EEG/SEEG SIGNAL MODELLING USING FREQUENCY AND

FRACTAL ANALYSIS

Vairis Caune

, Juris Zagars

and Radu Ranta

Ventspils University College, Inzenieru iela 101a, LV-3601, Ventspils, Latvia

CRAN UMR 7039, Nancy Université - CNRS, Vandoeuvre les Nancy, France

Keywords:

EEG Modelling, Fractal Dimension, Frequency Analysis.

Abstract:

EEG (Electroencephalography) is used to measure the electrical activity of a human brain. It is widely used to

analyse both normal and pathological data, because of its very high temporal resolution. Different algorithms

were proposed in the literature for EEG signal processing, but a difﬁcult issue is their validation on real

signals. An important goal is thus to realistically simulate EEG data. The starting point of this research was

the model proposed by Rankine et al. for the surface newborn EEG signal generation. The model, based on

both statistical, fractal and classical frequency modelling, has parameters estimated from the real data. A ﬁrst

objective is to validate and parametrize this model on adult surface EEG. A second and more important goal

is to parametrize it and to apply it to depth EEG measurements (SEEG). The ﬁrst results presented in this

communication show that the proposed model can be applied in both cases (surface and depth adult EEG),

although the parameters are slightly different. As expected, seizures cannot be modelled using this approach.

1 INTRODUCTION

Electroencephalography (EEG) is the most widely

used method to record electrical activity of the hu-

man brain. This data can be used to analyse the be-

haviour of the normal brain, as well as to diagnose

different pathologies, as for example epilepsy. Since

our knowledge about the generators of the electrical

activity in brain is still on a fairly basic level, most of

the signal processing algorithms developed for EEG

signals can be validated only by medical expertise. In

order to have reliable results, we need to use large

datasets for testing. Since EEG recording is time con-

suming and problematic (because of the high variabil-

ity of the signals), consistent large data sets are quite

difﬁcult to obtain. Simulated realistic datasets would

help to build more consistent algorithms and test them

more properly.

Depth EEG (called further as SEEG – Stereoelec-

troencephalography) uses the same principle of elec-

trical activity recording like EEG, but electrodes are

surgically inserted into the brain. As expected, be-

cause of the invasiveness of the technique, SEEG data

is even less frequent than EEG data. Because of their

acquisition method, the SEEG signals supposedly di-

rectly record brain sources, while the surface EEG

is a mixture of source signals. Simulated signal can

be useful both for SEEG dedicated studies and for

forward/inverse problem applications: with a realis-

tic source modelling, one can expect more realistic

scalp EEG modelling. Moreover, in an inverse prob-

lem setup, simulated SEEG can be compared to the

one obtained by the source estimation algorithms and

thus used to validate them.

2 EEG MODELS

There are several different approaches to model and

simulate EEG signals, depending on the purpose of

their applications. The most popular of them are:

• Source modelling from EEG signals (inverse

problem, source separation)(Delorme et al., 2007)

• Biological neurocomputing(Robinson et al.,

2003; Wendling et al., 2005)

• EEG/SEEG modelling mimicking real signals

(Rankine et al., 2008)

Following (Rankine et al., 2008; Stevenson et al.,

2005), we focus in this paper on third approach. Sig-

nal imitation is made using real signal characteris-

tics. Datasets of real EEGs are analysed, in order to

obtain these characteristics. Depending on the model,

249

Caune V., Zagars J. and Ranta R..

EEG/SEEG SIGNAL MODELLING USING FREQUENCY AND FRACTAL ANALYSIS.

DOI: 10.5220/0003780302490253

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2012), pages 249-253

ISBN: 978-989-8425-89-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

different supplementary assumptions are made, and

validation is performed against large real datasets.

Rankine et al. separate two models having differ-

ent characteristics: seizure model and background

model, aiming to characterize different new-born real

EEGs (Figs. 1 and 2). We focus here on the (Rank-

50 100 150 200 250

Real EEG

Time (samples)

Figure 1: Background EEG signal.

50 100 150 200 250

Real EEG

Time (samples)

Figure 2: Seizure EEG signal.

ine et al., 2008) background EEG model, our aim be-

ing to ﬁnd if it can be applied on adult surface and

depth data. We will asses its validity for both back-

ground and seizure signals. The different steps of the

cited model and employed methodological tools, are

described in more detail in the next section.

