Minimum Overlap Component Analysis (MOCA) of EEG/MEG data for more than two sources.

Journal of Neuroscience Methods 183 (2009) 72–76
Contents lists available at ScienceDirect
Journal of Neuroscience Methods
journa l homepage: www.e lsev ier .com
Minim A) o
data fo
Guido No
a Fraunhofer FI
b Department o
c Institute for A
d Cuban Neuro
a r t i c l
Article history:
Received 5 Ma
Received in re
Accepted 3 Jul
Keywords:
EEG
MEG
Inverse metho
Connectivity
Decompositio
MOCA
to an
ces. W
stinc
rce sp
istrib
raliza
ource
. The
1. Introduction
Electroe
(MEG) data
are simulta
the non-un
source estim
help of add
1993). How
cluded from
example, ev
of the full
infer locati
algorithm (
Similarly, s
ponent Ana
(SSP) (Tesch
nal space fr
assumption
or covarian
phies order
While it ca
∗ Correspon
E-mail add
(L. Marzetti), p
tions that the first few PCA topographies are a linear combination
0165-0270/$ –
doi:10.1016/j.jncephalographic (EEG) or magnetoencephalographic
consist of a superposition of many brain sources which
neously active and are eventually interacting. Due to
iqueness of the inverse problem (Helmholtz, 1853), final
ates from measured data can only be found with the
itional assumptions (Baillet et al., 2001; Hamalainen,
ever, in many cases important properties can be con-
the data without making additional assumptions. For
ent related potentials (ERPs) typically span a subspace
sensor space which can, in a second step, be used to
ons of dipolar sources with the MUSIC or RAPMUSIC
Mosher et al., 1992, 1998; Mosher and Leahy, 1999).
ignal decomposition strategies such as Principal Com-
lysis (PCA) (Jolliffe, 1986) or signal space projection
e et al., 1995) aim at identifying topographies in sig-
om prominent features in the data without additional
s. In particular, PCA decompositions of cross-spectra
ce matrices of spontaneous activity contain topogra-
ed with respect to relevance for the measured data.
n be concluded without making additional assump-
ding author.
resses: nolte@first.fraunhofer.de (G. Nolte), lmarzetti@unich.it
eter@cneuro.edu.cu (P. Valdes Sosa).
of the topographies of the most relevant sources, it is in gen-
eral unclear how these PCA topographies can be demixed into
source topographies. A well-known method for such a decompo-
sition is the Independent Component Analysis (ICA) (Hyvarinen
et al., 2001; Vigario et al., 2000) which, however, is only appli-
cable if the sources are statistically independent. ICA additionally
requires that the source activations contain a substantial amount of
nonlinearity and/or have different spectral content. Especially sta-
tistical independence is a dubious assumption if interacting brain
systems are studied and, even more, if the interaction is the focus of
the analysis. Semi-blind ICA algorithms have been developed that
replace statistical independence constraint by prior knowledge of
quite accurate information on some parameters of the signals to
be separated, e.g. prior knowledge of the autocorrelation function
of a source (Papathanassiou and Petrou, 2002), rough knowledge
of the time-course of a functional magnetic resonance imaging
(fMRI) source (Lu and Rajapakse, 2005; Calhoun et al., 2005) or MEG
source (Barbati et al., 2006). Nevertheless, none of these methods
is designed for interacting brain system studies. In the framework
of interaction analysis, we propose here to decompose subspaces of
the sensor space exploiting only spatial assumptions without using
a parametric model. Furthermore, since apart from the prior step
of finding the subspaces themselves no further dynamical assump-
tions are made in the decomposition, the method is equally valid for
correlated and uncorrelated sources as long as the subspace itself
is accurate. This method is a generalization of the MOCA-algorithm
see front matter © 2009 Elsevier B.V. All rights reserved.
