Experiments and Analysis for Fractional Order Modelling of an EEG

Recording Process

Guillaume Becq, Alina Voda, Gildas Besançon, Pierre-Olivier Amblard and Olivier Michel

Univ. Grenoble Alpes, GIPSA-Lab, F-38000 Grenoble, France

CNRS, GIPSA-Lab, F-38000 Grenoble, France

Keywords:

Constant Phase Element, CPE, EEG, Electroencephalogram, Fractional Order System, Frequency-based

Identiﬁcation, Head Phantom, Impedance Spectroscopy.

Abstract:

In this study, an original experimental setup is proposed to analyze the transfer function of an electroen-

cephalographic (EEG) measurement chain: this setup allows to control electrodes inserted into an electrolytic

medium, and observe the electrical potentials at different points with electrodes connected to an electroen-

cephalographic recording system. Experimental transfer functions are obtained that enable to characterize the

inﬂuence of the electrolyte, the electrode-electrolyte interfaces, and the medium itself. A theoretical model

containing so-called constant phase elements is then proposed and analysed to understand the behaviour of

the transfer function in Bode or Nyquist representations. Experimental and theoretical systems are ﬁnally

compared.

1 INTRODUCTION

Electroencephalographic recorders (EEG) are widely

used in hospital or experimental protocols and

their basic principles are described in several text-

books (Cooper et al., 1969), (Nunez and Srinivasan,

2006). They are nowadays simple to use and make

possible the recordings of huge amounts of data nec-

essary to observe neural activities, diagnose diseases

or understand brain functions. They have a good tem-

poral resolution and can be used to have a precise spa-

tial location, as in intracranial stereotaxic recordings

for example. However, depending on the electrodes

used to measure neural activities, such as silver elec-

trodes on the scalp, or platinum electrodes on the in-

tracranial recordings for instance, the recorded signals

are not the same in terms of amplitudes and dynamics.

Several studies have emphasized a modiﬁcation of

the amplitude according to laws varying with non in-

teger powers of the frequency (Boonstra et al., 2009),

(Miller et al., 2009), (Logothetis et al., 2007), (Bé-

dard and Destexhe, 2009), (Dehghani et al., 2010),

(Ragheb and Geddes, 1990), (Becq et al., 2008), (Ma-

gin et al., 2011) by incriminating different parts of the

observed systems: some of them show that it is due to

neuronal activities, other ones that this is due to the

metallic electrodes, or ﬁnally that this can come from

the propagation medium.

In order to understand the inﬂuence of each part of

the system, experimental rigs can be tested. Among

them, so-called head phantoms have been developed

to mimic EEG measurements of brain signals inside

a head. As presented in (Leahy et al., 1998), several

systems such as cadaver heads, gelatin ﬁlled skulls, or

other chemical materials have been used to reproduce

the media of the head; dipoles are proposed to gener-

ate the signals mimicking the activity of a brain area

and currents of 1 s, 10 Hz sine waves are driven into

the conductor to evaluate the recordings. In (Baillet

et al., 2001), a similar setup is used with a 5 Hz si-

nusoid. In (Collier et al., 2012), an even more real-

istic head phantom is used with a 10 Hz, 30 s, 6.2V

peak-to-peak sine signal applied. In all these stud-

ies, the devices enable to study the performances of

source localization algorithms. However, the dynam-

ical properties of the different parts of the system

are not taken into account, and since impedances of

the biological tissues depend on frequencies, as de-

scribed in (Schwan and Foster, 1980) for example, it

is hard to evaluate the transfer function of the whole

system. Besides, the impedances of the electrode-

electrolyte interfaces, which are of high importance

in electrophysiology (Geddes, 1997), is not taken into

account. Modelisations of such systems with frac-

tional order systems are proposed in (Magin and Ova-

dia, 2010) and (Magin et al., 2011). Finally, nothing

is said about the effect of using elevated potentials

on the electrodes, that leads to reduction-oxidation re-

Becq, G., Voda, A., Besançon, G., Amblard, P-O. and Michel, O.

Experiments and Analysis for Fractional Order Modelling of an EEG Recording Process.

DOI: 10.5220/0006417905590566

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 559-566

ISBN: 978-989-758-263-9

559

actions, sources of physical phenomenons which can

transform electrodes and electrolytes.

