Recent advances in the analysis of biomagnetic signals

The M**in**d and Bra**in** III :Audition, Language, Communication

Dubrovnik, Croatia, April, 2003

Neuromagnetic Source Reconstruction and

Inverse Model**in**g

Part I: Introduction to adaptive spatial filter techniques

Kensuke Sekihara

Tokyo Metropolitan Institute **of** Technology

This talk:

•formulates **the** neuromagnetic source reconstruction

problem us**in**g spatial filters.

•**in**troduces non-adaptive and adaptive spatial filter

techniques.

•focuses on **the** adaptive spatial filter technique (adaptive

beamformer).

Magnetoencephalography

(Neuromagnetic measurements)

•can provide a high temporal resolution.

•cannot provide (adequate) **in**formation

on **the** source spatial configuration.

⇓

Efficient numerical algorithms for estimat**in**g source

configuration are need to be developed.

(Source localization problems)

Source localization problem

•Dipole model**in**g approach

•Image reconstruction approach

Tomographic reconstruction

Spatial filter

Tomographic reconstruction

•Assume pixel grids **in** **the** region **of** **in**terest.

•Assume a source at each grid.

•Estimate **the** moment **of** each source by

least-squares fitt**in**g to **the** measured data.

Spatial filter technique

•Form spatial filter weight wr ( ) that focuses **the**

sensitivity **of** **the** sensor array at a small area at r.

•Scan this focused area over **the** region **of** **in**terest

to obta**in** source reconstruction.

Focused region

r

Right posterior tibial nerve stimulation

measured by a 37-channel sensor array

Hashimoto et al., NeuroReport 2001

7 mm

Right median nerve stimulation

measured by a 160-channel whole-head sensor array

Hashimoto et al., J. Cl**in**ical Neurophysiology submitted for publication

Def**in**itions

⎡b () t

1 ⎤

⎢ ⎥

⎢b () t

2 ⎥

• data vector: b(t) = ⎢

⎥

⎢ ⎥

⎢b

() t ⎥

⎣ M ⎦

b () t

1

b () t

2 b () t

M

T

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

• source magnitude: s( r, t)

• source orientation: η( r,t ) = [ ηx( r,t ), ηy( r,t ),

ηz(

r,t)] T

x

x l x l 3

2

l 1

l M

x

l 1

y l 2

y l 3

y

l M

y

z

l l z z

l

2 3

1

l M

z

z

s x

y

z

s y

y

z

s z

y

x

x

|s x

|=|s y

|=|s z

|=1

x

Lead field vector for **the** source orientation η()

r

x y z

⎡l1 () r l1() r l1()

r ⎤

⎢

⎥

⎡η

⎤

x y z

x()

r

⎢l2() r l2() r l2()

r ⎥

⎢ ⎥

Lr ( ) = ⎢

⎥, lr ( ) = Lr ( ) ⎢ηy

( r)

⎥

⎢

⎥

⎢ ⎥

⎢ x y z

⎢η

⎥

() () () ⎥

()

⎣

z

r

l ⎦

⎣

M

r lM r lM

r

⎦

Basic relationship

b () t = ( )( )

j ∫ l r s r,t dr

j

or

b() t = ∫ L( r)( s r,t)

dr

Problem **of** source localization:

Estimate sr ( , t) from **the** measurement b( t)

Estimate

Spatial filter

⎡b () t ⎤

1

⎢ ⎥ M

T

sr ˆ( ,) t = w ()() rbt = [ w (), r …, w ()] r

w () ()

1

⎢ ⎥ = ∑

M

m

r bm

t

m = 1

⎢ ⎥

⎢b

() t

⇑

⇑ ⎣ M ⎥⎦

estimate **of** sr (,) t weight vector

How to evaluate an appropriateness **of** **the** weight ?

b = ∫ L( r) s( r)

dr

T

} → ˆ sr ( ) = ( ) ( ') ( ')

d '

T

∫ w rLr sr r

ˆs( r)= w ( r)

b

R(

r, r ')

⇑

Resolution kernel

sr ˆ( ) = ∫ R( rr , ') sr ( ') dr'

