Meg preprocessing

Magnetoencephalography
Preprocessing and Noise
Reduction Techniques
Eliezer Kanal
2/20/2012
MEG Basics Course

1

About Me
• 2005 -! 2009!!!
! ! !
!
!
University of Pittsburgh
PhD, Bioengineering

• 2009 -! 2011!!!
! ! !
!
!
Carnegie Mellon University
Postdoctoral fellow, CNBC

• 2011 -! current! !!
! ! ! !
PNC Financial Services
Quantitative Analyst, Risk Analytics

2

Dealing with Noisy Data
• Overview of MEG Noise
• Noise Reduction
- Averaging, thresholding, frequency ﬁlters

- SSP

- SSS/tSSS

• Source Extraction
- PCA

- ICA

3

Biological Noise

Vigário, Jousmäki, Hämäläinen, Hari, & Oja (1997)

10

Line Noise
50 Hz Line Noise
(60 Hz in USA)

Subject

Empty Room

11

Bad Channels

Find the bad one:

12

Noise from nearby construction

13

Noise Reduction
Techniques

• Averaging, thresholding, frequency ﬁlters
• SSP
• SSS/tSSS

14

Averaging
• Removes non-timelocked noise
• Requires:
- Time-locked block paradigm design

- Temporal or low-frequency analyses

15

Thresholding
• Discarding trials/channels with maximum signal
intensity greater than some user-deﬁned value

• Removes most “data blips”
• Rudimentary, better technique is to simply examine
each trial/channel

16

Frequency Filter
Filter Removes…
High-pass Lower frequencies
Low-pass Higher frequencies
Band-pass Outside specified band
Notch All except specified

• Very good first step, remove data you won’t analyze
(don’t waste time cleaning what you won’t examine)

• Use more advanced techniques for specific noise signals
17

Signal Space Projection

20

• Overview: SSP uses the difference between source
orientations and locations to differentiate distinct
sources.

• Theory: Since the ﬁeld pattern from a single source is
1) unique
2) time-invariant,
we can differentiate sources by examining the angle
between their “signal space representations”, and
project noise signals out of the dataset.

21

• In general,
M
X
m(t) = ai (t)si + n(t)
i=1

24

• In general,
M
X
measured i=1
signal

24

• In general, source i
M
X M = Total number of channels
measured i=1
signal

24

source
• In general, amplitude
source i
M
measured i=1
signal

24

source
source i
M
m(t) = ai (t)si + n(t) noise
measured i=1
signal

24

source
source i
M
measured i=1
signal

• SSP states that s can be split in two:
- s‖ ! = signals from known sources

- s⟂ ! = signals from unknown sources
s k = Pk m
s ? = P? m

24

source
source i
M
measured i=1
signal


known s k = Pk m
sources MEG signal
s ? = P? m
unknown
sources
Projection
operators

24

source
source i
M
measured i=1
signal


known s k = Pk m
sources MEG signal
s ? = P? m
unknown
sources
Projection Worth mentioning that sk + s? = s
operators

24

How ﬁnd P‖ and P⟂?

25


• Ingenious application of the magic
1 technique of
Singular Value Decomposition (SVD)

1 Not really magic

25


• Ingenious application of the magic technique of
1

a matrix of all known sources

• Let K = {s , s , . . . , s } 2 s . Using SVD, we ﬁnd a basis
1 2 k k
for s‖, and therefore P‖.2

1 Not really magic

25


• Ingenious application of the magic technique of
1

a matrix of all known sources

• Let K = {s , s , . . . , s } 2 s . Using SVD, we find a basis
1 2 k k
for s‖, and therefore P‖.2

1 Not really magic
2 Let K = U⇤VT. By the properties of the SVD, the first k columns of U form an
orthonormal basis for the column space of K, so we can define
Pk = U k U T k since s + s = P m + P m = s
k ? k ?
P? = I Pk

25

M
X
• Recall m(t) = i=1
ai (t)si + n(t) . To find a(t), invert s‖:

m(t) = a(t)sk
a(t) = sk 1 m(t)
ˆ
1
a = V⇤
ˆ UT m(t)

• In practice, soften consists of known noise signals
‖
specific to a particular MEG scanner. The final step is
simply to project those out of m(t), leaving only
unknown (and presumably neural) sources in s.

26

M
X
• Recall m(t) = i=1
ai (t)si + n(t) . To find a(t), invert s‖:

m(t) = a(t)sk
a(t) = sk 1 m(t)
ˆ Recall that K = {s1 , s2 , . . . , sk } 2 sk

a = V⇤ 1 UT m(t)
ˆ = U⇤VT
| {z }

• In practice, soften consists of known noise signals
‖
specific to a particular MEG scanner. The final step is
simply to project those out of m(t), leaving only
unknown (and presumably neural) sources in s.

