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Bayesian inference in EEG/MEG
Workshop on Bayesian approaches in neuroscience – Karolinska Institute – 2nd of Frebruary 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)
Previously…“What I cannot create, I cannot understand”
Richard Feynman (Nobel laureate in Physics, 1918-1988)
• Bayesian or Probabilistic modelling: accounts for knowledge/uncertainty in the process of interest
• Generative or phenomenological models
𝒙 = 𝒇 𝒙, 𝜽𝒆𝒗𝒐𝒍, 𝒖
𝒚 = 𝒈 𝒙, 𝜽𝒐𝒃𝒔 + 𝜺
Hidden states Unknown parameters
Observations(data features)
Inputs
Ex. Dynamic causal models (DCM)
evolution
observation Fixed
Variable
Data
Y
𝜺𝜽𝒐𝒃𝒔
𝒙
𝜽𝒆𝒗𝒐𝒍
f
g
graphicalrepresentation
Previously…“All models are wrong but some are useful”
George E. P. Box (statistician, 1919-2013)
• Bayesian or Probabilistic modelling: accounts for knowledge/uncertainty in the process of interest
• Generative or phenomenological models
• Inference on models
Model M : your scientific hypothesis put into predictive equations (functions f and g + prior assumptions)
• Usually approximated (FE, AIC, BIC)• Trades accuracy and complexity• Model comparison through Bayes Factors and model posteriors• Models can only be compared on the same data
MYP
MPMYPMYP
,,
Likelihoodpriors
Evidence
Previously…
• Bayesian or Probabilistic modelling: accounts for knowledge/uncertainty in the process of interest
• Generative or phenomenological models
• Inference on models
• Inference on model parameters
MYP
MPMYPMYP
,,
Most plausible model or model family
Posterior
• Inference at the group level
Outline
• 1 - EEG/MEG Source reconstruction
• 2 - Inferring learning mechanisms from EEG/MEG responses
• 3 - Online Bayesian inference and design optimization
1 – EEG/MEG source reconstructionAn ill-posed inverse problem
Inverse Problem
Y
Likelihood Prior
),|( MYp )|( Mp
Generative model M
Data
Parameters
No unique solution!
Additional informationis needed
Posterior Evidence
)|( MYp),|( MYp
Bayesian inference
1 – EEG/MEG source reconstructionTypical model families
• Biophysical model
Y: sensor data (incl. noise)
),|( MYp
Geometrical and physical propertiesof the head tissues
Static (forward) model family
• Source model
Equivalent current dipole (ECD) model family Distributed sources or Imaging model family
: dipole locations,orientations & intensity
)|( Mp : number of dipoles,initial location
: dipole intensities
)|( Mp : number of dipoles,fixed locations
1 – EEG/MEG source reconstructionDetailed example: the distributed source model
: dipole intensities
𝒀 = 𝑲. 𝜽 + 𝜺
Evoked response: Nsens x Time
Forward operator or lead-field matrix: Nsens x Nsources
Unknown source dynamics: Nsources x Time
Additive noise
Cortical mesh (3D surface)
• Likelihood
𝒑 𝒀 𝜽,𝑴 = 𝑵(𝑲. 𝜽, 𝑪𝜺)
𝒑 𝜺 𝑴 = 𝑵(𝟎, 𝑪𝜺)
The proba. of a largenoise is small