(neglect**in**g **the** explicit time notation)

Non-adaptive weight

wr ( )

is data **in**dependent

Adaptive weight

wr ( )

is data dependent

Data-**in**dependent (non-adaptive) weight

m**in**imum-norm estimate (Hamala**in**en and Ilmoniemi)

The weight

wr ( ) is obta**in**ed by

[ R − − ] 2

m**in** ∫ ( rr , ') δ ( r r′ ) dr′

⇓

T T 1

w ( r) L ( r) G −

T

= , where Gij ,

= ∫li( r) lj

( r)

dr

Gram matrix

T −1

Inverse solution: sr ˆ( ) = L( rG ) b

⇑

This is erroneous

Gram matrix G is usually calculated by **in**troduc**in**g pixel grid

b = ∫ L( r) s( r) dr = ∑ L( r ) s( r)

N

j = 1

⎡sr

( ) ⎤

1

⎢ ⎥

= ⎡Lr ( ), ,

Lr ( ) ⎤ ⎢ ⎥ = Ls

⎣ 1

⎦ ⎢ ⎥

L

( )

N ⎢sr

⎣ N ⎥

⎦

s

j

j

N N N

N

r j

Therefore

G

=

L L

N

T

N

and

w T ( r ) = L T ( r )( L L T )

N

N

−1

Resolution kernel for non-adaptive (m**in**imum-norm) method

One example

•Auditory-evoked field were measured us**in**g 148-channel

whole-head sensor array (Magnes 2500).

Stimulus: 1-kHz pure tone applied to subject’s left ear

The number **of** pixels: 12940 po**in**ts

9

The condition number **of** G: ∼ 10

Property **of** **the** gram matrix G

G

= ∫ l ( r) l ( r)

dr

i,j i j

Biomagnetic **in**struments

X-ray computed tomography

Overlaps **of** sensor lead fields is large

G is poorly conditioned

G ≈

unit matrix

G is poorly conditioned

•Apply regularization when calculat**in**g G -1

Baysian-type approach

•Do not use G

m**in** F : F

sˆ

= b − Lsˆ

+

modified G

2 2

ˆ ( )

Hsˆ

T

T −1

s = HL LHL

N N

+ γ I b

Adaptive beamform**in**g technique

Adaptive spatial filter

M**in**imum-variance beamformer

⎡b1()

t ⎤

⎢ ⎥ M

T

sˆ( r,) t = w ()() r b t = [ w1(), r …, wM()] r ⎢ ⎥ = ∑ wm() r bm()

t

⎢ ⎥

m=

1

⇑ ⎢bM

() t

⎣ ⎥⎦

weight vector

m**in** T T

w Rw subject to

w l () r

=

1

w

ˆ( ,)

1

() r R l()

r

2

s r t =

T −1

l

T

⇒ w () r =

l

T −1

l () r R

r R l r

T −1

() ()

Assumption that source activities are uncorrelated

T

With constra**in**t: w ( r ) l( r ) 1,

p p

=

T

2 2 T

w ( r ) Rw( r ) = s( r , t) + ∑ s( r, t) w ( r ) l(

r )

p p p q p q

q≠p

⇑

s( r , t) s( r, t) = 0 when p ≠ q

p

q

m**in**

w

T

T

⎡⎣w ( rp) Rw( rp) ⎤⎦

⇒ w ( rp) l(

rq)=0,

q ≠ p

Therefore, this m**in**imization gives **the** weight satisfy**in**g

w

T

( rp) l( rq) = 1 for p = q

= 0 for p ≠ q

Spatial filter technique

•Form spatial filter weight wr ( ) that focuses **the**

sensitivity **of** **the** sensor array at a small area at r.

Focused region

r

Adaptive beamformer sensitivity pattern: plot **of** wr (

0) lr ( )

The density **of** **the** colors is propotional to wr (

0) lr ( ).

wr ( ) lr ( ) > 0

0

po**in**t**in**g location r 0

wr ( ) lr

( ) < 01

0

wr ( ) lr ( ) = 0

0

The weight sets null-sensitivity at regions where sources exist.

