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Bioelectrode Models with Fractional CalculusEmmanuel A. GONZALEZ, L’ubomír DORCÁK, and Celso B. CO
Abstract—The introduction of fractional calculus has spurred alot of possibilities in biomedical engineering and bioengineeringresearch and development, especially in the modeling, design,and analysis of bioelectrodes. With the advent of memristorsand fractional calculus, it is seen that these concepts wouldhave several advantages and applications in such fields. In thisshort note, we present various bioelectrode equivalent circuitmodels including their impulse and frequency responses. Theequivalent models presented utilize the concepts of memristorsand fractional calculus.
Index Terms—Fractional calculus, fractor, memristor, bioelec-trode.
I. INTRODUCTION
Recording and stimulating bioelectrodes is a very importantpart in the field of bioengineering as it deals with the optimaldesign of such components for electrode/tissue interface appli-cations. The design of such components would, however, startwith the development of a mathematical model and possiblyan equivalent circuit model for the electrical impedance of theinterface. A lot of efforts have also been directed toward thedevelopment of accurate impedance models for electrophys-iological recording and electrical simulation [11], [13], [16],[18].
Such bioelectrodes can be modeled using passive electriccomponents, i.e. resistors, inductors, and capacitors. However,most of time, models only lie with resistors and capacitorsdue to practical reasons. The resistance in the model ofan electrode usually represents the intrinsic resistance ofthe electrolyte, while the capacitance is associated with theinterface. By making appropriate assumptions for the values ofthe resistance and capacitance, the impedance then determinesthe response of the electrode in time and frequency domain—in which, most electrodes especially designed for electrocar-diogram (ECG) systems, cell suspensions, and whole tissuesare of the first-order [10], [12], [20], [19].
Although there is a great amount of literature involvingthe use of integer-order dynamic models for bioelectrodes,there are still some cases in which the agreement between
E. A. Gonzalez (corresponding author) is with the Department of ComputerTechnology, College of Computer Studies, De La Salle University Manila,2401 Taft Ave., Malate Manila 1004, Philippines, and with the Schoolof Electrical Engineering, Electronics and Communications Engineering,and Computer Engineering, Mapua Institute of Technology, Muralla St.,Intramuros Manila, Philippines. He is also with Jardine Schindler ElevatorCorporation, 8/F Pacific Star Bldg., Sen. Gil Puyat Ave. cor. Makati Ave.Makati City, Philippines (e-mail: [email protected])
L. Dorcák is with the Department of Applied Informatics & Process Controlas well as a B.E.R.G. faculty at the Technical University of Kosice, B. Nem-covej 3, 042 00 Kosice, Slovak Republic (e-mail: [email protected])
C. B. Co is with the Department of Electronics, Computer, and Commu-nications Engineering, School of Social Science and Engineering, AteneoDe Manila University, Katipunan Ave., Quezon City, Philippines (e-mail:[email protected])
the theoretical model of the bioelectrode’s impedance andthe observed impedance seems to be questionnable especiallyif the angular frequency of characterization follows a non-integer-order power [14]. In other words, the mathematicalmodel describing the bioelectrode may not follow an integer-order dynamics, consequently leading to fractional calculusthat is able to describe fractional dynamics—dynamics ofsystem having non-integer-order representations.
II. FRACTIONAL CALCULUS
Fractional calculus is a topic that has been dealt for morethan three centuries up to this date, and researches in thetheories and applications of this field are still rapidly growing.The term fractional calculus is actually a misnomer sincethe actual designation is integration and differentiation ofarbitrary order, which is rather approriate in the currentsetting. It started in 1695 when L’Hospital raised a questionto Leibniz as to the meaning of dny/dxn if n = 1/2, that wasanswered: “...this is an apparent paradox from which, one day,useful consequences will be drawn... d1/2x will be equal tox√dx : x...”
The derivative of abitrary real order α has the notationaD
αt f (t), where a and t are the limits of operation, and α
is the order of differentiaton or integration. For fractionaldifferentiation, it is assumed that the order α is positive., i.e.α > 0. If α < 0, then the operation becomes a fractionalintegration having the general notation aD
−αt f (t).
