Identifying true brain interaction from EEG data using the imaginary part of coherency.

Identifying true brain interaction from EEG data using
the imaginary part of coherency
Guido Nolte*, Ou Bai, Lewis Wheaton, Zoltan Mari, Sherry Vorbach, Mark Hallett
Human Motor Control Section, NINDS, NIH, 10 Center Drive MSC 1428, Bldg 10, Room 5N226, Bethesda, MD 20892-1428, USA
Accepted 17 April 2004
Available online 10 July 2004
Abstract
Objective: The main obstacle in interpreting EEG/MEG data in terms of brain connectivity is the fact that because of volume conduction,
the activity of a single brain source can be observed in many channels. Here, we present an approach which is insensitive to false connectivity
arising from volume conduction.
Methods: We show that the (complex) coherency of non-interacting sources is necessarily real and, hence, the imaginary part of coherency
provides an excellent candidate to study brain interactions. Although the usual magnitude and phase of coherency contain the same
information as the real and imaginary parts, we argue that the Cartesian representation is far superior for studying brain interactions. The
method is demonstrated for EEG measurements of voluntary finger movement.
Results: We found: (a) from 5 s before to movement onset a relatively weak interaction around 20 Hz between left and right motor areas
where the contralateral side leads the ipsilateral side; and (b) approximately 2–4 s after movement, a stronger interaction also at 20 Hz in the
opposite direction.
Conclusions: It is possible to reliably detect brain interaction during movement from EEG data.
Significance: The method allows unambiguous detection of brain interaction from rhythmic EEG/MEG data.
q 2004 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.
Keywords: Coherence; EEG; Connectivity; Motor control; Volume conduction; Time-lag
1. Introduction
The millisecond temporal resolution of electroencepha-
lography (EEG) and magnetoencephalography (MEG)
measurements make these techniques ideal candidates to
study the brain as a dynamic system. Recently, much
attention has been paid to interpreting rhythmic EEG/MEG
activity in terms of brain connectivity. Probably the simplest
and most popular measure of ‘interaction’ at a specific
frequency is coherence, a generalization of correlation to the
frequency domain (Nunez et al., 1997, 1999). Coherence is
almost always studied as a relation between EEG or MEG
channels while one is interested in relations between brain
sites. Since the activity of a single generator within the brain
is typically observable in many channels outside the head,
with details of this mapping depending on the volume
conductor (Sarvas, 1987), it is likely that a relation between
channels is rather a trivial ‘volume conduction artefact’ than
a reflection of an underlying interacting brain (Nunez et al.,
1997). Mathematically similarly, for EEG one always needs
a reference. If this reference is the same for the electrode
pairs being studied, it can contribute significantly to the
coherence, and thus, relative power changes may also affect
coherencies without reflecting a change in coupling (Fein
et al., 1988; Florian et al., 1998).
A plausible attempt to avoid artefacts from volume
conduction is first to apply an inverse method (see Baillet
et al. (2001) for an overview) to the data and then to
calculate coherence or any other measure of interaction of
the estimated source amplitudes (Gross et al., 2001). The
problem is that a fully satisfactory inverse method does not
exist and cannot exist (Sarvas, 1987). Any inverse method is
based on prejudices about the underlying sources. If these
prejudices are wrong (and sometimes even if they are
correct) the separation of channel amplitudes into source
amplitudes will be wrong (incomplete), and again it is likely
Clinical Neurophysiology 115 (2004) 2292–2307
www.elsevier.com/locate/clinph
1388-2457/$30.00 q 2004 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/j.clinph.2004.04.029
* Corresponding author. Tel.: þ1-301-435-1578.
E-mail address: nolteg@ninds.nih.gov (G. Nolte).

that artefacts of volume conduction will be misinterpreted
as brain interaction.
In this paper, we interpret brain interaction from a
quantity, namely the imaginary part of coherency, which
itself cannot be generated as an artefact of volume
conduction as will be shown below. The fundamental
assumption we make to arrive at this conclusion is that the
the quasi-static approximation holds for EEG, i.e. that an
observed scalp potential has no time-lag to the underlying
source activity, which is indeed widely accepted (Stinstra
and Peters, 1998). The imaginary part of coherency is only
sensitive to synchronizations of two processes which are
time-lagged to each other. If volume conduction does not
cause a time-lag, the imaginary part of coherency is hence
insensitive to artifactual ‘self-interaction’.
It is conceivable that in many experiments this imaginary
part is very small or even vanishing if the time-lag between
two processes is small or even vanishing. Therefore, it is
likely that our approach misses parts and in the worst case
all of the brain interaction. However, because of this special
property of being inconsistent with non-interacting sources,
in our opinion, it deserves special attention. If we find a non-
vanishing imaginary part, the interpretation as a reflection of
true brain interaction is almost immediate.
