Adaptive beamformer source reconstruction:

Our recent developments toward spatio-temporal reconstruction of brain activities

Kensuke Sekihara1, Srikantan S. Nagayajan2

1Department of Engineering, Tokyo Metropolitan Institute of Technology2Biomagnetic Imaging Laboratory, University of California, San Francisco

ISBET 2003Santa Fe, November, 2003

Right median nerve stimulationmeasured by a 160-channel whole-head sensor array

Hashimoto et al., “Muscle afferent inputs from the hand activate human cerebellum sequentially

through parallel and climbing fiber systems”, Clin. Neurophysiol. Nov;114, pp.2107-17, 2003 .

Right posterior tibial nerve stimulation

measured by a 37-channel sensor array

Hashimoto et al., “Serial activation of distinct cytoarchitectonic areas of the human {SI} cortex after

posterior tibial nerve stimulation,” NeuroReport 12, pp1857-1862, 2001

7 mm

( )

( ) data vector: (t) =

( )

1

2

M

b t

b t

b t

⎡ ⎤⎢ ⎥⎢ ⎥

• ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

b

Definitions( ) 1b t

( ) 2b t ( ) Mb t

data covariance matrix: ( ) ( )Tt t• =R b b

= [ , ,

source magnitude: ( , )

source orientation: ( ) ( ) ( ) ( )]

•

•

r

r r r rη Tx y z

s t

,t ,t ,t ,tη η η

( ) ( ) ( )

( ) ( ) ( )( ) =

( ) ( ) ( )

η

η

η

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

1 1 1

2 2 2

( )

, ( ) ( ) ( )

( )

x y z

xx y z

y

zx y zM M M

l l l

l l l

l l l

r r rr

r r rL r l r L r r

rr r r

y

x

sx

l1x l2

x l3x

lMx

z

y

x

l1z l2

z l3z lM

z

z sz

y

x

l1y l2

y l3y lM

y

z sy

|sx|=|sy|=|sz|=1

Lead field vector for the source orientation ( )rη

Adaptive spatial filter

1

11

( )

(̂ , ) ( ) ( ) [ ( ), , ( )] ( ) ( )

( )

MTM m m

m

M

b t

s t t w w w r b t

b t=

⎡ ⎤⎢ ⎥

= = =⎢ ⎥ ∑⎢ ⎥⎢ ⎥⎣ ⎦

r w r b r r…

1

1

( )( )( ) ( )

TT

T

−

−⇒ =l r Rw rl r R l r

Minimum-variance beamformer

21

1(̂ , )( ) ( )Ts t −=r

l r R l r

subject to ( ) 1min T =T

ww Rw w l r

Spatial filter

Following problems arise when applying minimum-variance beamformer to MEG/EEG source reconstruction.

(1) Output SNR degradation.

(2) Vector source detection.

(3) Statistical significance evaluation.

Minimum-variance beamformer is very sensitive to errors in forward modeling or errors in sample covariance matrix.

Because such errors are inevitable in neuromagnetic measurements, minimum-variance beamformer generally provides noisy spatio-temporal reconstruction results.

Introducing eigenspace projection

⇓

⇓

Output SNR degradation.

and

1

1 1

0 0

0 00

, [ , , | , , ]0

0 0

0 0

ST TP P P M

NS N

M

+

⋅⎡ ⎤⎢ ⎥

⋅⎢ ⎥⎡ ⎤⎢ ⎥= = =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⋅

⎢ ⎥⎢ ⎥⋅⎣ ⎦

ΛΛ

… …R U U U U U e e e eE E

λ

λ

λ

Output SNR ∝+

2[ ( ) ( )][ ( ) ( ) ]

TS

TS

TN

l r l rl r l r

Γε Γ εΓ

error in estimating ( )l r

Even when is small, may not be small,because noise level eigenvalueλ +≈ ←22 /

TN

TN p j

ε ε εε ε ε

ΓΓ

Also, −= =1 1T TS S S S N N

−Ν ΝΓ Ε Λ Ε Γ Ε Λ Ε,

Some definitions:

Eigenspace projection

Extension to eigenspace projection beamformer

where or , ,TS S x y zµw E E wµ µ= =

∝2[ ( ) ( )]

[ ( ) ( )]

TS

TS

l r l rl r l r

ΓΓ

Output SNR

Output SNR ∝+

2[ ( ) ( )][ ( ) ( ) ]

TS

TS

TN

l r l rl r l r

Γε Γ εΓ

⇓

(non-eigenspace projected)

(eigenspace projected)

The error term arises from the noise subspace component of ( ).TNε Γ ε w r

Application to 37-channel auditory-somatosensory recordingeigenspace-projection results

Application to 37-channel auditory-somatosensory recordingNon-eigenspace projected results

The minimum-variance beamformer formulation should be extended to incorporate the vector nature of sources.

The electromagnetic sources are three dimensional vectors.

Two-types of extensions has been proposed: scalar and vector formulations.

