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The Mode Mixing Problem and its Inﬂuence in the

Neural Activity Reconstruction

Maximiliano Bueno-López, Eduardo Giraldo, Marta Molinas and Olav Bjarte Fosso

Abstract—This paper presents and discusses the challenge of

mode mixing when using the Empirical Mode Decomposition

(EMD) to identify intrinsic modes from EEG signals used

for neural activity reconstruction. The standard version of

the EMD poses some challenges when decomposing signals

having intermittency and close spectral proximity in their

bands. This is known as the Mode Mixing problem in EMD.

Several approaches to solve the issue have been proposed in

the literature, but no single technique seems to be universally

effective in preserving independent modes after the EMD

decomposition. This paper exposes the impact of mode mixing

in the process of neural activity reconstruction and reports the

results of a performance comparison between a well known

strategy, the Ensemble EMD (EEMD), and a new strategy

proposed by the authors for mitigating the mode mixing

problem. The comparative evaluation shows a more accurate

neural reconstruction when employing the strategy proposed

by the authors, compared to the use of EEMD and its variants

for neural activity reconstruction.

Index Terms—EEG signals, Empirical Mode Decomposition,

Mode Mixing.

I. INTRODUCTION

THE EEG is an indicator of neural activity and is

used to study cognitive processes, physiology, emotion

recognition and complex brain dynamics [1], [2]. Due to

the non-linear and non-stationary nature of EEG signals,

they are difﬁcult to analyze in the time and frequency

domain. However, some important characteristics can be

extracted to assist early detection of different disorders by

using advanced signal analysis techniques [3], [4]–[6]. In

recent years, the Hilbert Huang Transform (HHT) has been

increasingly used in the analysis of such signals. However,

in some applications, the extraction of information has been

hampered by the mode mixing problem that appears in

the Empirical Mode Decomposition (EMD) when frequency

components are relatively close or exhibit intermittency.

A mode mixing problem appears when an Intrinsic Mode

Function (IMF) either consists of signals of widely disparate

Manuscript received July 31, 2018; revised January 21, 2019.

This work was carried out under the funding of the Departamento

Administrativo Nacional de Ciencia, Tecnología e Innovación (Colciencias).

Research project: 111077757982 "Sistema de identiﬁcación de fuentes

epileptogénicas basado en medidas de conectividad funcional usando

registros electroencefalográﬁcos e imágenes de resonancia magnética en

pacientes con epilepsia refractaria: apoyo a la cirugía resectiva".

M. Bueno-López is with the Department of Electrical

Engineering, Universidad de La Salle, Bogotá, Colombia, e-mail:

maxbueno@unisalle.edu.co.

E. Giraldo is with the Department of Electrical Engineering, Universidad

Tecnológica de Pereira, Pereira, Colombia, e-mail: egiraldos@utp.edu.co.

M. Molinas is with the Department of Engineering Cybernetics,

Norwegian University of Science and Technology, Trondheim, Norway,

e-mail: marta.molinas@ntnu.no

O. B. Fosso is with the Department of Electric Power Engineering,

Norwegian University of Science and Technology, Trondheim, Norway,

e-mail: olav.fosso@ntnu.no

scales, or a signal of a similar scale resides in different IMF

components.

The presence of mode mixing can hamper the physical

interpretation of the process which is intended to be

described by the individual IMFs [7], [8]. The mode mixing

problem has been studied in different applications, for

example in [9], where the authors discuss how the mode

mixing inﬂuences the EMD-based methods for hydrocarbon

detection. They use mode-mixing elimination methods,

speciﬁcally ensemble EMD (EEMD) and complete ensemble

EMD (CEEMD), as tools for identiﬁcation of the peak

amplitude above average volume and the peak frequency

volume. In [10], a method was proposed based on the

morphological ﬁlter to remove the noise and the revised blind

source separation to deal with the mode mixing. The method

was tested with vibration signals from a mechanical system.

In [11], a sinusoidal-assisted EMD (SAEMD) for efﬁcient

and effective HHT computation to solve mode-mixing

problems was proposed. The new tool was tested using the

Global Sea Surface Temperature (GSST) application from

1856 to 2003.

