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Automatic Selection of Frequency Bands for Electroencephalographic

Source Localization

Pablo Andr´

es Mu˜

noz1,2, Eduardo Giraldo2, Maximiliano Bueno L´

opez3and Marta Molinas4

Abstract— This paper shows a method to locate actives

sources from pre-processed electroencephalographic signals.

These signals are processed using multivariate empirical mode

decomposition (MEMD). The intrinsic mode functions are

analyzed through the Hilbert-Huang spectral entropy. A cost

function is proposed to automatically select the intrinsic mode

functions associated with the lowest spectral entropy values and

they are used to reconstruct the neural activity generated by

the active sources. Multiple sparse priors are used to locate the

active sources with and without multivariate empirical mode

decomposition and the performance is estimated using the

Wasserstein metric. The results were obtained for conditions

with high noise (Signal-to-Noise-Ratio of -5dB), where the

estimated location, for ﬁve sources, was better for multiple

sparse prior with Multivariate Empirical Mode Decomposition,

and with low noise (Signal-to-Noise-Ratio of 20dB), where the

estimated location, for three sources, was better for multiple

sparse prior without MEMD.

I. INTRODUCTION

Electroencephalographic (EEG) Source Localization (ESL)

has been widely used in different medical ﬁelds (neuroscience

studies or clinical applications) for its high temporal resolution

that allows to measure the changes of neural activity in time

intervals of the order of milliseconds. The main drawback of

ESL is to solve the neuromagnetic inverse problem which is

ill-posed and it does not have an unique solution. Therefore,

to obtain an approximated locations of neural current sources

from EEG, it is necessary to solve the inverse problem

using some a priori information or applying some constraints

over the source space [

1

], [

2

]. Nowadays, spatio-temporal

constraints have been used in different works, in [

1

] was

included, to improve the spatial resolution, a basis set for

smoothing the source space (localized areas that could be

potentially active brain regions) and based on a Markovian

assumption applied at each sample time to estimate the brain

activity, the time resolution was improved. Another spatio-

temporal constraints were incorporated as a small and locally

patches to reconstruct sparse brain activity; to smooth the

solution over the time, temporal constraint was imposed for

penalizing the difference between consecutive time points

[2].

Currently, some research have focused their studies to analyze

the neural activity in frequency bands, in this way, they have

1

Universidad del Quind

´

ıo, Colombia

pamunoz@uniquindio.edu.co

2

Universidad Tecnol

´

ogica de Pereira, Colombia

egiraldos@utp.edu.co

3

Universidad de la Salle, Colombia

maxbueno@unisalle.edu.co

4

Norwegian University of Science and Technology, Norway

marta.molinas@ntnu.no

found it e.g. some theta-band activities of low-amplitude

desynchronised were associated to visual areas when they

were compared among motion stimuli and static stimuli [

3

].

Besides, in [

4

], the authors focused the research in alpha-band

oscillations because they are the main frequency components,

associated to neural activity, present in EEG signals. Recently,

some works have proposed a method whose structure is

based on data-driven analysis. An example for this kind of

analysis is the empirical mode decomposition (EMD) and

some applications in brain activity reconstruction are shown

in [

5

], [

6

]. One of the results shown in [

6

] was the way how

the neural activity was split in frequency bands which can be

seen in the intrinsic mode functions (IMFs). Similar results

can be seen in [

5

], but these results were analyzing according

to the retained energy and the amount of entropy in each

IMFs. Despite of the relevant results, some issues associated

to EMD method were regarded in the full reconstruction,

namely, mode mixing and mode splitting.