Fractional differentiation and integration come in manydefinitions as they are studied and developed throughout theyears. One of the most-common definition is the Grünwald-Letnikov (GL) differintegration that is defined by
GLa Dα
t f (t) = limh→0;nh=t−a
h−αn∑r=0
(−1)r
(αr
)f (t− rh) ,
(1)which represents the derivative of order m if α = m and them-fold integral if α = −m. The values of a and t are thelimits of operation. Definition (1) is derived from the well-known definition of the nth-order derivative
f (n) (t) = limn→0
h−nn∑r=0
(−1)r
(nr
)f (t− rh) , (2)
where n is the order of differentiation and(nr
)is the usual
binomial coefficient notation. In the particular case of (1), thebionmial coefficient results in the expression(
αr
)=
Γ (α+ 1)
Γ (r + 1) Γ (α− r + 1). (3)
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMSVOL. 17, NO. 1, MARCH 2012, 7-13
Another definition of fractional differintegration is theRiemann-Liouville (RL) definition given by
RLa Dα
t f (t) =1
Γ (n− α)
dn
dtn
ˆ t
a
f (τ)
(t− τ)α−n+1 dτ, (4)
for n− 1 < α < n.However, especially in the frequency domain, both the RL
and GL definitions lead to initial conditions containing limitvalues that may not be physically interpretable. In spite ofhaving some fractional calculus intial value problems that aresolved mathematically through RL and GL definitions, theirsolutions are practically useless as there may be no physicalinterpretations for such types of initial conditions. This prob-lem is solved by M. Caputo by allowing the formulation ofinitial conditions for intial-value problems for fractional-orderdifferential equations in a form involving only integer-orderlimit values. Caputo’s approach is defined by
CaD
αt f (t) =
1
Γ (n− α)
ˆ t
a
f (n) (τ) dτ
(t− τ)α−n+1 , (5)
for n− 1 < α ≤ n.Looking into Caputo’s definition, the Laplace transform of
the Caputo fractional derivative of order n− 1 < α ≤ n is
L{CaD
αt f (t)
}= sαF (s)−
n−1∑k=0
sα−k−1f (k) (0) . (6)
If initial conditions are not considered, especially in impedancestudies, then the Laplace-transform of (5) would simply resultto
L{CaD
αt f (t)
}= sαF (s) . (7)
A thorough discussion of fractional calculus and fractionaldifferential equations can be found in [17].
III. THE FRACTOR
A fractor [8], [15] is defined as either a capacitive orinductive element that is following fractional-order properties.Fractors were most probably formally introduced by G. E.Carlson and C. A. Halijak, which is also presented by S.Westerlund and L. Ekstram, who obviously are unaware aboutCarlson and Halijak’s work [1], [2], [3], [4], [21].
Fractors are considered to be a special case of memristors—a theoretical concept introduced by L. O. Chua in 1971 [5],[6], [7], which was then discovered by S. Williams and hisresearch group of scientists in HP Labs. Fractors can bededuced as memristive devices, having properties equivalentto real electrical elements with fractional order mathematicalmodels generally described as
Z (s) = Ksk, (8)
where K ∈ R+ is a constant representing a physical elementthat be a resistor, capacitor, inductor, or a generic memris-tor, and k ∈ R is the order of the memristive device. Ifk ∈ (−1, 0), then the memristive device is considered aspartially-capacitive. On the other hand, if k ∈ (0, 1), then thememristive device is considered as partially-inductive. Havingk = 0 simply means that the memristive device is purely
resistive, and having k equal to 1 and −1 would make thedevice purely inductive, and purely capacitive, respectively.
One thing to be noted is that fractors can physically repre-sent components having fractional-order properties, and maybe the actual physical representation of some bioelectrodespreviously tested.
IV. FRACTIONAL BIOELECTRODE MODELS ANDPROPERTIES
In this section, we present a class of bioelectrode modelsthat can be used for bioelectrode benchmarking purposes. Themodels presented in this section are of the fractional-ordertype that can be physically implemented using memristors, aswell as resistors and capacitors.
