Robustly Estimating the Flow Direction of Information in Complex Physical Systems
Guido Nolte,1 Andreas Ziehe,2 Vadim V. Nikulin,3 Alois Schlo¨gl,1 Nicole Kra¨mer,2
Tom Brismar,4 and Klaus-Robert Mu¨ller1,2
1Fraunhofer FIRST.IDA, Berlin, Germany
2Technical University of Berlin, Computer Science, Machine Learning Laboratory, Berlin, Germany
3Department of Neurology, Charite; Bernstein Center for Computational Neuroscience, Berlin, Germany
4Karolinska Institutet, Clinical Neurophysiology, Karolinska Hospital, Stockholm, Sweden
(Received 10 December 2007; published 10 June 2008)
We propose a new measure (phase-slope index) to estimate the direction of information flux in
multivariate time series. This measure (a) is insensitive to mixtures of independent sources, (b) gives
meaningful results even if the phase spectrum is not linear, and (c) properly weights contributions from
different frequencies. These properties are shown in extended simulations and contrasted to Granger
causality which yields highly significant false detections for mixtures of independent sources. An
application to electroencephalography data (eyes-closed condition) reveals a clear front-to-back infor-
mation flow.
DOI: 10.1103/PhysRevLett.100.234101 PACS numbers: 05.45.Tp, 05.45.Xt, 87.19.le
To understand interacting systems it is of fundamental
importance to distinguish the driver from the recipient, and
hence to be able to estimate the direction of information
flow. The direction can be estimated with a temporal argu-
ment: the driver is earlier than the recipient implying that
the driver contains information about the future of the
recipient not contained in the past of the recipient while
the reverse is not the case. This argument is the conceptual
basis of Granger causality [1] which is probably the most
prominent method to estimate the direction of causal in-
fluence in time series analysis. Granger causality was
originally developed in econometry, but is applied to
many different problems in physics, geosciences (cause
of climate change), social sciences, and biology [2–5].
The difficulty in realistic measurements in complex
systems is that asymmetries in detection power may as
well arise due to other factors, specifically independent
background activity having nontrivial spectral properties
and eventually being measured in unknown superposition
in the channels. In this case the interpretation of the
asymmetry as a direction of information flow can lead to
significant albeit false results [6]. The purpose of this
Letter is to propose a novel estimate of flux direction which
is highly robust against false estimates caused by con-
founding factors of very general nature.
More formally, we are interested in causal relations
between a signal of interest consisting of two sources
with time series xi�t� for i � 1, 2. The measured data y�t�
are assumed to be a superposition of these sources and
additive noise ��t� in the form
y �t� � x�t� � B��t�; (1)
where ��t� is a set of M independent noise sources which
are mixed into the measurement channels by an unknown
2�M mixing matrix B.
The proposed method is based on the slope of the phase
of cross-spectra between two time series with cross spectra
defined for channels i and j as
Sij�f� � hy^i�f�y^�j �f�i; (2)
where h�i denotes expectation value.
The idea behind using phase slope is that interactions
require some time, and if the speed at which different
waves travel is similar, then the phase difference between
sender and recipient increases with frequency and we
expect a positive slope of the phase spectrum. This is
most easily seen if we assume that the interaction is merely
a delay by a time �, i.e., y2�t� � ay1�t� �� with a being
some constant. In the Fourier domain this relation reads
y^2�f� � a exp��i2�f��y^1�f�. For the cross spectrum be-
tween the two channels one has S12�f� exp�i2�f��
exp�i��f��. The phase-spectrum ��f� � 2�f� is linear
and proportional to the time delay �. The slope of ��f� can
be estimated, and the causal direction is estimated to go
from y1 to y2 (y2 to y1) if it is positive (negative).
