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

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) ]

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) ]

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) 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)? Random Field Theory 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

Taylor & Henson (2008) Biomag 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... 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}

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

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) ]

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)

(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

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, (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 = … ….

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

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)

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:)

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, QsQs Q s,Q v 500 simulations Q s,Q i Q s,Q i,Q v 500 simulations

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)

The “model log-evidence” is bounded by the free energy: Model Evidence Friston Et Al (2007) Neuroimage, 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”:

Model Comparison (Bayes Factors) Mattout Et Al (2006) Neuroimage, Log- Evidence Bayes Factor QsQsQsQs (1/9899) (1/9899) Q s,Q v Q s,Q v,Q i (Q s,Q i ) QsQs QvQv QiQi Multiple constraints: Smooth sources (Q s ), plus valid (Q v ) or invalid (Q i ) focal prior

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)

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 ~

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

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)

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,

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

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,

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)

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 …

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 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)

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 (submitted-a) Separate Error Covariance components for each of i=1..M modalities (C i (e) ): Data and leadfields scaled (with m i spatial modes):

Fusion of MEG/EEG Henson Et Al (submitted-a) Magnetometers (MEG)Gradiometers (MEG)Electrodes (EEG) + Fused…

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) ]

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

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

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

A “Canonical” Cortical Mesh Henson Et Al (submitted-b) 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

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

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 ms with >95% probability

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

fMRI spatial priors Flandin Et Al (in prep) 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

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

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:

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

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

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)

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

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

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

y1y1 y2y2 A C B 2. Observer Model Measurements assumed to reflect currents in (large) pyramidal cells ( x 0 ) Kiebel et al. (2006) NeuroImage One option is a small number of “equivalent current dipoles” (ECDs) Fix their locations, but allow orientations to be estimated as 3 parameters (  )

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

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

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

The End (Really)

If want to try… SPM5 Manual (…/spm5/man/manual.pdf) SPM5 Manual (…/spm5/man/manual.pdf)

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

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

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

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

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 ~

Can be shown that expected energy for one trial in a time-frequency window defined by W: Localising Power (eg induced) Similarly, can extend over trials as well as samples, such that given i=1..N trials, Y i : 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 Y = [Y 1 Y 2... Y N ] = concatenated data, n x tN

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

Henson Et Al (2008) Neuroimage 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) SPM of F-statistic for EEG condition effect across subjects (NB: MEG data requires sensor-level realignment, e.g SSS)