A novel method for reliable and fast extraction of neuronal EEG/MEG oscillations on the basis of spatio-spectral decomposition.

ct
iver
Article history:
Available online 27 January 2011
Keywords:
Oscillations
Synchronization
EEG
MEG
een shown to underlie various cognitive, perceptual and motor functions in the
ese oscillations is notoriously difficult with EEG/MEG recordings due to a massive
Neuronal oscillations constitute a major operational mode of brain used algorithms are based on independent component analysis (ICA)
NeuroImage 55 (2011) 1528–1535
Contents lists available at ScienceDirect
NeuroIm
j ourna l homepage: www.e lactivity. Many studies have shown that they underlie perceptual,
cognitive and motor functions (for reviews see: Buzsáki and Draguhn,
2004; Varela et al., 2001). Neuronal oscillations are ubiquitous, they
are generated in almost any area of the cortex as well as in the
subcortical structures, such as basal ganglia and thalamus (Steriade,
2001). Yet, exactly because there are so many sources of neuronal
oscillations, as well as a massive amount of background neuronal
noise, it is difficult to extract distinct sources of oscillatory activity.
This holds not only for non-invasive electrophysiological measures,
such as EEG/MEG, but also for the invasive approaches, such as
not requiring inverse modeling calculations. ICA extracts independent
sources by exploiting non-Gaussianity of the sources (Hyvärinen et al.,
2001), e.g., through the maximization of kurtosis, hyperbolic tangent,
skewness (FastICA algorithm, Hyvärinen and Oja, 1997) or maximiz-
ing information (infomax approach, Bell and Sejnowski, 1995).
Another class of ICA approaches utilizes temporal structure/spectral
differences between the sources (SOBI approach, Belouchrani et al.,
1997; Tang et al., 2005; or TDSEP/ffdiag approach, Ziehe et al., 2004).
Importantly, as noted by Hyvärinen et al. (2010) similar spectra
indicate similar (statistically) temporal correlations and vice versaelectro-corticographic recordings in patients
Abbreviations: CSP, common spatial patterns; IC
analysis; SOBI, second-order blind identification alg
decomposition; SNR, signal-to-noise ratio.
⁎ Corresponding author at: Department of Neurology
Berlin, D-12200 Berlin, Germany. Fax: +49 30 8445 426
E-mail address: vadim.nikulin@charite.de (V.V. Niku
1053-8119/$ – see front matter © 2011 Elsevier Inc. Al
doi:10.1016/j.neuroimage.2011.01.057Different decomposition techniques have been proposed for
extracting neuronal oscillations. In EEG/MEG research frequentlyIntroductionAlpharecordings. The method is based on a linear decomposition of recordings: it maximizes the signal power at a
peak frequency while simultaneously minimizing it at the neighboring, surrounding frequency bins. Such
procedure leads to the optimization of signal-to-noise ratio and allows extraction of components with a
characteristic “peaky” spectral profile, which is typical for oscillatory processes. We refer to this method as
spatio-spectral decomposition (SSD). Our simulations demonstrate that the method allows extraction of
oscillatory signals even with a signal-to-noise ratio as low as 1:10. The SSD also outperformed conventional
approaches based on independent component analysis. Using real EEG data we also show that SSD allows
extraction of neuronal oscillations (e.g., in alpha frequency range) with high signal-to-noise ratio andwith the
spatial patterns corresponding to central and occipito-parietal sources. Importantly, running time for SSD is
only a few milliseconds, which clearly distinguishes it from other extraction techniques usually requiring
minutes or even hours of computational time. Due to the high accuracy and speed, we suggest that SSD can be
used as a reliable method for the extraction of neuronal oscillations from multi-channel electrophysiological
recordings.