Solutions for reducing the mode mixing have been highlighted

in [

7

] e.g. Noise aided EMD computation (EEMD) and

multivariate empirical mode decomposition (MEMD). In this

paper is presented a method, based on data-driven analysis

using MEMD, to improve the localization of the actives

sources in the brain and for reducing the mode mixing

problem. To separate the frequency bands is used MEMD

method and the relevant IMFs are chosen from marginal

Hilbert-Huang spectrum (MHHS) and entropy analysis. A cost

function based on entropy is proposed to dismiss the IMFs

with Hilbert-Huang Spectral Entropy (HHSE) greater than

the estimated HHSE threshold, and those IMFs with lower

HHSE are chosen to located the actives sources. The proposed

method is evaluated by comparing ESL using multiple sparse

prior (MSP) with and without MEMD, and the performance is

measured with Wasserstein metric on simulated brain activity.

II. MATERIAL AND METHODS

A. Multivariate Empirical Mode Decomposition (MEMD)

Signals represented in multivariate form should have a

coherent treatment to obtain a suitable time-frequency esti-

mation, because these signals contain generalized oscillations

(joint rotational modes). Therefore, it is important to remark

that when the single EMD is applied (channel by channel)

to multichannel signals, this approach is obstructed by [7]:

•

Nonuniformity. Each channel would not be decomposed

with the same number of IMFs using standard EMD.

•

Scale alignment. It is possible that the scales across data

channels do not have the same-index.

•

Nature of IMFs. It is not convenient to enforce the

same number of IMFs for each channel, because the t-f

estimation could be affected, as such IMFs are typically

not monocomponent.

Common mode alignment and nonuniqueness have been the

greatest obstacles for application of the EMD in studies where

is necessary same-index IMFs within of the same scale for the

corresponding information (synchrony, causality, data/image

fusion), being a problem in applications data/image fusion

[

7

]. For multivariate signals, the local maxima and minima

can not be calculated directly and, the notion of ”oscillatory

modes” to deﬁne an IMF is confuse in this case [

8

]. This

method proposes to take a signal projections along of multiple

directions that have been distributed in a uniform way within

of a n-dimensional space to obtain multiple envelopes which

are averaged and then, interpolated (using cubic spline) their

extrema to estimate the local n-dimensional mean. Especial

attention is required to choose a suitable set of directions

from the signal projections taken in the n-dimensional space

[7].

The following algorithm summarizes how the MEMD

works [7]:

1)

Using the Hammersley sequence, as a uniformly sam-

pling a n-dimensional sphere, generate a P-point.

2)

Projections

qθp(tk)

of the signal

y(tk)

must be calcu-

lated in the same direction vector

xθp

, for

p= 1, ..., P

and then to obtain a set of projections {qθp(tk)}P

p=1.

3)

Find the time instants

{ti

θp}P

p=1

that correspond to

the maxima of the set of projections of signals

{qθp(tk)}P

p=1.

4)

Interpolate [

ti

θp

,

s

(

ti

θp

)] to obtain the envelope curves

{eθp(tk)}P

p=1.

5)

Calculate the mean of the P multidimensional envelopes

m(tk) = 1

P

P

X

p=1

eθp(tk)(1)

6)

Extract the “detail”

d(tk) = s(tk)−m(tk)

. If

d(tk)

fulﬁlls the stoppage criterion for a multivariate IMF,

apply the above procedure to

s(tk)−d(tk)

, else repeat

for d(tk).

B. Hilbert-Huang Spectral Entropy

Spectral entropy can be deﬁned as a measure of the amount

of disorder and this deﬁnition is based on the spectrum of

a signal. The Hilbert-Huang Epectral Entropy (HHSE), for

non-stationary signals, is calculated from Hilbert spectrum

following these steps [9]:

1)

The signal

x(t)

is decomposed into a series of IMFs

(IMFj).