In the speciﬁc case of EEG signals, the EMD and

the Hilbert Huang Transform (HHT) have been used

to obtain a better signal representation and to detect

instantaneous frequencies [12], [13]. In [14], an approach

that combines EEMD and ICA for selection of artifactual

components and concentration of artifacts was presented. The

effectiveness of the proposed approach was examined using

semi-simulated data purposely contaminated with selected

artifacts. In [15], the authors quantiﬁed the interaction

between different electrodes using a nonlinear measure

known as synchronization likelihood (SL) which effectively

measures the synchronization between non-stationary signals

like EEG. The empirical mode decomposition (EMD)

is applied to decompose the EEG signal into intrinsic

oscillatory modes. In other cases, the EMD is used for

classiﬁcation of electroencephalogram signals. The intrinsic

mode functions generated have been used as an input to

classiﬁers as least squares support vector machine (LS-SVM)

[16], [17]. Based on the above, it is arguable that the use

of EMD in applications involving EEG signals needs to be

combined with other techniques to handle the Mode Mixing

problem [13], [18]–[24].

The purpose of this paper is to contribute to a better

understanding of the challenges that EMD poses when

applied to EEG signals and to discuss solutions to the mode

mixing problem in this speciﬁc area. With this application,

the goal is to ﬁnd an accurate brain reconstruction from

the EEG bands decomposition. This paper is organized as

follows: Section II gives an introduction to the essential

concepts of EMD. The mode mixing problem and some

established solutions are given in section III, and in section

IV we propose a new methodology to solve this problem.

Illustrative examples are shown in Section V, while the

discussion of the results is presented in Section VI. Finally,

conclusions are given in Section VII.

II. EMPIRICAL MO DE DECOMPOSITION

The Empirical Mode Decomposition has been proposed

as an adaptive time-frequency data analysis method [25].

The EMD does not require any restrictive assumption on the

underlying model (no basic function) of the process/system

under analysis and is able to handle both non-linear and

non-stationary signals. However, the algorithm has shown to

have some limitations in identifying closely spaced spectral

tones and components appearing intermittently in the signal.

The aim of the EMD method is to decompose the nonlinear

and nonstationary signal y(tk)into a sum of intrinsic mode

functions (IMFs) that satisﬁes two conditions [26]:

1) Symmetric upper/lower envelopes (zero mean).

2) The numbers of zero-crossing and extrema that are

either equal or differ by exactly one.

The EMD algorithm for the signal y(tk)can be

summarized as follows:

1) Identify all extrema (maxima and minima) in y(tk).

2) Interpolate between minima and maxima, generating

the envelopes el(tk)and em(tk).

3) Determine the local mean as m(t)=(el(tk)+em(tk))/2.

4) Obtain the residue r(tk)=y(tk) − m(tk)

5) Decide whether r(tk)is an IMF or not based on the

two basic conditions for IMFs mentioned above.

6) Repeat step 1 to 4 until r(tk)will be monotonic.

Empirical Mode Decomposition is applied over y(tk)to

obtain γi(tk)being ithe intrinsic mode function (IMF), and

y(tk)=

N

Õ

i=1

γi(tk)+r(tk)(1)

where Nis the number of IMFs and r(tk)a residual.

Recently, some optimization techniques have been proposed

to improve the performance of the EMD [27], [28].

Having obtained the intrinsic mode function components,

the Hilbert transform can be applied to each component and

the instantaneous frequency is found using equation (2).

fi(t),1

2π·dθi(t)

dt ,(2)

In (2), θi(t)is the instantaneous phase of each IMF calculated

from the associated analytical signal [29]. Finally, the

instantaneous frequency can be observed in the Hilbert

Spectrum.