© 2011 Elsevier Inc. All rights reserved.. due to Wiener–
showed that dec
different spectra
ICA approaches m
oscillatory com
relatively narrow
fairly Gaussian
oscillatory sourc
A, independent component
orithm; SSD, spatio-spectral
, Charité - University Medicine
4.
lin).
l rights reserved.neuronal oscillations from multi-channel EEG/MEG/LFPRevised 13 December 2010
Accepted 20 January 2011overlap of activity frommultiple sources and also due to the strong background noise. Here we present a novel
method for the reliable and fast extraction ofReceived 6 October 2010
Neuronal oscillations have b
brain. However, studying thA novel method for reliable and fast extra
the basis of spatio-spectral decomposition
Vadim V. Nikulin a,b,⁎, Guido Nolte c, Gabriel Curio a,b
a Neurophysics Group, Department of Neurology, Campus Benjamin Franklin, Charité - Un
b Bernstein Center for Computational Neuroscience, Berlin, Germany
c Fraunhofer FIRST, D-12489 Berlin, Germany
a b s t r a c ta r t i c l e i n f oion of neuronal EEG/MEG oscillations on
sity Medicine Berlin, D-12200 Berlin, Germany
age
sev ie r.com/ locate /yn imgKhinchin theorem. Moreover, Ziehe et al. (2000)
orrelation in time domain is equivalent to exploiting
l content of the sources in the Fourier domain. These
ight meet intrinsic limitations for regular EEG/MEG
ponents which not only represent a mixture of
-band signals with very similar spectra but are also
distributed due to the amplitude modulation of
es (Hyvärinen et al., 2010). Moreover, the majority

of ICA methods are based on numerical approaches, which do not
always guarantee adequately converging solutions, and in addition,
computing ICA can be time consuming.
Here we introduce a novel method for the separation of oscillatory
sources from a set of mixed signals, which is a common problem in
EEG/MEG research. This method is based on a linear decomposition of
the recorded EEG/MEG signals guided by the objective to maximize
power at a peak frequency of oscillations and simultaneously
minimize the power at the neighboring frequency bins. We show
that such decomposition is equivalent to the maximization of signal-
to-noise ratio (SNR) and that it produces components with spectral
profiles characteristic for oscillatory sources. We refer to this method
as spatio-spectral decomposition (SSD). We present below a theoret-
ical basis for its efficiency to extract oscillatory activity, validate the
method with simulations, and present a first exemplary analysis of
real EEG data.
Material and methods
Motivation
1529V.V. Nikulin et al. / NeuroImage 55 (2011) 1528–15355 10 15 20 25 30
Frequency [Hz]
Po
w
er
, l
og
noise
Fig. 1. A schematic explanation of spatio-spectral decomposition algorithm. The main
idea of SSD is to find linear filters which maximize the power in the frequency band of
studied neuronal oscillations while minimizing the power at the neighboring “flanking”
frequency bins. Such procedure results in the extraction of the components, which haveOften signal and noise have overlapping frequency content and
thus their separation becomes difficult. A frequently used approach is
to record EEG/MEG activity in two different time windows
corresponding to a predominant presence of either noise or signal
of interest (Koles, 1991). However, there is not always a possibility to
record different sessions with either noise or signal. Therefore from
both the algorithmical and practical point of view it would be
desirable to estimate noise and signal parameters from the same
measurement.
An important assumption of our approach is that noise sources
produce signals with a relatively broad frequency range, e.g., from few
Hz to tens of Hz. Inmany cases the noise sources aremodeled either as
white or 1/f noise, the latter being typical for many experimentally
obtained LFP and EEG/MEG recordings. Because the noise is spectrally
extended we can use frequency bins around the frequency range of
interest in order to enhance the SNR by estimating the noise in the
spectral neighborhood of the frequency range of interest. Algorith-
mically, one minimizes the variance of the noise around the spectral
peak of interest while simultaneously maximizing the variance of the
signal at the peak frequency (Fig. 1). In the Fourier domain this leads
to a typical “peaky” profile of the spectrum with large power P at a
signalcharacteristic “peaky” spectrum typical for oscillatory signals.central peak flanked by considerably smaller power at the neighbor-
ing frequency bins, i.e., P(f−Δf)bP(f ) and P(f )NP(f+Δf), represent-
ing the characteristic shape of a spectrum for signals with good SNR.
