Localizing and estimating causal relations of interacting brain rhythms.

Frontiers in Human Neuroscience www.frontiersin.org November 2010 | Volume 4 | Article 209 | 1
HUMAN NEUROSCIENCE
REVIEW ARTICLE
published: 22 November 2010
doi: 10.3389/fnhum.2010.00209
From the cross-spectra S( f ) one can construct coherency matri-
ces C( f ), which are a normalized version of S( f ), as
C f
S f
S f S f
ij
ij
ii jj
( )
( )
( ) ( )
.=

(2)
In contrast to the imaginary parts of the cross-spectra, ( C ( f ))
also depends on independent sources through the denomina-
tor in Eq. 2. However, independent sources can only lead to a
decrease of ( C ( f )) and hence also ( C ( f )) reflects true inter-
action even though the physiological interpretation is not trivial
especially when interpreting differences of ( C ( f )), e.g., between
different tasks.
Based on these observations we suggested a series of meth-
ods to identify and localize brain interactions (Meinecke
et al., 2005; Nolte et al., 2006; Stam et al., 2007; Marzetti et al.,
2008; Nolte et al., 2009). Additionally, we proposed a method
to identify causal structures of the dynamical system under
study (Nolte et al., 2008). We here give a brief review of some
of these methods (Nolte et al., 2006; Marzetti et al., 2008; Nolte
et al., 2008) to identify interacting brain sources and to estimate
causal relationships. All the methods will be demonstrated using
simulated data whose characteristics are defined in the following
section.
2. Simulated interacting neural data
We simulated a seminal case with four dipolar sources as shown
in Figure 1, in which the dipoles have all a parallel orientation
and are spatially well separated. The sources on the right (left) are
interacting with each other but not with the sources on the left
(right). We thus considered t wo inte rac ting s ubsy ste ms . For both
subsystems the source in the back served as driver while the activity
1. introduction
Electroencephalography (EEG) can directly measure ongoing
brain activity with very high temporal but low spatial resolution.
In the past decades the main focus was the analysis of event related
potentials, i.e., the average brain response to a given stimulus. More
recently, the variability of brain activity and especially its inter-
pretation as signatures from the brain as a dynamical network has
attracted many researchers (Daglish et al., 2005; Womelsdorf and
Fries, 2006; Buckner and Vincent, 2007; Damoiseaux and Greicius,
2009; Fries, 2009; Miller et al., 2009).
Studying brain connectivity using noninvasive electrophysio-
logical measurements like EEG or MEG faces the challenge that the
data are largely unknown mixtures of activities of brain sources.
To address this issue, we suggest to construct estimates of brain
connectivity from quantities that are unbiased by non-interacting
sources. For zero mean data1 the linear statistical signal properties
can be determined by the cross-spectral matrices S( f ) defined as
S f x f x fij i j( ) ( ) ( )
*= 〈 〉

(1)
where x
m
(f) are the Fourier transforms at frequency f in channel m for
a given segment or trial and 〈·〉 denotes the expectation value which
is typically approximated by an average over the segments or trials.
It is straight forward to show that noninteracting sources do
not contribute systematically, i.e., apart from random fluctuations
around zero to the imaginary part of the cross-spectra, (S( f )),
regardless of the number of sources and details of the forward map-
ping (Nolte et al., 2004). The reason is that the forward mapping
is essentially instantaneous and does not induce phase delays to
excellent approximation (Stinstra and Peters, 1998) which would
be necessary to yield a nonvanishing imaginary part of S( f ).
Localizing and estimating causal relations of interacting
brain rhythms
Guido Nolte1* and Klaus-Robert Müller 2
1 Intelligent Data Analysis Group, Fraunhofer FIRST, Berlin, Germany
2 Machine Learning Group, Technical University Berlin, Berlin, Germany
Estimating brain connectivity and especially causality between different brain regions from
EEG or MEG is limited by the fact that the data are a largely unknown superposition of the
actual brain activities. Any method, which is not robust to mixing artifacts, is prone to yield
false positive results. We here review a number of methods that allow for addressing this
problem. They are all based on the insight that the imaginary part of the cross-spectra cannot
be explained as a mixing artifact. First, a joined decomposition of these imaginary parts into
pairwise activities separates subsystems containing different rhythmic activities. Second,
assuming that the respective source estimates are least overlapping, yields a separation of
the rhythmic interacting subsystem into the source topographies themselves. Finally, a causal
relation between these sources can be estimated using the newly proposed measure Phase
Slope Index (PSI). This work, for the first time, presents the above methods in combination; all
illustrated using a single, simulated data set.
Keywords: EEG, volume conduction, causality, interaction, PISA, MOCA, PSI
Edited by:
Kai J. Miller, University of Washington,
USA
Reviewed by:
Pedro Valdes-Sosa, Cuban
Neuroscience Center, USA
Andreas Daffertshofer, VU University
Amsterdam, Netherlands
*Correspondence:
Guido Nolte, Intelligent Data Analysis
Group, Fraunhofer FIRST, Kekul estr. 7,
12489 Berlin, Germany.
e-mail: nolte@first.fhg.de
1In an event related design the mean can be subtracted.