III. MOD E MI XI NG P ROB LE M AN D IT S SO LU TI ON

Mode mixing, observed in the context of the EMD caused

by either intermittency of a signal component or by spectral

proximity, is a well recognized challenge of the method [7],

[9], [30]. In [8] and [31], the authors address the issue of

One or Two Frequencies?, and deﬁne a set of conditions that

must exist between the frequency components of a signal to

ensure that they can be recognized as independent modes

in the EMD decomposition. The mode mixing problem has

been analyzed in different areas. In [9], an application for

hydrocarbon detection is presented, while in [10], the authors

show an application in mechanical systems. In this paper, two

previously known methods and one new method proposed by

the authors in [23] are introduced to handle the mode mixing

in the detection of signal sources from different regions of

the brain. The signals are collected using EEG.

A. Masking Signal

The masking signal method reduces the problem of mode

mixing when frequency components are closely spaced (in

the same octave). The masking signal approach was proposed

in [30]. The basic idea is to add a new signal to the analyzed

signal. This signal will prevent lower frequency components

from being mixed together with the higher frequencies in

the same IMFs. Since the masking signal is known, it can

be removed from the IMF following the procedure indicated

below:

1) Construct a masking signal s(n), from the frequency

information of the original data, y(tk).

2) Perform EMD on y+(tk)=y(tk)+s(n)to obtain the

IMF z+(n). Similarly obtain z−(n)from y−(tk)=y(tk)−

s(n).

3) Deﬁne the IMF as z(n)=(z+(n)+z−(n))/2.

The challenge with this method is the choice of the masking

signal s(n). How to select the frequency and amplitude of

this new signal? According to [30], an appropriate choice

would be to have each frequency within the signal separated

by at least a factor of 2, is s(n)=a0sin(2πfst). Although

some indications are given on how to choose a0and fs, this

process is usually empirical and the experience must guide

the selection of parameters for a particular problem.

In [23], a strategy to calculate a0and fswas proposed,

based on the relation between the frequencies and amplitudes

of each IMF. This new approach exploits the properties of

the boundary map shown in Figure 1, which was previously

developed in [8].

Fig. 1. Mode mixing boundary conditions map reproduced from [8] used

for deﬁning parameters of the masking signal

B. Ensemble Empirical Mode Decomposition

Wu and Huang proposed a noise-assisted data analysis

(NADA), the Ensemble Empirical Mode Decomposition

(EEMD), which deﬁnes the true IMF components as the

mean of an ensemble of trials, each consisting of the signal

plus a white noise of ﬁnite amplitude [7]. The EEMD is

described as:

1) Add a white noise series to the data base y(tk).

2) Decompose the data with added white noise using

EMD to obtain the IMFs.

3) Repeat step 1 and step 2 again, but with different white

noise series each time.

4) Obtain the (ensemble) means of corresponding IMFs

of the decompositions as the ﬁnal result.

The main effect of the decomposition using the EEMD

is that the added white noise series cancel each other

in the ﬁnal mean of the corresponding IMFs. Modiﬁed

versions of the EEMD have been recently proposed. In [32],

the authors proposed one variation of EEMD, a Complete

Ensemble Empirical Mode Decomposition with Adaptive

Noise (CEEMDAN), and an improvement of CEEMDAN can

be found in [33].

IV. NEW MASKING SI GN AL STRATE GY

The new method applied is based on the combination

of the technique presented by Kaiser and the Boundary

Conditions Map presented by Flandrin [8]. The properties

of the map of boundary conditions between well separated

tones and mixed tones (Figure 1) guides the choice of the

masking signal’s frequency and amplitude. To be able to

extract a frequency by applying this principle, the ratio of

that frequency to the frequency of the masking signal, must

be located in the red area of the Boundary Map (mode mixing

area is higher than 0.67 and approaches 1.0), while the ratio

of the next frequency should be located in the blue area of

the map, where mode mixing does not occur. The amplitude

ratios need to be adopted from the map, with a ratio that will

ensure that the above conditions are preserved. It is therefore

necessary to operate with a frequency sufﬁciently close to the

ﬁrst IMF mode and sufﬁciently distant from the next IMF

mode, to be successful. The map illustrates well how closely

spaced spectral tones attract each other in a mode mixing, the

colors representing the mode mixing degree; red for mode

mixing and blue for No-mode mixing. This same property

is exploited in this new masking method for constructing

effective masking signals to separate closely spaced spectral

tones.