Such simultaneous minimization of surrounding noise and maximi-
zation of the signal at the peak frequency can be achieved
straightforwardly with diagonalization of two covariance matrices
corresponding to the signal of interest and the surrounding “flanking”
noise. Below we elaborate on the details of our approach.
Spectral ratios
We assume an additive model for the signal s and noise n such that
the measured activity m is expressed as:
m = s + n ð1Þ
SNR in one sensor for frequency f is defined as:
SNR = Ps fð ÞPn fð Þ
ð2Þ
where Ps and Pn are the powers at the frequency f for the signal and
noise, respectively.
In general, we are interested in extracting oscillatory activity and
thus the spectrum of s is restricted to a small frequency range. In
addition we also assume that the noise spectrum is relatively
monotonous/smooth and in general its power is proportional to 1/fα
(for up to tens of Hz), whereα=0 andα=1 correspond to white and
1/f types of noise, respectively, the latter being the most common
noise encountered in EEG and MEG recordings.
Usually SNR cannot be reliably assessed from the recordings due to
theoverlapof s andn at the frequency f. Instead,weshowbelow thatone
can express SNR using different frequency bins, i.e., f−Δf and f+Δf.
If frequencies f−Δf and f+Δf are outside of the spectral peak of s
and Δf is relatively small then:
Pn fð Þ≈ Pm f–Δfð Þ + Pm f + Δfð Þ½ � = 2: ð3Þ
This is a linear approximation of the noise spectrum in the
frequency neighborhood of f.
Recall that:
Fm fð Þj j
2
D E
= Fs fð Þj j
2
D E
+ Fn fð Þj j
2
D E
+ 2ℜ Fs fð ÞF
�
n fð Þ
� �
ð4Þ
where Fm(f), Fs(f) and Fn(f) are the Fourier transforms of the
signals m, s and n, respectively.
If s and n are uncorrelated then ℜ〈Fs(f)Fn*(f)〉=0 and thus Eq. (4)
is reduced to:
Fm fð Þj j
2
D E
= Fs fð Þj j
2
D E
+ Fn fð Þj j
2
D E
ð5Þ
or
Pm fð Þ = Ps fð Þ + Pn fð Þ: ð6Þ
Taking into account Eqs. (3) and (6) we can write:
Pm fð Þ
Pm f−Δfð Þ + Pm f + Δfð Þ½ �
≈
Ps fð Þ + Pn fð Þ
2Pn fð Þ
= 0:5
Ps fð Þ
Pn fð Þ
+ 1
� �
: ð7Þ
This indicates that the increase of the ratio in the left part of Eq. (7)
should also lead to the increase in SNR defined in Eq. (2).
Algorithm
Taking into account the considerations provided above, we show
how to find spatial filters which would maximize SNR for the EEG/

1530 V.V. Nikulin et al. / NeuroImage 55 (2011) 1528–1535MEG recordings. We start with t × k measurement matrix M, where t
is the number of samples and k is the number of channels. Assuming
additivity of noise and signal the matrix can be written as:
M = S + N ð8Þ
where S and N correspond to signal and noise components,
respectively.
We then proceed with the filtering of each column in M in two
different ways. 1) M is filtered around signal frequency f and leads to the
matrix Ms. 2) Next we filter M separately around the frequencies f−Δf
and f+Δf and sum the filtered signals, yielding the matrix Mn.
Alternatively, Mn can be obtained by filtering M in the frequency range
[f−Δf:f+Δf]with the following subtraction ofMs . In additionMn canalso
be obtained by filtering M in the frequency range [f−Δf:f+Δf] and then
performing band-stop filtering around frequency f. In principle Δf should
be small, in order to get a better linear estimation of SNR. However, due to
the spectral leakage caused by band-pass filtering, realistic values for Δf
can be within 1–2 Hz.
