M/EEG source analysis · M/EEG source analysis SPM for MEEG course –London –May 2016 Jérémie Mattout Lyon Neuroscience Research Center “Will it ever happen that mathematicians

M/EEG source analysis

SPM for MEEG course – London – May 2016

Jérémie MattoutLyon Neuroscience Research Center

“Will it ever happen that mathematicians will know enough about the physiology of the brain, andneurophysiologists enough of mathematical discovery, for efficient cooperation to be possible”

Jacques Hadamard (mathematician, 1865-1963)

Introduction

?

EEG

MEG

Jacques Hadamard (mathematician, 1865-1963)Well-posed inverse problem:

- a solution exists

- the solution is unique

- the solution is stable


IntroductionIll-posed inverse problemWhy a Bayesian approach?

Generative modelsSourcesFrom sources to sensorsSensors

Bayesian inferenceECD modelImaging modelsSetting priorsEmpirical BayesComparing modelsGroup inferenceEEG/MEG fusion

ExampleMMN studyGroup multimodal inference

reconstruction / inversion

Introduction

?

EEG

MEG

reconstruction / inversion

likelihood / predictive / forward

Bayesian inference enables:

- to incorporate priors on the solution

- to account for uncertainty through probabilitic distributions

- to yield a unique and optimal solution given a likelihood model andpriors over model parameters






Generative models

EEG

MEG

likelihood / predictive / forward

A particular generative model is fully defined by:

- A data likelihood density function

- A prior distribution over source parameters 𝜽

𝜽

𝒀

𝒑 𝒀 𝜽

𝒑 𝜽






Sources

• Observable from scalp:the synchronous and additive activities of numorous neighbouring neurons

Cortical macro-column Current dipole

𝜽 :- Dipole location (x, y, z)- Dipole orientation (Ox, Oy, Oz)- Dipole strength






• Equivalent Current Dipole (ECD)- Only a few activated sources- Each source corresponds to a fairly large brain area- Each source activity is modelled by one current dipole

e.g. early response to auditory stimulus

e.g. MRI-derived cortical mesh

• Distributed or imaging approach- The whole brain/cortex may be active- The source space is discretized using a grid over the whole brain (voxels)

or a cortical mesh (nodes)- Each voxel or node is the location of a dipolar source- Each dipole models the activity of a small brain region

Only a few parameters 𝜃 to be estimated(location, orientation and strength)

Many parameters 𝜃 to be estimated(strength only)






Sources

From sources to sensors

MEG

EEG

• Predicting the sensor data Y from known source parameters 𝜽:- requires solving the Maxwell’s equations in a quasi-static regime- amounts to solving a well-posed forward problem- involves approximations

?𝜽𝒀

James Clerk Maxwell(1831 - 1879)






• EEG is sensitive to bothradial and tangential sources

• MEG is barely sensitive to radial sources

• EEG is sensitive to conductivities • MEG is barely sensitive to conductivities







Simple Head Model Realistic Head Model

Concentric spheres Boundary element method (BEM)

Pros :

Cons :

Fast analytic solution Realistic geometry

Heads are not spherical Slow approximate numerical solutions

Scalp (skin-air boundary)

Outer Skull (bone-skin boundary)

Inner Skull (CSF-bone boundary)







• Automated extraction of individual meshes

Individual MRI

Estimate Spatial Transform

Template

Deformation

Template meshes in MNI space

Inverse normalziation

Canonical (individual) meshMattout et al., Comp. Int & Neuro, 2007







• Data features 𝒀 to be fitted/explained- Evoked response- Induced response- Steady-state response

• Accounting for noise 𝜺 in the data

𝒀 = 𝒈 𝜽 + 𝜺

𝒑 𝜺 = 𝑵(𝟎, 𝑪𝜺)

Gaussian noise

The proba. of a largenoise is small

𝒑 𝒀 𝜽 = 𝑵(𝒈 𝜽 , 𝑪𝜺)

Data likelihood






Sensors

Bayesian inference

MYP

MPMYPMYP

,,

EEG

MEG

Likelihood Prior

),|( MYp )|( Mp

Generative model M

𝜽𝒀

Posterior Evidence

)|( MYp),|( MYp

Bayesian inference






ECD model

j

r

Y

j r

• A Bayesian model for Equivalent Current Dipole (ECD) solutions- Enables to put priors on source parameters- Enables formal model comparison (e.g. on number of sources or initial conditions)