Frontiers in Human Neuroscience www.frontiersin.org November 2010 | Volume 4 | Article 209 | 2
Nolte and Müller Causal relations of brain rhythms
of the more frontal sources appeared merely identical to the ones
of the drivers but the activity was delayed by 20 ms. The activity
of the right driver was given as
u t u t u t t1 1 1 10 35 1 0 7 5( ) . ( ) . ( ) ( )= − − − + ξ (3)
where ξ
1
(t) is white Gaussian noise with standard deviation 1. Similarly,
the activity of the driver on the left side was simulated via
u t u t u t t2 2 2 20 35 1 0 7 4( ) . ( ) . ( ) ( )= − − − + ξ (4)
We defined a single time step to equal 10 ms, i.e., we consid-
ered a sampling rate of 100 Hz, by which the time series u
1
(t)
and u
2
(t) displayed pronounced spectral peaks at around 8 and
12 Hz, respectively, and had roughly identical magnitudes. Both
time series also have (weak) higher harmonics at 24 and 36 Hz,
respectively.
The frontal sources, v
1
(t) and v
2
(t) for right and left side, respec-
tively, are merely delayed versions of the drivers:
v t u ti i( ) ( )= − 2 (5)
corresponding to a delay of 20 ms. In total, we modeled 200 s of
EEG data.
The activities of the four dipolar sources were mapped into 118
EEG channels equally distributed on the scalp. As volume conductor
we assumed a three-shell realistic model calculated from the MRI data
containing brain, skull, and scalp with equal conductivities for brain
and scalp and 50:1 conductivity ratio between scalp and skull. The
Maxwell equations were solved using a semianalytic expansion of the
electric lead fields (Nolte and Dassios, 2005). An accurate forward
model is important but difficult. For the sake of simplicity we here
assumed that the forward model is correct, i.e., for the inverse methods
we used the same forward model as for the forward simulation.
To the activities of the sources of interest we superimpose spatially
correlated and temporally white noise generated as the activity of a
collection of dipoles placed on a 1 cm grid within the entire brain. All
components of all dipoles were modeled as iid Gaussian noise leading
FIGurE 1 | Four dipolar sources overlayed on MrI-slices.

Frontiers in Human Neuroscience www.frontiersin.org November 2010 | Volume 4 | Article 209 | 3
Nolte and Müller Causal relations of brain rhythms
least when focussing on the current discussion. These assumptions
can be expressed for an even number of N channels as a model for
the imaginary part of the cross-spectra:
ℑ = −( )
=
∑( ( )) ( )
/
S f p f
k
N
k k k
T
k k
T
1
2
a b b a

(6)
For each k the set of topographies (a
k
and b
k
) and the “interac-
tion spectrum” P
k
( f ) form a – what we call – PISA component. We
note that this model is only unique up to linear mixing of the two
topographies for each k. In other words, the model only identifies
the 2D-subspace spanned by the two topographies and not the
individual components.
The model is found by joined diagonalization (cf. Ziehe et al.,
2004) of (S( f )) in the complex domain: we find a demixing matrix
W such that W(S( f ))W† is diagonal. It can be shown that real
and imaginary parts of the columns of the mixing matrix A = W−1
span the same subspaces as the pairs of topographies a
k
and b
k
. For
technical details we refer to Nolte et al. (2006).
Results of the PISA decomposition for the simulated data set are
shown in Figure 4, where we show the largest three components.
Only the first and the second component revealed a significant
interaction spectrum corresponding to the two interacting sub-
systems in the left and right hemisphere, respectively.
3.2. minimum overlap component analySiS (moca)
In order to uniquely decompose the 2D-subspaces found by the
PISA method into contributions from individual sources we must
introduce further spatial constraints on the nature of the sources.
to highly correlated noise in the EEG electrodes. The noise level was
chosen such that the average of power over all channels and frequen-
cies was 20 times higher than the respective average of the signal of
interest. In “good” channels and at peak frequencies the power of the
signal of interest was still around 10 times higher than the noise.
Power (imaginary part of coherency) over all channels (pairs
of channels) are shown as function of frequency in Figure 2. The
spatial distribution of the imaginary part of coherency at 10 Hz, i.e.,
between the peaks and with contributions from both interacting
subsystems, is shown in Figure 3.
3. methodS
3.1. pairwiSe interacting component analySiS (piSa)
In general, EEG data are a superposition of many subsystems
including (effectively) independent sources but also interacting
rhythmic sources of various physiological content. To separate these
systems we assumed that (a) all interactions are pairwise and that
(b) there are not more interacting sources than channels. These two
assumptions are a clear simplification of the true brain dynamics,
but they yield a unique decomposition of the data and may capture
the most relevant aspects of the interaction observed in EEG data, at
FIGurE 2 | Left: Power over all channels. Right: Imaginary part of coherency
over all pairs of channels.
FIGurE 3 | Imaginary part of coherency at 10 Hz. Each small circle
corresponds to one row of the coherency matrix.
FIGurE 4 | Each row displays the result for one PISA Component. Left
and middle columns show the respective topographies. Panels in the right
column show the interaction as a function of frequency.