Assume a signal with the two frequencies f1and f2

(f1>f2), where the ratio between them will cause mode

mixing due to spectral proximity. A masking signal of

frequency fmlarger than f1will attract f1if the ratio f1/fm

falls into the attraction region of the map (red color). If the

ratio between f2/fmfalls in the region where there is no

attraction (bluecolor), adding a positive masking signal of

frequency fmwill separate the two signals f1and f2and the

ﬁrst IMF will have a controlled mode mixing of the signals

f1and fm. To separate f1and fma negative masking signal

may be added, and by averaging the two ﬁrst IMFs, the new

IMF will be a signal of frequency f1. However, depending on

how close the two frequencies f1and f2are, some amplitude

modulation may be observed between the signals f1and f2.

A way of identifying the frequencies involved in the original

signal, is to process a Fast Fourier Transform (FFT) of the

signal. In [23], a technique has been developed to identify the

involved instantaneous frequencies and amplitudes, to assist

in choosing the right masking signal. Assume a signal x

deﬁned by:

x=Asin(2πf1t)+Bsin(2πf2t)(3)

After EMD is performed, these two signals will be

mixed into one IMF if they are sufﬁciently close in

frequency. A Hilbert-transform of the mode mixed IMF

(s=x+jy)followed by an amplitude and an instantaneous

frequency calculation, will provide the required information

for identifying the amplitudes and frequencies of the two

signals involved. The instantaneous frequency used here is

deﬁned by:

f=1

2π

∂φ

∂t,(4)

where tan φ=(y/x)and φ=arctan(y/x).

Using this deﬁnition, the equations for the amplitudes and

the instantaneous frequencies can be derived. The amplitudes

of the Hilbert-transformed signal are given by:

K=qA2+B2+2AB cos(2π(f1−f2)t)(5)

From this equation, the following expressions for the

extreme values can be derived:

Kmin =pA2+B2−2AB =(A−B)(6)

Kmax =pA2+B2+2AB =(A+B)(7)

Similarly, the equation for the instantaneous frequencies

are given by:

f=A2f1+B2f2+AB(f1+f2)cos(2π(f1−f2)t)

A2+B2+2AB cos(2π(f1−f2)t)(8)

The expressions obtained for the extreme values are:

Fmin =A∆f

(A+B)+f2(9)

Fmax =A∆f

(A−B)+f2(10)

where, Kmin is the Minimum value of the amplitude plot,

Kmax is the Maximum value of the amplitude plot, Fmin

is the Minimum value of the instantaneous frequency plot,

Fmax is the Maximum value of the instantaneous frequency

plot and ∆fis the difference between the two frequencies

(f1−f2).

It is also demonstrated that ∆fis equal to the number

of peaks/second in the instantaneous frequency and the

amplitude plots. The frequencies f1and f2can now

be calculated and may be further validated with a FFT

calculation. In the case of synthetic signals, these calculations

are accurate and in principle the signal components

could have been obtained directly by applying the above

presented analytical technique. However, for real signals the

instantaneous amplitude and frequency functions are less

smooth, but still this procedure will reveal information about

the amplitudes and frequencies involved in the different

periods of a mode mixed signal. From this information,

an optimal masking signal based on the boundary map

can be constructed. The process of extracting the IMFs

is illustrated in the ﬂowchart of Figure 2. An iterative

procedure is used where the IMFs are extracted one by

one by using the previously described procedure for optimal

design of a masking signal. After the ﬁrst IMF is extracted

and accepted as an intrinsic mode, the residue (sn+1(t)) of

the signal is used in the next iteration. The acceptance of

the IMF is decided by observation of the obtained ﬁrst IMF

instantaneous frequency proﬁle and depends on the user

experience and his a-priory knowledge of the studied system.

The next iteration consists on performing a new EMD and

again observing the ﬁrst IMF instantaneous frequency proﬁle.

