canonical Cortical mesh and Group Analyses [ Dynamic Causal Modelling (DCM) ]

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

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 Spatial Normalisation Normalised MRI Ashburner & Friston (2005) Neuroimage Warps… Template MRI (in “MNI” space)

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

A “Canonical” Cortical Mesh But warps from cortex not appropriate to skull/scalp, so use individually (and easily) defined skull/scalp meshes… Free Energy/104 N=9 Canonical Cortex Individual Skull Individual Scalp CanInd Statistical tests of model evidence over N=9 MEG subjects show: MSP > MMN BEMs > Spheres (for CanInd) (7000 > 3000 dipoles) (Normal > Free for MSP) Henson Et Al (2009b), Neuroimage

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

Group Analyses in 3D Taylor & Henson (2008), Biomag 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… N=19, MNI space, Pseudowords>Words 300-400ms with >95% probability Smoothed, Interpolated J Taylor & Henson (2008), Biomag

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

fMRI spatial priors Human Brain Mapping Henson Et Al (2010) 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 … Human Brain Mapping Henson Et Al (2010) … Prior covariance components

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

Group-based source priors Concatenate data and leadfields over i=1..s subjects… …projecting data and leadfields to a reference subject via A: Common source-level priors: Subject-specific sensor-level priors: Litvak & Friston (2008), Neuroimage; Henson et al (2011), Frontiers

Group-based source priors Litvak & Friston (2008), Neuroimage; Henson et al (2011), Frontiers

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

Summary 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: effectively automatically “regularises” in principled fashion allows for multiple constraints (priors), valid & invalid allows model comparison, or automatic relevance detection… …to the extent that multiple (100’s) of sparse priors possible also offers a framework for MEG+EEG fusion SPM can also inverse-normalise a template cortical mesh that: obviates manual cortex meshing allows use of fMRI priors in MNI space allows using group constraints on individual inversions

Overview Random Field Theory for Space-Time images Empirical Bayesian approach to the Inverse Problem A Canonical Cortical mesh and Group Analyses Dynamic Causal Modelling (DCM)

Functional vs Effective Connectivity Correlations: A B C 1 0.49 1 0.30 0.12 1 B and C correlated because of common input from A, eg: A = time-series B = 0.5 * A + e1 C = 0.3 * A + e2 B 0.49 0.31 Effective connectivity Functional connectivity A -0.02 2=0.5, ns. C Effective connectivity is model-dependent… Real interest is changes in effective connectivity induced by (experimental) inputs

Basic DCM Approach 1. Neural dynamics 2. Observer function Design experimental inputs 1. Neural dynamics Define likelihood model 2. Observer function Specify priors Inference on models Inference on parameters Invert model Make inferences

inhibitory interneurons 1. Neural Dynamics Extrinsic forward connections Extrinsic backward connections Intrinsic connections Extrinsic lateral connections Jansen & Rit (1995) spiny stellate cells inhibitory interneurons pyramidal cells David et al. (2006) NeuroImage

B A C 2. Observer Model y1 y2 Kiebel et al. (2006) NeuroImage Measurements assumed to reflect currents in (large) pyramidal cells (x0) B A One option is a small number of “equivalent current dipoles” (ECDs) C Fix their locations, but allow orientations to be estimated as 3 parameters (q) y1 y2 Kiebel et al. (2006) NeuroImage

EEG example: MisMatch Negativity (MMN) standards deviants Standards (1kHz) Deviants (2kHz) HEOG VEOG “MMN” = deviants – standards A1 STG IFG Doeller et al., 2003 =>Seed 5 ECDs Garrido et al. (2007) NeuroImage

EEG example: MisMatch Negativity (MMN) log evidence subjects Forward (F) Backward (B) Forward and Backward (FB) IFG IFG IFG - STG STG STG STG STG STG A1 A1 A1 A1 A1 A1 input input input Forward Forward Forward Backward Backward Backward Lateral Lateral Lateral Garrido et al. (2007) NeuroImage

EEG example: MisMatch Negativity (MMN) MisMatch reflects changes in forward and backward connections: Invalid top-down predictions fail to suppress bottom-up prediction error? Group-based posterior densities of connections in FB model: IFG 0.93 (55%) 1.41 (99%) STG STG frontal cortex auditory cortex prediction error prediction 2.41 (100%) 4.50 (100%) 1.74 (96%) 5.40 (100%) A1 A1 Forward Backward Lateral input Garrido et al. (2007) NeuroImage

The End (Really)

If want to try… http://www.fil.ion.ucl.ac.uk/spm SPM5 Manual (…/spm5/man/manual.pdf) http://www.fil.ion.ucl.ac.uk/spm/data/mmfaces.html

Future Directions Variational Bayes (VB), relaxing Gaussian assumptions e.g, VB for ECD (Kiebel et al, Neuroimage, 2007) Dynamic Causal Modelling (DCM), for ECD or MSP Multi-level heirarchical models (e.g, across subjects) Nonstationary hyperparameters Proper Data Fusion (single forward model from neural activity to both M/EEG and fMRI)

“Optimal” Regularisation Single hyper-parameter for a coherent (smoothness) constraint on sources (cf. LORETA) L-curve EM 500 simulations C(e) C(j) Mattout Et Al (2006) Neuroimage, 753-767

Where is an effect in time-frequency (2D)? Kilner Et Al (2005) Neuroscience Letters 374, 174–178

Expectation Maximisation How estimate J and λ simultaneously…? Maximise the “free energy” (F): Y = Data, n sensors x t time-samples λ = hyperparameter(s) q(J) = any distribution over J using EM algorithm, where E-step: Friston Et Al (2006) Neuroimage, 20, 220-234 and M-step is: For Gaussian distributions, equivalent to ReML objective function: In practice, this gives ReML estimates of λ, which can then be used to give MAP estimates of J (via Cj and Ce):

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

Localising Power (eg induced) Similarly, can extend over trials as well as samples, such that given i=1..N trials, Yi: Y = [Y1 Y2 ... YN] = concatenated data, n x tN Can be shown that expected energy for one trial in a time-frequency window defined by W: W = time-frequency contrast matrix and total energy (induced and evoked) across i=1..N trials becomes: Friston Et Al (2006) Human Brain Mapping, 27:722–735

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

Where is an effect in time-sensor space (3D)? SPM of F-statistic for EEG condition effect across subjects (NB: MEG data requires sensor-level realignment, e.g SSS) Henson Et Al (2008) Neuroimage