2.1 Background EEG Modelling

According to (Rankine et al., 2008) and the references

cited therein, the power spectrum of a background

surface EEG approximately follows a power law:

S( f) ≈

|f|

(1)

where c is constant, f is frequency and γ is the power

law exponent

. If one wants to generate a simulated

EEG signal x(t), the ﬁrst step is to express S( f) as

X( f)X

∗

( f), with X( f ) being the amplitude spectrum

of x(t), obtained by the Fourier transform:

X( f) =

√

|f|

jθ( f)

, (2)

where θ( f) is the phase of the Fourier transform. In

order to obtain a more realistic signal, (Rankine et al.,

2008) proposes to generate several X

( f) using differ-

ent phase vectors θ

( f). Several x

(t) can be obtained

by inverse Fourier transform from X

( f), and the ﬁnal

simulated background EEG signal is generated as

x(t) =

∑

−1

( f)) (3)

Since real EEGs are non-stationary, γ is considered

constant for every epoch of 4 seconds (assuming a quasi-

stationary signal during one epoch).

As it can be seen, this model needs three parameters:

c, γ and θ(f). The amplitude c is of secondary impor-

tance, so we will focus only on the last two param-

eters. In order to use realistic values, they must be

extracted from real data.

2.1.1 Parameter Estimation

The method used in (Rankine et al., 2008) to estimate

the power law exponent γ exploits the linear relation-

ship between γ and the fractal dimension FD of a sig-

nal (Wornell and Oppenheim, 1992), expressed by:

FD =

5−γ

(4)

This step is useful because the FD can be estimated

from the real EEGs using one of the fractal dimension

estimation methods. Different fractal dimension esti-

mators such as Box-counting, Information and Corre-

lation dimensions (Ott, 2000) can be used, with quite

similar results on classical fractals. Higuchi’s FD es-

timation (Higuchi, 1988) is a particular example of

fractal dimension derived from box-counting. This al-

gorithm works directly in the time domain (analysing

the geometrical form of signal), so it can be used for

relatively short signal lengths (recall that EEG’s are

assumed stationary on short time intervals).

As said previously, in order to simulate realistic

signals, the needed parameters (FD and θ( f)) must

respect real signals characteristics. As in (Rankine

et al., 2008), we have estimated them using the fol-

lowing procedure, applied to a database of real adult

background EEG/SEEG signals:

• compute the FD and the phase for each signal

• assume that, over the database, FD follows a beta

distribution and estimate the distribution param-

eters (method of moments (NIST/SEMATECH,

2011)). Probability density function of a beta dis-

tribution with two parameters, α and β can be ex-

pressed as

f(x;α,β) =

Γ(α+ β)

Γ(α)Γ(β)

α−1

(1−x)

β−1

, (5)

where x ∈ [0,1] and Γ(z) =

∞

z−1

−t

dt is the Γ

function.

• assume that the phase θ follows a uniform distri-

bution in [−π,π]

• test (Kolmogorov-Smirnov) the empirical distri-

butions against theoretical distributions generated

using the previously estimated parameters.

BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing

250

2.1.2 Signal Simulation

Assuming that estimated realistic probability distribu-

tions have been obtained for both the fractal dimen-

sions FD and for the phases θ( f), a realistic simu-

lated background EEG can be generated by randomly

choosing a value for FD and a phase vector θ( f) and

introducing them in (4),(2) and (3).

In order to validate the approach, (Rankine et al.,

2008) suggests to extract FD and θ( f) from a real

EEG measurements and to use the described method

to generate a synthetic signal: if the method is correct,

then the original signal and the simulated one should

be similar (correlated). The correlation index, noted

further on as ρ, can be computed in time domain (ρ

as well in frequency (after computing the Welch peri-

odogramm, ρ

) and in time-frequency (spectrograms

after short-time Fourier transforms, ρ

t f

3 RESULTS

The described model was applied to different classes

of EEG signals: surface and depth, background and

seizure. The database contained 400 signal fragments

from 3 different patients, 4 seconds each. Seizure pe-

riods were pointed out by neurologists beforehand.