In this article, an original experimental rig is pro-

posed to characterize the transfer function of the

medium and its coupling with the electrodes at differ-

ent frequencies. To that end, electrical potentials are

applied on electrodes inserted into a medium. Electri-

cal potentials are chosen in a range that do not in-

duce reduction-oxidation reactions at the electrode-

electrolyte interfaces. Signals are collected on an

EEG recorder, via other electrodes also inserted into

the medium. In this article, the inﬂuence of two differ-

ent media, an electrolyte or a sponge wet with an elec-

trolyte, are compared. An electrical circuit includ-

ing so-called Constant Phase Elements (CPE) (Diard

et al., 2013), (Jorcin et al., 2006), (Zoltowski, 1998)

in order to capture the non-integer order of the sys-

tem is proposed and its theoretical transfer function

and properties are calculated. This enables to ﬁt the

experimental data and identify the parameters of the

theoretical proposed system, from the arc paths of the

transfer functions observed in the Nyquist representa-

tions.

The article is constructed as follows: ﬁrst the ex-

perimental setup and related empirical transfer func-

tion estimates are presented in section 2. Then, the

theoretical transfer function based on a proposed elec-

trical circuit is studied in section 3 and compared with

the experimental recordings. Results are discussed in

section 4 before concluding.

2 EXPERIMENTAL SETUP,

MEASURES & ANALYSIS

2.1 Description of the Experimental

Setup

The general setup proposed for testing the EEG

recorder is described in Fig. 1: in short, signals are

simulated on a computer (Simu) and sent to a digital-

to-analog converter (DAC) as voltage inputs applied

to electrodes. These inputs electrodes (Elec in) are in-

serted into a medium (Med). Output electrodes (Elec

out) are also inserted into the medium to record some

signals. Input and output values on the medium are

recorded with an EEG recorder (EEG rec). Accord-

ing to linear system theory, the system is reduced to

Fig. 1. e) with different transfer functions H

from the

simulation into an input electrode X

to a recording

electrode X

The DAC is a National Instruments CRIO-9263,

voltage analog output module with 4 outputs, 16 bits,

100 KS/s per channel, mounted on a cDAQ-9181 eth-

ernet chassis. Signals are simulated in Python with

Scipy modules (Jones et al., 2016). They are binded

to an own-purpose library in C linked to the National

Instruments library interfacing the DAC. Python and

C codes are available in (Becq, 2016).

Home-made pure platinum electrodes are used in

this study in order to avoid reduction–oxidation re-

actions at the contact of the electrodes with elec-

trolytic solutions for potentials lower than E

◦

= 1.2V

at 25

◦

C (Didier, 1984). Each electrode consists of a

thin cylinder of platinum with a diameter of 0.5 mm

and a length of 1 cm, soldered on a copper wire. The

solder junction is covered by a thin layer of epoxy

resin for insulation.

Two media are tested to compare their inﬂuences

on the system: an electrolytic solution; a sponge fully

wet with an electrolytic solution. The electrolyte so-

lution is obtained from the dissolution of a tablet of

phosphate buffered saline (PBS) into pure water. A

synthetic tiler foam grout sponge, with dimensions

15.5 by 10.5 by 5.2 cm is used to test the inﬂuence

of a propagation medium. The sponge is washed with

pure water between experiments. The sponge, or the

solution, is contained in a home-made basin made

of polyethylene terephthlate glycol-modiﬁed (PETG)

slices stuck together with a silicon glue, as seen in

Fig. 1. This setup is simple, cheap and very con-

venient for working with liquids. PBS is a P4417

Sigma-Aldrich reference. The basic dissolution is one

tablet in 200ml of pure water. The dissolution used in

this study is 1/2 tablet in 1 l of pure water, so as to ob-

tain a weakly conductive solution. The temperature of

the room is checked with a thermometer and is about

24 to 25

◦

Electrodes are inserted into the sponge or liq-

uid on a depth of approximatley 0.5 cm. Electrodes

are linearly spaced on 6 locations with d

1,2

= 1 cm,

1,3

= 3 cm, d

1,4

= 5 cm; d

1,Re f

= 6 cm; d

1,GND

= 8 cm

with the input electrode in 1; the output electrodes are

in 2, 3 and 4; the ground electrode (GND) is common

to the EEG recorder and the ADC; the reference elec-

trode of the EEG recorder is set in Ref; electrode 1 is

also connected to the EEG recorder.