This proﬁle will provide a hint about the presence of mode

mixing. Once the frequencies involved in the mode mixing

are identiﬁed, a new masking signal is designed according

to the procedure above described. After the masking method

is applied, the second IMF of this iterative process will be

obtained. Once this second IMF is accepted, a new residue of

the signal will be calculated by subtracting the two accepted

IMFs from the original signal. This new residue will then

be used in the next iteration. Iterations will be stopped once

mode mixing is not detected anymore. The last residue in

that case will contain the last IMF and/or the ﬁnal residue

of the EMD process.

V. IL LU ST RATI VE EX AM PL ES

In order to evaluate the behavior of the methods presented

in section III and IV, three databases for EEG generation

are used. For the ﬁrst and second databases the objective

is to be able to observe in each IMF, neural activities at

different frequencies. For the third database the objective is

to obtain in one IMF the relevant activity (alpha band). The

use of simulated databases for EEG source localization is a

common approach for evaluation of brain mapping methods

since the underlying source activity is known, and therefore,

having a benchmark it is possible to evaluate the quality of

the brain activity estimation.

The ﬁrst simulated database (DB-1) contains neural

activity x(tk)by considering temporal localized sinusoidal

signals with two different frequencies (6Hz and 8Hz) by

using a Gaussian window, with sampling rate of 100Hz.

In this case, two sources randomly located into the brain

are selected, where the activity in each source is generated

according to the following expression:

xi(tk)=e−1

2tk−ci

σ2

sin (2πfitk)(11)

being cithe center of the windowed signal in seconds,

and fithe frequency of the signal, with i=1,2. The ciis

selected in the following ranges ci:[0.5,3.5]seconds. The

second simulated database (DB-2) uses the same model

shown in (11) but with three different frequencies (4Hz,

8Hz and 10Hz). In DB-1 and DB-2, we have considered

information from 30 channels.

The third simulated database (DB-3) contains neural activity

generated by two sources randomly located into the brain

with activity x(tk)in the range of 8Hz to 13Hz, with

sampling rate of 100Hz. The time of the two distinguished

Original Signal

Mode

Mixing?

Empirical Mode

Decomposition

EMD

Applied Masking

signal

The first IMF

has a good

decomposition?

=

YES

NO Final

NO

YES

n: number of iterations

: original signal

Fig. 2. Flowchart of the iterative process for IMFs extraction in the presence

of Mode Mixing

sources in the vector x(tk)are modeled using bivariate

linear auto-regressive (AR) models with time-delayed

linear inﬂuences from one source to another. Sources are

bandpass-ﬁltered in the alpha band using a causal third-order

Butterworth ﬁlter with zero phase delay. The generated

time series therefore represent alpha oscillations that are

either mutually statistically independent or characterized by

a clearly deﬁned sender-receiver relationship. In addition,

500 mutually statistically independent brain noise time

series characterized by 1 / f-shaped (pink noise) power and

random phase spectra are generated, and placed randomly at

500 locations sampled from the entire cortical surface [34].

In DB-3, we have considered information from 108 channels.

For DB-1 and DB-2, the EEG signals are obtained as

a linear combination of the underlying neural activity as

follows:

y(tk)=M x(tk)+(tk)(12)

being y(tk) ∈ Rdthe EEG signal measured at each electrode

on the scalp, and x(tk) ∈ Rnthe neural activity or amplitude

of ncurrent dipoles (distributed sources inside the brain),

with tk=kh the time at sample kbeing k=1, . . . , Tthe

total number of samples, hthe sample time and M∈Rd×n

the lead-ﬁeld matrix that relates the neural activity with the

EEG. Speciﬁcally, each row of the lead-ﬁeld matrix describes

the current ﬂow for a given electrode through each dipole

position [35]. Here, the Gaussian additive noise is deﬁned

by (). A Signal-to-Noise-Ratio (SNR) of 10dB is used for

both databases.

For the DB-1 and DB-2, the lead ﬁeld matrix is described

in [36] based on a Boundary Element Method with a high

number of distributed sources n=20484 and d=30

electrodes over the scalp according to a 10-20 standard

BIOSEMI [37]. Positions of electrodes and sources are

shown in Fig. 3.