Surface EEG signal was ﬁltered with cut-off frequen-

cies at 0.5 and 30Hz whereas source SEEG signal was

ﬁltered with low-pass ﬁlter at 128Hz (no assumptions

on SEEG signal spectral behaviour was made). Con-

sequently, surface EEG signals contained 256 sam-

ples and source SEEG signals contained 1024 sam-

ples for every 4 seconds window.

3.1 Adult Surface EEG

At ﬁrst, the simulation method was applied to adult

surface EEG data. For generality, we tested the model

both to background and seizure EEG, downsampled

to 64Hz (as in (Rankine et al., 2008)).

The power spectral density (PSD) was computed

for several time windows of 4s length, to ﬁnd out if it

exhibitsa power law (1) process behaviour (ﬁgure 3).

Under this hypothesis, the fractal dimension FD can

be estimated using (4).

3.1.1 Parameter Extraction for Background

EEG Data

Fractal dimension (thus γ) and phase spectrum were

calculated for every time window from the database

and empirical distributions were estimated as de-

scribed previously. Results are shown in ﬁgure 4. γ

0 10 20 30

Frequency (Hz)

Magnitude

0 10 20 30

Frequency (Hz)

Magnitude

Figure 3: Adult scalp EEG power spectra: background (left)

and seizure (right).

1.3 1.4 1.5 1.6 1.7

0.05

0.1

0.15

0.2

0.25

(a) γ

−4 −2 0 2 4

0.02

0.04

0.06

0.08

(b) θ

Figure 4: Empirical distributions of the power coefﬁcient γ

and of the phase θ for adult scalp background EEG. Theo-

retical distributions (beta and uniform respectively) are rep-

resented by dotted lines.

was found to follow a beta distribution with α = 1.936

and β = 2.975. θ was found to follow uniform distri-

bution in [−π,π]. These hypothesis were conﬁrmed

using Kolmogorov-Smirnov test at a 5% signiﬁcance

level.

3.1.2 Parameter Extraction for Seizure EEG

Data

The same procedure could be applied also for seizure

signals. Still, as seen in Fig. 3, the PSD does not

display a power law process behaviour: because of

rhythmic seizure activity, a peak in the seizure fre-

quency band might be observed. Consequently, eq.

(4) does not hold and other modelling techniques

must be applied (see also (Rankine et al., 2008)).

3.1.3 Validation

In order to validate the approach, the second proce-

dure described previously was used: starting from a

real signal, FD is estimated and thus γ. Its phase spec-

trum was computed (θ) as well as its power (used to

estimate c). These parameters were used to generate

a particular synthetic signal that was later compared

with the real one using the validation procedure de-

scribed previously (3 correlations ρ

, ρ

t f

Background EEG. The obtained modelling results

are rather similar between new-born and adult data

(see table 1). Adult modelled signals show a better

EEG/SEEG SIGNAL MODELLING USING FREQUENCY AND FRACTAL ANALYSIS

251

Real EEG

0 50 100 150 200 250

Simulated EEG

Time (samples)

Figure 5: Real and simulated background EEG signals.

Real EEG

0 50 100 150 200 250

Simulated EEG

Time (samples)

Figure 6: Real and simulated seizure EEG signals.

Table 1: Correlations (mean and sd) for background EEG.

ρ new-born (Rankine et al., 2008) adult

0.795 (0.081) 0.675 (0.075)

0.716 (0.131) 0.803 (0.150)

t f

0.817 (0.113) 0.705 (0.075)

correlation than new-borns in the frequency domain,

but correlation in time and time-frequency domains

are lower. Globally, it seems that the model proposed

for newborns by (Rankine et al., 2008) can be used

also for adult surface background EEG modelling.

Seizure EEG. Same analysis was performed for

seizure EEGs. We compare the correlations of our

FD-based model with Rankine’s et al.seizure model

(developed using a time-frequency approach).

Results from table 2 indicate that, unlike in the

previous case, in the time domain this method give

better results than (Rankine et al., 2008). On the con-

trary, in frequency domain correlations are very low.

This might be related to the power spectrum density

of surface seizure EEG that does not follow power

law. Still, due to the high result in the time domain,

we think that after an appropriate power spectrum

density estimation (i.e. different from 1/ f process),

this model could be used also for adult seizure EEGs.