Throughout this section, the following notations will beused: R to denote the internal resistance value, C to denote theinternal capacitance value, L to denote the internal inductancevalue, M to denote the memristance value. The order of valueof α in most cases positive, is expected to be non-integer invalue.
A. Fractor (F) Class
The simplest class of a bioelectrode model that is derivedfrom fractional calculus has the impedance of the form
ZF (s) = Msα (9)
where M > 0 and α /∈ Z. However, one should be care-ful in generalizing the order α. In particular cases whereα = 1, 2, · · · , this class would just be equivalent to a puredifferentiator of order 1, 2, · · · , and magnitude responses willresult in a slope of 20dB per decade (20dB/dec) per puredifferentiator. For example, if α = 1, then the model willsimilarly result in a pure inductor where M = L being theequivalent value of inductance. For a particular case where0 < α < 1, the model in (9) will result in a partially-inductiveelement. On the other hand, having α = −1,−2, · · · wouldmake (9) equivalent to a pure integrator of order 1, 2, · · · , thushaving magnitude responses equivalent to -20dB/dec per pureintegrator, and M = 1/C, where C being the equivalent valueof capacitance.
The magnitude spectrum of this class is
|ZF (jω)| = |M (jω)α| = Mωα (10)
which will have a slope of 20αdB/dec in the entire frequencyspectrum. If α = 1/2, then the model will be a semi-differentiator having a slope of 10dB/dec for the entire fre-quency spectrum. This can be graphically proven by plottingthe magnitude spectrum (10) in a logarithmic frequency scaleand a magnitude scale with dB as the unit of measure (seeFig.1).
The impulse repsonse of (9) is determined to be
hF (t) =Mt−(α+1)
Γ (−α)(11)
in the restriction that α is not an integer. In (11), the termΓ (·) is known as the Euler’s gamma function.
8 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL.17, NO. 1, MARCH 2012
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30
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50
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Bode Diagram
Frequency (rad/sec)
Figure 1. Magnitude and phase responses of a semi-differentiator, i.e.,Z (s) = s1/2. The slope of the magnitude is 10dB/dec.
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4
6
8
10
Time, t
Am
plitu
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Impulse Response for Class F Systems
alpha = 1.5
alpha = −1.0
alpha = −0.5
alpha = 0.5
Figure 2. Impulse response (11) with α = {−0.5, 0.5, 1.0, 1.5}.
Looking further into the impulse response (11) and lettingM = 1 without loss of generality, one can see that theinitial value of the impulse response would depend on thevalue of order α since the term Γ (−α) dictates if the entireresponse is positive of negative. If k − 1 < α < k fork = 1, 3, 5, · · · , then the impulse response becomes negative.Otherwise, if k = 2, 4, 6, · · · , then the impulse responsebecomes positive. However, in the case where the model iscapacitive, i.e., −1 < α < 0, the impulse response becomespositive. When α ≤ −1, the model becomes unstable sincelimt→∞ ‖h (t)‖ 6= 0, where ‖·‖ is any vector norm. Theimpulse response (11) with M = 1 and for different values ofα are shown in Fig. 2.
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Frequency (rad/sec)
Z1
Z2
Z1Z2
Figure 3. Magnitude and phase responses of the SFR class with transferfunctions Z1 (s) = s0.3 + 1 and Z2 (s) = s0.8 + 1.
B. Series Fractor-Resistor (SFR) Class
The series combination of a resistor and a fractor has theimpedance form of
ZSFR+ (s) = R+Msα, (12)
where R,M > 0, for α > 0 s.t. α /∈ Z+, and
ZSFR− (s) =Rsα +M
sα, (13)
for α < 0 s.t. α /∈ Z− . This class is treated separately forpositive and negative values of α.
1) α > 0: The magnitude response of (12) is given by
|ZSFR+ (jω)| = |R+M (jω)α|
=
√R2 +M2ω2α + 2RMωα cos
(4n+ 1)απ
2, (14)
for n = 0, 1, 2, · · · . The Bode plots for this class are depictedin Fig 3. As the value of frequency increases from its cut-off,i.e., ω >> ωC , the magnitude response increases by a slopeof 20αdB/dec. However, when frequency decreases from itscut-off, i.e., ω << ωC , the magnitude reponse tends to be aconstant with a slope of approximately 0dB/dec.