Fig. 3. Positions of electrodes and distributed sources for DB-1 and DB-2

Figure 4 shows the simulated DB-1 neural activity and

source locations for the two different frequencies (6Hz

and 8Hz) at two different positions into the brain and the

corresponding EEG, with f1=6Hz, f2=8Hz, c1=1s,

c2=2s and σ=0.12.

Fig. 4. Neural Activity of simulated DB-1 with their corresponding source

locations and simulated EEG.

Figure 5 shows the Fourier spectrum of one channel for

DB-1 and DB-2. In this ﬁgure it is possible to see that

different frequencies appear but they are not appropriate to

determine at which instant of time they occur.

Fig. 5. Fourier spectrum for one channel using DB-1 and DB-2.

For the DB-3, the lead-ﬁeld matrix is obtained from the

so-called New York Head model as used in [34], which

combines a highly detailed magnetic resonance (MR) image

of an average adult human head with state-of-the-art ﬁnite

element electrical modeling. In particular, the New York

Head model holds n=2004 sources and d=108 electrodes.

Figure 6 shows the simulated DB-3 neural activity and

source locations for the two different sources and their

locations into the brain with their corresponding EEG.

Figure 7 shows the Fourier spectrum of one channel of

DB-3.

The brain activity estimation is performed by using a

dynamic inverse problem considering that only the EEG y(tk)

and the lead-ﬁeld matrix Mare known. The dynamic inverse

problem of brain activity estimation bx(tk)can be formulated,

according to [36], as:

bx(tk)=arg min

x(tk)ky(tk) − M x(tk)k2

2+λkkx(tk) − bx(tk−1)k 2

2

+αkkx(tk)k1

(13)

where λkand αkare the regularization parameters computed

by generalized cross validation. It can be noticed that as

Fig. 6. Neural Activity of simulated DB-3 with their corresponding source

locations and simulated EEG.

Fig. 7. Fourier spectrum for one channel using DB-3.

a result of the EMD decomposition, the brain activity

estimation of x(tk)can be obtained for each one of the

resulting IMFs. That means that according to (1), the

dynamic inverse problem of (13) is solved for γi(tk)instead

of y(tk), as described in [38], as follows

bχi(tk)=arg min

χ(tk)kγi(tk) − M χi(tk)k2

2

+λkkχi(tk) − bχi(tk−1)k2

2

+αkkχi(tk)k1

(14)

where χi(tk)is the neural activity estimation for the

corresponding IMF γi(tK). In addition, as shown in (12),

the forward problem of EEG generation is a linear problem,

then it can be seen that x(tk)can be rewritten as a linear

combination of the estimated neural activity for each IMF

x(tk)=ÍN

i=1χi(tk). Therefore, equation (14) is a sub-band

brain mapping based on an EMD decomposition. Finally,

Fig. 8 shows the methodology used, which starts with the

decomposition of the EEG using three different methods to

ﬁnally do the mapping of the neural activity.

Multi Channel

EMD

EEMD

Masking Signal

IMF1

2

B i M ppi g

IMFs Selection

Fig. 8. Schematic representation of the methodology with its different

stages

A. Analysis for DB-1

The IMFs obtained for 1 of the 30 channels with the EMD

proposed in [25] and the EEMD proposed in [7] are shown in

Fig. 9. The decomposition obtained for one channel with the

novel method proposed in this paper is shown in Figure 10.

The noise standard deviation of EEMD is set to 0.1 in this

work. According to the experiments carried out, the noise

standard deviation is the most relevant parameter in this

algorithm. The number of iterations had been set to 1000.

This process is repeated for each channel and in this way,

the IMFs are obtained for the entire database.