The EEG recorder is a g.tec g.USBamp, with an

input voltage range of ±250 mV , no highpass ﬁlter,

a lowpass at 6.6 kHz, an input impedance > 10

Ω,

an analogic to digital conversion resolution of 24 bits

with a maximal sampling frequency of 38.4kHz per

channel, and 16 channels. Signals are recorded at

4.8kHz in this study.

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560

a) b) c)

Simu

DAC Elec

Med

Elec

out

EEG

rec

Simu

EEG rec

d) e)

Figure 1: Illustration of the proposed experimental setup for EEG recorder testing with different representations. a) Overall

view of the setup; b) Electrolytic solution and electrode positioning; c) Sponge experiment and positioning; d) Scheme of the

setup. e) Scheme of one equivalent system H

for one input electrode X

and one recording electrode X

2.2 Experimental Transfer Functions

Estimates and Analysis

Empirical transfer function estimates (ETFE) (Ljung,

1987) are obtained by impedance spectroscopy (Al-

Nazer et al., 2013) either by generating with the com-

puter a sequence of sine waves with amplitude vary-

ing from -0.1 to 0.1 V at different frequencies, [0.1,

0.125, 0.25, 0.5, 1, 2, 4, 8, 10, 16, 32, 64, 100, 128]

Hz, or by using a reproducible sequence of white

Gaussian noise with mean 0 V and standard deviation

0.1 V. For all methods the ETFE is the ratio between

the output signal over the input signal at different fre-

quencies. When working with pure frequencies, se-

quences of at least 10 periods are recorded and the

evaluation is made after removing transients, gener-

ally the ﬁrst cycle of the trial for low frequencies. The

ETFE is obtained by dividing the Fourier coefﬁcients

of the signals at the different frequencies of interest:

( f ) =

( f )

When working with white noise, a sequence of

100 s is used. Half a second at the beginning and end

of the sequence are removed for data processing. A

method similar to Welch windowing (Welch, 1967)

with non overlapping windows of 1 s, 4800 samples,

is applied to estimate the Fourier coefﬁcients X

( f ) of

each signal x

(t) for each frequency of interest. The

ETFE

H( f ) is the mean of the ratio of these Fourier

coefﬁcients obtained on windows indexed by k:

( f ) =

∑

k=1

i,k

( f )

in,k

( f )

The empirical transfer functions obtained with

impedance spectroscopy on an electrolyte (e) or on

the sponge wet with the electrolyte (s) are represented

in the Bode and Nyquist representations of Fig. 2.

On all electrodes the transfer functions show be-

haviors of high pass ﬁlters. For the electrolyte alone

(e, red color), differences are observed on the Bode

amplitudes that decrease with distances from the elec-

trodes to the input. There is no modiﬁcation of the

Bode phase with distances. Arcs are observed on

the Nyquist representation with varying centers and

radii. The same behaviors are observed with the

sponge wet with an electrolyte (s, blue color). On

the Bode representation, amplitudes are higher with s.

The white noise estimates show less variance with s.

On the Nyquist plot, dispersion of point clouds seems

less important with s. On the Bode representation,

there is a small shift in phase angle between the two

media. In the Nyquist representation and especially

in (d), there are slight differences in the tangents at

the origin at low frequencies, and always differences

in the real part at high frequencies.

These behaviors looks like transfer functions of

a ﬁrst order high pass ﬁlters but with several differ-

ences: slopes of the attenuations of the amplitude at

low frequencies on the Bode representations are less

than -20 dB / dec; the limits of the phases at low fre-

quencies in the Bode representations tend to a value

smaller than

; the path in the Nyquist representation

is an arc but the center of this arc is not on the real

axis; equivalently, the path at the origin is not tangent

to the imaginary axis. In order to understand these be-

haviors, a theoretical model based on constant phase

elements is proposed in the next section.

Experiments and Analysis for Fractional Order Modelling of an EEG Recording Process

561

a) b)

c) d)

Figure 2: Empirical transfer functions estimated at different frequencies for the electrolyte alone (e) or the sponge wet with

the electrolyte (s): a) b) with sinusoid; c) d) with white noise; a) c) Bode plot of the transfer function for the electrodes 2, 3

and 4 at distances 1, 3, and 5 cm from electrode 1. Magnitudes (up) and phases (bottom) of the transfer functions are given

in function of the frequencies in Hz. b) d) Nyquist representation of the transfer function in arbitrary unit (a. u.). Fits, with

theoretical paths presented in section 3, are represented with black curves on d). All systems show general behaviors of high

pass ﬁlters.

