Modeling of generation and propagation of cardiac action potential using fractional capacitance

IOSR Journal of Biotechnology and Biochemistry (IOSR-JBB)
e-ISSN: XXXX-XXXX, p-ISSN: XXXX-XXXX, Volume 1, Issue 2 (Jan. – Feb. 2015), PP 01-11
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Modeling of generation and propagation of cardiac action
potential using fractional capacitance
1
Harsh Wardhan*, 2
Siddharth Singh
Center for VLSI and Embedded Systems Technology, International Institute of Information Technology
Hyderabad, Hyderabad-500032, Telangana, India
Abstract: This paper proposes modeling of canine cardiac action potential using fractional differential
equations (FDE). The movement of ions across cardiac myocytes (heart muscle) generates a voltage signal
which propagates along the body of the cell as action potential. The complex behavior of such physiological
events can be understood by developing mathematical models that describe the biological phenomena.
Typically, ordinary linear differential equations are used for constructing these mathematical models. In this
paper we have shown that a non-integer order differential equation model can capture these events more
precisely. Using FDE we are able to model cardiac action potential with a higher degree of accuracy compared
to ordinary differential equations (ODE).
Keywords: Action potential (AP), Cardiac arrhythmia, Cardiac myocytes, Electrical model, Fractional
differential equations (FDE), Ion exchange mechanism
I. Introduction
Heart beat is triggered by a series of electrical events taking place in the tissues of the heart. These
electrical events are characterized by changes in electric potential between the interior and exterior of the cells
of the cardiac tissue, leading to the generation of cardiac action potential. This action potential is a spike
generated by the difference of potential between the interior and the exterior of each cardiac cell and follows a
consistent trajectory [1]. Such spikes are generated due to the movement of ions through the transmembrane ion
channels in the cardiac cells [2]. The period during which the action potential spike lasts is called the absolute
refractory period. No other action potential can be fired during this period [3]. The absolute refractory period is
followed by the relative refractory period. During this period a stronger than usual stimulus is required to fire
another spike [3].
The abnormal conduction or irregular formation of an action potential can disrupt the normal rhythm of
the heart and may lead to conditions such as cardiac arrhythmia, tachycardia or bradycardia [7, 8]. Such
alterations in the normal rhythm of the heart, if not diagnosed properly, can prove to be fatal in many cases [8,
9]. A cardiac myocyte model captures the ionic phenomenon at the cellular level with a mathematical
description and describes the generation and propagation of the action potential [1]. This modeling of the
cardiac action potential can be extremely helpful in analyzing the rhythmicity of heart. Hodgkin and Huxley, in
1952, proposed an electrical model of the giant squid axon (nerve cell) [10]. Much of the mathematics of cardiac
cell modeling is inspired by the Hodgkin-Huxley (HH) model [11,12]. This analogy has been drawn because of
the fact that neural and cardiac cells have similar ion exchange mechanisms [1].
The complexity of mathematical models capturing the dynamics of cardiac myocytes has increased
over the past few decades [13, 14]. The discovery of new ion channels and ionic phenomena in addition to the
advancement in voltage-clamp techniques has contributed towards increasing this complexity [14-22].
In the resting phase, also known as the polarized state, the interior of the cardiac myocyte has higher
concentration of K+
ions as compared to the exterior [23]. Phosphate and the conjugate bases of organic acids
are the major anions inside the cell [23]. Outside the cell, Na+
ions and Cl-
ions have higher concentration [23].
In cardiac myocytes, at the beginning of the depolarization phase, the influx of Ca2+
ions through voltage-gated
calcium channels on the sarcolemma induces release of Ca2+
ions from the sarcoplasmic reticulum. This
phenomenon is called calcium-induced calcium release [4] and increases the free Ca2+
ions concentration in the
myoplasm causing muscular contraction. At the same time Na+
ions enter the cell, further depolarizing the cell.