neumeth.2009.07.006um Overlap Component Analysis (MOC
r more than two sources
ltea,∗, Laura Marzettib,c, Pedro Valdes Sosad
RST.IDA, Berlin, Germany
f Clinical Sciences and Bioimaging, Gabriele D’Annunzio University, Italy
dvanced Biomedical Technologies, Gabriele D’Annunzio University Foundation, Italy
science Center, Havana, Cuba
e i n f o
y 2009
vised form 3 July 2009
y 2009
ds
n
a b s t r a c t
In many situations various methods
spanned by potentials of a set of sour
a subspace into contributions from di
finding the respective subspace in sou
transformation such that the source d
The corresponding algorithm is a gene
Analysis’ (MOCA) to more than two s
almost never trapped in local minima/ locate / jneumeth
f EEG/MEG
alyze EEG/MEG data result in subspaces of the sensor space
e propose a general model free method to decompose such
t sources. This unique decomposition can be achieved by first
ace using a linear inverse method and then finding the linear
utions are mutually orthogonal and have a minimum overlap.
tion of the recently presented ‘Minimum Overlap Component
s. The computational cost is negligible and the algorithm is
method is illustrated with results for alpha rhythm.
© 2009 Elsevier B.V. All rights reserved.

G. Nolte et al. / Journal of Neuroscience Methods 183 (2009) 72–76 73
(Marzetti et al., 2008), in which all source interactions are assumed
to be pairwise, to take into account systems composed by more than
two interacting sources.
2. Method
2.1. Pairwis
Useful q
parts of cro
sources (No
pairwise on
set of all im
the subspac
2008; Nolte
sensor spac
linear inver
topographie
s
i
= G(x
i
)
where s
i
= s
brain voxel
tributions d
within the
tions of the
s
i
=
∑
j
A
ij
q
The mixing
assumption
1. The sour
〈q
i
, q
j
〉 ≡
2. The sour
L(q1, q2)
This cost
moment
els. It van
(i.e. disjo
overlap.
orthogon
lap allow
long as th
bias towa
To solve
sˆ
i
=
∑
j
W
ij
with W = V
V
ij
=
∑
m,k
s
i
(
In a second
(
q1
q2
)
=
(
and find the angle � by minimizing the cost function defined in (4).
This minimization can be done analytically in closed form and leads
to the solution:
tan−
( )
(
∑
(
∑
k
m
(
s sol
n an
ting
ith a
k th
OCA
mally
d. W
ey ar
A
ij
q
the
ing a
sour
q
j
〉 ≡
sour
L(q
i
,
L de
n th
e tec
be s
ative
s iter
arate
lutio
ns ha
ther
o eac
s wh
ght t
s be
nges
n. To
. For
i0 a
fors
e MOCA
uantities to study interacting sources are imaginary
ss-spectra since these are not biased by non-interacting
lte et al., 2004). If one assumes that all interactions are
e can construct two-dimensional subspaces from the
aginary parts of the cross-spectra which coincide with
es of the corresponding pairs of sources (Marzetti et al.,
et al., 2006). Now, assume that the two topographies in
e x
i
for i = 1, 2 span such a subspace. Then we apply a
se operator, e.g. a minimum norm solver G, such that the
s are mapped into distributions s
i
of the source field:
(1)
i
(m, k) is a three-dimensional vector field calculated on
s m = 1, . . . , M and in directions k = 1, . . . , 3. The dis-
o not represent the actual sources of the brain but are,
accuracy of the inverse method, unknown superposi-
actual brain sources, q
i
, involved in this interaction:
j
(2)
matrix A can be calculated uniquely with the following
s
ces are orthonormal:
∑
m,k
q
i
(m, k)q
j
(m, k) = ı
ij
(3)
ces have minimum overlap:
≡
∑
m
(
∑
k
q1(m, k)q2(m, k)
)2
= min (4)
function first squares the scalar product of two dipole
s at each voxel and then sums these squares over all vox-
ishes if the two dipole distribution have disjoint support
int regions of non-vanishing activity), thus measuring
It also vanishes if the orientations at each voxel are
al and therefore corresponds to a weaker form of over-
ing in principle also activities at the same location as
e orientations are sufficiently different. Thus, a strong
rds remote interaction is removed.