Low-rank signal assumption

Consider a easiest case where we know locations and orientations

**of** all Q sources

weight

by solv**in**g

wr ( )(conta**in****in**g M unknowns)

1

a set **of** Q l**in**ear equations:

can be obta**in**ed

T

w ( r)( l r) = w ( r) l( r) + … + w ( r) l ( r)

=

1 1 1 1 1 1 M 1 M 1

T

w ( r)( l r) = w ( r) l ( r) + … + w ( r) l ( r)

=

1 2 1 1 1 2 M 1 M 2

T

w ( r)( l r ) = w ( r) l ( r ) + … + w ( r) l ( r ) =

1 Q 1 1 1 Q M 1 M Q

T

when Q > M, **the**re is no solution for w ( r)

1

1

0

0

Low-rank signal

Number **of** sensors M > Number **of** sources Q

⎡λ1 0 ⋅ 0 ⎤

⎢

⎥

⎢ 0 0 ⋅ ⎥

⎢

⎥ ⎡Λ

S

0 ⎤

T

R = U λQ

⎢

⎥ U = U ⎢ ⎥ U

⎢ 0 Λ

N

⎢

⎥

⋅ 0 0 ⎥ ⎣ ⎦

⎢

⎥

⎢ 0 ⋅ 0 λ

M

⎥

⎣

⎦

T

U = [ e1, …, eQ | eQ+

1, …, eM] = [ ES | EN]

E E

S

N

T

− T

Γ = Ε Λ Ε and Γ = Ε Λ Ε ⇒ R = Γ + Γ

− 1 − 1 − 1 1 − 1 − 1 − 1

S S S S Ν N Ν N S Ν

Ε

Orthogonality pr**in**ciple

−1

lr ( ) = Γ lr ( ) = 0 at any source location r

T

N q N q q

M**in**imum-variance spatial filter output:

sˆ( r)

2

1 1

= =

l r R l r l r l r l r l r

T −1 T − 1

T −1

() () () Γ

S

() + () ΓN

()

2

sˆ( r)

source location

small

←

T

l () r Γ l()

r

N

→

large

Non-adaptive spatial filter

Adaptive spatial filter

assumed source waveform

148-channel sensor array

latency (ms)

latency (ms)

-5

}

z (cm)

-10

-15

-5 0 5

y (cm)

M**in**imum-norm

reconstruction

M**in**imum-variance

spatial filter reconstruction

MUSIC reconstruction

Adaptive-beamformer techniques were orig**in**ally developed **in**

**the** fields **of** array signal process**in**g, **in**clud**in**g radar, sonar, and

seismic exploration.

Two major problems arise when apply**in**g m**in**imumvariance

beamformer to MEG source localization.

(1) Vector source detection.

(2) Output SNR degradation.

Vector source detection

The neuromagnetic sources are three dimensional vectors.

The m**in**imum-variance beamformer formulation

should be extended to **in**corporate **the** vector nature **of**

sources.

⇓

Two-types **of** extensions has been proposed: scalar and

vector formulations.

Scalar MV beamformer formulation

w

T

(, r η)

=

T −1 T T −1

l (, r η) R η L () r R

=

T −1 T T −1

l (, r η) R l(, r η) η L () r R L()

r η

The weight depends not only on r but also on η.

Vector MV beamformer formulation

[ w ( r), w ( r), w ( r)] = [ L ( r) R L( r)] L ( r)

R

x y z

T T −1 −1 T −1

The three weight vectors detect x, y, and z components.

Calculation **of** **the** weight does not require η.

Scalar MV beamformer formulation

w

T

(, r η)

=

T −1 T T −1

l (, r η) R η L () r R

=

T −1 T T −1

l (, r η) R l(, r η) η L () r R L()

r η

The weight depends not only on r but also on η.

Vector MV beamformer formulation

[ w ( r), w ( r), w ( r)] = [ L ( r) R L( r)] L ( r)

R

x y z

T T −1 −1 T −1

The three weight vectors detect x, y, and z components.