As the cell potential reaches a threshold, fast Na+
ion channels open on the sarcolemma and an action potential
(AP) is fired. As the cell is depolarized and reaches a positive potential with respect to the exterior of the cell,
K+
ion channels open. The K+
ions leave the cell while Ca2+
ions are still entering the cell. No new AP spike can
be fired during this period. This period is known as the absolute refractory period. When the absolute refractory
period subsides, Ca2+
ion channels close while flow of K+
ions out of the cell continues, causing repolarization
of the cell. The cardiac muscle cells are tightly bound such that when one of these cells is excited the action
potential propagates to all of them [2, 4, 5]. The selective permeability of these ion channels leads to the
generation and propagation of the action potential [6]. Mutations in cardiac ion channels may lead to

Modeling of generation and propagation of cardiac action potential using fractional capacitance
deformation in the action potential [6].
Models of cardiac cell can provide useful insights into the underlying ion exchange mechanism [73].
Several models of cardiac myocytes have been proposed over the past few decades [73]. The Beeler-Reuter,
Luo-Rudy, Jafri et al., Winslow et al., Fox et al., Greenstein-Winslow, Cabo-Boyden, Hund-Rudy and Flaim et
al. models are some of the most significant ones [69,14-22].
Luo and Rudy, in 1991, proposed a mathematical model of the electrical activity of mammalian
ventricular myocyte [68] based on the Beeler and Reuter model [69]. Their aim was to improve the Beeler and
Reuter model by incorporating advanced empirical data available with the improvement in experimental
techniques in subsequent years. The Luo-Rudy model focuses on the phenomena of depolarization and
repolarization in cardiac action potential. This model was limited to modeling the phenomena related with the
fast inward Na+
current and the outward K+
currents. Later they improved the model by incorporating
electrophysiological phenomena that regulate intracellular Ca2+
ion handling. This model is known as the Luo-
Rudy 2 (LR2) model (1994) [18]. Jafri et al. (1998) improved upon the Luo-Rudy model by adding more
detailed descriptions of the Ca2+
ions. They incorporated calcium-induced calcium release from Sarcoplasmic
Reticulum (SR) in their model. Based on the guinea pig ventricular model proposed by Jafri et al. [15], Winslow
et al. (1999) developed the first canine ventricular myocyte model for normal and infarcted cells [16]. In this
model the data was taken from O’Rourke et al [70]. The Winslow et al. model predicts the mechanism of
increase in action potential duration during heart failure. This phenomenon is related to the change in the release
of Ca2+
ions from the SR.
The models discussed above, do not model the alternans of canine ventricular action potential duration
completely. To overcome this limitation, Fox et. al (2002) [13] developed a model incorporating ionic currents
from the Luo-Rudy [18], Winslow et al. [16] and Chudin et al. [17] models to describe the alternans occurring
at fast pacing rates.
The ionic currents in the infarcted cardiac myocytes are significantly different from those in the non-
infarcted cells. Cabo and Boyden (2003) developed a model to study the change in ionic currents in the infarcted
and normal cells and the response of the action potential of infarcted cells to antiarrhythmic agents [19]. This
model is based on the LR2 model.
Hund and Rudy [14], in 2004, developed an elaborate model of the canine ventricular epicardial cells
to study the relationship between pacing rate, action potential properties, and Ca2+
ion channels. This model was
based on the hypothesis that the action potential duration and Ca2+
transient current depend on the pacing rate of
cardiac myocytes. Change in the pacing rate could lead to cardiac arrhythmias that may even result in death.
This model incorporates more detailed description of calcium channels than many previous models.
Greenstein and Winslow (2002) [20] improved upon the Winslow et al. canine ventricular model
adding more details about the intracellular Ca2+
ion handling. The model captures the calcium induced calcium
release in the SR by using multiple calcium release subunits. This results in a very complex model with several
thousand variables in each simulation making it computationally very intense. Greenstein et. al [21], in 2006,
published a simplified version of the above model by using the method described by Hinch et. al [71].
Based on the simplified Greenstein et al. model, Flaim et al. (2007) [22] published a model to
investigate the contribution of sustained components of INa and IKv43 in shaping the canine ventricular action
potential and to study their impact towards increasing the risk of cardiac arrhythmia.