• Prior𝒑 𝜽 𝑴 = 𝑵(𝟎, 𝑪𝜽)
1 – EEG/MEG source reconstructionDetailed example: the distributed source model
: dipole intensities
𝒀 = 𝑲. 𝜽 + 𝜺
Evoked response: Nsens x Time
Forward operator or lead-field matrix: Nsens x Nsources
Unknown source dynamics: Nsources x Time
Additive noise
Cortical mesh (3D surface)
• Likelihood
𝒑 𝒀 𝜽,𝑴 = 𝑵(𝑲. 𝜽, 𝑪𝜺)
𝒑 𝜺 𝑴 = 𝑵(𝟎, 𝑪𝜺)
The proba. of a largeintensity is small
• Prior𝒑 𝜽 𝑴 = 𝑵(𝟎, 𝑪𝜽)
1 – EEG/MEG source reconstructionCommonly used priors
• Priors𝒑 𝜽 𝑴 = 𝑵(𝟎, 𝑪𝜽)
Nsources x Nsources
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)
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
Lead-field based(Beamformer or MSP)
Philips et al. 2005Mattout et al. 2005 & 2006Wipf and Nagarajan 2009
1 – EEG/MEG source reconstructionComparing priors
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
n ed
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 priors
𝒀 : observed data
𝜽 : estimatedsource intensities
𝒀 : predicted data
0
20
40
Log-evidence
1 – EEG/MEG source reconstructionEmpirical Bayes: multiple sparse priors (MSP)
𝑸𝟏 𝑸𝟐 𝑸𝟑
…
𝑸𝒏
𝑪𝜽 = 𝝀𝟏. 𝑸𝟏 + 𝝀𝟐. 𝑸𝟐 + 𝝀𝟑. 𝑸𝟑 +⋯+ 𝝀𝒏. 𝑸𝒏
Model (Hyper)parameters
0 100 200 300 400 500 600 700-8
-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
estimated response - condition 1
at 46, -31, 4 mm
time ms
PPM at 229 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.93 (65.54)
log-evidence = 8361406.1
0 100 200 300 400 500 600 700-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
estimated response - condition 1
at 46, -31, 4 mm
time ms
PPM at 229 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.87 (65.51)
log-evidence = 8388254.2
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
estimated response - condition 1
at 46, -31, 4 mm
time ms
PPM at 229 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.90 (65.53)
log-evidence = 8389771.6
AccuracyFree EnergyCompexity
Inference (iterative) process
Friston et al. 2008
1 – EEG/MEG source reconstructionModel comparison on other dimensions
Mattout et al. 2007
Spherical head model Realistic surfacic model (BEM)
Biophysics
Anatomy
Mesh resolution (High / Low)Dipole oreintation (fixed / free)
Sources
Henson et al. 2009
Henson et al. 2007Model comparison on real MEG data
• 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
1 – EEG/MEG source reconstructionGraphical representation of MSP
Fixed
Variable
Data
...
Y
,η Ω
...
Directed Acyclic Graph (DAG) representation
𝑸𝟐𝑸𝟏
𝝀𝒊𝑪𝜽 𝑪𝜺𝝀𝒊𝜺
𝑸𝟏𝜺 𝑸𝟐
𝜺
𝜺
K
𝜽
Friston et al. 2008
1 – EEG/MEG source reconstructionModel extension to group level inference
Fixed
Variable
Data
...
,η Ω
...
Directed Acyclic Graph (DAG) representation
𝑸𝟐𝑸𝟏
𝝀𝒊𝑪𝜽 𝑪𝟏,𝜺𝝀𝟏,𝒊𝜺
𝑸𝟏,𝟏𝜺 𝑸𝟏,𝟐
𝜺
𝜽𝟏
Litvak and Friston 2008
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
1 – EEG/MEG source reconstructionModel extension to EEG-MEG data fusion
Fixed
Variable
Data
...