Table 2: Correlations (mean and sd) for seizure EEG.

ρ new-born (Rankine et al., 2008) adult

0.345 (0.176) 0.661 (0.705)

0.799 (0.093) 0.494 (0.178)

t f

0.901 (0.056) 0.680 (0.090)

3.2 Adult Depth EEG (SEEG)

The main difference from a methodological point of

view between applying the same approach on EEG

and SEEG data is that, since the frequencies contained

in the SEEG might be higher, ﬁltering and downsam-

pling are not applied. Examples of power spectra are

given Fig. 7.

0 10 20 30

Frequency (Hz)

Magnitude

0 10 20 30 40

Frequency (Hz)

Magnitude

Figure 7: PSD of an adult background (left) and seizure

(right) SEEG signal.

3.2.1 Parameter Extraction for Background

SEEG

Fractal dimension (and thus γ) and phase spectrum

were estimated for every time window. Results are

shown in Fig. 8.

1.6 1.8 2

0.1

0.2

0.3

(a) γ

−4 −2 0 2 4

0.02

0.04

0.06

0.08

0.1

(b) θ

Figure 8: Empirical distributions of the power coefﬁcient γ

and of the phase θ for adult depth background SEEG. Theo-

retical distributions (beta and uniform respectively) are rep-

resented by dotted lines.

According to power spectrum density (Fig. 7),

we can see that SEEG could be considered as a 1/ f

process. The fractal dimension γ distribution was

found to follow beta distribution with α = 1.578 and

β = 2.945 (note that the values are quite different

from the surface EEG). This hypothesis was tested

with Kolmogorov-Smirnov test and could not be re-

jected at the 5% signiﬁcance level. Meanwhile θ dis-

tribution was not uniform (Fig. 8(b)), so other dis-

tributions models should be used to model the phase

(Gaussian mixtures for example).

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252

3.2.2 Parameter Extraction for seizure SEEG

The same procedure has been applied also for seizure

SEEGs. As expected, the PSD does not display a 1/ f

behaviour, and phase distribution as well is far from

the uniform distribution: the described approach is

not appropriate for a reasonable simulation of seizure

SEEG data.

3.2.3 Validation

Background SEEG. As before, for every particu-

lar signal of background SEEG a synthetic signal was

generated using the extracted parameters.

According to Table 3, the simulated and real sig-

nals are moderately correlated (a higher value for the

time-frequency correlation though). Still, as shown

in Fig. 9, the modelling gives rather visually correct

results when compared to real data.

Real EEG

0 50 100 150 200 250

Simulated EEG

Time (samples)

Figure 9: Real and simulated background SEEG signals.

Table 3: Correlation (mean and sd) for background SEEG.

ρ adult SEEG

0.587 (0.064)

0.582 (0.201)

t f

0.720 (0.049)

Seizure SEEG. For consciousness, the same proce-

dure was applied for adult SEEG seizures. As ex-

pected, the obtained signals show very low corre-

lations results both in time and frequency domains.

Again, this is probably due to the speciﬁc frequency

content of epileptic seizures.

4 CONCLUSIONS AND FUTURE

RESEARCH

The goal of the research presented in this paper was

to explore if an existing model of surface new-born

background EEG (Rankine et al., 2008) can be used

for adult EEGs (background and seizure, surface and

depth). According to our results, it seems that it is

possible (although slightly less reliable) to generate

an adult background EEG than a newborn EEG. Sim-

ilarly, it is harder (but possible, mainly if a more

realistic phase model is used) to mimic background

SEEG signals than surface EEGs. On the contrary,

seizure EEG/SEEG signals cannot be reliably gener-

ated, probably due to the model assumption on the

spectral behaviour (1/ f).

A ﬁrst immediate perspective is to conﬁrm the

presented ﬁndings on a larger database. It might be

useful to introduce some categorisation in order to

have more speciﬁc classes of EEG signals to work

with (depending on the actual cerebral activity or on

the recording site). Finally, it could be interesting to

apply differentmodels for the power spectrum estima-

tion (besides 1/ f, clearly not appropriate for seizure

data) and for the phase (not necessarily following a

uniform distribution, as seen in the SEEG case).

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