The impulse response of (12) is determined to be
hSFR+ (t) = Rδ (t) +Mt−(α+1)
Γ (−α), (15)
where δ (t) is the well-known Dirac-delta or impulse function.Furthermore, it is seen in (15) that the value of resistancedoesn’t affect much the impulse response as it only providesa spike at t = 0. When R = 0, the impulse responseresults in something similar to a simple fractor, i.e., F class.When k − 1 < α < k for k = 1, 3, 5, · · · , the impulseresponse becomes negative. Furthermore, for k = 2, 4, 6, · · · ,the response becomes positive.
GONZALEZ et al: Bioelectrode Models with Fractional Calculus 9
0
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60
80M
agni
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(dB
)
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10−2
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102
104
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−30
0
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se (
deg)
Bode Diagram
Frequency (rad/sec)
Z1
Z2
Z1
Z2
Figure 4. Magnitude and phase responses of the SFR class with transferfunctions Z1 (s) = s−0.3 + 1 and Z2 (s) = s−0.8 + 1.
2) α < 0: The magnitude response of (13) is given by
|ZSFR− (jω)| =∣∣∣∣R (jω)
α+M
(jω)α
∣∣∣∣=
1
ωα
√M2 +R2ω2α + 2MRωα cos
(4n+ 1)απ
2, (16)
for n = 0, 1, 2, · · · . The magnitude and phase responses forthis type are depicted in Fig 4. It can be seen in Fig. 4that the magnitude reponse tends to be constant as the valueof ω increases from its cut-off. However, below cut-off, themagnitude changes with frequency.
The impulse response of (13) is determined by
hSFR− (t) = Rδ (t) +Mt(|α|−1)
Γ (|α|). (17)
Similarly to the previous type where α > 0 s.t. α /∈ Z+, theimpulse response also has a spike generated by the resistancevalue R at t = 0. When −1 < α < 0, the impulse responsebecomes positive. However, when α ≤ 1, the model becomesunstable, having a phenomenon similar to that of the F class.
C. Parallel Fractor-Resistor (PFR) Class
The parallel combination of a resistor and a fractor has theimpedance form of
ZPFR+ (s) =Rsα
sα + RM
, (18)
when α > 0 s.t. α /∈ Z+, and
ZPFR− (s) =M
sα + MR
, (19)
when α < 0 s.t. α /∈ Z−. In both cases, M,R > 0. However,each type must be considered separately.
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Frequency (rad/sec)
Z1
Z2
Z2
Z1
Figure 5. Magnitude and phase responses of the PFR class with transferfunctions Z1 (s) = s0.3/
(s0.3 + 1
)and Z2 (s) = s0.8/
(s0.8 + 1
).
1) α > 0: The magnitude spectrum of this type is
|ZPFR+ (jω)| =
∣∣∣∣∣ R (jω)α
(jω)α
+ RM
∣∣∣∣∣=
Rωα√ω2α + 2M
R ωα cos (4n+1)απ2 +
(RM
)2 , (20)
for n = 0, 1, 2, · · · . With high values of ω >> ωC , where it isassumed that ωC is the cut-off frequency, the magnitude tendsto be constant with a magnitude of R. On the other hand, atlower frequencies, ω << ωC , the magnitude tends to increasewith frequency. This would then result in a magnitude responsethat is similar to a high-pass filter, except that the slope atlower frequencies is not necessarily a multiple of 20dB/dec.The Bode plots of (20) for the case where M,R = 1 andα = {0.3, 0.8} are shown in Fig. 5.
