Measuring Activation and Causality using multiple Prior Information Pedro A. Valdés-Sosa Cuban Neuroscience Center

Measuring Activation and Causality using multiple Prior Information

Pedro A. Valdés-Sosa

Cuban Neuroscience Center

Mapping

Brain Maps

Multimodal T1-Nissl-cryotomy-PET-myelin stain

Localization versus Connectivity

Hiparcus ( II BC)

Jackson

US air Traffic

Luria

Localization versus Connection

Anatomical Physiological

Localization Morphometry:•Voxel based•Region based •Cortical thickness

Activation:•EEEG/MEG•fMRI

Connection Anatomical Connectivity•Diffusion Weighted

Functional Connectivity

Effective Connectivity

Brain Tomographies

Physical Model of some brain characteristic

Prediction of measurement

Direct Problem

Image

of some brain characteristic

measurement

Inverse Problema Priori Information

EEG/MEG Forward Problem

K

Primary

current j

EEG/MEG

v

,P Hv j

EEG inverse Problem

1 1 21 2j jv k k

1vBayesian Inference!!

1 1 1N N p p N y X β e

2

2 2 2expc p y Xβ β

Methods for Regression Data

• VARETA

• LORETA

• ICA

• Non Negative Matrix Factorization

• In fact can be unified or combined

L0 Norm “Sparsness”

0 0 0( ) expp c β β

0 0 0ln ( ) lnp c β β

22 1 0

( ; )f β y y Xβ β

AIC, BIC, TIC, RIC

“subset selection” “Matching Purusit” “Dipoles”

L1 Norm “Sparseness”

1 1 1( ) expp c β β

1 1 1ln ( ) lnp c β β

22 1 1


“Lasso” “Basis Pursuit” “FOCUSS”

Connection with ICA

Fast LARS Algorithm (Friedman, Hastie, Tibshirani)

Regularization path for diabetes data

22 1 1

y Xβ β

L2 Norm “Minimum Norm”

2

1 2 2( ) expp c β β

2

2 2 2ln ( ) lnp c β β

222 1 2


“Ridge” “Frames” “Minimum Norm”

Simplest EEG inverse Problem

1 1 21 2j jv k k

1vBayesian Inference!!

Multiple Priors

Sparseness Minimal Norm

Non smooth Dipoles=FOCUSS Minimum Norm

Smooth VARETA LORETA

222 1 2

y Xβ β

222 1 2

y Xβ Lβ

22 1 1

y Xβ β

22 1 1

y Xβ Lβ

Which inverse solution to choose?: let the data decide combining all solutions

Bayesian Model Averaging

1M

2M

kM

1kM

NM

For 69 compartments2010N

3M 2kM

1

,N

k kk

E E M P M

j v j v v

Simulations with Bayesian Model Averaging

BMA during concurrent EEG/fMRI

Combining Priors

22 1 31 1

( ; )f β y y Xβ β Lβ

222 1 2 21

( ; )f β y y Xβ Lβ Lβ

Fused Lasso

VARETA-LORETA

Combining penalties

(L1,I) (L1,L) (L1,I) (L1,L)

Between LORETA and VARETA

LORETA

VARETA

Solution Chosen

Further Combination: Multiple Priors plus (semi) Non Negative Matrix Factorization

• Non Negative Matrix Factorizations used for data reduction

• Equivalent to Cluster Analysis

Multiple Priors plus (semi) Non Negative Matrix Factorization

22 1 2 11

( , ; )f

M B y y XMB M LM

M 0

0

N T N p p k k t N T

Y X M B E

M

Fast Non-negative LARS Algorithm (Morup)

Regularization paths for diabetes data

Results for a Simulation 64 Channels, 1 Patch complex time series

BIC

Regularization path

Results of a Simulation








Effective vs. Functional Connectivity(Karl Friston)

Statistical Analysis of Causal Modeling

"Beyond such discarded fundamentals as 'matter' and 'force' lies still another fetish amidst the inscrutable arcana of modern science, namely, the category of cause and effect.“ Karl Pearson (1911)

Granger (Non) Causality for TWO time series

1, 11 1, 1 2, 1 1,

2, 21 1, 1 22 2,

12

1 1,

t t t t

t t t t

y a y y e

y

a

a y a y e

1

2

0 12 2 1: 0 0H a I

1

2

Granger Non Causality

t t-1

t t-1

t =1,…,N

Granger Causality of EEG signals

3 4

4 3

0

0C C

C C

I

I®

®

>

=

Freiwald et al. (1999) J. Neurosci. Methods. 94:105-119

C3

C4

t t-1

{ }3, 4C CW=

What happens when you have a LOT of time series?

1

2

…

p

…

t t-1

1, 1,1 1,2 1, 1, 1 1,

2, 2,1 2,2 2, 2, 1 2,

, ,1 ,2 , , 1 ,

t p t t

t p t t

p t p p p p p t p t

y a a a y e

y a a a y e

y a a a y e

1t t ty A y e t =1,…,N

Long history: Bressler, Baccala, Kaminski, Eichler, Goebel

Problems with the Multivariate Autoregressive Model for Brain Manifolds

1, 1,1 1,2 1, 1, 1 1,

2, 2,1 2,2 2, 2, 1 2,

, ,1 ,2 , , 1 ,

t p t t

t p t t

p t p p p p p t p t

y a a a y e

y a a a y e

y a a a y e

1t t ty A y e

p→∞ 22 ( )

2

p pg r p

+= × +# of parameters

1,t tp y y Alikelihood

Regions of Interest

Alemán-Gómez Y. et al. PS0103

( )1

, ,g

G

gg

ROIg ty y s t ds

=

W

W= W

= òòò

U

Point influence Measures

s uI ® ( )0 : , 0H a s u =

,s u Î W

is the simple test

38

Spike and Wave

39

Spike and Wave

What happens when you have a LOT of time series?