Thus, the experimental question is whether an imaginary
part of coherency can be observed in real data. We will
exemplify this for EEG measurement of voluntary left and
right-hand movements, which has been the objective of
many EEG coherence studies (Andrew and Pfurtscheller,
1999; Florian et al., 1998; Gerloff and Andres, 2002; Ginter
et al., 2001; Leocani et al., 1997; Manganotti et al., 1998;
Mima et al., 2000; Pfurtscheller and Andrew, 1999). While
an increased coherence at around 20 Hz (beta-activity)
between electrodes over the left and right motor areas
during movement and a decrease after movement is well
known, it has been argued that these effects are essentially
artefacts of volume conduction and that beta-activity is not
involved in the connection between left and right motor
areas (Andrew and Pfurtscheller, 1999; Pfurtscheller and
Andrew, 1999). Regardless of whether this is right or wrong,
of special interest to this paper is the debate itself.
Coherence (as an absolute value) is ambiguous: any
outcome is perfectly consistent with non-interacting
sources. In contrast, such a debate is unnecessary when
analyzing the imaginary part of coherency. We show here
that in fact beta-activity is involved in the communication
between left and right motor areas, but that the timing of
these interactions differs from the one expected from
‘standard’ coherence analysis.
This paper is organized as follows. In Section 2.1 we
recall the definition of coherence. A coherence matrix
contains a large amount of information. Our solution as to
how to visualize this information is presented in Section 2.2.
In Section 2.3, we discuss the special role of the imaginary
part of coherency, and in Section 2.4, we present the
statistics of coherency including a method to control for
multiple comparisons. In Section 3, we give the results for a
simple motor task. The paradigm is explained in Section 3.1
and the results of standard coherence and power analysis are
presented in Section 3.2. Our main results are shown in
Section 3.3, which contains the results for the imaginary
part of coherency. Finally, we discuss our findings in
Section 4.
2. Theoretical aspects of coherency
2.1. Definition of coherency/coherence
Coherency between two EEG-channels is a measure of
the linear relationship of the two at a specific frequency.
Here, we recall the basic definitions (Nunez et al., 1997).
Let xiðf Þ and xjðf Þ be the (complex) Fourier transforms of
the time series x^iðtÞ and x^jðtÞ of channel i and j; respectively.
Then the cross-spectrum is defined as
Sijðf Þ ; kxiðf Þxpj ðf Þl ð1Þ
where p means complex conjugation and k l means
expectation value. In practice, the expectation value can
only be estimated as an average over a sufficiently large
number of epochs. Coherency is now defined as the
normalized cross-spectrum:
Cijðf Þ ;
Sijðf Þ
ðSiiðf ÞSjjðf ÞÞ1=2
ð2Þ
and coherence is defined as the absolute value of coherency
Cohijðf Þ ; lCijðf Þl ð3Þ
We note that the terminology varies in different papers.
Since it is the major objective of this paper to exploit phase
structure as shown below, we use the two terms ‘coherency’
and ‘coherence’ to distinguish the full complex information
from its magnitude.
In the case of ‘event-related coherence’, we are
interested in the dependence of coherency as a function of
the time relative to a given stimulus. We then divide a long
epoch (typically in the order of a few seconds) into segments
of length T (typically between 250 ms and 1 s) which are
small enough for the desired time-resolution, given by T
itself, and large enough for the desired frequency resolution,
given by 1=T : Coherency then becomes a function of both
frequency and time
Cijðf Þ! Cijðf ; tÞ ð4Þ
where t indicates the time of the center of the respective
segment.
Coherency essentially measures how the phases in
channel i and j are coupled to each other. In the following,
frequency dependence is implicitly understood. If we
write the Fourier transformed signals as xi ¼ ri expðiFiÞ
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–2307 2293

and xj ¼ rj expðiFjÞ then the cross-spectrum becomes
Sijðf Þ ¼ krirj expðiDFÞl ð5Þ
where DF ¼ Fi 2Fj is the phase difference between the
signals in channel i and j at a specific frequency. The cross-
spectrum is the average of expðiDFÞ weighted with the
product of the amplitudes ri and rj: For coherency, we
merely normalize with respect to the ‘global amplitudes’
kr2i l1=2 and kr2j l1=2: If the signals in the two channels are
independent, DF is a random number and the coherency is
zero.
It is worthwhile to compare coherency with (1:1) ‘Phase-
Locking’ or ‘Phase Synchrony’ defined as an unweighted
average
P ¼ kexpðiDFÞl ð6Þ
Lachaux et al. (1999) gave two arguments why phase
synchrony is preferable to coherency.1 First, it is argued that
coherency can only be applied to stationary signals, and
second, it is noted that phase synchrony specifically
quantifies phase relationships. We disagree with the first
argument. Coherency is a characteristic quantity of a
stationary as well as a non-stationary process. Only if we
interpret it as a parameter of a stationary process we actually
assume stationarity. Similarly, we do not assume that the
processes are linear by calculating a linear measure. We
only look at the linear properties.