2)

The hilbert transform is applied to

IMFj

(

1≤j≤n

)

to obtain YIM Fj

3) The analytical signal is calculated for each IMFj:

ZIM Fj(t) = I M F j(t) + iYI M Fj(t) = aj(t)eiθj(t)

(2)

where

aj(t)=[I M F 2

j(t) + Y2

IM Fj(t)] 1

2(3)

and

θj(t) = arctan( YIM Fj

I M F j(t))(4)

4)

The instantaneous frequency is calculated for

IMFj

(1≤j≤n):

ωj(t) = dθj(t)

dt ,(5)

The time series is expressed as:

x(t) =

n

X

j=1

aj(t)exp(iZωjdt)(6)

The equation 6 represents, as function on time, the

amplitude and the instantaneous frequency, therefore, this

equation corresponds to the Hilbert Transform

H(ω, t)

. The

Hilbert spectrum is the energy-time-frequency distribution

over the signal

x(t)

y HHSE es calculated using the frequency

marginal by integrating the Hilbert spectrum over the time-

axis.

C. Neuromagnetic Inverse Problem

The neural activity can be generated through the following

model of EEG generation:

y(tk) = M x(tk) + (tk)(7)

being the EEG at sample time

tk

termed

y(tk)∈Rd×1

,

the lead ﬁeld matrix

M∈Rd×n

and the neural activity

x(tk)∈Rn×1

. The forward problem indicated in

(7)

, allows

to deﬁne that the estimation of the neural activity can be

obtained by solving the inverse problem based on the EEG

measurements

y(tk)

and the knowledge of the lead ﬁeld

matrix

M

. Besides, to get an unique solution, it is necessary

to consider some spatio-temporal dynamics of EEG signals,

which can improve the approximated location of the active

sources [

1

]. The MSP method was proposed by [

10

] and this

method apply a hierarchical or empirical Bayes model as

spatio-temporal constraints to reconstruct the inverse problem

in a distributed way, and multiple cortical sources with a

spatial support, speciﬁed in terms of empirical priors, are

automatically selected.

III. EXP ER IM EN TAL SE TU P

Studies in neuroscience have set ﬁve frequency bands,

namely: delta-band (0-4 Hz), theta-band (4-8 Hz), Alpha-band

(8-14 Hz), beta-band (14-30 Hz) and gamma-band (30-150Hz)

[

3

]. The aim was to simulate brain activity for three sources

and ﬁve sources, these sources were randomly located in three

(delta, alpha and beta bands) and ﬁve (delta, theta, alpha, beta

and gamma bands) different frequency bands, they were also

located randomly in different areas in the brain. The activity

in each source was simulated using the following expression:

xi(tk) = e−1

2tk−ci

σ2

sin (2πfitk),(8)

being

ci

the center of the windowed signal in seconds

(

1

,

3

and

5

seconds for three sources and

1

,

2

,

3

,

4

and

5

seconds for ﬁve sources), the frequency of the signal (

fi

)

was chose randomly within of the ranges according with the

frequency bands mentioned above and

σ= 0.2

. In this work

were simulated 30 trials for Signal-to-Noise-Ratio (SNR) of

20

dB,

10

dB,

0

dB and

−5

dB using the model of generation

(7).

After applying the HHSE to each trial and each noise level,

It was possible to ﬁnd that the lowest spectral entropy values

were associated to the IMFs where the simulated activity was

observed in the frequency bands. For this reason, the subset

of of IMFs whose entropy was under a threshold

τe

were

chosen to locate the active sources.

The proposed entropy function is the following:

ej=−X

k

kI M F j(t)k2

2log(kI M F j(t)k2

2)(9)

It is applied over each

I M F j(t)

and where

ej

is the entropy

of each IMF, and

e= [e1. . . eN]

. The estimated EEG signal

˜

y(t)

from IMFs with lowest entropy (chosen automatically)

is rebuilt according to the measured entropy ei.

˜

y(t) = X

i∈O

I M F j(t)(10)

Access to a standard EEG database is important because

it is necessary to know the underlying source activity to

evaluate the methods for solving the inverse problem. We

used a model with

n= 8,196

sources and

32

electrodes for

simulation, as described by ([1]).

IV. RES ULTS

After to analyze all of trials in the four noise levels, it

could be found that the most suitable threshold, to choice

the relevant IMFs for locating the active sources, was the

IMFs with lowest spectral entropy and the chosen IMFs

were those whose sum did not exceed 40 percent of the

normalized HHSE for all IMFs. To show the results, for this

work were done two simulations with controlled conditions,

especially with respect to the location of the active sources,

which were located for a clear visualization.