,η Ω
Directed Acyclic Graph (DAG) representation
𝑸𝟐𝑸𝟏
𝝀𝒊𝑪𝜽
Henson et al. 2011
𝑲𝑬𝑬𝑮 𝑲𝑴𝑬𝑮𝒀𝑬𝑬𝑮 𝒀𝑴𝑬𝑮
𝜽 𝜺𝑴𝑬𝑮
...𝑸𝑴𝑬𝑮,𝟏𝜺 𝑸𝑴𝑬𝑮,𝟐
𝜺
𝑪𝑴𝑬𝑮,𝜺𝝀𝑴𝑬𝑮,𝒊𝜺
𝜺𝑬𝑬𝑮
...𝑸𝑬𝑬𝑮,𝟏𝜺 𝑸𝑬𝑬𝑮,𝟐
𝜺
𝑪𝑬𝑬𝑮,𝜺𝝀𝑬𝑬𝑮,𝒊𝜺
1 – EEG/MEG source reconstructionAnother model family
Dynamic causal model (DCM) family
• Source model
• Biophysical model
1
2
4
3
24
u
Golgi Nissl
internal granular
layer
internal pyramidal
layer
external pyramidal
layer
external granular
layer
EP
EI
IIModel parameters:- Source intrinsic parameters
(e.g. synaptic gain)- Extrinsic coupling- Source orientation
𝒙 = 𝒇 𝒙, 𝜽𝒆𝒗𝒐𝒍, 𝒖
𝒚 = 𝒈 𝒙, 𝜽𝒐𝒃𝒔 + 𝜺
1 – EEG/MEG source reconstructionAn MMN example
Characterizing a group response in terms of modulations of effective connectivity
……S S S D S S
Classical auditory oddball paradigm
S: standard toneD: (frequency) deviant tone
rIFG
rSTG
rA1
lSTG
lA1
/
Garrido et al. 2009
Boly et al. 2011
Ctrls: ControlsMCS: Minimally Conscious StateVS: Vegetative State
2 – Inferring learning mechanisms from EEG/MEG data
2 – Inferring learning mechanisms from EEG/MEG data
• In line with the Bayesian brain hypothesis and its neuronal instantiation (predictive coding), it has been suggested that evoked responses reflect prediction error
Friston 2005
• Deviance responses should be modulated by predictability
• Single trial modulations of evoked responses should reflect perceptual learning
Neurophysiological correlates of implicit perceptual learning
2 – Inferring learning mechanisms from EEG/MEG data
Lecaignard et al. 2015
Effect of predictability
US
PS
US - PS
USPS
US: Unpredictable sequencePS: Predictable sequence
• Deviance responses should be modulated by predictabilityNeurophysiological correlates of implicit perceptual learning
2 – Inferring learning mechanisms from EEG/MEG data
• Single trial modulations of evoked responses should reflect perceptual learning
M1
M2
M3
M4
M5
Bayesian learnerwith different forgetting
parameter value
M6
M7
M8
Change detection
models
Null model (noise)
Alternative models
Meta-Bayesian approach
Ostwald et al. 2012
Data features: individual single-trial source activity in clusters estimated from EEG-MEG group inversion
Lecaignard et al. in prep (a)
𝒙 = 𝒇 𝒙, 𝜽, 𝒖 𝒚 = 𝒈 𝒙,𝝋 + 𝜺
2 – Inferring learning mechanisms from EEG/MEG data
• Single trial modulations of evoked responses should reflect perceptual learning
Preliminary results
Model comparison in a single subject, single block (condition US)
time time
Relative Free-energies or Bayes Factors w.r.t null model (M8)
M1
M2
M3
M4
M5
Bayesian learnerwith different forgetting
parameter value
M6
M7
M8
Change detection
models
Null model (noise)
Lecaignard et al. in prep (b)VBA toolbox, Daunizeau et al. 2014
3 – Online Bayesian inference and design optimization
3 – Online Bayesian inference and design optimization
Fine neurobiological and/or computational models of evoked responses and their modulations may tell us more about information processing in the brain
Given such models, we may be in a position to disentangle between alternative hypothesis about ongoing perceptual inference and learning in a given subject/patient
However, what design to choose? How to optimize it for each individual?