The idea here is now to define an average phase slope in
such a way that (a) this quantity properly represents rela-
tive time delays of different signals and especially coin-
cides with the classical definition for linear phase spectra,
(b) it is insensitive to signals which do not interact regard-
less of spectral content and superpositions of these signals,
and (c) it properly weights different frequency regions
according to the statistical relevance. This quantity is
termed ‘‘phase-slope index’’ (PSI) and is defined as
~� ij � =
�X
f2F
C�ij�f�Cij�f� �f�
�
; (3)
where Cij�f� � Sij�f�=
������������������������
Sii�f�Sjj�f�
q
is the complex coher-
ency, S is the cross-spectral matrix, �f is the frequency
resolution, and =��� denotes taking the imaginary part. F is
PRL 100, 234101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending13 JUNE 2008
0031-9007=08=100(23)=234101(4) 234101-1 © 2008 The American Physical Society
the set of frequencies over which the slope is summed. To
see that the definition of ~�ij corresponds to a meaningful
estimate of the average slope it is convenient to rewrite it as
~�ij �
P
f2FjCij�f�jjCij�f� �f�j sin���f� �f� ���f��.
For smooth phase spectra, sin���f��f����f��
��f�
�f����f� and hence ~� corresponds to a weighted aver-
age of the slope. We emphasize that since ~� vanishes if the
imaginary part of coherency vanishes it will be insensitive
to mixtures of noninteracting sources [7,8].
Finally, it is convenient to normalize ~� by an estimate of
its standard deviation
� � ~�=std� ~��; (4)
with std� ~�� being estimated by the jackknife method. In
the examples below we always show normalized measures
of directionality, and we consider absolute values larger
than 2 as significant.
We emphasize, that � indicates the temporal order of
two signals, which is then interpreted as a driver-responder
relation. For bidirectional (or unknown) coupling a finding
that, e.g., A drives B does not imply that B has no impact on
A. Rather, one cannot make a statement about the reverse
direction.
Estimations of cross spectra is standard [7,9] but tech-
nical details may differ. Here, we first divide the whole data
into epochs containing continuous data (4 s duration), then
we divide each epoch further into segments of time T, here
of 2 s duration corresponding to a frequency resolution of
�f � 0:5 Hz, multiply the data for each segment with a
Hanning window (a cosine, raised by 1, with first minima
at the edges of the segment), Fourier-transform the data,
and estimate the cross spectra according to Eq. (2) as an
average over all segments. The segments have 50% overlap
within each epoch but not across epochs. To apply the jack-
knife method, for each pair of channels we calculate ~�k
from data with the kth epoch removed for all k. The stan-
dard deviation of ~� is finally estimated for K epochs as����
K
p
� where � is the standard deviation of the set of ~�k
[10].
Our new method is compared to Granger causality using
autoregressive (AR) models both for wide and narrow band
analysis [11] with analogous normalization by the esti-
mated standard deviation. To estimate the parameters of
the model we here use the Levinson-Wiggens-Robinson
[12] algorithm available in the open Biosig toolbox [13].
Granger causality is defined as the difference between the
flux from channel 1 to 2 and the flux from channel 2 to 1
normalized to unit estimated standard deviation.
We illustrate typical results for two simple cases in
Fig. 1. To study a more general class of signals we simu-
lated data with structure
y �t� � �1� ��x�t�
Nx
� �B��t�
N�
: (5)
Here, the signal x�t� contains truly unidirectional informa-
tion flux and ��t� is undirected noise. Both the signal part
and the mixed noise part [B��t�] are normalized by the
Frobenius norms of the respective data matrices (Nx and
N�) and added with a parameter � controlling for the
relative strength.
The data were generated using AR models of order P �
5 for two channels. An AR model is defined as
z �t� �
XP
p�1
A�p�z�t� p� � ��t�; (6)
where A�p� are the AR matrices up to order P and ��t� is
white Gaussian noise with covariance matrix � chosen
here to be the identity matrix. For computing Granger
causality, the AR model was fitted with order P � 10.
All entries of AR matrices were selected randomly as in-
dependent Gaussian random numbers with A21�p� � 0 for
the signal part x�t�, corresponding to unidirectional flow
from the second to first signal, and A12�p��A21�p��0 for
the noise part ��t�, corresponding to independent sources.