In this case, the results of EMD shows clear mode mixing

in all the IMFs. In the ﬁrst and second IMF, it is possible

to observe the two frequency components (6Hz in t=1s and

8Hz in t=2s). In the ﬁrst IMF, the frequency component of

6Hz appears more clearly. Normally, we would expect to

ﬁnd the highest frequency component,in this case 8Hz, in

the ﬁrst IMF. For the EEMD, it is possible to observe the

frequency component for 6Hz and 8Hz in the ﬁrst and second

IMF respectively. The instantaneous frequency shows similar

behavior with the EMD and the EEMD. In the instantaneous

frequency corresponding to the ﬁrst IMF of the EMD, it

is possible to distinguish one frequency component around

t=1s and other frequency component around t=2s. In the

EEMD, it is possible to distinguish one frequency component

around t=2s. However, although it is possible to identify the

frequencies of our interest, some other elements are observed,

especially in the IMF1; these elements correspond to noise

components. The results obtained with the new masking

signal method, allow us to observe a better decomposition

of the signal. In Figure 10, it is possible to see how the

signal of one-channel EEG is decomposed correctly in 3

IMFs. At the top of the ﬁgure, the original signal appears.

In the ﬁrst IMF, the noise component is well isolated and in

the remaining IMFs, the expected two frequency components

are clearly identiﬁed. Figure 11 shows a comparison among

the proposed methods by using the neural activity mapping

for the ﬁrst two IMFs for DB-1. It can be seen that in the

Fig. 9. IMFs and Instantaneous Frequency with EMD and EEMD using DB-1

Fig. 10. IMFs with EMD+Mask using DB-1

ﬁrst IMF, the EMD with a Mask has less spurious activity

than the standard EMD and EEMD in comparison with the

Ground truth.

B. Analysis for DB-2

The IMFs obtained for 1 of the 30 channels with the EMD

proposed in [25] and the EEMD proposed in [7] are shown

in Fig. 12. Finally the decomposition obtained with the novel

method proposed in this paper is shown in Figure 13.

In this case, the results of EMD shows clear mode mixing

in all the IMFs. In the second and third IMF, it is possible to

observe the three frequency components (4Hz in t=1s, 8Hz

in t=2s and 10Hz in t=3s). The ﬁrst IMF corresponds to the

noise of the signal. For the EEMD is possible to observe

all the information of interest in the second and third IMFs,

where the second IMF is one clear case of mode mixing.

The instantaneous frequency has a similar behavior with

the EMD and the EEMD. In the instantaneous frequency

corresponding to the second IMF of the EMD, it is possible

to distinguish three frequency components around t=1s, t=2s

and t=3s. In the EEMD, it is possible to distinguish two

frequency components around t=2s and t=3s in the second

IMF. The results obtained with the new masking signal

method, allow us to observe a better decomposition of the

signal. In Figure 13, it is possible to see how the signal of

one-channel EEG is decomposed correctly in 3 IMFs. At the

top of the ﬁgure, the original signal appears. In the ﬁrst IMF,

the noise component is well isolated and in the remaining

IMFs the expected three frequency components are clearly

identiﬁed.

Figure 14 shows a comparison among the proposed

methods by using the neural activity mapping for the ﬁrst

two IMFs for DB-2. It can be seen that in the ﬁrst IMF the

EMD with a Mask has less spurious activity than the standard

EMD and EEMD in comparison with the Ground truth.

C. Analysis for DB-3

The IMFs obtained for 1 of the 108 channels with the

EMD proposed in [25] and the EEMD proposed in [7] are

shown in Fig. 15. For this case, the decomposition obtained

with the new masking signal is shown in Fig. 16.

The EMD and the EEMD have clear mode mixing in

all the IMFs. However, the frequency components of our

interest, 8 Hz and 13 Hz, appear in 2 of the 7 IMFs. The

IMF1 in both cases corresponds to noise. In IF2 and IF3,

it is possible to see two red lines that corresponds to the

frequencies of 8 Hz and 13 Hz.

Fig. 17 shows a comparison among the proposed methods

by using the neural activity mapping for the IMF2 and IMF3

for DB-3 (according to the IMFs instantaneous frequencies

the IMF2 and IMF3 hold the relevant information). The

reconstruction of neural activity in this case was very close

to the ground truth.