26

Signal Space Separation
(SSS)

27

• Overview: Separate MEG signal into sources (1)
outside and (2) inside the MEG helmet

• Theory: Analyzing the MEG data using a basis which
expresses the magnetic field as a “gradient of the
harmonic scalar potential” (defined below) allows the
field to be separated into internal and external
components.

By simply dropping the external component, we can
significantly reduce the MEG signal noise.

28

MEG data – raw

29

MEG data – SSP

30

MEG data – SSS

31

• Begin with Maxwell’s laws:
⇤⇥H=J (1)
⇤ ⇥ B = µ0 J (2)
⇤·B=0 (3)

32

⇤⇥H=J (1)
magnetic ⇤ ⇥ B = µ0 J sources (2)
field
⇤·B=0 (3)

32

⇤⇥H=J (1)
field i. e., nos!
ce
⇤·B=0 sour (3)
• Note that on surface of sensor array, J = 0. As such,
⇥ H = 0 on array surface

Taulu et al, 2005

32

⇤⇥H=J (1)
field i. e., nos!
ce
⇤·B=0 sour (3)
• Deﬁning H = ∇Ψ, we obtain the identity ∇ × ∇Ψ = 0 in (1).
This term (∇Ψ) is called the “scalar potential.”
• “Scalar potential” has no physical correlate.

• Often written with a negative sign (–∇Ψ) for convenience.

• H = –∇Ψ → B = –μ0∇Ψ… used interchangeably

Taulu et al, 2005

32

⇤⇥H=J (1)
field i. e., nos!
ce
⇤·B=0 sour (3)
• Deﬁning H = ∇Ψ, we obtain the identity ∇ × ∇Ψ = 0 in (1).
This term (∇Ψ) is called the “scalar potential.”
• “Scalar potential” has no physical correlate.

• Often written with a negative sign (–∇Ψ) for convenience.

• H = –∇Ψ → B = –μ0∇Ψ… used interchangeably

• Substituting scalar potential into (3) we obtain the Laplacian:
⇥ ·⇥ = ⇥2 =0
Taulu et al, 2005

32

• Substituting the scalar potential into (3), we obtain the
Laplacian:
⇥·B=0
⇥ ·⇥ = ⇥2 =0

33

Laplacian:
⇥·B=0
⇥ ·⇥ = ⇥2 = 0
|{z}
 ✓ ◆ ✓ ◆
1 @ @ @ @ 1 @2
sin ✓ r2 + sin ✓ + + K2 =0
r2 sin ✓ @r @r @✓ @✓ sin ✓ @ 2

• We can express the scalar potential using spherical
coordinates ( Ψ(Φ, θ, r) ), separate the variables
( Ψ(Φ,θ,r) = Φ(φ)Θ(θ)R(r) ), and solve the harmonic
to obtain
⇥ l ⇥ l
lm (⇥, ⌅)
lm (⇥, ⌅)
l
B(r) = µ0 lm µ0 lm r
rl+1 l=0 m= l
l=0 m= l

⇥ B (r) + B (r)
internal external
signal signal

33

Laplacian:
⇥·B=0
⇥ ·⇥ = ⇥2 = 0
|{z}
 ✓ ◆ ✓ ◆
1 @ @ @ @ 1 @2
sin ✓ r2 + sin ✓ + + K2 =0
r2 sin ✓ @r @r @✓ @✓ sin ✓ @ 2

• We can express the scalar potential using spherical
coordinates ( Ψ(Φ, θ, r) ), separate the variables
( Ψ(Φ,θ,r) = Φ(φ)Θ(θ)R(r) ), and solve the harmonic
to obtain
⇥ l
lm (⇥, ⌅)
B(r) = µ0 lm
internal
rl+1
l=0 m= l

⇥ B (r)
internal
signal

33


34

Temporally-extended
(tSSS)

35

Temporally-extended Signal Space Separation

Conceptually very simple:

36



• Recall that the SSS algorithm ends with two signal
components – Bα(r) and Bβ(r), or Bin(r) and Bout(r) –
and we discard the Bout(r) component
- Rationale: signals originating outside MEG sensor helmet
cannot be brain signal

36



• Recall that the SSS algorithm ends with two signal
components – Bα(r) and Bβ(r), or Bin(r) and Bout(r) –
and we discard the Bout(r) component
- Rationale: signals originating outside MEG sensor helmet
cannot be brain signal

• tSSS looks for correlations between B out(r) and Bin(r)
and projects those correlations out of Bin(r)
- Rationale: Any internal signal correlated with the external
noise component must represent noise that leaked into the
Bin(r) component

36


• From the
original article:

37


• From the original article:

38


• Without tSSS:

39


• With tSSS:

40

Source Separation
Algorithms

41

Primary Component
Analysis (PCA)

42

• Ordinary Least
Squares (OLS)
regression of X
to Y

Following ﬁve plots from http://stats.stackexchange.com/a/2700/2019

43

• Ordinary Least
Squares (OLS)
regression of Y
to X

44

• Regression lines
are different!

