measuring activation and causality using multiple prior information pedro a. valdés-sosa cuban...
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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
( ; )f β y y Xβ β
“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
( ; )f β y y Xβ β
“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
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
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)
Cuban Neuroscience Center
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
Cuban Neuroscience Center
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/
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