Frontiers in Human Neuroscience www.frontiersin.org November 2010 | Volume 4 | Article 209 | 4
Nolte and Müller Causal relations of brain rhythms
3.3. phaSe Slope index (pSi)
We finally want to estimate causal structures between the esti-
mated sources. Since the combination of PISA and MOCA
resulted in a complete basis of topographies we can find the
source activities by applying the inverse of the respective matrix
onto the data.
The “Phase Slope Index” (PSI) estimates the causal structure
between any two source activities. It is defined as Nolte et al.
(2008)
Ψij
f F
ij ijC f C f f= ℑ +



∈
∑ * ( ) ( )δ

(11)
where C
ij
( f ) is the complex coherency between sources i and j, as
given in Eq. 2, and δf is the frequency resolution of the coherency.
F is the set of frequencies over which the slope is summed. Usually,
F contains all frequencies, but it can also be restricted to a specified
band for rhythmic activities.
To see that the definition of
Ψij corresponds to a meaningful
estimate of the average slope it is convenient to rewrite it as
Ψ & &ij ij ij
f F
f f f f f f= + +
∈
∑A A δ δ( ) ( ) ( ( ) ( ))si n (12)
with C
ij
( f ) = α
ij
( f )e x p (iΦ( f )) and α
ij
( f ) = | C
ij
( f )| being frequency
dependent weights.
For smooth phase spectra, sin(Φ(f + δf ) − Φ( f )) ≈ Φ(f + δf ) − Φ
( f ) and hence
Ψ corresponds to a weighted average of the slope.
We list the most important qualitative properties of Ψ:
1. For an infinite amount of data and for arbitrary instanta-
neous mixtures of an arbitrary number of independent
sources, Ψ is exactly zero, because mixtures of independent
sources do not induce an imaginary part of coherencies
(Nolte et al., 2004) which in turn is necessary to generate a
non-vanishing Ψ. For finite data, Ψ will then fluctuate in this
case around zero within error bounds. A special case of this
are phase jumps from 0 to ±π which can arise also for mixtu-
res of independent sources.
To this end we apply a linear inverse operator, e.g., a minimum
norm solver G onto the topographies denoted here for any fixed k
as x
1
= a
k
and x
2
= b
k
, such that the topographies are mapped into
distributions s
i
of the source field
s Gi i� ( )x (7)
where s
i
= s
i
(m,k) is a three dimensional vector field calculated in
brain voxels m = 1,.., M and in directions k = 1,..,3. The distribu-
tions do not represent the sources of the brain, denoted as q
i
, but
are, within the accuracy of the inverse method, a yet unknown
superposition of them:
s H qi
j
ij j=
=
∑
1
2

(8)
for i = 1,2. The 2 × 2 mixing matrix H can be calculated uniquely
under the following constraints
1. The sources are orthonormal:
� �y =∑q q q m k q m ki j
m k
i j ij, ( , ) ( , )
,
D

(9)
2. The sources have minimum overlap:
L q q q m k q m k m in
m k
( , ) ( , ) ( , )1 2 1 2
2
y ¥
§¦
´
¶µ
=∑ ∑