The second argument is more subtle. Phase synchrony is
indeed a clearer measure of the phase relationship only, and
if DF is statistically independent of the amplitudes, there is
no reason to weight with respect to amplitudes. However,
the question is whether this is the case, and, if not, whether
the weights result in statistically more robust estimators of
phase relationships. Note, that independence of phases and
amplitudes leads to
Cij ¼ kexpðiDFÞl
krirjl
ðkr2i lkr2j lÞ1=2
ð7Þ
and since
krirjl
ðkr2i lkr2j lÞ1=2
# 1 ð8Þ
we find
lCijl # lPijl ð9Þ
In our experience, this inequality is (slightly) violated in
real data examples and the results for coherency are
(slightly) more robust than for phase synchrony. To some
extent, this is surprising since phase synchrony is obviously
less sensitive to outliers. However, this result indicates a
dependence of the phase difference on the amplitudes: if the
signal is weak it is more likely that noise destroys the phase
structure. We want to emphasize that we do not claim that
this is always the case but we do believe that the question of
what quantity is preferable is ultimately a statistical and not
a conceptual one.
2.2. Visualising coherency
A coherence/coherency matrix contains an enormous
amount of information. In order to assess this information,
looking at all connections in one plot is very helpful. Our
solution to this problem is presented in Fig. 1 where we
show coherence in the alpha band at rest. The single large
circle represents the whole scalp. At each electrode position,
we place a small circle also representing the scalp and
containing the coherence of the respective electrode with all
other electrodes, i.e. the ith small circle contains the ith row
of the coherence matrix lCijl: In Fig. 1, we observe strong
coherence with all neighboring electrodes which is (at least
qualitatively) consistent with a uniform distribution of
independent sources in the brain.
In order to avoid overlapping circles, the positions have
been slightly shifted. This transformation is shown in Fig. 2.
Since it is very tedious to do this manually, we developed an
(quite heuristic) algorithm to shift the electrodes: we regard
them as a set of particles with a strong short-range repulsive
force (to avoid overlaps) and a weak long-range force (to
keep the circles together) attracting the circles to the overall
center (typically at CZ). In an iterative procedure, the
electrodes ‘move’ a small step proportional to the force until
a satisfactory solution is found. The coordinate transform-
ation only affects the position of the small circles within the
large one but not the electrode locations within the small
circles. Within each small circle, we placed a very small
black dot marking the original electrode location. Without
the transformation, the relative position of a dot within the
small circle is the same as the relative position of that small
circle within the large one. By looking at the positions of the
small dots we can then qualitatively assess the impact of the
transformation.
2.3. The special role of the imaginary part of coherency
We intend to interpret coherency between EEG channels
as reflecting an interaction between different brain sites.
Probably the biggest problem for that is the fact that the
activity of a single source is measurable in many channels.
This is usually referred to as ‘volume conduction’.
Especially close-by electrodes are highly coherent which
reflects redundancies in the measurement rather than brain
interaction. Formally speaking, any coherence matrix is
consistent with non-interacting sources. Differentiating true
brain dynamics from artifactual results caused by volume
conduction is therefore a highly non-trivial task.
A way to avoid this problem could be to first estimate the
activity at the brain sources instead of EEG/MEG channels
using an appropriate inverse method. However, the inverse
1 In that paper phase synchrony is defined as the absolute value of P
which, however, is irrelevant for this discussion.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–23072294

problem is not solvable in principle. To end up with a
mathematically unique solution, one has to impose a large
number of constraints reflecting the researcher’s prejudices
about the underlying source rather than the unknown
properties of the actual true source.
Here, we pursue a different approach. We isolate that part
of coherency which necessarily reflects interaction and that is
given by the imaginary part of the coherency. This does not
replace the ultimate goal to also localize the sources of inter-
action, but the interpretation of observed synchronizations
Fig. 1. Coherence in the alpha range for one subject.
Fig. 2. Original electrode locations (left) are slightly shifted to avoid overlapping spheres (right).
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–2307 2295

as brain interactions does not depend on the validity of the
inverse method. To see this, let us assume that the signals in
channel i and j arise from a linear superposition of K
independent sources skðf Þ
xiðf Þ ¼
XK
k¼1
aikskðf Þ ð10Þ
and similarly for xjðf Þ: We further assume that mapping of
sources to sensors is instantaneous, implying that the phases
are not distorted resulting in real coefficients aik and ajk:
We then have for the cross-spectrum
Sijðf Þ ¼ kxiðf Þxpj ðf Þl ¼
X
kk0
aikajk0 kskðf Þspk0 ðf Þl
¼
X
k
aikajkkskðf Þspkðf Þl ¼
X
k
aikajklskðf Þl2 ð11Þ
which is real. Since the normalization is also real, it follows
immediately that coherency is also real.
For the derivation, we assumed a linear superposition of
sources which is certainly justified because the Maxwell
equations are linear. The assumption that the mapping
between sources and sensors is free of phase shifts is less
trivial to assess. Note, that a phase shift in the frequency
domain corresponds to a time-lag in the time domain. The
imaginary part of coherency is insensitive to artifactual
‘self-interaction’ caused by volume conduction because a
signal is not time-lagged to itself.
In fact, our assumption that volume conduction does not
cause phase shifts follows from the validity of the quasi-
static approximation of the forward problem, stating that
one can ignore time-derivatives in the Maxwell equations,
which actually depends on the frequencies under study.
Plonsey and Heppner (1967) estimated the quasi-static
approximation to be valid below 2 kHz. In a detailed study,
Stinstra and Peters (1998) found no phase shifts for frequen-
cies below 100 Hz for both EEG and MEG (higher
frequencies were not analyzed). Since we are interested in
frequencies in typical EEG bands ð, 50 HzÞ; we believe
that our assumption is justified.