⇓

Vector source detection

=1 1

1 1

( , ) ( )( , )( , ) ( , ) ( ) ( )

T T TT

T T T

− −

− −=l r η R η L r Rw r η

l r η R l r η η L r R L r η

Scalar MV beamformer formulation

uses a single weight vector, but it depends not only on but also on .r η

S. E. Robinson et al., Recent Advances in Biomagnetism, Tohoku University Press, 1999

1 1 1[ ( ), ( ), ( )] [ ( ) ( )] ( )T T Tx y z

− − −=w r w r w r L r R L r L r R

Vector MV beamformer formulation

uses three weight vectors which detect , , and source components. x y z

M. E. Spencer et al., 26th Annual Asilomer Conference on Signals, Systems, and Computers, 1992B. D. van Veen et al., IEEE Trans. Biomed. Eng., 1997

max max21

11

1(̂ , )( ) ( )

1( ( ) ( ) )min

T T

T T

min

s t

γ

−

−−

=

⎡ ⎤= =⎢ ⎥⎣ ⎦

η η

η

rη L r R L r η

η L r R L r η

minimum eigenvalue of 1: [ ( ) ( )]Tminγ −L r R L r

Scalar formulation:

max22

1 1

ˆ ˆ ˆ ˆmax( , ) [ ( ), ( ), ( )]

1max[ [ ( ) ( )] ]min

x y z

T T

s t s s s

γ− −

=

= =

ηη

η

r r r r η

η L r R L r η

Vector formulation:

Two types of formulations give the same output power.

Output power

Asymptotic output SNR

2

2 2200

ˆ ˆ ˆ[ ( ), ( ), ( )] 1max( )( )

x y zoptV

min

s s sZ

ασσ= =

η

r r r ηr

W r η

max12 2

2 2 1 220 00

(̂ , , ) 1 ( ) ( ) 1( ) min( ) ( )( , )

T Topt

nT T

miS

s tZ

ασ σσ

−−

−

⎡ ⎤= = =⎢ ⎥

⎣ ⎦η η

r η η L r R L r ηrη L r R L r ηw r η

Scalar beamformer

Vector beamformer

minimum eigenvalue of 11 2: ( ) ( ) ( ) ( )T T

minα−− −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦L r R L r L r R L r

Two types of formulations give the same asymptotic output SNR.

2

2 2200

ˆ ˆ ˆ[ ( ), ( ), ( )] 1max( )( )

x y zoptV

min

s s sZ

σ ασ= =

η

r r r ηr

W r η

Asymptotic output SNR for vector beamformer22 2

22 220

ˆ ˆ ˆ( ) ( ) ( )( )

( ( ) ( ) ( ) )x y zconv

V

x y z

s s sZ

σ

+ +=

+ +

r r rr

w r w r w r

(orientation optimized)

(conventional)

optVZ

convVZ

MV beamformer

−

−=

1

2

( )( )

( ) ( )

η L r Rw r

η L r R L r η

T ToptT

T Topt opt

11 2( ) ( ) ( ) ( )T Topt min optα

−− − =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦L r R L r L r R L r η ηwhere

1

1

( )( )

( ) ( )

−

−=η L r R

w rη L r R L r η

T ToptT

T Topt opt

1( ) ( ) γ− =⎡ ⎤⎣ ⎦L r R L r η ηTopt min optwhere

Weight normalized MV (Borgiotti-Kaplan) beamformer

Orientation optimized weight

Orientation optimized results always gives the best SNR

(̂ , ) ( ) ( ) ( ) ( ) ( ) ( )T T TSs t t t t= = +r w r b w r b w r n

(̂ , ) ( , ) ( ) ( , ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( )

T T TS S

T T

s t t t t

t t

∆

∆

= = +

+ +

r w r b w r b w r b

w r n w r n

η η η

η η

( ) ( ) ( )St t t= +b b nData model (signal plus additive noise):

Beamformer reconstruction:

Parametric method

Thus, if , then 22 2

0 0ˆ( ) (0, ) ( , ) ( ( ) ( ), ( ) )T TSt N s t N tσ σn r w r b w r∼ ∼

Problem:

This is because the weight is obtained using a sample covariance.

Evaluation of the statistical significance:

Test this hypothesis at each voxel using a non-parametric statistics

The null hypothesis: no signal source

Signal sources: sources existing in the test (task) period but not in the control period.

Evaluation of the statistical significance:Our proposed method

assumed source waveform sensor array

} sources

y (cm)z

(cm

)

0-5 5

-5

-10

-15

computer-generated magnetic field spontaneous magnetic recordings

+

(̂ )js t

source waveform

source magnitude waveform

0(̂ )s t

use this distribution as an empirical distribution to test H0 at each (x,y,z).⇐

histogram of prestimulus period

(̂ )∈

j

j

s tt

If is significant. 0 0ˆ ˆ( ) , ( )> ths t s s t

prestimulus poststimulus

95% ths

(̂ )s t

⇓

where 22 2

( ) ( )( ) ( ) ( )

j jj j j

s t s tT t s t s tσ

σ

−= = −

Multiple comparison procedure

histogram of ( )js t histogram of ( )jT t

indicates the average over the prestimulus period.⋅

⇒

Standardization of the distribution

Maximum statistics

histogram of max( , )T x y

1 1( , )maxT x y

2 2( , )maxT x y

3 3( , )maxT x ythmaxT95%

Statistical thresholdingcross sectional view

( , ) ( , ) ( , )Σ σ= +thmaxx y T x y s x y

Threshold:

( , )x yΣ

2( , )s x y

( , )x yσ

Application to auditory-button-press measurements

Summary

My talk has addressed the problems caused from:

•output SNR degradation•vector source detection•statistical significance evaluation

and described our solutions for them:

•eigenspace projection•orientation optimized weight•a novel nonparametric statistical method.

(requires SPM 2)

will be soon available from UCSF.

MEG-MRI Toolbox

http://www.tmit.ac.jp/~sekihara/

Visit

The PDF version of this power-point presentation as well as PDFs of the recent publications are available.

Collaborators

Kanazawa Institute of TechnologyHuman Science Laboratory

Dr. Isao Hashimoto

University of MarylandLinguistics and Cognitive Neuroscience Laboratory

Dr. David Poeppel

Massachusetts Institute of Technology,Department of Linguistics and Philosophy

Dr. Alec Marantz