Noise was mixed into channels with a random 2� 2 mix-
ing matrix B. The time constant implicit in the AR model
was assumed to be 10 ms, and we generated 60 000 data
points for each system and channel. This corresponds to a
Nyquist frequency of 50 Hz and to 10 min measurement.
We analyzed systems for all � in the range [0, 1] with step
0.1. For each � we analyzed 1000 randomly selected stable
systems with both methods and both for wide band (using
all frequencies) and narrow band analysis. For the narrow
band, we used a band of 5 Hz width, centered this band
around the spectral peak of the (known) signal of interest
0 10 20 30
−0.01
0
0.01
A
m
pl
itu
de
[a
.u.
]
Unidirectional flux
−50
0
50
G Ψ
0 10 20 30−0.02
−0.01
0
0.01
Time [bins]
A
m
pl
itu
de
[a
.u.
]
Correlated noise
−10
0
10
G Ψ
FIG. 1 (color online). Upper panels: Strong interaction from
second (dashed line in left panels) to first (full line in left panels)
signal estimated from 2000 data points generated with a simple
AR model of order 1. Both methods detect this direction cor-
rectly. Lower panels: mixture of pink and white noise. In contrast
to PSI, Granger causality erroneously still detects a significant
direction. The error bars in the right panels indicate estimated
95% error margins corresponding to 2 standard deviations. Time
series in the left panels were upsampled to 400 Hz.
PRL 100, 234101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending13 JUNE 2008
234101-2
and analyzed only cases where the band includes at least
60% of the total power of the signal of interest.
The fractions of significant false and significant correct
detections as a function of � are shown in Fig. 2. We
observe that for increasing noise level the fraction of
significant false detections for Granger causality comes
close to 50% while PSI rarely makes significant detections
at all. For PSI, the worst case observed is at � � 0:8 for the
wide band with 6% significant false detections. This level
can be reduced to about 3.5% if we increase the frequency
resolution to 0.25 Hz. However, the price is some loss in
statistical power and it is important to show that also the
proposed method might fail, even if it is unlikely in the
sense of the present simulation.
We observe similar significant correct detection rates for
small and moderate noise levels. For high noise level
Granger causality shows a much larger fraction of signifi-
cant correct detections which, however, is meaningless
given the large fraction of significant false detections.
We now apply the PSI to real electroencephalography
(EEG) data in rest condition [14]. For this, 88 healthy
subjects were recruited randomly by the aid of the
Swedish population register. During the experiment, which
lasted for 15 min, the subjects were instructed to relax and
keep their eyes closed. Every minute the subjects were
asked to open their eyes for 5 s. EEG was measured with
a standard 10–20 system consisting of 19 channels. Data
were analyzed using the linked mastoids reference. The
protocol was approved by the Hospital Ethics Committee.
The most prominent feature of this measurement is the
alpha peak at around 10 Hz. This rhythm is believed to
represent a cortico-cortical or thalamo-cortical interaction.
The direction of this interaction is an open question. While
it is mostly believed that this rhythm originates in occipital
(back part of the brain) or thalamic (which is deep) areas,
and spreads to other (more frontal) areas this view has also
been challenged [17,18].
For illustration we show results for PSI for one selected
subject in Fig. 3. The power (upper right panel), averaged
over the two occipital channels (O1 and O2), shows a very
strong peak at 9.5 Hz. PSI values were calculated for all
channel pairs with frequency resolution 0.5 Hz using a
frequency band of 5 Hz width centered around frequency
f. In the lower right panel we show the net information flux
at f � 9:5 Hz defined for the ith channel by �net�i; f� �P
j�ij�f�=std�
P
j�ij�f��. We clearly observe that frontal
channels are net drivers (�net > 0) and occipital channels
net recipients (�net < 0).