for A we first whiten the distributions s
i
to fulfill (3):
s
j
(5)
−1/2 and
m, k)s
j
(m, k) (6)
step we rotate sˆ
i
as
cos(�) sin(�)
− sin(�) cos(�)
)(
sˆ1
sˆ2
)
(7)
�0 =
1
4
with
a =
∑
m
b=
∑
m
c =
1
4
∑
Variou
functio
alterna
tions w
and pic
2.2. M
For
forwar
that th
s
i
=
∑
j
Again,
follow
1. The
〈q
i
,
2. The
∑
i<j
with
As i
(5). Th
cannot
an iter
Thi
all sep
lytic so
rotatio
In o
ative t
source
first si
overlap
all cha
functio
K steps
indices
p
i
= q
i1 b
a − c
(8)
k
sˆ1(m, k)sˆ2(m, k)
)2
sˆ1(m, k)sˆ2(m, k)
∑
k
(sˆ1(m, k)sˆ1(m, k) − sˆ2(m, k)sˆ2(m, k))
)
∑
k
(sˆ1(m, k)sˆ1(m, k) − sˆ2(m, k)sˆ2(m, k))
)2
.
utions arise due to the various branches of the tan−1
d differ by multiples of �/4. Minima and maxima are
, and we only have to calculate two neighboring solu-
ngles �max and �min for the maximum and minimum,
e one referring to the minimum out of these two.
for N sources
, the generalization of MOCA to N sources is straight
e now have N distributions s
i
for i = 1, . . . , N and assume
e unknown superpositions of N sources q
i
:
j
(9)
mixing matrix A can be calculated uniquely with the
nalogous assumptions
ces are orthonormal:
∑
m,k
q
i
(m, k)q
j
(m, k) = ı
ij
(10)
ces have minimum overlap:
q
j
) = min (11)
fined in (4).
e case for two source we first whiten the sources with
hnical difficulty is now that the cost function in (11)
olved analytically and must be solved numerically with
procedure.
ative procedure chosen here consists of minimizing over
pairs of source distributions using the respective ana-
n and repeating these pairwise minimizations until all
ve converged.
words, in each iteration we only rotate two sources rel-
h other and minimize the overlap between these two
ich can be done analytically. This procedure appears at
o be rather naive because such a rotation also affects
tween other pairs of sources. The key point here is that
between other pairs indeed cancel out in the total cost
show this we define q
i
to be the estimated sources after
the K + 1 th iteration we only rotate the sources with
nd j0 and define new sources p
i
as
i /= i0, j0 (12)

74 G. Nolte et al. / Journal of Neuroscience Methods 183 (2009) 72–76
and
(
p
i0
p
j0
)
=
(
cos(�) sin(�)
− sin(�) cos(�)
)(
q
i0
q
j0
)
(13)
Then for any i /= i0, j0 one has
L(p
i
, p
i0
) + L(p
i
, p
j0
) =
∑
m
(
∑
k
q
i
(m, k)(cos �q
i0
(m, k) + sin �q
j0
(m, k))
)2
+
∑
m
(
∑
k
q
i
(m, k)(cos �q
j0
(m, k) − sin �q
i0
(m, k))
)2
=
∑
m
(
∑
k
q
i
(m, k)q
i0
(m, k)
)2
+
∑
m
(
∑
k
q
i
(m, k)q
j0
(m, k)
)2
= L(q
i
, q
i0
) + L(q
i
, q
j0
)
(14)
and hence
∑
i<j
L(p
i
, p
j
) = L(p
i0
, p
j0
) + c (15)
where c is a constant independent of the angle �. Thus, any pairwise
minimization necessarily also reduces the total cost function.
We note that this property, termed ‘semi-invariance’, was found
similarly in (Matsuda and Yamaguchi, 2004) in the context of ICA.
Our approach differs with respect to the actual physical meaning
of the quantities (we assume spatial rather than temporal non-
overlap) an
than scalar
is identical
We foun
usually suffi
Trivially, fo
different so
the sources
improbable
Out of thou
we could no
result in th
lar topograp
minima in v
3. Illustrat
3.1. Simulat
First, we
of spatially
able. We si
inside the brain and with random orientation. Each dipole pair leads
to two electric potentials, say v
i
(j) with i = 1, 2 and j = 1 . . . J for J
sensors, with respective source estimates q
i
(m, k) at voxel m and
direction k. Angles, �, are defined according to the respective scalar
products, i.e.