Kiebel et al., NeuroImage, 2008






• A Bayesian model for Equivalent Current Dipole (ECD) solutions- Enables to put priors on source parameters- Enables formal model comparison (e.g. on number of sources or initial conditions)

Kiebel et al., NeuroImage, 2008






ECD model

𝒀 = 𝑲. 𝜽 + 𝜺

Evoked EEG response: Nsens x Time

Forward operator or lead-field matrix: Nsens x Nsources

Unknown source dynamics: Nsources x Time

noise

• A Bayesian model for Distributed / Imaging solutions- Many dipoles with fixed location and orientation- Dipole strength ? -> linear model

𝒑 𝒀 𝜽 = 𝑵(𝐊. 𝜽, 𝑪𝜺)






Imaging models

• A Bayesian model for Distributed / Imaging solutions- Many dipoles with fixed location and orientation- Dipole strength ? -> linear model

The proba. of a largeintensity is small

𝒑 𝜽 = 𝑵(0, 𝑪𝜽)

Prior over dipole strength

𝜃






Setting priors

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

i.i.d or Minimum norm

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Single dipole

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Smoothness (like LORETA)

• Alternative priors correspond to alternative prior covariance matrices

𝒑 𝜽 = 𝑵(0, 𝑪𝜽)

Ndip x Ndip

• Typical priors






Setting priors

• Alternative priors correspond to alternative prior covariance matrices

𝒑 𝜽 = 𝑵(0, 𝑪𝜽)

Ndip x Ndip

• More advanced priors

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

fMRI based

246810121416

2

4

6

8

10

12

14

160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Data or Lead-field basede.g. (Beamformer or MSP)

Henson et al., Hum. Brain Map., 2010 Mattout et al., NeuroImage, 2005






Setting priors






𝑸𝟏 𝑸𝟐 𝑸𝟑

…

𝑸𝒏

𝑪𝜽 = 𝝀𝟏. 𝑸𝟏 + 𝝀𝟐. 𝑸𝟐 + 𝝀𝟑. 𝑸𝟑 +⋯+ 𝝀𝒏. 𝑸𝒏

Empirical Bayes

Model (Hyper)parameters0 100 200 300 400 500 600 700-8

-6

-4

-2

0

2

4

6

8

estimated response - condition 1

at 36, -18, -23 mm

time ms

PPM at 229 ms (100 percent confidence)

512 dipoles

Percent variance explained 99.15 (65.03)

log-evidence = 8157380.8

0 100 200 300 400 500 600 700-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


at 46, -31, 4 mm

time ms


512 dipoles



0 100 200 300 400 500 600 700-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


at 46, -31, 4 mm

time ms


512 dipoles



0 50 100 15010

2

104

106

108

Iteration

Free energy

Accuracy

Complexity

0 100 200 300 400 500 600 700-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


at 46, -31, 4 mm

time ms


512 dipoles



AccuracyFree EnergyCompexity

Inference (iterative) process

Philips et al., NeuroImage, 2005Mattout et al., NeuroImage, 2006Friston et al., NeuroImage, 2008Lopez et al., NeuroImage, 2014

• Multivariate Sparse Priors (MSP)

• Comparing priors using log-evidence (free energy)

5010

015

020

025

030

035

040

0-0

.2

-0.1

5

-0.1

-0.0

50

0.050.

1

0.15

estim

ated

res

pons

e -

cond

ition

1

at -

54, -

27, 1

0 m

m

time

ms

PP

M a

t 188

ms

(100

per

cent

con

fiden

ce)

512

dipo

les

Per

cent

var

ianc

e ex

plai

ned

11.2

1 (1

0.26

)

log-

evid

ence

= 5

3.1

5010

015

020

025

030

035

040

0-0

.25

-0.2

-0.1

5

-0.1

-0.0

50

0.050.

1

0.150.