(10)
This cost function first squares the scalar product of two dipole
moments at each voxel and then sums these squares over all voxels.
It vanishes if the two dipole distributions have disjoint support (i.e.,
disjoint regions of non-vanishing activity), thus measuring overlap.
It also vanishes if the orientations at each voxel are orthogonal
and therefore corresponds to a weaker form of overlap allowing in
principle also activities at the same location as long as the orienta-
tions are sufficiently different. Thus, a strong bias toward remote
interaction is removed.
The minimization in Eq. 10 can be realized analytically
(Marzetti et al., 2008). If the concept is generalized to more
than two topographies the minimization requires a numerical
approach, which, however, is surprisingly fast and robust (Nolte
et al., 2009). We note that the spatial constraints (Eqs 9 and 10)
and the methods to solve the minimization are similar to those
used in ICA in the context of fMRI data analysis (McKeown and
Sejnowski, 1998; Matsuda and Yamaguchi, 2004) with the major
difference that we here decompose vector fields rather than scalar
ones. In particular, the orthogonality constraint in Eq. 9 corre-
sponds, mutatis mutandis, to “sphering” as is used in most ICA
methods also used for EEG/MEG data analysis: for simplicity,
the data are transformed to be exactly uncorrelated while inde-
pendence in higher statistical orders is only forced to be as good
as possible.
For the present data set we further assumed the sources to be
located on the cortex but allowed for arbitrary orientation. Source
estimates of the first two PISA components for the simulated data
set are shown in Figure 5. We observe that each of the topographies,
decomposed from the PISA results using MOCA, corresponds to
one of the simulated dipoles.
FIGurE 5 | Left and middle panels: estimated sources of the PISA
components. Right panels: causal structure as function of function. Positive
results indicate that the sources shown in the left panels drive those shown in
the middle panels.

Frontiers in Human Neuroscience www.frontiersin.org November 2010 | Volume 4 | Article 209 | 5
Nolte and Müller Causal relations of brain rhythms
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Conflict of Interest Statement: The
authors declare that the research was
conducted in the absence of any com-
mercial or financial relationships that
could be construed as a potential conflict
of interest.
Received: 31 May 2010; accepted: 09
October 2010; published online: 22
November 2010.
Citation: Nolte G and Mueller KR (2010)
Localizing and estimating causal rela-
tions of interacting brain rhythms. Front.
Hum. Neurosci. 4:209. doi: 10.3389/
fnhum.2010.00209
Copyright © 2010 Nolte and Mueller. This
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sive license agreement between the authors
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2. Ψ is expressed in terms of coherencies, only. The standard
deviation of a coherency is approximately constant and only
depends on the number of averages which is equal for all fre-
quencies. Thus, large but meaningless phase fluctuations in
frequency bands containing essentially independent signals
are largely suppressed.
3. If the phase Φ( f ) is linear in f and provided that the frequency
resolution is sufficient (i.e., δf is sufficiently small), the argu-
ment in the sum has the same sign across all frequencies and
then Ψ will have the same sign as the slope of Φ( f ).
It is convenient to normalize Ψ by an estimate of its standard
deviation
Ψ Ψ
Ψ
�

std( )
(13)
with std( )Ψ being estimated by the Jackknife method, which we
validated in own simulations. In the examples below we consider
absolute values of each larger than 2 as significant.
It is important to point out that the phase of coherency itself is
not interpreted in terms of causality. For example, a phase of π/2
switches to −π/2 if the sign of one of the signals is reversed, but the
PSI measure is invariant with respect to the sign of the signals. Rather
than on phase, PSI is based on the slope of the phase as a function of
frequency. Note, that a sign change adds a constant to the phase and
has no effect on the slope. The method assumes that the studied fre-
quency range properly covers the dynamical range. For purely periodic
signals, any causality estimate would be dubious. In that case Ψ would
be insignificant because negative and positive slopes cancel.
Results for the causal structure of the sources estimated from
the simulated data are shown in the right panels of Figure 5. To
calculate PSI we chose segments of length 2 s corresponding to a
frequency resolution of δf = 0.5 Hz. We observe that in both cases
the source in the back is estimated as the driver.
4. concluSion
Accurately measuring the interaction of oscillatory brain sources
from EEG/MEG is a challenge. Due to the well-known effects of
volume conduction, it is easy and not uncommon to detect spuri-
ous interaction and thus reach spurious neuroscientific insight.
The present review has assembled three data analytical techniques
that avoid such erroneous conclusions as they are based on the
imaginary parts of the cross-spectra S( f ) that – as outlined above –
immunizes analysis against volume conduction artifacts. To clarify
this basic message we have used simulated EEG data from inter-
acting neural systems and took the reader through three essential
analysis steps (a) discovering interacting sources by PISA, (b) local-
izing them under constraints by MOCA and (c) estimating their
causal relationship by PSI.
Future research will extend the studies on causal relations of
interacting sources also for high noise situations (cf. Nolte et al.,
2010) and nonstationary processes (cf. von Bünau et al., 2009) and
the broad application of the presented computational methods in
the neurosciences.
acknowledgmentS
The authors gratefully acknowledge financial support by EU, BMBF,
and DFG. We would like to thank our co-authors for their permis-
sion to use material from joint publications.