∑
j
v1(j)v2(j) ≡
⎛
⎝
∑
j
v1(j)v1(j)
⎞
⎠
1/2⎛
⎝
∑
j
v2(j)v2(j)
⎞
⎠
1/2
cos �
(16)
for the electric potentials and
∑
q1(m, k)q2(m, k) ≡
(
∑
q1(m, k)q1(m, k)
)1/2
sour
ig. 1
tima
d ori
nrea
the a
al da
dec
eig
from
ertz
occ
al oc
ust b
Fig. 1 ace (ld by the fact that we here decompose vector fields rather
fields. However, this special property ‘semi-invariance’
for both problems.
d that around five sweeps across all pairs of indices are
cient to reach a stable minimum with high accuracy.
r each pair 4 physically equivalent but mathematically
lutions exist corresponding to different order and sign of
. Apart from these equivalent solutions it is surprisingly
that the algorithm gets trapped into a local minimum.
sands of simulations with 3–10 random dipolar patterns
t find a single case where two random mixtures did not
e same decomposition. Only when we replaced dipo-
hies by random numbers were we able to detect local
ery rare cases.
ion
ion: distribution of angles
will test whether the assumption that source estimates
distinct sources are approximately orthogonal is reason-
mulated 20,000 pairs of dipoles with random location
m,k
for the
In F
and es
tion an
is an u
while
90◦.
3.2. Re
We
largest
10 Hz
(Blank
by the
bilater
but m
. Histogram of distribution of angles of topographies of random dipoles in signal spm,k
×
(
∑
m,k
q2(m, k)q2(m, k)
)1/2
cos � (17)
ce estimates.
we show the histograms of the angles for potentials
ted sources calculated from dipoles with random loca-
entation. We observe that orthogonality in sensor space
sonable assumption with respect to this simulation
ngles in source space are clearly concentrated around
ta
ompose PCA eigenvectors corresponding to the four
envalues of the real part of the cross-spectrum at
EEG data measured during imagined foot movement
et al., 2003). The cross-spectrum at 10 Hz is dominated
ipital alpha rhythm which is activated primarily in
cipital cortices. This rhythm is not induced by the task
e considered as ongoing activity also present under
eft), and of the respective minimum norm estimates (right).

G. Nolte et al. / Journal of Neuroscience Methods 183 (2009) 72–76 75
eyes open condition. The data also contain some amount of central
alpha also at 10 Hz due to event related synchronization—in this
case induced primarily by the absence of imagined hand move-
ment, which has been the task in non-analyzed trials. EEG was
measured in 118 channels and 70 trials, each of 3.5 s duration. The
cross-spectrum at 10 Hz is measured with a frequency resolution
of 2 Hz. Channel locations were matched on a realistic standard
head model taken from the program CURRY (Neuroscan, Hamburg,
Germany). As an inverse method we used the weighted minimum
norm solution as outlined in (Marzetti et al., 2008).
The eigenvectors corresponding to the four largest eigenvectors
are shown in Fig. 2. The topographies are essentially symmetric
Fig. 2. Topographies of eigenvectors corresponding to the largest four eigenvalues
of the real part of cross spectrum at 10 Hz.
Fig. 3. MOCA
shown in Fig. 2
Fig. 4. Source
to the four PCA
Fig. 5. Source strength of weighted minimum norm inverse solutions correspond-
ing to the four MOCA topographies shown in Fig. 3. No threshold was used for
visualization.
and are apparently determined by the PCA orthogonality constraint
rather than stemming from separate sources.
The MOCA decomposition of the space spanned by the four
eigenvectors is shown in Fig. 3. We observe simple structures which
qualitatively suggest to originate from focal sources.