At this point, we would like to make clear what we
precisely mean by stating that the imaginary part of
coherency is insensitive to artifacts of volume conduction.
Since we still measure at channels and since the source
amplitudes are ‘volume conducted’ to the electrodes, this
volume conduction affects where we measure what
interaction. Furthermore, the coherency is normalized with
respect to the diagonal elements of the cross-spectrum
which belong to the real part of it and are also affected by
non-interacting sources. Adding non-interacting sources
(e.g. noise) causes a decrease in the imaginary part of
coherency. However, it can never cause an increase und thus
it cannot ‘create’ a non-vanishing imaginary part of
coherency. The situation is slightly different for the cross-
spectrum itself. While there, too, volume conduction affects
what signal is observable at what channel, non-interacting
sources do not affect the result at all-apart, of course, from
random fluctuations which vanish in the mean.
The above result is, in our opinion, a relatively trivial
observation which just has not been exploited so far.
Magnitude and phase of coherency are common measures of
connectivity in many studies. Since the real and imaginary
parts of coherency are just a different representation
(Cartesian instead of polar) of the complex coherency, we
do not calculate different quantities but rather we look at
coherency from a different viewpoint.
Although magnitude and phase contain the same
information as the real and imaginary parts, there are subtle
but important advantages/lack of disadvantages to look at
the imaginary part instead of the phase: (1) Non-interacting
sources do not lead to small but rather to random phases. We
cannot interpret a phase without having an estimate of its
significance at the same time. (2) One usually calculates
coherency with respect to a baseline (a rest condition). Since
in the individual coherencies the real parts are typically
much larger than the imaginary parts, the phase flips by p
depending on whether the real part of coherency is larger in
the rest or active condition. The interesting structure is
easily obscured by this rather meaningless effect. (3) Phase
is usually regarded as an additional information about time
delay between two processes. However, volume conduction
strongly affects the real part but does not create an
imaginary part. Processes can appear to be synchronized
with almost vanishing time delay while it is only the volume
conducted copies of the signals which do not have a time
delay.
An illustration of the imaginary part of the same data as
for Fig. 1 is given in Fig. 3. In contrast to the absolute value
of coherency, we observe interesting structure. Although
very blurred, we can see interactions between occipital and
left parietal electrodes. The imaginary part of coherency
between parietal and occipital electrodes is positive (the
central circles are red in occipital regions) which means the
central activity precedes the occipital one.
For comparison, we also show the phase of coherency in
Fig. 4. The structure now looks very different. Significant
deviation from zero phase can be seen for very remote (here:
frontal and occipital) pairs of electrodes. However, this does
not necessarily reflect an interaction. In fact, similar patterns
can be seen for all frequencies, and they are likely to be a
consequence of anti-correlated ‘self-interaction’ due to
opposite polarity of the electric potential of neuronal dipoles
in remote electrode pairs. If the imaginary part is negligible,
then anti-correlation leads to phases fluctuating near ^p.
Interestingly, although the imaginary part of coherency is
easily calculated from coherence and phase, we see this
occipital-parietal interaction neither in the coherence (Fig. 1)
nor in the phase (Fig. 4) themselves which are both
dominated by the effects of volume conduction. Finally, we
note that the shown plots are actually based on one-quarter
of the available data for this subject. The same plots for
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–23072296

Fig. 3. Imaginary part of coherency in the alpha range for the same subject as in Fig. 1.
Fig. 4. Phase of coherency in the alpha range for the same subject as in Fig. 1.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–2307 2297

the other quarters are almost undistinguishable by eye from
the ones shown.
2.4. Statistics
2.4.1. The statistics of a single pair of channels
The statistic for coherence is well known and described
in detail; e.g. by Rosenberg et al. (1989) or by Amjad et al.
(1997). If c is a coherence calculated from N Gaussian
distributed observations, then for large N; the Fisher’s Z
transform arctanhðcÞ is approximately Gaussian distributed
with a standard deviation of approximately 1=
ffiffiffiffi
2N
p
: This is a
very good approximation unless c is very close to zero. In
this case, c is distributed like a radius of a two-dimensional
Gaussian distribution and the standard deviation is
overestimated.
We are interested in the statistic of (complex) coherency.
The stabilizing Z-transform of a coherency Cij then becomes
a scale transformation in the complex plane:
Cij !
Cij
lCijl
arctanhðlCijlÞ ; ~Cij ð12Þ
and ~Cij is approximately Gaussian in the two-dimensional
complex plane. To discuss the covariance matrix, let us
first assume that the true phase is zero and later perform
a rotation to arbitrary phases. In this case, the imaginary
part fluctuates randomly around zero and must be
uncorrelated with the real part. For the variance of the
real part, one obtains the usual result 1=2N: In contrast to
coherence, the real part of coherency is not constrained
to non-negative values and the approximation is also
valid for real parts close to zero. The variance of the
imaginary part is, in general, reduced due to the presence
of a non-vanishing real part. The non-transformed
imaginary parts have variance ð12 lCijl2Þ=2N which
follows immediately from the variance of the phase
approximately given by ð1=lCijl2 2 1Þ=2N (Mima et al.,
2000). Due to the transformation, the imaginary part is
additionally scaled by a factor arctanhðlCijlÞ=lCijl and the
total variance reads
varðImðCijÞÞ ¼
ð12 lCijl2Þ
2N
arctanh2ðlCijlÞ
lCijl2
ð13Þ
For general phases, the same arguments as for the real
and imaginary parts apply for the coordinates in the
direction of the phase and orthogonal to it, respectively.