The impulse response of (18) is
hPFR+ (t) = R·Rα,α(− RM, t
)= R
∞∑k=0
(− RM
)ktα(k−2)−1
Γ (αk),
(21)where Rq,v (a, t) is known as the Generalized R function (seepage 30 of [9]) defined by
Rq,v (a, t) =∞∑k=0
akt(k+1)q−1−v
Γ (q (k + 1)− v)(22)
having the Laplace transform of sv/ (sq − a).2) α < 0: The magnitude spectrum of this type is
|ZPFR+ (jω)| =
∣∣∣∣∣ M
(jω)α
+ MR
∣∣∣∣∣=
M√ω2α + 2M
R ωα cos (4n+1)απ2 +
(MR
)2 , (23)
for n = 0, 1, 2, · · · . Unlike in the previous type where α > 0,this type is similar to a low-pass filter. When the operating
10 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 17, NO. 1, MARCH 2012
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Frequency (rad/sec)
Z1
Z2
Z2
Z1
Figure 6. Magnitude and phase responses of the PFR class with transferfunction Z1 (s) = 1/
(s0.3 + 1
)and Z2 (s) = 1/
(s0.8 + 1
).
frequency is much greater than its cut-off, i.e., ω >> ωC ,its magtniude decreases with frequency. However, when theoperating frequency is much less than its cut-off, i.e., ω <<ωC , its magnitude remains at constant equal to R. The Bodeplots of (23) with M,R = 1 and α = {0.3, 0.8} are shown inFig. 6.
The impulse response of (19) is
hPFR− (t) = M ·Fα(−MR, t
)= M
∞∑k=0
(−MR
)ktα(k+1)−1
Γ (α (k + 1)),
(24)where Fq (a, t) is known as the Rabotnov-Hartley function [9]defined by
Fq (a, t) =
∞∑k=0
akt(k+1)q−1
Γ (q (k + 1))(25)
having the Laplace transform of 1/ (sq − a).
D. Parallel Fractor-Capacitor (PFC) Class
This model depicts an intrinsic capacitor connected inparallel with a fractor. The impedance of the PFC class isdefined by
ZPFC+ (s) =1
C
sα
sα+1 + 1MC
, (26)
where M,C > 0 and α > 0 s.t. α /∈ Z+,
ZPFC−,1 (s) =1/C
s|α|(sβ + 1
MC
) , (27)
where −1 < α < 0 and |α|+ β = 1, and
ZPFC−,2 (s) =1/C
s|α|(sγ + 1
MC
) , (28)
where α < −1 s.t. α /∈ Z− and |α| − γ = 1.
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45
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se (
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Bode Diagram
Frequency (rad/sec)
Z1 Z2
Z1 Z2
Figure 7. Magnitude and phase responses of the PFC class with transferfunction Z1 (s) = s0.05/
(s1.05 + 1
)and Z2 (s) = s0.1/
(s1.1 + 1
).
1) α > 0: The magnitude response of (26) is defined by
∣∣Z(PFC)+ (jω)∣∣ =
∣∣∣∣∣ 1
C
(jω)α
(jω)α+1
+ 1MC
∣∣∣∣∣=
1
C
ωα√ω2(α+1) +
(2
MC
)ωα+1 cos (4n+1)(α+1)π
2 +(
1MC
)2 .(29)
The shape of the magnitude response of such type woulddepend on the value of α. As α approaches 0, the responsebecomes similar to a low-pass filter. However, as α increases,the shape becomes similar to that of a narrow band-pass filterwith the left and right slopes having different values. Fig. 7shows that Bode plots of such type with α = {0.05, 0.1},while Fig. 8 show the Bode plots of such type with α ={0.8, 1.2}. Without loss of generality, it is assumed that M =1 and C = 1.
The impulse response of (26) is
hPFC+ (t) =1
C· Rα+1,α
(− 1
MC, t
)=
1
C
∞∑k=0
(− 1MC
)ktk(α+1)
Γ (k (α+ 1) + 1), (30)
where Rq,v (a, t) is known as the Generalized R functiondiscussed in the previous section.