VI. DISCUSSION

The mode mixing problem has been studied in different

ﬁelds and in most cases the suggested solution has been to

address the problem by using EEMD or to a lesser extent

a masking signal (the idea is to separate two components

whose frequencies are close). Noteworthy is the fact that

while EEMD has been conceived to solve the mode mixing

problem caused by intermittency, it has been generally used

to solve all types of mode mixing problems regardless

of their origin. The new masking method presented in

this paper is speciﬁcally designed to reduce the mode

mixing problem caused by the presence of modes with

close spectral proximity. The main challenge when using

the masking signal is the selection of its amplitude and

Fig. 11. Brain mapping comparison of the proposed methods for each IMFs for DB-1.

Fig. 12. IMFs and Instantaneous Frequency with EMD and EEMD using DB-2

Fig. 13. IMFs with EMD+Mask using DB-2

frequency, but after selecting adequate values, the method

shows to work very well. Another alternative that has been

used to reduce mode mixing was to combine EMD or

EEMD with some statistical method that allows to establish

a criterion for selection of the IMFs according to the

application. For example, in [39], the authors decompose

EEG signal segments using EEMD, and then they extract

statistical moment based features from the resulting IMFs.

In [40], they proposed one technique to separate the sources

from single-channel EEG signals by combining EEMD and

Independent component analysis (ICA). For the case study

in this paper, the objective is to show that it is possible to

achieve an accurate mapping of neuronal activity using the

IMFs obtained directly from the EMD with masking.

In Fig. 12, it is evident that the IMFs from the EMD

have mode mixing. This is detected by observing different

oscillations in the same IMF. As shown in Figures 11 and 14,

it can be seen that the sub-band reconstruction effectively

splits the brain activity into frequency bands. In fact, for

signals with low SNR, it is clear that the noise is adaptively

ﬁltered in a speciﬁc IMF. However, a drawback of the

method can be observed in the reconstructions obtained at

low frequencies, where the activity is identiﬁable in several

IMFs. On the other hand, the Instantaneous Frequency (IF)

shown in Fig. 9, 12 and 15 is useful to detect instantaneous

variations of the frequency. For EEG signals resulting from

Fig. 14. Brain mapping comparison of the proposed methods for each IMFs for DB-2.

Fig. 15. IMFs and Instantaneous Frequency with EMD and EEMD using DB-3

Fig. 16. IMFs with EMD+Mask using DB-3

Evoked-Related-Potentials or Epilepsy, the IF can be used

to automatically detect the neural response to an external

stimulus, or the beginning of an epileptic seizure. To improve

the results with the new masking signal proposed, it is

possible to use some statistical methods and combine the

results of both strategies.

VII. CONCLUSIONS

In this paper, we have proposed a method for decomposing

an EEG signal into a set of intrinsic mode functions (IMF)

for mapping and reconstruction of the neuronal activity

of the brain. The use of EMD is motivated by the fact

that EEG signals are non-stationary and EMD is a data

dependent method exhibiting a better adaptability towards the

non-stationary nature of the EEG signals. The decomposition

IMF3 IMF3

IMF3

Fig. 17. Brain mapping comparison of the proposed methods for the IMF-4 for DB-3.

allows to identify behaviors in each of the frequency

bands. Although the reconstruction has some undesirable

components, the results show clearly the two sources where

the activity was added in DB-1 and DB-2. For DB-3, it

was possible to identify the frequency range of interest. Due

to the mode mixing problem of EMD, it is possible that

one frequency component appears in different bands, which

is one of the drawbacks when using EMD. However, the

results obtained show a signiﬁcantly improved reconstruction

of neural activity compared to conventional methods used for

the same purpose. Future research efforts will be dedicated

to the development of tools for mitigating the problem

of mode mixing in EEG signals to obtain an even better

sub-bands separation. The proposed methodology shows a

clear potential to separate EEG signals in sub-bands that

will facilitate the analysis of brain activity reconstruction,

by adaptively separating noise from signals. In addition,

based on the obtained results by applying a sub-band neural

reconstruction, it can be seen that the reconstructed neural

activity can be used for functional connectivity analysis by

sub-bands, which implies a great potential of the proposed

methodology as a tool for assisted diagnosis in brain related

disorders. The case studies shown in this paper have allowed

us to demonstrate that it is not always necessary to combine

EMD or EEMD with some statistical method (energy or

entropy function) to obtain a good reconstruction of the

neuronal activity.

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