In this case, the variances of real and imaginary parts are
found by projection and read
varðReðCijÞÞ ¼
1
2N
ðgðCijÞsin2ðFÞ þ cos2ðFÞÞ ð14Þ
varðImðCijÞÞ ¼
1
2N
ðgðCijÞcos2ðFÞ þ sin2ðFÞÞ ð15Þ
with the abbreviation
gðxÞ ¼ ð12 lxl2Þ arctanh
2ðlxlÞ
lxl2
ð16Þ
To obtain an averaged coherency (over subjects) or a
difference of coherencies, the average/difference is taken for
the Z-transformed coherencies and the variances add
accordingly. An exception to this rule is an average over
time which is done for the cross-spectra since we can regard
that as an increased number of epochs. A P-value is
calculated from the number of standard deviations the
measured quantity differs from zero. The averaged/differ-
enced coherency is finally transformed with the inverse
~Cij !
~Cij
l ~Cijl
tanhðl ~CijlÞ ð17Þ
2.4.2. Correction for multiple comparisons using the false
discovery rate
To assess whether a specific spatial pattern in coherency
is significant, we must correct for multiple comparisons.
Because coherencies are largely redundant in neighboring
channels with dense electrode settings, a Bonferroni
correction is surely overconservative. Here, we adopt the
False Discovery Rate (FDR), well established in functional
magnetic resonance imaging (Benjamini and Hochberg,
1995; Benjamini and Yekutieli, 2001; Genovese et al.,
2002).
FDR controls for the rate of true detections out of all
detections. The general procedure is as follows: For N
comparisons, the P-values are sorted in ascending order
ðpi; i ¼ 1;…;NÞ and one finds the maximum of the ðpiÞ
which satisfies the relation
pi ,
ai
NcðNÞ ð18Þ
where a is the level we control at. All P-values lower or
equal to this maximum are regarded as significant detection.
The function cðNÞ should be set according to the statistics of
the P-values: for general distributions cðNÞ ¼ PNi¼1 1=i and
for positively correlated P-values, it is sufficient to set
cðNÞ ¼ 1: Here, positively correlated P-values mean that an
increased deviation from the null-hypothesis in one
electrode-pair does not lead to decreased deviations in
other pairs. Regardless of whether one uses the ‘pessimistic’
ðcðNÞ ¼ P1=iÞ or ‘optimistic’ ðcðNÞ ¼ 1Þ variant of FDR,
the lowest P-value is always compared to the Bonferroni
level a=N:
Setting, e.g. a ¼ 0:05 means that on average, 95% of all
detections are true detections. The ratio true detections/all
detections is defined as always being one if the denominator
(and hence also the numerator) is zero. This means that if
there is no true effect in (at least) 95% of hypothetical
repetitions of the experiment, one does not detect anything
significant.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–23072298

3. Interhemispheric connectivity in voluntary
finger movement
3.1. Paradigm and preprocessing
In this experiment, nine right-handed subjects were
asked to perform brisk voluntary movement with either left
or right-hand fingers lasting approximately 1 s with an
interval of about 10 s between movements. The subjects
were supposed to switch randomly between left and right-
hand finger movements which, in practice, turned out to be
dominated by alternating movements.
The EEG was measured continuously in 122 channels at
a sampling rate of 1 kHz with reference set to the right
earlobe. Impedances were kept below 5 kV. Four occipital
channels were taken out either because they were corrupted
by artefacts or because different analogue filter settings were
chosen, which significantly affects the analysis of the
imaginary part of coherency. EMG and EOG were measured
simultaneously. The remaining EEG channels were divided
into 20 s epochs with movement onset set to 10 s. Each
epoch was divided into 80 non-overlapping segments of
250 ms duration. Since the epochs are very long, there were
eye-blinks in almost all of them. Therefore, we corrected for
eye-blinks (and other artefacts) segment-wise by taking out
an artefact if the maximum of the detrended signal is above
100 mV in any of the EEG or EOG channels. Coherency was
calculated for each segment and for all electrode pairs by
applying a Fourier-transform to Hanning windowed data.
3.2. Coherence and power
In this motor experiment, we expect to find coherence/
coherency especially between electrodes over central areas
close to motor regions such as C3 and C4 which is the usual
approach in the literature. Let us first look at the coherence
between C3 and C4 as a function of time and frequency
which is presented in Fig. 5 for left and right-hand finger
movement and with and without subtraction of a baseline.
