Granger Causality on Spatial Manifolds: applications to Neuroimaging

Pedro A. Valdés-SosaCuban Neuroscience

Centre

Multivariate Autoregressive Model for EEG/fMRI

1

2

…

p…

t t-1

t =1,…,Nt

{ }1, , pW= L

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

Point influence Measures

s uI ® ( )0 : , 0H a s u =

,s u Î W

is the simple test

Granger Causality must be measured on a MANIFOLD

( ) ( ) ( ) ( )1

, , , ,r

kk

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

= - +å òòò

surface of the brainW=

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

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kk

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= - +å òòò

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;

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é ùê úê úê úê ú= ê úê úê úê úê úë û

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( )( ), ,

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i ts

y y u t duD

= òòò

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t k t k tk

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= +åy A y e

Discretization of the Continuos AR Model -I

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T Tr

T TN N r- -

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y y

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= +Z XB E

Multivariate Regression Formulation

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r N+

=

é ù= ë û

B A A

Z y y

K

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( ) { }1

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i i ik j k

p ir

vec b

é ù é ùê ú ê úê ú ê ú= = =ê ú ê úê ú ê úê ú ê úë û ë û

B

L L

2 2ˆ arg min arg min= - = -Σ

B BB Z XB Z XB

1ˆ ( )T T-=B X X X Z 1ˆ ( ) ii T T-= X X z X

( ),

,

,

ˆ

ˆ

ik ji

k j ik j

tSE

b

b= { }, , 1

ik i k j i pI t®W £ £

=

ML Estimation and detection of Influence fields

Problemas 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→∞ t =1,…,N

22 ( )

2

p pg r p

+= × +# of parameters

( ) ( )( ) ( )( )11 1

1; , , , , . exp

M

M M m mm

P P C Pp -

== Õ - L

( )2 1

1

ˆ arg minM

m mm

P -

== - + å

BB Z X B

( )2 1Ttr -=X X X

( ) ( )( )

1 l

length

m ml

P p w=

= åx

w

Prior Model on Influence Fields

Priors for Influence Fields

x BI ® Are of minimum norm, or maximal smoothness, etc.

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

Penalty Functions

11 1

ˆ ˆ( ( ))i T i Tk k i

-+ += +X X D X z

1

( ) ( ( ) / )M

i i im l l

m

diag p w w¢

=

=åD

| |

,0

( ) ( )m m

pp p dt

t

ql

e q q ee

= -+ò

( )1

( ) ( ( ) / )M

i i im l l

m

diag p w we e¢

=

= +åD

Estimation via MM algorithm

Penalty Covariance combinations

( )21,rp

L I

( )22,rp

L I

( )21,rp

L L

( )22,rp

L L

( )( )2 21, 2,rp rp

L LI I

( )( )2 21, 1,rp rp

L LI D

( )( )2 22, 2,rp rp

L LI D

( )( )( )( )2 2 2 21, 1, 2, 2,rp rp rp rp

L L L LI L I L ?

“Ridge Fusion”

Fused Lasso

Elastic Net

Spline (“LORETA”)

Data Fusion

FramesRidge

Basis PursuitLASSO

Known as to wavleteers as

Name in statisticsModel

spa

rsen

ess

smo

oth

nes

sb

oth

Simulated “fMRI”

1t t ty A y e

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

Correlations of the EEG with the fMRI

Martinez et. al Neuroimage July 2004