Calculation **of** **the** weight does not require η.

max

How to derive **the** optimum η **in** scalar formulation?

⇓

Choose η that gives **the** maximum power output

2 1

T T −1

sˆ( r,) t = max

=

⎡

( () ())

⎤

T T −1

m**in** η L r R L r η

η L () r R L()

r η ⎢ ⎣

⎥ ⎦

η η η

T −1

T

Eigendecomposition: [ L ( r) R L( r)]

= ∑ γ v v , ( γ > γ > γ )

⇓

T T −1

m**in**( η L ( r) R L( r) η)

= γ

3

η

↑

m**in**imum eigenvalue

3

j = 1

j j j

1 2 3

−1

max

η

2

sˆ( r,) t 1/γ 3

= =

S

opt

Vector beamformer formulation

Each weight vector, wx(), r wy(), r or wz()

r is obta**in**ed by us**in**g

**the** follow**in**g multiple constra**in**ts.

T T T T

x x x x x y x z

m**in** w Rw subject to w lre (, ) = 1, w lre (, ) = 0, w lre (, ) = 0

T T T T

y y y x y y y z

m**in** w Rw subject to w lre (, ) = 0, w lre (, ) = 1, w lre (, ) = 0

T T T T

z z z x z y z z

m**in** w Rw subject to w lre (, ) = 0, w lre (, ) = 0, w lre (, ) = 1

e , e , e : unit vectors **in** **the** , , directions.

x y z xyz

⇓

[ w ( r), w ( r), w ( r)] = [ L ( r) R L( r)] L ( r)

R

x y z

T T −1 −1 T −1

(The weight is not equal to **the** scalar weight with η = e , e ,or e .)

x y z

van Veen et al., 1996 Spencer et al., 1992

In vector formulation, η that gives **the** maximum power output

can be obta**in**ed us**in**g

max

η

2

2

T T −1 −1

= max

x y z

= max

η

η

sˆ(,) r t [ sˆ(), r sˆ(), r sˆ()] r η η [ L () r R L()]

r η

max

η

⇓

2

ˆ( r,) = 1/γ

3

=

s t S

opt

Ei**the**r types **of** formulations atta**in** S opt

when **the** beamformer

po**in**t**in**g direction is optimized.

⇓

Two types **of** formulations are ma**the**matically equivalent.

148-channel sensor array

-5

1st

2nd

z (cm)

-10

3rd

(reconstruction region)

-15

scalar formulation

-5 0 5

y (cm)

vector formulation

Output SNR degradation.

M**in**imum-variance beamformer is very sensitive to errors **in**

forward model**in**g or errors **in** sample covariance matrix.

⇓

Because such errors are almost **in**evitable **in** neuromagnetic

measurements, m**in**imum-variance beamformer generally

provides noisy spatio-temporal reconstruction results.

⇓

Introduc**in**g eigenspace projection

assumed source waveform

37-channel sensor array

latency (ms)

z (cm)

-5

-10

-15

}

three sources

reconstruction region

relative magnitude

latency (ms)

generated magnetic field

-5 0 5

y (cm)

Add very small amount **of** noise

to obta**in** **the** simulated MEG record**in**gs

Spatio-tempotal reconstruction

Recall some def**in**itions:

⎡λ1

0 ⋅ 0 ⎤

⎢

⎥

⎢0 0 ⋅ ⎥

⎢

⎥ ⎡ΛS

0 ⎤

T

T

R = U λP

⎢

⎥U = U ⎢ ⎥U , and U = [ e1, …, eP | eP+

1, …, eM]

⎢ 0 ΛN

⎢

⎥

⋅ 0 0 ⎥ ⎣ ⎦

ES

EN

⎢

⎥

⎢0 ⋅ 0 λM

⎥

⎣

⎦

Also,

Γ Ε Λ Ε , Γ Ε Λ Ε

−1 −1 T −1 −1

T

S

=

S S S Ν

=

N Ν N

Output SNR

∝

[ l ( r) Γ l( r)]

T −1 2

S

T −2

T −2

[ l ( r) ΓS

l( r) + ε Γ ε ]

N

overall error **in** estimat**in**g lr ( )

T −2

Even when ε is small, ε Γ ε may not be small,

N

2 2 2

N

/ λp+

j

T −

because ε Γ ε ≈ ε

← noise level eigenvalue

Eigenspace projection

T −2

The error term ε Γ ε arises from **the** noise subspace component **of**

N

wr ().