Here (and for all the following plots), we averaged over all 9
subjects. If we subtract a baseline, calculated from a time-
average between 27.5 and 25 s, we observe a strong
increase of coherence during movement and a decrease of
coherence after movement (left column) which is well
known in the literature. It is instructive to also look at the
same coherence without subtracting a baseline (right
column). Apparently, coherence is very low in the beta-
range almost during the whole cycle but returns to ‘normal’
during movement. This figure suggests that an observed
increase of coherence with respect to baseline during
movement is due to the absence of beta-activity during
activity which itself is just less coherent than background
noise. In view of the later analysis of the imaginary part of
coherency, we also plotted the difference of coherences
between left and right-hand movement. Although the main
effects (of course) cancel out, slight and temporarily blurred
differences remain.
That the absence or presence of activity in the beta range
is a factor contributing to the observed increased or
decreased coherences is supported by Fig. 6. We observe
synchronization and desynchronization at the same times as
we observe changes in coherence. Interestingly, the power
also changes in the alpha range with less pronounced change
in coherence. This discrepancy between alpha and beta
indicates that a mere change in the strength of the rhythm is
probably too simple an explanation for the observed
changes in coherence. For a more detailed discussion of
phenomena and possible explanations for coherence in the
alpha range, we refer to the work of Florian et al. (1998). We
emphasize that we do not claim that coherence is only a
consequence of volume conduction. It is just difficult to
interpret: we do not know what part reflects real interaction
and what part is due to volume conduction.
For completeness, we present the spatial pattern of power
in the beta range during and after movement in Fig. 7. These
findings are well known. Desynchronization during move-
ment is essentially bilateral while synchronization after
movement is clearly contralateral.
Finally, we show the full coherence during (at t ¼ 125 ms)
and after (averaged from 2 to 4 s) movement for left hand
finger movement in Figs. 8 and 9, respectively. During
movement we observe three rather than two foci of activity:
left and right motor area and a slightly frontal area. While left
and right motor areas are common candidates for coherence
analysis for this type of paradigm, the frontal area is
somewhat a surprise. Again, whether the observed coherence
reflects a true interaction is difficult to deduce from coherence
itself. Even more difficult is interpreting the spatial pattern of
coherence after movement which also shows somewhat
central, but at least spatially asymmetric structures. It is
conceivable that the relevant activities stem from SMA but a
detailed source analysis is beyond the scope of this paper.
3.3. The imaginary part of coherency
The imaginary part of coherency between C3 and C4 as a
function of time and frequency is displayed in Fig. 10. The
most prominent feature is a burst approximately 2–4 s after
movement. For left hand finger movement, this imaginary
part is positive and for right-hand movement, it is negative.
This is most clearly seen in the left column where a baseline is
subtracted but it is even visible in the coherency without
baseline subtraction, although a relatively strong background
activity is present mainly in the alpha but also in the beta
range.
In general, if the imaginary part of Cðx; yÞ is positive,
then x and y are interacting and x is earlier than y; indicating
that information is flowing from x to y: At specific
frequencies, however, ‘earlier’ and ‘later’ are ambigious;
e.g. at 10 Hz 10 ms earlier is the same as 90 ms later. For the
present interpretation we assumed that the smaller delay in
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–2307 2299

Fig. 5. Coherence as a function of time and frequency for right and left hand finger movement. In the left column, a baseline was subtracted consisting of the
coherence time-averaged in the interval ½27:5 s;25 s�:
Fig. 6. Relative power as a function of time and frequency in C3 and C4. Displayed is logðP=PrestÞ where Prest is calculated from a time average between 27.5
and 25 s.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–23072300

absolute value is the more probable explanation; e.g. in the
above example we would favor ‘10 ms earlier’ over ‘90 ms
later’. Note, that we can make this interpretation just from
the sign of the imaginary part of coherency without actually
calculating a delay for which we would need a reliable real
part of coherency.
The signs for the post-movement activity indicate that the
interaction is directed from the ipsilateral to the contralateral
side. This might be considered as surprising since, generally,
the contralateral hemisphere is expected to control move-
ments. However, over a relatively long period of approxi-
mately 5 s prior to movement, we also observe the opposite
Fig. 7. Spatial pattern of relative power in the beta-range during movement (at t ¼ 125 ms) and after movement (averaged from 2 to 4 s) for left and right-hand
finger movement. Displayed is logðP=PrestÞ where Prest is calculated from a time average between 27.5 and 25 s.
Fig. 8. Coherence in the beta range between all channel pairs during left hand finger movement.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–2307 2301

Fig. 9. Coherence in the beta range between all channel pairs after left hand finger movement.
Fig. 10. Imaginary part of coherency as a function of time and frequency for right and left hand finger movement. In the left column a baseline was subtracted
consisting of the coherency time-averaged in the interval [27.5 s, 2 5 s].
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–23072302

behaviour: the imaginary part of coherency between C3 and
C4 is largely negative. This can only be seen if we do not
subtract a baseline and most clearly if we look at the
difference between left and right-hand finger movement.