2) −1 < α < 0: The magnitude response of (27) is definedby
|ZPFC−,1 (jω)| =
∣∣∣∣∣∣ 1
C
1
(jω)|α|(
(jω)β
+ 1MC
)∣∣∣∣∣∣
=1
C
1
ω|α|√ω2β +
(2
MC
)ωβ cos (4n+1)βπ
2 +(
1MC
)2 , (31)
where β = 1 − |α|. When α approaches 0, the magnituderesponse becomes similar to that of a low-pass filter. However,
GONZALEZ et al: Bioelectrode Models with Fractional Calculus 11
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180
270
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Bode Diagram
Frequency (rad/sec)
Z1
Z1
Z2
Z2
Figure 8. Magnitude and phase responses of the PFC class with transferfunction Z1 (s) = s0.8/
(s1.8 + 1
)and Z2 (s) = s1.2/
(s2.2 + 1
).
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Z1
Z2
Z1
Z2
Figure 9. Magnitude and phase responses of the PFC class with transfer func-tion Z1 (s) = 1/s|−0.1| (s0.9 + 1
)and Z2 (s) = 1/s|−0.9| (s0.1 + 1
).
when α approaches −1, the magnitude response becomessimilar to that of an integrator. This observation is depicted inFig. 9, where without loss of generality, M = 1, C = 1 andα = {−0.1,−0.9}. The impulse response of (27) is
hPFC−,1 (t) =1
C· Rβ,1/|α|
(− 1
MC, t
)=
1
C
∞∑k=0
(− 1MC
)ktβ(k+1)−1−1/|α|
Γ (β (k + 1)− 1/ |α|). (32)
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90
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Frequency (rad/sec)
Z1
Z2
Z1
Z2
Figure 10. Magnitude and phase responses of the PFC classwith transfer function Z1 (s) = 1/s|−1.1| (s0.1 + 1
)and Z2 (s) =
1/s|−6.1| (s5.1 + 1).
3) α < −1: The magnitude response of (28) is defined by
|ZPRC−,2 (jω)| =
∣∣∣∣∣ 1
C
1
(jω)|α| (
(jω)γ
+ 1MC
) ∣∣∣∣∣=
1
C
1
ω|α|√ω2γ +
(2
MC
)ωγ cos (4n+1)γπ
2 +(
1MC
)2 , (33)
where γ = |α| − 1. As α approaches −∞, the slope of themagnitude response increases negatively. The slope of themagnitude response before and after the cut-off frequency,however, are different. This can be seen in Fig. 10 forα = {−1.1,−6.1} with M = 1 and C = 1 without lossof generality.
The impulse response of (28) is
hPFC−,2 (t) =1
C· Rγ,1/|α|
(− 1
MC, t
)=
1
C
∞∑k=0
(− 1MC
)ktγ(k+1)−1−1/|α|
Γ (γ (k + 1)− 1/ |α|). (34)
E. Addition of a Delay Factor
In addition to the models above, if a dynamic impulseresponse of the actual bioelectrode in measure exhibits a delaycompared to the fractional model, then a delay factor intoimpedance can be added, thus resulting in the model
Z (s) = ZX (s) e−τs, (35)
where τ > 0 is the time delay in seconds, and ZX (s) is theimpedance model discussed in the previous subsections. It isalso important to note that the addition of this delay factordoes not change the magnitude response of the model. Phasereponse, however, is modified by a factor of ωτ , which alsocorresponds to an impulse response h (t− τ).
12 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 17, NO.1, MARCH 2012
V. CONCLUSION
In this short note, we present the use the fractional calculusin the mathematical modeling of bioelectrodes with the com-bination of the traditional passive elements and memristivedevices, i.e., fractors. Models presented in this paper includeimpedances represented by 1) a single fractor, 2) series combi-nation of a fractor and a resistor, 3) parallel combination of afractor and a resistor, and 4) parallel combination of a fractorand a capacitor. The impulse and frequency responses of theimpedances of these models are also presented and discussedin detail. Finally, we also show indirectly that fractionalcalculus is being ubiquitous and its use in modeling andsimulation of electric circuits and their equivalent bioelectrodemodels must also be considered.
ACKNOWLEDGEMENT
This work is supported under a research grant by JardineSchindler Elevator Corporation, 8/F Pacific Star Building.,Sen. Gil Puyat Ave. cor. Makati Ave., Makati City, Philippines,and under grant VEGA 1/0404/08 from the Slovak GrantAgency of Science.
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