3 THEORETICAL MODELLING

WITH CONSTANT PHASE

ELEMENTS

In the rest of the paper, X(p) and X (ω) are used to

represent the Laplace and Fourier transform of signal

x(t) with time t.

3.1 Constant Phase Elements and

General Models

Let us start with the simple model of impedance Z(ω)

of a constant phase element, as given in (Diard et al.,

2013), (Jorcin et al., 2006), (Zoltowski, 1998):

Z(ω) =

Q( j ω)

Qω

exp(− j α

) =

Qω

− j s

)

with c

= cos





and s

= sin





The phase of this component is φ

= −α

, is con-

stant, and explains its name. We replace resistors with

CPE components in voltage dividers generally used to

obtain ﬁrst order high pass ﬁlters. Other complex cir-

cuits with CPE can be found in (Diard et al., 2013).

The general electrical scheme for a potential or

voltage divider is given in Fig. 3. a).

It is supposed that the input resistance when mea-

suring X

is high as compared to Z

. In this case, the

impedance Z(p) of the equivalent circuit from X

(p)

and the transfer function H(p) =

(p)

are related by:

H(p) = Z

(p) Z

−1

(p), or Z(p) = Z

(p) H

−1

(p)

If Z

(p) = R, the relation between Z or H, depend-

ing on the measurement made on the system is given

by:

|Z(p)| =

|H(p)|

Z(p)

= exp (− j φ

(p))

|H(p)| =

|Z(p)|

H(p)

= exp (− j φ

(p))

(1)

For Z

(p) = R and Z

(p) = 1/( jC ω), the system

is a high pass ﬁlter of the ﬁrst order.

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562

i ≈ 0

X1 X2

a) b)

Figure 3: Examples of voltage divider: a) general model, b) resistive and high pass ﬁlter with CPE.

3.2 Introducing a Constant Phase

Element in a High Pass Filter

The proposed circuit to obtain a high pass ﬁlter with

a CPE is given in Fig. 3 b). This model is proposed to

take into account a resistive component in the medium

and a CPE that summarizes the electrode-electrolytic

interfaces for the input electrode X

and output elec-

trode X

. The development of the transfer function

leads to:

H(ω) =

+ R

Q( j ω)

+ R

)Q( j ω)

+ 1

Let A(ω) = R

Qω

= A ω

with A = R

Q and

∗

(ω) = (R

+ R

)Qω

= A

∗

H(ω) =

A(ω)(c

+ j s

)

∗

(ω)(c

+ j s

) + 1)

H(ω) =

A(ω) (A

∗

(ω) + c

)

∗

(ω)

+ j

A(ω)s

∗

(ω)

∗

(ω) = 1 + 2 A

∗

(ω)c

+ A

∗

(ω)

H(ω) = C(ω) (A

∗

(ω) + c

) + jC(ω)s

(2)

Where C(ω) = A(ω)/B

∗

(ω)

Let f be the frequency such that ω = 2 π f . On one

hand, if the real part of the transfer function is denoted

by x and the imaginary part by y, which corresponds

to the respective axes in the Nyquist representation,

the path of the transfer function H( f ) can be given in

its parametric form with:



x( f ) = C( f )(A

∗

( f ) + c

)

y( f ) = C( f )s

(3)

On a second hand, the equation of a circle centered at

C with coordinates (x

) and radius r is given by:

− 2 x

x + x

+ y

− 2 y

y + y

− r

= 0 (4)

Let a = R

Q(2π)

, a

∗

= (R

+ R

)Q(2π)

, and u =

. Combinations of the terms of the parametrization

(3) yields:

+ y

1 + 2 a

∗

+ a

∗

−2x

x − 2 y

y = −2

au(a

∗

u + c

) + y

(aus

)

1 + 2 a

∗

+ a

∗

Substituting these expressions in the circle equation

(4) gives:

− 2 x

au(a

∗

u + c

) − 2 y

(aus

) +

+(x

+ y

− r

)(1 + 2 a

∗

+ a

∗

) = 0

Developing and reordering the terms in function of

the order of u yields:



− 2 a a

∗

+ a

∗

+ y

− r

)



+ ···

··· + (−2 a c

− 2 a s

+ ···

··· + (x

+ y

− r

)2a

∗

)u + ···

··· + (x

+ y

− r

) = 0

If parametrization (3) follows a circle, this must be

satisﬁed for all u. This leads to the resolution of the

system:

− 2 a a

∗

+ a

∗

+ y

− r

) = 0 (5)

−2a c

− 2 a s

+ (x

+ y

− r

)2 a

∗

= 0 (6)

+ y

− r

) = 0 (7)

Using Eq. (7) in Eq. (5) gives: a

− 2aa

∗

= 0

which implies:

∗

(8)

Using Eq. (7) in Eq. (6) and Eq. (8) gives:

+ s

= 0

= −

∗

= −

tan(α

)

(9)

Finally, Eq. (8) and (9) in Eq. (7) yield:

r =

∗

(10)

This proves that the parametrization (3) follows a cir-

cle. One can notice that the ratio ρ =

∗

does

not depend on Q, and that the location of x

depends

only on this ratio. In addition, we also have the simple

rules x

= s

r and y

= −c

r. At low frequencies,

the circle goes to the origin (for f = 0, x(0) =

0, y(0) = 0), and at high frequencies, it reaches the

Experiments and Analysis for Fractional Order Modelling of an EEG Recording Process

563

x = Re(H(f))

y = Im(H(f))

|α = 1

|α

|α = 1/2

(0,0)

(1 − α)

Figure 4: Theorical path of the transfer function for a high

pass circuit with a CPE (circle with radius and center de-

pending on the ratio of the resistors in the circuit and on the

α parameter of the CPE).

point (ρ,0) (for f → ∞, x( f ) →

∗

, y( f ) → 0).

These special features are emphasized in the Nyquist

plot of Fig. 4.

The angle α

is the angle between the radius

crossing the x-coordinate at x

and the radius crossing

the origin. By simple geometric relations, this angle

is also obtained between the tangent of the circle at

the origin and the x-coordinate axis.

3.3 Comparison of Fitted Theoretical

Paths With Experimental Data

Parameters of the models can be optimised to ﬁt ex-

perimental data. Examples of ﬁtted paths are pro-

posed in black curves in Fig. 2 d). Table 1 gives the

parameters used to ﬁt experimental data.

Table 1: Possible ﬁts for the parameters according to the

data given in Fig. 2.

α ρ α ρ

e 1 cm 0.77 0.28 s 1 cm 0.68 0.22

e 3 cm 0.77 0.13 s 3 cm 0.68 0.12

e 5 cm 0.77 0.04 s 5 cm 0.68 0.06

The coefﬁcient Q is set to one to describe com-

pletely the circle but appropriate ﬁts can be made if

necessary, for example by ﬁtting special points on the

path depending on the frequency.

With these parameters, it seems that the introduc-

tion of the sponge changes the order of the constant

phase element as observed in Fig. 2 d). This corre-

sponds to the shift between the electrolyte setup and

the sponge setup, observed in the Bode representation

of the phase angle in Fig. 2 a).

Figure 5: Experimental setup to test easily some electrodes

with a sponge wet with electrolyte. Slits are cut into the

sponge and electrodes are inserted inside them. The elastic-

ity of the sponge enables to maintain electrodes in position.

A PETG basin maintains the sponge and electrolyte.

4 DISCUSSION

Using platinum electrodes and low potentials enable

to obtain the results presented in this study. Indeed, an

important part of the study was realized with weakly

golden pins for electrodes and lead to the conclusion

that platinum or nobel metals with high electrochem-

ical potentials are necessary to avoid random shifts in

voltage due to reduction-oxidation reactions and lead

to observable and understandable behaviors. How-

ever, in reality, EEG electrodes are sometimes weakly

covered or defectuous, and this generates artefacts

that are hard to correct. Understanding the model

could precise the effect of these artefacts and the cor-

rections to apply.

Concerning the choice of the medium, it is a long

time since sponge are studied as elementary nervous

systems, even if it is only a muscle system encap-

sulated in a complex body (Parker, 1919) or a more

complex system in current biology (Leys et al., 1999).

The conduction into these organisms is not done on a

wide range, say somes centimeters. Here we tested

only the conductivity induced by the structure of such

an organism with synthetic sponges, not for biolog-

ical purpose, but for practical one. Indeed, sponges

are good for practical usage to test electroencephalo-

graphic recordings. One easy way to use them is to

cut some slits with a cutter on a 1 or 2 cm depth, put

some electrodes inside and ﬁll with conductive water.