Matrices Ms and Mn are of the same dimensions as M, and
correspond to the frequency ranges of: 1) signal plus noise and 2)
noise alone, respectively. The columns in each matrix are also
centered to have a zero mean. Next, we estimate the time-averaged
covariance matrices for Ms and Mn, respectively:
Cs =
MTs Ms
t ð9Þ
Cn =
MTnMn
t ð10Þ
Now one should find spatial filters which would be related to high
spectral peak (variance) at the frequency of interest and low variance
of the noise in the surrounding frequency bins. Denoting such spatial
filter as w→ the problem is hence to maximize:
SNR w→
� �
=
w→
T
Cs w
→
w→
T
Cn w
→
ð11Þ
which is a standard problem and leads to the generalized eigenvalue
decomposition (Fukunaga, 1990):
Cs w
→= λCn w
→
: ð12Þ
For the later discussion we reformulate the problem using a
coordinate transformation. Defining ‘whitened’ filters and signal
covariance as:
v→≡C1 = 2n w
→
ð13Þ
Ds≡C
−1 = 2
n CsC
−1 = 2
n : ð14Þ
Then, Eq. (11) reads:
SNR v→
� �
=
v→
T
Ds v
→
v→
T
v→
ð15Þ
and its maximization leads to the eigenvalue equation:
D v→= λ v→: ð16ÞsWith k solutions v→i and λi for i=1…k. Since Ds is a hermitian
matrix, all eigenvectors are orthogonal, i.e. v→Ti v
→
j = δij, where δij is a
Kronecker's delta or, equivalently, for V = v→1; ⋯; v
→
k
� �
we have:
V−1 = VT : ð17Þ
If we now assume that the signal is generated from k uncorrelated
sources the signal covariance matrix Cs has the form:
Cs = ∑
k
i=1
αi a
→
i a
→T
i ð18Þ
where a→i is the topography and αi the variance of the i-th source. In
the whitened space this covariance has the form
Ds = ∑
k
i=1
αi b
→
i b
→T
i ð19Þ
with b
→
i = C
−1 = 2
n a
→
i which is identical to the eigenvalue decompo-
sition provided that the b
→
i are orthogonal. In this case b
→
i is i-th
eigenvector of Ds, i.e. b
→
i = v
→
i and hence v
→
i = C
−1 = 2
n a
→
i. We
emphasize here that the abovementioned holds only if b
→
i are
orthogonal. Whether one can assume such orthogonality depends
on the case considered. With realistic and hence correlated noise the
whitening corresponds to a spatial high pass and dipolar patterns are
mostly orthogonal in the whitened space (this also relates to source
PCA andMinimumOverlap Component Analysis (Marzetti et al., 2008,
please see Appendix A).
We finally go back to the original space to derive some relations
between the spatial filter w→i and the corresponding topography a
→
i,
the former being one of the generalized eigenvectors of Eq. (12). Since
v→i = C1 = 2n w
→
i and v
→
i = C−1 = 2n a
→
i, for the latter assuming orthogo-
nality of the topographies in the whitened space as discussed above, it
follows that
a→i = Cn w
→
i ð20Þ
If we define W = w→1; ⋯; w
→
k
� �
and A = a→1; ⋯; a
→
k
� �
we can express
this as
A = CnW: ð21Þ
Recalling that V=Cn1/2W is orthogonal we get:
W−1 = C−1=2n V
� �
−1
= VTC1 = 2n = CnWð Þ
T = AT : ð22Þ
In other words, the topographies can be calculated either by the
inversion of W or by applying the noise covariance on it, and are
contained in the respective rows.