45

• PCA minimizes
error orthogonal
to the model line

(Yes, this is a different dataset)
46

Primary Component Analysis

• “Most accurate”
regression line
for the data

(Yes, this is another different dataset)
47

PCA – Formal Deﬁnition

48

PCA – Formal Deﬁnition

http://stat.ethz.ch/~maathuis/teaching/fall08/Notes3.pdf

49

PCA shortcomings
• Will only detect
orthogonal signals

“A Tutorial on Principal Component Analysis”, Jonathon Shlens, April 2009

•
• Cannot detect
polymodal distributions

Appl. Environ. Microbiol. May 2007 vol. 73 no. 9 2878-2890

50

Independent Component
Analysis (ICA)

51

Independent Component Analysis
• Assumptions: Each signal is…
1. Statistically independent
2. Non-gaussian

• Recall Central Limit Theorem:
! “Given independent random variables x + y = z, z is
! more gaussian than x or y.”

• Theory: We can ﬁnd S by iteratively identifying and
extracting the most independent and non-gaussian
components of X

52

ICA in FieldTrip package

53

ICA – Mixing matrix

54


s2
s1

54


s2
s1

x2
x1

54

x1 = a11 s1 + a12 s2
⌘ x = As
x2 = a21 s1 + a22 s2

s2
s1

x2
x1

54

x1 = a11 s1 + a12 s2
⌘ x = As
x2 = a21 s1 + a22 s2

s2
s1

x2
x1 Goal: Separate s1 and s2 using
information from x1 and x2

54

• Consider the general mixing equation:
9
x1 = a11 s1 + . . . + a1n sn >
=
.
. .
. = .
. >
⌘ x = As
;
xn = an1 s1 + . . . + ann sn

55

9 mixing
x1 = a11 s1 + . . . + a1n sn > matrix
=
.
. .
. = .
. >
⌘ x = As
; sources
xn = an1 s1 + . . . + ann sn
sensors

55

9 mixing
x1 = a11 s1 + . . . + a1n sn > matrix
=
.
. .
. = .
. >
⌘ x = As
; sources
xn = an1 s1 + . . . + ann sn
sensors

• If we could ﬁnd one of the rows of A (let’s call that
-1

vector w), we could reconstruct a row of s.
Mathematically:
X
T
w x= w i xi = y
i

55

9 mixing
x1 = a11 s1 + . . . + a1n sn > matrix
=
.
. .
. = .
. >
⌘ x = As
; sources
xn = an1 s1 + . . . + ann sn
sensors

-1

Mathematically:
X
T
w x= w i xi = y
i

w
Some ro-1
from A

55

9 mixing
x1 = a11 s1 + . . . + a1n sn > matrix
=
.
. .
. = .
. >
⌘ x = As
; sources
xn = an1 s1 + . . . + ann sn
sensors

-1

Mathematically:
e ICs
X One of th mponents)
t co
wT x = w i xi = y ( independen ake up S
i that m

w
Some ro-1
from A

55

X
T
w x= w i xi = y
• Working through the math… let x = As
i

z = AT w

56

X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i

mixing matrix Some row fr -1
T
z=A w

56

X
T
w x= w i xi = y
let x = As
i

mixing matrix z = A w Some row fr
-1
T

• So, y = w x
T

= wT As
= zT s

56

X
T
w x= w i xi = y
let x = As
i

-1
T

• So, y = w x
T

One of = wT As
the ICs = zT s

56

X
T
w x= w i xi = y
let x = As
i

-1
T

• So, y = w x
T

One of = wT As
the ICs = zT s

• y (an IC) is a linear combination of s, with weights z .T

56

X
T
w x= w i xi = y
let x = As
i

-1
T

• So, y = w x
T

One of = wT As
the ICs = zT s

• y (an IC) is a linear combination of s, with weights z . T

zT is more gaussian than any of si, and is least gaussian
when equal to one of the si.

56

X
T
w x= w i xi = y
• Working through the math… let
T
x = As
i

z=A w
• So, y = w xT
We want to take w as a vector that
T

maximizes the nongaussianity of
One of = wT As wTx, ensuring that wTx = zTs
the ICs = zT s

• y (an IC) is a linear combination of s, with weights z . T

zT is more gaussian than any of si, and is least gaussian
when equal to one of the si.

56

• How can we find w Tso as to maximize the
nongaussianity of wTx?

• Numerous methods:
- Kurtosis

- Negentropy

- Approximations of Negentropy

• Once find, similar to PCA… find w , remove, find next
T

best wT, remove, repeat until no more sensors
available.

57

ICA in Fieldtrip (2)

58

Mantini, Franciotti, Romani, & Pizzella (2007)

59


1


61

ICA – Method Comparison

Zavala-Fernández, Sander, Burghoff, Orglmeister, & Trahms (2006)

62

Summary
• Examine your data in as many ways as possible
• Use SSS & tSSS to best clean data
• Use ICA to ﬁnd speciﬁc artifacts
• Always check your data!

63

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Meg preprocessing