2

0.25

estim

ated

res

pons

e -

cond

ition

1

at 5

0, -

26, 1

1 m

m

time

ms

PP

M a

t 188

ms

(100

per

cent

con

fiden

ce)

512

dipo

les

Per

cent

var

ianc

e ex

plai

ned

99.7

8 (9

1.33

)

log-

evid

ence

= 1

12.8

5010

015

020

025

030

035

040

0-0

.03

-0.0

2

-0.0

10

0.01

0.02

0.03

estim

ated

res

pons

e -

cond

ition

1

at 5

7, -

26, 9

mm

time

ms

PP

M a

t 188

ms

(100

per

cent

con

fiden

ce)

512

dipo

les

Per

cent

var

ianc

e ex

plai

ned

99.1

8 (9

0.78

)

log-

evid

ence

= 9

9.9

Predicted

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Explained variance

Predicted

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Predicted

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

246810121416

2

4

6

8

10

12

14

160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Predicted

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

5010

015

020

025

030

035

040

0-4-3-2-101234

x 10

-3es

timat

ed r

espo

nse

- co

nditio

n 1

at 6

5, -

21, 1

1 m

m

time

ms

PP

M a

t 188

ms

(100

per

cent

con

fiden

ce)

512

dipo

les

Per

cent

var

ianc

e ex

plai

ned

98.4

9 (9

0.16

)

log-

evid

ence

= 9

3.1

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Measured

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

11 % 96% 97% 98%

Alternative priors𝒀 : observed data

𝜽 : estimatedsource intensities

𝒀 : predicted data

0

50

Free-energy

𝑭 ≅ 𝒑 𝒀 𝑴M/EEG source analysis





Comparing models






• Any assumption (part of model M) can be formally tested on real datausing Bayesian model comparison

Spherical head model Realistic surfacic model (BEM)

Biophysics

Anatomy

Mesh resolution (High / Low)Dipole oreintation (fixed / free)

Sources

• No evidence in favor of individual vs. Inverse-norm mesh• Evidence in favor of BEM head model• Evidence in favor of high + fixed vs. low + free

Mattout et al., NeuroImage, 2007

Henson et al., NeuroImage, 2007Henson et al., NeuroImage, 2009

Comparing models






• MSP based source reconstruction for a single subject

Fixed

Variable

Data

...

Y

,η Ω

...𝑸𝟐𝑸𝟏

𝝀𝒊𝑪𝜽 𝑪𝜺𝝀𝒊𝜺

𝑸𝟏𝜺 𝑸𝟐

𝜺

𝜺

K

𝜽

Friston et al., NeuroImage, 2008

Group inference






• MSP based source reconstruction for multiple subjects

Fixed

Variable

Data

...

,η Ω

...𝑸𝟐𝑸𝟏

𝝀𝒊𝑪𝜽 𝑪𝟏,𝜺𝝀𝟏,𝒊𝜺

𝑸𝟏,𝟏𝜺 𝑸𝟏,𝟐

𝜺

𝜽𝟏

𝑲𝟏 𝑲𝟐

𝜺𝟏 𝜽𝟐 𝜺𝟐

𝒀𝟏 𝒀𝟐

...𝑸𝟐,𝟏𝜺 𝑸𝟐,𝟐

𝜺

𝑪𝟐,𝜺𝝀𝟐,𝒊𝜺

A two-step procedure:- Estimating the group prior variance- Estimating the individual source intensities

Litvak and Friston, NeuroImage, 2008

Group inference






EEG/MEG fusion• MSP based source reconstruction for multimodal data

Fixed

Variable

Data

...

,η Ω

𝑸𝟐𝑸𝟏

𝝀𝒊𝑪𝜽

𝑲𝑬𝑬𝑮 𝑲𝑴𝑬𝑮𝒀𝑬𝑬𝑮 𝒀𝑴𝑬𝑮

𝜽 𝜺𝑴𝑬𝑮

...𝑸𝑴𝑬𝑮,𝟏𝜺 𝑸𝑴𝑬𝑮,𝟐

𝜺

𝑪𝑴𝑬𝑮,𝜺𝝀𝑴𝑬𝑮,𝒊𝜺

𝜺𝑬𝑬𝑮

...𝑸𝑬𝑬𝑮,𝟏𝜺 𝑸𝑬𝑬𝑮,𝟐

𝜺

𝑪𝑬𝑬𝑮,𝜺𝝀𝑬𝑬𝑮,𝒊𝜺

Henson et al., NeuroImage, 2011

Acknowledgements

Gareth BarnesAnne CaclinJean DaunizeauGuillaume FlandinKarl FristonRik HensonStefan KiebelFrançoise LecaignardVladimir LitvakChristophe Philips

Documents

M/EEG source analysis · M/EEG source analysis SPM for MEEG course –London –May 2016 Jérémie Mattout Lyon Neuroscience Research Center “Will it ever happen that mathematicians