Simulations
We simulated EEG recordings for 64 channels, whichwere fitted to
the outermost layer of the standard Montreal Neurological Institute
head (Evans et al., 1994). The head model was based on a three
compartment realistic volume conductor and was used for calculation
of EEG forward solutions (Nolte and Dassios, 2005). The sources were
modeled as multiple current dipoles located in the triangularly
tessellated cortical mantle. For the generation of noise we used 500
uncorrelated dipoles producing 1/f type of noise. The dipoles had
random spatial location and orientation. Importantly, this type of
noise produces spatial correlations in the sensor space. The signal of
interest was band-pass filtered white noise (10–12 Hz, frequency
range as for the alpha oscillations). The simulations were performed
for 5 alpha dipoles with random placement in the cortical grid and

random orientation. The duration of the simulated data was 125 s and
sampling frequency 200 Hz. For simulations we used the following
SNRs: 5, 1, 0.5, 0.25, 0.01, 0.05, and 0.01. For each SNR we performed
100 simulations with 5 dipoles. The SNR was calculated as the ratio
between the mean variance across channels for each projected alpha
dipole and the mean variance of additive 1/f noise (produced by all
noise dipoles) in the 10–12 Hz frequency range. In order to obtain the
matrix Ms the data were filtered in the frequency range 10–12 Hz (we
elaborate in the Discussion on the basic principles related to defining
frequency bands when analyzing neuronal data). Matrix Mn was
obtained by band-pass filtering data in the range 8–14 Hz and then
performing in addition a band-stop filtering in the range 9–13 Hz (the
cut-off frequencies for such band-stop filtering were selected to avoid
spectral leakage from the signal part in 10–12 Hz range). Usually the
filters do not have a box-car shape and there is a certain roll-off
indicating that frequencies close to the edges of the filter contain non-
zero power. Because of this there might be some residual oscillations
present from the band-pass range in the flanking frequency bins and
vice versa. Thereforefilters should be designed to be sufficiently steep in
order to prevent power leakage. In the present studyweused band-stop
filtering extending ±1 Hz with respect to the band-pass filter edges. If
the flanking frequency bins would contain part of the signal then SSD
would not produce spatialfilters optimal for the extraction of oscillatory
sources, since the flanking covariance matrix Mn should ideally include
only noise component. For a comparisonwe alsoutilized a second-order
blind identification algorithm (SOBI, Belouchrani et al., 1997), which is
creating multiple time-delayed covariance matrices. We tested FastICA
(Hyvärinen and Oja, 1997), with kurtosis or hyperbolic tangent as
contrast functions, and found that it often did not converge and we
dropped this algorithm from the present comparison. In addition we
also tried to use ICA on the basis of the infomax approach (Bell and
Sejnowski, 1995; Delorme and Makeig, 2004, authors' default settings).
However, as in case of FastICA, infomax did not provide a satisfactory
convergence of the solutions. Such performance of FastICA and infomax
is most likely due to the very similar amplitude distributions of the
band-pass filtered components.
The error between recovered and original pattern was calculated
according to:
Err = 1−
aTorar
�
�
�
�
�
�
‖aor‖⋅‖ar‖
ð23Þ
where aor are original patterns and ar are patterns recovered through the
SSD or ICA. Since there is no exact ordering of the recovered patterns we
performed a pair-wise greedy search by first taking the pair with the
smallest error, excluding this pair and continuing the pair matching. For
both SSD and ICA we selected ten components with the largest spectral
ratio, R:
R =
Pf
Psf
ð24Þ
sho
ases
1531V.V. Nikulin et al. / NeuroImage 55 (2011) 1528–1535onevariantof ICAdecompositionoftenused for theanalysis of EEG/MEG
signals. Themain idea of SOBI is tominimize cross-correlations between
the components at different time-lags. Algorithmically the method is
based on simultaneous (approximate) diagonalization of many covari-
ance matrices each corresponding to a specific delay. Such diagonaliza-
tion procedure leads to temporally decorrelated components. For this
algorithm we also performed a band-pass filtering in the 8–14 Hz
frequency range and used 50 time-delayed covariance matrices with
time delays of 5, 15, 25...495 ms. Thus, both SSD and SOBI algorithms
used the same information in the relatively broad frequency range
8–14 Hz. However, further transformations with the data were
performed differently for each algorithm. While SSD was based on
forming two complementarily filtered data sets, SOBI proceeded with
Fig. 2. An example of recovering patterns for simulated dipoles (SNR=1). The upper row
algorithms, respectively. A global sign of patterns in SSD or SOBI was flipped in some c
arbitrary units.where Pf is amean spectral power in the10–12 Hz frequency range and Psf
is the combinedmeanspectralpower in the8–10and12–14 Hz frequency
ranges (similar to the idea presented in Fig. 1).