For both the PCA and the MOCA topographies we calculate
inverse solutions constrained on the cortical surface as shown in
Figs. 4 and 5, respectively. For both cases we find sources to be
distributed mainly in the occipital region. However, only for the
decomposition we find sources restricted to right and left
heres.
clusion
proposed a method to decompose N-dimensional subspaces
ignal space based on the assumption that the true sources of
ubspaces are spatially distinct and can roughly be found by
inverse methods. The new aspect here is the generalizationdecomposition of the subspace spanned by the four topographies
.
MOCA
hemisp
4. Con
We
of the s
these s
linearstrength of weighted minimum norm inverse solutions corresponding
topographies shown in Fig. 2. No threshold was used for visualization.

76 G. Nolte et al. / Journal of Neuroscience Methods 183 (2009) 72–76
of the analytically solvable case of the two-dimensional decompo-
sition, presented recently in (Marzetti et al., 2008) as ‘Minimum
Overlap Component Analysis’ (MOCA), to the N-dimensional case.
The most important technical message is that this problem can be
solved with minimal computational cost in a very robust way.
The two-dimensional problem naturally appeared in the con-
text of analyzing interacting brain sources from imaginary parts of
cross-spectra, which can be shown to be not affected in the mean by
independent sources (Nolte et al., 2004), and are hence a well suited
quantity to study interactions. For example, the eigenvectors of the
imaginary part of a cross-spectrum are necessarily complex with
real and imaginary part representing unknown superpositions of
potentials of interacting sources. However, the interpretation of the
two-dimensional subspace of the sensor space spanned by this real
and imaginary part requires a couple of assumptions. First, it must
be assumed that all interactions are pairwise, i.e. the interacting
brain, as measured within a given experiment, decays into two-
dimensiona
Second, the
orthogonal.
‘Pairwise In
et al., 2008
transforme
Analysis’ (s
than PCA in
follows from
Howeve
interaction
The goal of t
is conceivab
tem of four
is a reasona
subspace in
Althoug
tions it sho
occurs also
are generat
tive time co
‘statistically
principle, b
Similarly, a
the real pa
contributin
ceptually, t
parametric
Another
oped by Val
the inverse
source spat
and sparse.
and nonneg
tions are as
of each vox
nality requires that at each voxel at most one of the sources can
be active. The possible connection between MOCA and STONNICA
is a problem the authors are currently investigating since mini-
mally non overlapping sources are achieved by minimization of two
apparently different objective functions.
Acknowledgement
This work was supported in part by the Bundesministerium für
Bildung und Forschung (01GQ0415).
References
Baillet S, Mosher JC, Leahy RM. Electromagnetic brain mapping. IEEE Signal Process
2001;18:14–30.
Barbati G, Sigismondi R, Zappasodi F, Porcaro C, Graziadio S, Valente G, et al. Func-
tional source separation from magnetoencephalographic signals. Hum Brain
Mapp 2006;27:925–34.
z B, D
ting b
mand
2003;
VD, A
tilizin
5;25:5
nen M
ns to
;65:4
ltz H.
icher
endo
en A, K
.
. Prin
japak
orks
Y, Ya
comp
t confe
JC, Le
io-tem
JC, Le
lizatio
JC, Le
) mus
i L, D
by i
8;42(1
Bai U
n from
4;115:
Mein
y com
nassio
essing
D, Uu
e proj
onal s
osa P
es MA
tive
–910.
R, Sar
oach
0;47(5l interacting subsystems independent of each other.
topographies of different subsystems are mutually
The second assumption can be either dropped by using
teracting Source Analysis’ (Nolte et al., 2006; Marzetti
) which uses information across many frequencies, or
d into source space using ‘source Principal Component
PCA), which is more plausible for the present purposes
sensor space as shown in (Marzetti et al., 2008) and as
the distribution of angles shown here.
r, in all previous works the assumption that the brain
decays into pairwise interacting subsystems is essential.
his work is to also drop this assumption. For example, it
le that two two-dimensional subspaces represent a sys-
interacting sources. In this case, the presented method
ble tool to decompose the respective four-dimensional
to distinct sources.
h much of this discussion is focused on brain interac-
uld be noted that localization of subspaces frequently
in other circumstances. For example, evoked potentials
ed by an unknown number of sources. If the respec-
urses are linearly independent (not to be confused with
independent’) the strongest PCA components can, in
e decomposed into topographies of distinct sources.