In contrast to coherence, subtracting left and right
finger movement for the imaginary part of coherency
enhances the signal we are interested in. This is indeed a
consequence of the antisymmetry of the imaginary part
ðImðCijÞ ¼ 2ImðCjiÞÞ: If we have a paradigm where the left
and right hemispheres switch their roles, then the difference
is potentially a very useful quantity. Let us denote by CRij and
CLij coherencies of right and left hand finger movement,
respectively. If we assume that
CLC4;C3 < CRC3;C4 ð19Þ
we get for the difference
ImðCLC3;C4Þ2 ImðCRC3;C4Þ < ImðCLC3;C4Þ2 ImðCLC4;C3Þ
¼ ImðCLC3;C4Þ þ ImðCLC3;C4Þ ð20Þ
and we double the signal. Note, that in this difference,
anything which is not task-related cancels out and hence
the task itself serves as an almost perfect baseline. Similar to
the study of lateralized readiness potentials (Vidal et al.,
2003), we also miss interactions which are identical in left
and right-hand finger movement. However, since for the
imaginary part of coherency, this difference gives the
clearest signal, from now on we will focus on this
difference. We want to emphasize that for this construction
to be meaningful it is unnecessary that the hemispheres
exactly switch their roles between the left and right-hand
paradigm; rather, we enhance that part which does have this
property. In other words, we look for this property rather
than assume it; Eq. (20) is the reasoning behind taking the
difference rather than an assumed accurate property of brain
interaction.
In Fig. 11, we show the statistically significant imaginary
part of coherency between C3 and C4 in a time-frequency
plot under various notions of ‘significance’. The post-
movement interaction is very strong and survives any type
of correction for multiple comparisons including a Bonfer-
roni correction. The pre-movement interaction is somewhat
weaker. In the optimistic (see Section 2.4.2) FDR method,
we see remnants of it as a few blue spots. For these data,
the pessimistic FDR method gives the same result as the
Bonferroni correction: nothing is significant apart from the
post-movement interaction.
The spatial patterns of the imaginary part of
coherency before movement (averaged in the interval
[25 s,0 s]) and after movement (averaged in the interval
[2 s,4 s]) are shown in Figs. 12 and 13, respectively.
The activity before movement is very clear: the
electrodes over the left motor area are coherent with
the electrodes over the right motor area and vice versa.
The stronger post-movement activity appears to be more
complex. Apart from an involvement of the left and right
motor area, frontal areas also seem to be involved. We
emphasize that an area specification such as ‘left motor
area’ cannot be accurate without making an inverse
calculation. Here, the assignments arising from the nature
of the paradigm are meant to be descriptive rather than
quantitative.
Finally, we show in Fig. 14 the significant part of the
imaginary part of coherency for the pre-movement activity
using the pessimistic FDR method. Compared to the time-
frequency plot, the statistics are much better because we
also averaged 20 non-overlapping time segments. The
effective cut-off in coherency makes the pattern appear less
blurred but essentially we obtain the same figure as when we
look at the whole coherency.
4. Discussion
In this paper we explored the imaginary part of
coherency as a reflection of true brain interaction in
contrast to artefacts from volume conduction which mainly
dominate coherence, the absolute value of coherency.
The fact that a significant imaginary part of coherency is
inconsistent with non-interacting sources is, in our opinion,
simple and obvious. Rather, the interesting question is
whether we can find such a non-vanishing imaginary part
in (literally) real data. Indeed, we found too much of it in
the sense that the task-dependent imaginary part of
coherency was masked by on-going rhythmic interactions.
In order to suppress these ongoing interactions, we
subtracted coherency from left and right-hand finger
movements which revealed information transfer from
contra- to ipsilateral hemisphere before movement and
from ipsi- to contralateral hemisphere after movement. It
must be noted that this interpretation of coherency in terms
of information flow is based on the relative timing of two
signals which is necessarily ambiguous if one looks at a
specific frequency.
However, what is not ambiguous is that the imaginary
part of coherency does reflect true interaction. Therefore, we
strongly suggest looking for that in any type of coherence
analysis. Since the calculation of an imaginary part is
typically an intermediate step to obtain amplitude and phase
of coherency, which are standard measures of coherence
analysis, we can hardly call our procedure a new method.
Rather, it is a new look at an old method. The rationale
behind it is that first we rigorously eliminated ‘self-
interaction’ caused by volume conduction. This largely
facilitates interpreting our measures in terms of
an interacting brain. What we have to show is: (a) that
a non-vanishing imaginary part is significant, and (b)
that an observed structure in the imaginary part comes
from the brain. Although, in practice, significance may turn
out to be difficult to prove, the concepts are straightforward.
In case of doubt, one can always run more subjects to clarify
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Fig. 11. Lower right panel of Fig. 10 with non-significant values set to zero for various notions of ‘significant’.
Fig. 12. Imaginary part of coherency in the beta range between all channel pairs time-averaged between25 and 0 s. Right-hand finger movement was regarded
as a baseline for left hand finger movement.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–23072304

Fig. 13. Imaginary part of coherency in the beta range between all channel pairs time-averaged over the post-movement interval from 2 to 4 s.
Fig. 14. Imaginary part of coherency in the beta range between all channel pairs time-averaged between 25 and 0 s with non-significant values set to zero.
Right-hand finger movement was regarded as a baseline for left hand finger movement. Significance was estimated by the pessimistic FDR method.