1

2

…

p

…

t t-1

1, 1,1 1,2 1, 1, 1 1,

2, 2,1 2,2 2, 2, 1 2,

, ,1 ,2 , , 1 ,

t p t t

t p t t

p t p p p p p t p t

y a a a y e

y a a a y e

y a a a y e

1t t ty A y e t =1,…,N

Long history: Bressler, Baccala, Kaminski, Eichler, Goebel

a) Teat CG as a Random Field

Concept applied to correlation fields by Worsley

Usual SPM: RF is the brain

New Idea RF is Cartesian product of Brain by Brain

=

=X1,1 1,2 1,

2,1 2,2 2,

,1 ,2 ,

p

p

p p p p

a a a

a a a

a a a

1

1

p

a

a

a

Granger Causality must be measured on a MANIFOLD

surface of the brainW=

( ) ( ) ( ) ( )1

, , , ,r

kk

y s t a s u y u t k du e s t= W

= - +å òòò

Influence Measures defined on a Manifold

sI ®W0 :H ( ), 0a s u =

s Î W u Î W

An influence field is a multiple test and all for a given

( ) ( ) ( ) ( )1

, , , ,r

kk

y s t a s u y u t k du e s t= W

= - +å òòò

1;

;

; 1

t

i tt

p t p

y

y

y´

é ùê úê úê úê ú= ê úê úê úê úê úë û

y

M

M

( )( ), ,

i

i ts

y y u t duD

= òòò

1

r

t k t k tk

-=

= +åy A y e

Discretization of the Continuos AR Model

( ) ( )( ), ,

i i

ki j k i j

s ua a s u ds du¢ ¢

D ´ D¢ ¢= ò òL

( )0,t N~e

Influence Fields and Bayesian Estimation

x BI ® Influence field

x

B

1, 1,1 1, 1, 1 1,

, 2,1 2,

1,

2,

,

2, 1 2,

, ,1 , , 1 ,

t p t t

x t p

x

x

p x

t t

p t p p p p t p t

y a a y e

y a a y e

y a a

a

e

a

ya

1,t tp y y A

likelihood

p A

prior

Influence Fields

Outield Infield

Priors for Influence Fields

x BI ® maximal SMOOTHNESS

Valdés-Sosa PA Neuroinformatics (2004) 2:1-12Valdés-Sosa PA et al. Phil. Trans R. Soc. B (2005) 360: 969-981

1, 1,1 1, 1, 1 1,

, 2,1 2,

1,

2,

,

2, 1 2,

, ,1 , , 1 ,

t p t t

x t p

x

x

p x

t t

p t p p p p t p t

y a a y e

y a a y e

y a a

a

e

a

ya

x

B

Minimum norm I

Minimum spatial laplacian L

p A

prior

vs

FFA

Amigdala

Fear Static + Fear Dynamic Neutral

Neural basis of emotional expression processing

Emotional Network (Dipole)


Concurrent EEG-fMRI recordingsEKG

EOGV

EOGH

Fp1

Fp2

F3

F4

C3

C4

P3

P4

O1

O2

F7

F8

T7

T8

P7

P8

Iz

Cz

Pz

FC1

FC2

CP1

CP2

FC5

FC6

CP5

CP6

TP9

TP10

Eog200 µV

EKG

EOGV

EOGH

Fp1

Fp2

F3

F4

C3

C4

P3

P4

O1

O2

F7

F8

T7

T8

P7

P8

Iz

Cz

Pz

FC1

FC2

CP1

CP2

FC5

FC6

CP5

CP6

TP9

TP10

Scan Start Scan Start

50 µV

Fine time scale


Concurrent EEG-fMRI ( Rhythm)

-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100

2000

4000

6000

8000

10000

12000

Basis of concurrent EEG/MEG-fMRI analysis-voxel level Trujillo et al. IJBEM (2001)

BOLD

Vasomotor Feed Forward

Signal

VFFS

*h t

Ensemble of Postsynaptic Potentials

ePSP

net Primary Current Density

nPCD

EEG/MEG

* sK

EEG/MEG-fMRI-voxel Inverese solution Association

BOLD

VFFS

ePSPnPCD

EEG/MEG

*inverse s K *inverse h t

correlationlog BOLD-log j

First order Autoregressive Model for fMRI and EEG

10 20 30 40 50 60 70 80 90 100

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

EEGfMRI

r=-0.62

1

2

fff fst t t

ssf sst t t t

A Af f eA As s e

Estimated A for fMRI-EEG (f,s) using L1 regularizer

2

2 1Y = A X + A

ts

ts

tftf

ff fs

sf ss

A A

A A

EEG-fMRI influence Fields

Maximal Evidence

dipole

MN

non smooth smooth nonsmooth+smooth

dipole+MN

http://journals.royalsociety.org/content/md5e04y6bgm8/








Documents

Measuring Activation and Causality using multiple Prior Information Pedro A. Valdés-Sosa Cuban Neuroscience Center