s was done here, PCA decompositions of the full or
rt of cross-spectra lead to subspaces spanned by the
g sources which can be decomposed with MOCA. Con-
he most important property is that no dipole or other
model was used for these decompositions.
type of inverse solution, STTONNICA, has been devel-
des-Sosa et al. (2009). Here component separation and
problem are solved simultaneously by requiring that the
ial weight matrix be nonnegative, orthogonal, smooth
The special combination of weights to be orthogonal
ative is equivalent to vanishing overlap if the orienta-
sumed to be known since in that case contributions
el can contribute only positively and hence orthogo-
Blankert
Boos
com
Eng
Calhoun
for u
200
Hamalai
catio
1993
Helmho
perl
Pogg
Hyvarin
2001
Jolliffe IT
Lu W, Ra
Netw
Matsuda
dent
join
Mosher
spat
Mosher
loca
Mosher
(RAP
Marzett
data
200
Nolte G,
actio
200
Nolte G,
nois
Papatha
proc
Tesche C
spac
neur
Valdes-S
Bob
nega
1898
Vigario
appr
200ornhenge G, Schaefer C, Krepki R, Kohlmorgen J, Mueller KR, et al.
it rates and error detection for the classification of fast-paced motor
s based on single-trial EEG analysis. IEEE Trans Neural Syst Rehabil
11(2):127–31.
dali T, Stevens MC, Kiehl KA, Pekar JJ. Semi-blind ICA of fMRI: a method
g hypothesis derived time courses in spatial ICA analysis. Neuroimage
27–38.
S. Magnetoencephalography—theory, instrumentation, and appli-
non invasive studies of working human brain. Rev Mod Phys
13–97.
Ueber einige Gesetze der Vertheilung elektrischer Stroeme in koer-
Leitern mit Anwendung auf die thierisch-elektrischen Versuche.
rff Annalen 1853;89:211–33.
arhunen J, Oja E. Independent component analysis. New York: Wiley;
cipal component analysis. New York: Springer-Verlag; 1986.
se J. Approach and applications of constrained ICA. IEEE Trans Neural
2005;16:203–12.
maguchi, K. Semi-invariant function of Jacobi algorithm in indepen-
onent analysis. In: Neural networks, proceedings. IEEE international
rence; 2004.
wis PS, Leahy RM. Multiple dipole modeling and localization from
poral MEG data. IEEE Trans Biomed Eng 1992;39(6):541–57.
ahy RM, Recursive. MUSIC: a framework for EEG and MEG source
n. IEEE Trans Biomed Eng 1998;45(11):1342–54.
ahy RM. Source localization using recursively applied and projected
ic. IEEE Trans Signal Process 1999;47(2):332–40.
el Gratta C, Nolte G. Understanding brain connectivity from EEG
dentifying systems composed of interacting sources. Neuroimage
):87–98.
, Weathon L, Mari Z, Vorbach S, Hallet M. Identifying true brain inter-
EEG data using the imaginary part of coherency. Clin Neurophysiol
2294–307.
ecke FC, Ziehe A, Mueller KR. Identifying interactions in mixed and
plex systems. Phys Rev E 2006;73:051913.
u C, Petrou M. Incorporating prior knowledge in ICA. In: Digital signal
2002. IEEE 14th international conference, vol. 2; 2002. p. 761–4.
sitalo MA, Ilmoniemi RJ, Huotilainen M, Kajola M, Salonen O. Signal-
ections of MEG data characterize both distributed and well-localized
ources. Electroencephal Clin Neurophysiol 1995;95:189–200.
A, Vega-Hernandez M, Sanchez-Bornot JM, Martinez-Montes E,
. EEG source imaging with spatio-temporal tomographic non-
independent component analysis. Hum Brain Mapp 2009;30(6):
ela J, Jousmiki V, Haemaelaeinen M, Oja E. Independent component
to the analysis of EEG and MEG recordings. IEEE Trans Biomed Eng
):589–93.