Basic idea
To make use of our ability to process data in real-time in order to optimize design parameters online, for a given individual, in the aim of selecting the most likely model both accurately and fast
Wald 1947, Cavagnaro et al. 2009, Myung et al. 2013
Motivation for Adaptive Design Optimization
3 – Online Bayesian inference and design optimizationGeneral principle of Adaptive Design Optimization
Sanchez et al. 2014
3 – Online Bayesian inference and design optimizationA simple example: behavioural task of word memorization
Cavagnaro et al. 2009, Myung et al. 2013
Y (observations) = number of recalled words
u (design variable) = Lag time
Models of “forgetting”
Predictions(Low uncertainty about model parameters)
EXP
POW
Predictions(High uncertainty about model parameters)
Experimental context
Hypothesis
3 – Online Bayesian inference and design optimizationOptimization criterion
u = 40
u = 10
Y
p(Y|H1 ,u)
p(Y|H2 ,u)
p(e|u)u* = argmin {p(e|u)}
u
…P(Y|H)
H1 : EXP
H2 : POW
u = 1
Minimizing the risk of model selection error
Danizeau et al. 2011
3 – Online Bayesian inference and design optimizationExtension to more complex (DCM-like) models
Sanchez et al. 2014
Generative model of the
“world”
ERPs
>>
Evolution function f
),,( evoluxfx ),( obsxgyObservation function g
+
Extension to more complex, multidimensional models
Danizeau et al. 2011
3 – Online Bayesian inference and design optimizationDisentangling between learning models
Sanchez et al. 2014
A perceptual learning model family: hierarchical Gaussian filters
Evolution function f
),,( evoluxfx ),( obsxgy
Observer function g
+Category (S or D)
(x1 = u)
Probability of D(u = 1)
Volatility
x1(t)
x2(t)
ω
x3(t)
ϑ
κ
KL-divergencebetween prior and post beliefs
Mathis et al. 2011 & 2014
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Models ω valuesκ values
(if κ = 0, no 3rd level)ϑ values
Ability to track
events probabilities
Ability to track
environmental volatility
M1 -Inf 0 - No No
M2 - 5 0 - Low learning No
M3 - 4 0 - High learning No
M4 - 5 1 0.2 Low learning Yes
M5 - 4 1 0.2 High learning Yes
Deviant
Standard
Simulations: alternative models and predictionsDisentangling between learning models
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Disentangling between learning models
Models ω valuesκ values
(if κ = 0, no 3rd level)ϑ values
Ability to track
events probabilities
Ability to track
environmental volatility
M1 -Inf 0 - No No
M2 - 5 0 - Low learning No
M3 - 4 0 - High learning No
M4 - 5 1 0.2 Low learning Yes
M5 - 4 1 0.2 High learning Yes
Deviant
Standard
Simulations: alternative models and predictions
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Disentangling between learning models
Models ω valuesκ values
(if κ = 0, no 3rd level)ϑ values
Ability to track
events probabilities
Ability to track
environmental volatility
M1 -Inf 0 - No No
M2 - 5 0 - Low learning No
M3 - 4 0 - High learning No
M4 - 5 1 0.2 Low learning Yes
M5 - 4 1 0.2 High learning Yes
Deviant
Standard
Simulations: alternative models and predictions
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Disentangling between learning models
Models ω valuesκ values
(if κ = 0, no 3rd level)ϑ values
Ability to track
events probabilities
Ability to track
environmental volatility
M1 -Inf 0 - No No
M2 - 5 0 - Low learning No
M3 - 4 0 - High learning No
M4 - 5 1 0.2 Low learning Yes
M5 - 4 1 0.2 High learning Yes
Deviant
Standard
Simulations: alternative models and predictions
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Disentangling between learning models
Models ω valuesκ values
(if κ = 0, no 3rd level)ϑ values
Ability to track
events probabilities
Ability to track
environmental volatility
M1 -Inf 0 - No No
M2 - 5 0 - Low learning No
M3 - 4 0 - High learning No
M4 - 5 1 0.2 Low learning Yes
M5 - 4 1 0.2 High learning Yes
Deviant
Standard
Simulations: alternative models and predictions
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Disentangling between learning models
Static designs to be compared with ADO
Classic -stable
Classic -volatile
p(u=1) = 0.2 0.1 0.3 0.1 0.2
3 – Online Bayesian inference and design optimization
Sanchez et al. 2014
Results on Monte Carlo simulations
Model 1 Model 2 Model 3 Model 4 Model 5
0
20
40
60
80
100
Optimal
Volatile Classical
Stable Classical
% of non-conclusive experiments
Trials beforereaching conclusion
Model 1 Model 2 Model 3 Model 4 Model 5
0
50
100
150
200
250
300
350
Optimal
Volatile Classical
Stable Classical
Disentangling between learning models
Conclusion
• 1 - EEG/MEG Source reconstruction- Bayesian inference on models and parameters- Group level inference- EEG/MEG data fusion
• 2 - Inferring learning mechanisms through EEG/MEG responses- meta-Bayesian inference- Model comparison
• 3 - Online Bayesian inference and design optimization
Acknowledgments
Olivier BertrandGareth BarnesAnne CaclinJean DaunizeauKarl FristonRik HensonFrançoise LecaignardEmmanuel MabyGaëtan Sanchez