In the present work the Hund-Rudy dynamic (HRd) model has been considered based on canine
ventricular cell [14]. This model has been chosen for the following two reasons. Firstly, it incorporates the
dynamic Ca2+
/calmodulin-dependent protein kinase II [24] (CAMKII) activity, which helps in regulating ion
exchange in myocytes [25]. By autophosphorylation CAMKII enables detection of Ca2+
ion spike frequency and
determines the AP duration [26]. Secondly, the HRd model has low computational complexity and is
mathematically tractable. It has been empirically proven that the CAMKII regulatory pathway plays a role in the
force-frequency relation and rate-dependent CaT abbreviation [27-29]. Through the inclusion of the CaMKII
regulatory pathway this model represents an important advance in the physiologic representation of rate
dependent cell processes. It is one of the most appropriate cardiac cell models as it incorporates dynamic
CaMKII activity and regulation of intracellular calcium ions handling [13].
Hawkins et al. (1994) [30] have included “local control” Ca2+
ion release, in their rabbit AP model, as
proposed by Toyofuku et al. (1994) [31] which involves statistical activation of individual Ca2+
ion release units
in the Sarcoplasmic Reticulum. But, computational demands discourage the use of this model in modeling
cardiac arrhythmias. The HRd model, on the other hand, reproduces the local-control features by using a
macroscopic approach. This approach while reducing the computational demands, maintains the accuracy of the
Hawkins et al. model. It also incorporates late sodium ion current, dynamic chlorine ion concentration changes,
and calcium ion dependent outward transient current and novel Irel formulation. Ca2+
ion release from the
junction sarcoplasmic reticulum (JSR) constitutes the Irel current. Junction Sarcoplasmic Reticulum (JSR) is the
part of the SR that contains calcium release channels. These features of the model distinguish it from other

models where electrophysiological remodeling is done after heart failure [16] and cardiac infarction [19]. Since
the model has only 29 variables, it is computationally more tractable compared to the Greenstein et al. and
Flaim et al. models.
In the past two decades, fractional differential equations (FDE) are being used to model the dynamics
underlying complex biophysiological phenomena [32-37]. For example, in [32] it has been shown that the
dynamics of biological systems can be represented by FDE [39,40]. However, the ventricular cardiac cell
models, including the HRd model utilize ODE to model the bio-physiological dynamic phenomena. It has been
shown that FDE models the step response of a capacitor more accurately than ODE [41]. Thus the AP trajectory
generated by the HRd model does not model the refractory period accurately. In the current paper, a modified
HRd model has been proposed using FDE and compared the results with actual recorded canine ventricular
cardiac action potential data. In this paper it has been shown that the FDE model gives a more accurate cardiac
action potential compared to the conventional ODE model. An electrical circuit based on our fractional FDE
model has also been designed that gives an electrical abstraction of the transmembrane ion exchange
mechanism.
II. Theory
Ion exchange mechanism
There are essentially four phases of the cardiac action potential as shown in figure 1 [72]. The first
phase, phase 4 in figure 1, is the polarized or resting membrane potential. In this phase the cell is not being
stimulated [2]. The resting membrane potential in the ventricular epicardium is normally about -85 to -95 mV
[2]. In this phase there is a higher concentration of Na+
ions on the exterior of the cell membrane compared to
the interior. The concentration of K+
is higher on the interior of the membrane compared to the exterior. But the
net concentration of positively charged ions on the exterior of the cell is higher than the interior. This leads to a
negative potential inside the cell relative to the outside. This inequality in the distribution of ions is maintained
by the selectively permeable cell membrane. During the initial phase the membrane is highly permeable to K+
ions and relatively less impermeable to other ions. Thus K+
ions are instrumental in determining the resting
membrane potential. This resting state is disturbed when an adjacent cell stimulates the cell with an electric
current. As soon as the cell is stimulated, its permeability towards Na+
ions increases. This leads to a fast inward
movement of Na+
ions, as shown in figure 1, thus depolarizing the cell. This rapid depolarization triggers an
action potential [37, 38]. This phase is called phase 0. Once the interior of the cell reaches a peak positive
potential, the voltage-gated Na+
channels are closed. This is the phase 1 of the myocyte action potential (See
figure 1). In phase 2 of the cardiac action potential, the slow outward movement of K+
ions continues. A small
outward current is also contributed by the inward movement of the negatively charged Cl-
ions. This results in a
drop in potential inside the cell (Refer to figure 1). But at the same time voltage-gated Ca2+
ion channels open
and positively charged Ca2+
ions enter the cell. This leads to an equilibrium being formed between inward and
outward currents for about 150 ms leading to the formation of a “plateau” [2]. During this phase the potassium
ion permeability is reduced. This phase is followed by rapid repolarization. The voltage-gated Ca2+
ion channels
close, while the slow outward movement of K+
ions is maintained. This phenomenon maintains a net outward
current, and facilitates opening of more types of K+
ion channels. This phase is known as phase 3, as shown in
figure 1. This process continues till the interior of the cell reaches a potential of about -85 mV [43].