The elastic structure of the sponge will maintain elec-

trodes inside the slits and allows to test EEG record-

ings as in the scheme proposed in Fig. 5.

An experimental technical point that can be dis-

cussed here is that it is difﬁcult to wash a sponge

and it is hard to really clean it. A measurement of

the impedance may help but the composition of the

ionic elements inside the sponge is hard to evaluate.

A proper model and a speciﬁc wash design must be

conducted to elaborate something more precise.

Impedance spectroscopy with measurements at

different frequencies can be long, but enable to con-

trol the amplitude of the simulation in order to stay in

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

564

the voltage range out of the oxydo reductive reactions.

Using white noise simulations or multiple com-

binations of wavelength can shorten the duration of

experiments. Interestings readings about impedance

spectroscopy are pointed out in (Al-Nazer et al.,

2013).

The ETFE obtained from sine waves are very pre-

cise for high frequencies but are less precise for low

ones. This is possibly due to the weak amplitude in

the low frequencies and the presence of noise in this

band. It is then hard to differentiate the tangent at

the origin of the transfer function path in the Nyquist

plane for the setup with electrolyte alone and the setup

with the sponge. On the contrary, the ETFE obtained

from white noise are more precise for low frequencies

but are more contaminated by noise on high frequen-

cies. With these ETFEs, differences of the tangents

at the origin can be obtained. Some improvements of

the setups can be made to improve these results.

This study can not conclude clearly on the modi-

ﬁcation of the order of the CPE and that another al-

ternative of the proposed model can be made to ﬁt

the experimental data by taking into account a model

of the electrode by a CPE in parallel with a resis-

tor (R

). This model is well known in litterature but

with a capacitor and is proposed in (Robinson, 1968),

(Ragheb and Geddes, 1990) for example. This model

introduces a shift in the real part of the transfer func-

tion and the real part will tend to R

+ R

)

for low frequency. A possible ﬁt for the experimen-

tal data obtained with the sponge experiment, with a

shift and an α set to the same value as the experi-

ment with the electrolyte is given in Fig. 2 d) in black

dashed line. With this model, the introduction of the

sponge introduces a resistance in parallel with the

CPE. More works have to be followed to discrimi-

nate these two models. Other appropriate methods to

identify the system can be tested, with non integer ex-

ponents, like the ones proposed in (Djouambi et al.,

2007), (Djouambi et al., 2012) for example.

The differences in the dispersion of the data points

observed in the Nyquist plane in Fig. 2 between the

electrolyte setup and the sponge setup indicate that

the sponge improve the measures that are done. One

hypothesis is that the sponge improves the stability of

the electrodes in front of the electrolyte, in term of

movements and in term of the stability of the compo-

sition of the double layer of the metal electrolyte in-

terface. Complementary experiments must be done to

precise the inﬂuence of the medium in this improve-

ment of measures.

Another point is the equivalence of the proposed

model with the different part of the experimental

setup. If each electrode is modelised by a CPE, then

what we observed between the input and the output

of the system is the composition of two electrodes in

series, then two CPE. The equivalent circuit is given

in (Diard et al., 2013) and the equivalent impedance

is:

(p) =



1 +



where β = α

− α

with α

≥ α

. Either the order

is the same and β = 0, and an equivalent CPE of the

same order is obtained with Q

, either one

order is less than the second and this one is prepon-

derant over the other on a given range of frequencies.

Finally, we have proposed results and discussions

on the transfer functions of the system. It is pos-

sible to switch to results and discussions with the

impedances, of the system, often proposed in other

studies, by using Eq. (1) and making some hypothe-

ses or tests on the value of R.

5 CONCLUSION

In this study an original system that can test the be-

havior of an EEG recorder observing simulated sig-

nals injected into a random medium is presented. The

system is described with an electrical circuit contain-

ing a constant phase element. Equations of the trans-

fer functions are analyzed to understand the theoret-

ical paths followed by the transfer function in Bode

or Nyquist representations. Comparison with exper-

imental results show that the proposed system and

model are pertinent for evaluating the non integer or-

der of some constant phase elements observed in the

system. Perspective of this study is the parametric

recursive identiﬁcation of the system using methods

like in (Djouambi et al., 2012).

ACKNOWLEDGEMENTS

This project was funded in part from Institut Rhône

Alpin des Systèmes Complexes (IXXI). The authors

would like to thank Pierre Granjon, for excellent

discussions about impedance spectroscopy measure-

ments and analyses.

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