The ﬁrst one was simulated for three active sources with

f1= 2Hz

,

f2= 9Hz

and

f3= 22Hz

, the SNR was of

20dB

. In ﬁg. 1 are shown three of the six IMFs chosen by

entropy cost function; the simulated EEG ﬁg. (1A), IMF2

Fig. (1B) associated to frequency beta-band (

f3= 22Hz

),

IMF5 ﬁg. (1C) associated to frequency alpha-band (

f2=

9Hz

) and IMF8 ﬁg. (1D) associated to frequency delta-band

(f1= 2Hz).

Each IMF used to locate the active source can be seen in

ﬁg. 2 whose sum allows to obtain the full location for the

three active sources ﬁg. 2B. The Wasserstein metric for this

Fig. 1. Selected IMFs for 3 sources with SNR 20dB

estimation was the 3.1467 and the location without MEMD

was 3.2313 ﬁg. 2C, this measurements compared with the

ground truth ﬁg. 2A.

Fig. 2. Wasserstein metric with and without MEMD for 3 active sources

located with SNR of 20dB

The second simulation was done for 5 sources with

f1=

1.5Hz

,

f2= 4Hz

,

f3= 9Hz

,

f4= 20Hz

and

f5= 45Hz

,

the SNR was of

−5dB

. The high level of noise can be seen

in ﬁg. 3A and the another ﬁgures are shown 5 of the 6 IMFs

chosen. The advantage by using the MEMD is to be able

to separate the activity in different bands of frequency e.g.,

in ﬁg. 3B corresponds to gamma-band (

f5= 45Hz

) and

was decomposed in the IMF2 with some noise. In the IMF4

(ﬁg. 3C) was located the frequency associated to beta-band

(

f4= 20Hz

) and the same way, it can be seen in ﬁg. 3D the

IMF5 with the frequency in alpha-band (

f3= 9Hz

), in ﬁg.

3E the IMF7 with the frequency in theta-band (

f2= 4Hz

)

and in ﬁg. 3F the IMF9 with the frequency in delta-band

(f1= 1.5Hz).

In the ﬁg. 4 is presented that the Wesserstien metric for

MSP with MEMD (ﬁg. 4B) was lower than the metric for

MSP without MEMD (ﬁg. 4C), compared with the ground

truth ﬁg. 4A.

V. CONCLUSION

A method based on data-driven analysis, for improving the

accuracy for EEG source localization (ESL), was evaluated.

Fig. 3. Selected IMFs for 5 sources with SNR -5dB

Fig. 4. Wasserstein metric with and without MEMD for 5 active sources

located with SNR of -5dB

The MEMD was used in order to decomposed the EEG signal

in its main modes and separate the noisy components and

reaching to locate the active sources with a minimum noise.

It could also be seen that the EEG signal was decomposed

in IMFs according with frequency bands and each IMF was

associated a speciﬁc spectral entropy values. Those IMFs

with incorporated frequency band or source activity allow to

reconstruct the brain activity of that source. The cost function

of entropy was proposed for choosing the IMFs with lowest

spectral entropy (calculated by using HHSE) and up to a

maximum of the 40 percent, with this cost function, all the

active sources were located and the performance of MSP with

MEMD, according to the Wesserstein metric, was better both

SNR 20db and SNR -5dB than MSP without MEMD. In both

cases, the method for choosing the IMFs took in account

additional IMFs, because the mode splitting generated by

MEMD method is present.

ACKNOWLEDGMENT

This work was carried out under the funding of COL-

CIENCIAS. Research project: 111077757982: Sistema de

identiﬁcacin de fuentes epileptognicas basado en medidas de

conectividad funcional usando registros electroencefalogrﬁcos

e imgenes de resonancia magntica en pacientes con epilepsia

refractaria: apoyo a la ciruga resectiva.

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