Real EEG recordings
Seven subjects participated in the study (2 females). EEG recordings
were performed at rest with subjects seated comfortably in the chair
with their eyes open. The subjects were instructed to relax and to fixate
their eyes on a mark in front of them. EEG data were recorded with 96
Ag/AgCl electrodes, using BrainAmp amplifiers and BrainVision Recorder
software (Brain Products GmbH, Munich, Germany). The signals were
ws original patterns. The middle and lower rows—patterns obtained with SSD and SOBI
in order to facilitate the comparisons with the original patterns. The color-scale is in

recorded in a 0.1–250 Hz frequency range and digitized at 1000 Hz. For
the following offline analysis the EEG data were decimated to 200 Hz.
In order to obtain thematrixMs the datawerefiltered in the frequency
range 8–13 Hz, matrixMn was obtained by band-pass filtering data in the
range 6–15 Hz and then performing in addition band-stop filtering in the
range 7–14 Hz. The band-stop filter was designed to exclude the 8–13 Hz
part of the EEG signal. For the SOBI algorithm we performed a band-pass
filtering in the 6–15 Hz frequency range and used 50 time-delayed
covariance matrices with time delays of 5, 15, 25...495 ms. Initial filtering
in the 6–15 Hz frequency range ensured that both algorithms used as a
starting point the same amount of information.
Results
Simulations
We performed simulations with different SNRs and calculated
errors indicating how strongly the recovered topographies deviated
from the simulated patterns.
Fig. 2 (top row) shows the original patterns for five exemplary
dipoles with random locations and orientations while the second and
third rows contain solutions provided by SSD and SOBI, respectively.
The figure shows that SSD was capable of recovering sources with
substantially overlapped topographies. The quantitative estimation of
error across multiple simulations and for different SNRs is presented
in Fig. 3. The error for SSD was up to two times smaller than for the
SOBI approach. The histograms in Fig. 3 show also that in case of SOBI
decomposition the errors were quite widely distributed and often
exceeded 0.3. On the contrary, the errors obtained with SSD clustered
around very small values having a very shallow right tail of the
distribution. Overall SSD provided a satisfactory recovery of the
source-patterns (errorb0.1). The lower part of Fig. 3 shows also
averaged errors for different SNRs. With very small SNR, i.e., at 0.01,
the error in both decompositions became quite high and similar.
Real data
The SSD algorithm was also applied to the real EEG data recorded at
rest for the extraction of alpha oscillations in the 8–13 Hz frequency band.
SSD commonly resulted in clear patterns, corresponding to radial and
tangential sources, located over sensorimotor and occipito-parietal areas
(Fig. 4). Note that for the centrally-located sources the spectra have two
characteristic peaks corresponding to the main frequency and its
harmonic at twice higher frequency—a hallmark of the sensorimotor
mu oscillations.
SOBI also resulted in patterns corresponding to radial and tangential
dipoles. In some cases patterns obtainedwith ICA corresponded to those
obtained with SSD. However when choosing components with the
50
150
250
0 0.2 0.4 0.60 0.2 0.4 0.6
250
50
150
250
250
N
SNR 5 SNR 1
SNR 0.5 SNR 0.25
0.5
nt S
1532 V.V. Nikulin et al. / NeuroImage 55 (2011) 1528–1535Error
0 0.2 0.4 0.6
50
150
N
5 1
0.1
0.15
0.2
0.25
0.3
Er
ro
r
Fig. 3. The errors in recovering original patterns with SSD and SOBI algorithms for differe
scatter-plot shows averaged errors for SNR from 5 to 0.01. Error bars are standard error of0 0.2 0.4 0.6
50
150
Error
SSD
SOBI
0.25 0.1 0.05 0.01
SNR
NRs. The upper four panels show distributions of errors for the different SNRs. The lower
the mean.