Extension to eigenspace projection beamformer

w

T

= E E w , where µ = xy , or z

S S

µ µ

Output SNR ∝

Output SNR

∝

[ l ( r) Γ l( r)]

T −1 2

S

T −2

T −2

[ l ( r) ΓS

l( r) + ε Γ ε ]

N

⇓

T

[ l ( r) Γ l( r)]

T

[ l ( r) Γ l( r)]

−1 2

S

−2

S

(non-eigenspace projected)

(eigenspace projected)

Spatio-tempotal reconstruction with eigen-space projection

Application to 37-channel auditory-somatosensory record**in**g

eigenspace-projection results

Application to 37-channel auditory-somatosensory record**in**g

Non-eigenspace projected results

Application to 37-channel auditory-somatosensory record**in**g

eigenspace-projection results

Summary

•This talk reviews **the** application **of** adaptive beamformer to

reconstruction **of** bra**in** activities, with some reference to **the** nonadaptive

methods.

•Two types **of** extensions, scalar and vector extensions to **in**corporate

**the** vector nature **of** **the** sources, are shown to be ma**the**matically

equivalent.

•Eigenspace projection is shown to overcome **the** SNR degradation

problem.

•The implicit assumptions for **the** adaptive beamformer are that

sources are uncorrelated and that **the** signal is low rank. The cases

where **the**se assumptions are **in**validated will be discussed **in** Part II.

Summary

•This talk reviews **the** application **of** adaptive beamformer to

reconstruction **of** bra**in** activities, with some reference to **the** nonadaptive

methods.

•Two types **of** extensions, scalar and vector extensions to **in**corporate

**the** vector nature **of** **the** sources, are shown to be ma**the**matically

equivalent.

•Eigenspace projection is shown to overcome **the** SNR degradation

problem.

•The implicit assumptions for **the** adaptive beamformer are that

sources are uncorrelated and that **the** signal is low rank. The cases

where **the**se assumptions are **in**validated will be discussed **in** Part II.

Summary

•This talk reviews **the** application **of** adaptive beamformer to

reconstruction **of** bra**in** activities, with some reference to **the** nonadaptive

methods.

•Two types **of** extensions, scalar and vector extensions to **in**corporate

**the** vector nature **of** **the** sources, are shown to be ma**the**matically

equivalent.

•Eigenspace projection is shown to overcome **the** SNR degradation

problem.

•The implicit assumptions for **the** adaptive beamformer are that

sources are uncorrelated and that **the** signal is low rank. The cases

where **the**se assumptions are **in**validated will be discussed **in** Part II.

Summary

•This talk reviews **the** application **of** adaptive beamformer to

reconstruction **of** bra**in** activities, with some reference to **the** nonadaptive

methods.

•Two types **of** extensions, scalar and vector extensions to **in**corporate

**the** vector nature **of** **the** sources, are shown to be ma**the**matically

equivalent.

•Eigenspace projection is shown to overcome **the** SNR degradation

problem.

•The implicit assumptions for **the** adaptive beamformer are that

sources are uncorrelated and that **the** signal is low rank. The cases

where **the**se assumptions are **in**validated will be discussed **in** Part II.

Collaborators

University **of** California, San Francisco

Biomagnetic Imag**in**g Laboratory

Dr. Srikantan S. Nagarajan

University **of** Maryland

L**in**guistics and Cognitive Neuroscience Laboratory

Dr. David Poeppel

Massachusetts Institute **of** Technology,

Department **of** L**in**guistics and Philosophy

Dr. Alec Marantz

Kanazawa Institute **of** Technology

Human Science Laboratory

Dr. Isao Hashimoto

Visit

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

The PDF version **of** this power-po**in**t presentation as

well as PDFs **of** **the** recent publications are available.