To show preferred direction for all pairs of channel and
for all frequencies we calculate the respective contribution
to a given direction: for channels i and j with locations ri
and rj in the two dimensional plane (as shown in the lower
right panel), respectively, we calculate the normalized
difference vector �rij � �rj � ri�=jri � rjj and project it
onto u � ��1; 0�T for right-to-left direction and onto u �
�0;�1�T for front-to-back direction. We finally calculate
the contribution of �ij�f� to direction u as �i;j�f;u� �
�ij�f�u � �rij.
Results for all channel pairs and for all frequencies are
shown for right-left information flow (upper left panel) and
for front-back information flow (lower left panel). We do
not observe any preferred direction in the right-left flow. In
contrast, the information flow in the front-back direction
shows a clear positive plateau at the alpha frequency (in-
dicated with the letter ‘‘B’’) meaning that typically the
frontal channels are estimated as the drivers. We also
observe a positive and negative peak (indicated with letters
‘‘A’’ and ‘‘C’’) at frequencies around 7 and 12 Hz, respec-
tively. Note that these peaks differ by the width of the
frequency band. They are clearly artifacts due to inade-
quate settings of the band. Specifically, the alpha rhythm
has a preferred phase (for given channel pair) which must
be distinguished from the slope of the phase. E.g., PSI at
around 7 Hz estimates the slope on the range from 4.5 to
9.5 Hz. The right edge of this interval just includes the
0 0.5 1
0
50
100
%
Signific. false, wide band
Ψ
G
0 0.5 1
0
50
100
Signific. false, narrow band
%
0 0.5 10
50
100
Signific. correct, wide band
γ
%
0 0.5 10
50
100
Signific. correct, narrow band
γ
%
FIG. 2 (color online). Fraction of significant detections of
Granger causality and PSI as a function of noise level �.
FIG. 3 (color). Upper right: signal power as a function of
frequency averaged over the two occipital channels O1 and O2
showing a clear alpha peak at f � 9:5 Hz. Lower right: net
information flux at f � 9:5 Hz. Left panels: PSI for all channel
pairs and all frequencies projected on right-to-left and front-to-
back direction, respectively. Middle panels: Average of PSI as a
function of frequencies over all channel pairs.
PRL 100, 234101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending13 JUNE 2008
234101-3
rising part of the phase but not the descending part leading
to positive values for PSI if the phase is positive.
We found a similar structure in about 60% of the sub-
jects. An average over all subjects of � between all chan-
nels is shown in Fig. 4. We also found a substantial
intersubject variability, both with regard to PSI and actual
phase at the alpha peak. The origin is unclear and goes
beyond the scope of this Letter. Granger causality did not
yield any consistent spatial pattern.
Recent neuroimaging studies have challenged a simple
view on a rest condition by showing a presence of default
states in the cortex, which display complex patterns of
neuronal activation [19,20]. We here show that not only
specific areas are coactivated during rest state, but they also
demonstrate at a gross level a preferential ‘‘default’’ mode
of information flow in the cortex. Importantly, the drivers
of such flow are mostly situated in the frontal areas, from
where many top-down attentional influences are thought to
be originated [21]. This suggests that the maintenance of
vigilance is a process displaying a coordination of neuronal
activity with well-defined drivers and recipients of infor-
mation flow.
To conclude, we presented a new method to estimate the
direction of causal relations from time series’ based on the
phase slope of the cross-spectra. We here defined an aver-
age of the phase slope such that this average is insensitive
to arbitrary mixtures of independent sources. We verified
this for random linear systems also showing that Granger
causality is highly sensitive to mixtures of independent
noise sources. Additionally, we showed that in situations
with combined unidirectional flow and undirected noise
our method correctly distinguished the two phenomena.
Application on real EEG data underlined the versatility of
our method as a universal tool for estimating causal flow in
noisy physical systems.
We acknowledge partial funding from DFG, BMBF,
BBCN Berlin and EU.
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FIG. 4 (color). Phase-slope index for all pairs of channels
averaged over all subjects each at the peak of the alpha rhythm.
The ith small circle is located at the ith electrode position and is
a contour plot of the ith row of the matrix with elements �ij. The
red color in frontal circles indicates that the frontal electrodes are
estimated as the drivers.
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