G. Nolte et al. / Clinical Neurophysiology 115 (2004) 2292–2307 2305

an issue. That an observed structure originates from the
brain is usually quite obvious from the qualitative spatial
pattern of the interaction. One can obtain a more
detailed picture of the origin of the coherency from
inverse calculations which, however, is beyond the scope
of this paper.
The important question in the context of this paper is
whether the imaginary part of coherency is superior to other
measures of brain connectivity. In our opinion, this is
clearly the case if the citerion is how robust the method is to
artefacts from volume conductions. Recently, more com-
plicated methods have become popular. In the directed
transfer function (DTF) approach, a linear autoregressive
model is fitted to the data and interaction is deduced and
characterized from mixing coefficients (Baccala and
Sameshima, 2001; Cassidy and Brown, 2003; Hesse et al.,
2003; Kaminski et al., 2001; Korzeniewska et al., 2003;
Mima et al., 2001). DTF can be regarded as a special version
of the more general concept of Granger causality (Chavez
et al., 2003; Hesse et al., 2003; Kaminski et al., 2001). A
signal A is said to Granger cause signal B if the present and
past of signal A contains information about the future of
signal B not contained in the present or past of signal B
itself. Ideally, this solves the problem of volume conduction
because a copy of a signal does not contain additional
information. The problem, however, is additive noise. If A
contains a simple and predictable signal plus white noise,
then all we can expect in the prediction of the future is a
good estimate of the predictable part. This estimate will be
improved if we average the channels A and B, the latter, say,
containing the same signal plus white noise independent of
the noise in A.
A similar argument applies to estimating the direction
of information flux. This direction is usually estimated
from the asymmetry in the Granger causality: the
information flux goes from A to B if A provides more
useful information to predict B than vice versa. Asymme-
tries, however, can also be due to asymmetric noise
levels. Let us discuss an extreme case: if A is noise-free
and contains only a simple and predictable signal and B
contains the same signal plus white noise then B cannot
contribute additional information to predict the future of A
but knowing A is optimal to predict the future of B. In
practice, channels are not noise-free but noise levels or
signal-to-noise-ratios vary substantially between channels,
and it is conceivable that an estimated direction of
information flux using Granger causality merely indicates
the direction from ‘good’ to ‘bad’ channels.
Coherency is a linear measure: for a linear (and
stationary) system the cross-spectrum, on which coherency
is based, completely determines the dynamics. There is
nothing more to know. For non-linear systems, however,
coherency might yield only an incomplete picture and more
general non-linear measures might be preferable. However,
‘sensitive to more’ also means ‘potentially less robust’.
With regard to the difficulty to even detect non-linear
dynamics in human EEG (Stam et al., 1999; Theiler and
Rapp, 1996), it is questionable whether non-linear methods
are superior to linear ones, unless, of course, the non-
linearity of the dynamics is the objective of a study itself.
We believe that the imaginary part of coherency is a very
useful measure to study brain connectivity. Being incon-
sistent with non-interacting sources its presence necessarily
reflects a dynamical interaction in contrast to volume
conducted ‘self-interaction’. Since volume conduction is, in
our opinion, the main obstacle in interpreting EEG/MEG
data in terms of brain connectivity, the value of such a
property cannot be overemphasized. It is possible that in
many studies the imaginary parts of coherencies essentially
vanish. This could mean that there is no interaction
(measured by EEG/MEG) or that the interaction between
two sources is not delayed or rather symmetrically delayed
such that none of the sources leads the other one.
This was found by Roelfsema et al. (1997), where the
authors report ‘zero time-lag synchronization among
cortical areas’ in visuomotor integration studied in cats.
However, the title of that paper is rather sketchy, e.g. the
authors found a small (<2 ms) but non-zero time-lag
between area 18 and 21. A time-lag of 2 ms at 20 Hz can
induce an imaginary part of the coherency up to sinð2
ms £ 2p=50 msÞ < 0:25 which is much larger than what we
have observed. Indeed, the authors argue that the observed
synchronization is not mediated by volume conduction
because the time-lag is non-vanishing and because the
cross-correlation is asymmetric. Since a symmetric cross-
correlation is equivalent to a vanishing imaginary part of
coherency, which follows from the fact that the imaginary
part of the cross-spectrum is the Fourier-transform of the
antisymmetric part of the cross-covariance, our argument is
also equivalent.
Larger time-lags in the beta-range were found by
Tallon-Baudry et al. (2001) in visual areas during rehearsal
of an object in visual short-term memory studied with
intracranial EEG in epilepsy patients. The authors report for
two subjects stable time-lags of 5.4 and 12.4 ms in waves
with period 50 and 62.5 ms, respectively. Note, that the
time-lag in the second subject is almost a quarter period
giving rise to an almost vanishing real part of the coherency.
There is always a trade-off between how much we want
to say about a system and how sure we are that what we say
is correct. By looking at the imaginary part of coherency, we
take an extreme position. We see, at best, only half of the
picture. But that half is safe. In conclusion, we do not think
that our approach can replace ‘classical’ analysis, but we
recommend analyzing the imaginary part of coherency
separately in all coherence studies.
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