The duration between the firing of the action potential, phase 0, till the end of the plateau phase, phase
2, is known as the absolute refractory period. It is impossible to fire another action potential during this period.
The duration between the plateau phase, phase 2, till the end of the action potential, phase 4, is called the
relative refractory period. During this period a stronger-than usual stimulus is required to evoke another action
potential [3].
Hund-Rudy dynamic (HRd) model
The Hund Rudy Dynamic is a mathematical model of canine ventricular epicardial cells. This model
comprises several transmembrane currents, like fast Na+
ion current, fast and slow components of the delayed
rectifier K+
ion current, transient outward K+
ion current, inward rectifier K+
ion current, plateau K+
ion current,
L-type Ca2+
ion current, Na+
- Ca2+
exchanger ion current, Na+
-K+
ion pump current, Ca2+
ion current in
sarcolemma, late Na+
ion current, Ca2+
-activated Cl-
ion current, and background Ca2+
and Cl-
ion currents [14].
A salient feature of this model is that it includes the Ca2+
/calmodulin-dependent protein kinase (CaMKII).
CaMKII is an important chemical for regulating action potential and calcium ion transient (CaT) rate
dependence. This feature was not included in the previous models.

Figure 1: Ventricular action potential. In phase 0 there is rapid inflow of Na+
ions (INa) which leads to
depolarization of the cardiac myocyte. In phase 1, K+
and Cl-
ions start flowing out generating the transient
outward current, Ito1,2 and the fast Na+
ion channels are inactivated. A balance between outward movement of K+
ions through slow delayed rectifier potassium channels, IKs, and inward movement of Ca2+
ions (ICa) through L-
type calcium channels is maintained in phase 2. In phase 3 the slow delayed rectifier (IKs) K+
ion channels are
still open, while the L-type Ca2+
ion channels close. This net outward current facilitates the opening of other
types of K+
ion channels like rapid delayed rectifier K+
ion channels (IKr) and the inwardly rectifying K+
ion
current, IK1. This causes rapid repolarization of the cell. The delayed rectifier K+
ion channels close when the
membrane potential reaches about -85 mV, while IK1 remains conducting throughout phase 4, contributing to set
the resting membrane potential [72].
Even though the HRd model has improved upon the previous models, the absolute refractory period of
the cardiac action potential generated by this model is slightly less than the recorded action potential. The reason
for this is that the HRd model uses ordinary differential equations to describe the ionic processes that fail to
account for the dielectric losses in the membrane [44, 32]. Whereas, fractional differential equations, as already
discussed, model the step response of the capacitor more accurately [41].
Fractional differential equations (FDE)
The notion of fractional calculus comes from the idea that the order of differentiation can be
generalized to a real number [74]. Hence, in equation (1), q can be any real number [75]. And, when q takes an
integer value, Dq
acts as an ordinary integer order differential operator [45].
Dq
y = q
q
dx
yd
(1)
It is to be noted that for integer values of q, fractional derivative of a function at a given point depends
on the values of the function in the neighborhood of that point. Whereas, for integer values of q the derivative of
the function does not depend entirely on the neighborhood of that point [75]. In fact, the derivative shows long

range memory behavior [32, 52].
The idea of fractional derivative can be extended to the current voltage relationship of a capacitor. It
has been shown that all dielectrics are lossy in nature [46-48]. Hence, there can be no ideal capacitor. In fact, it
has been observed that, in general, dielectric materials encounter losses that can be best captured by a fractional
capacitor [49]. The theory of fractional capacitor model emerges from Curie’s empirical law [49]. This law can
be generalized to
q
q
f
dt
tud
Cti
)(
)(  (2)
where i(t) is the current across capacitor, Cf is the fractional capacitance, q is the order of derivative, and u(t) is
the excitation voltage.
III. Materials and methods
Fractional calculus (FC) implies integration and differentiation to an arbitrary order [51]. The order of
the differential or integral operator, for example q in equation (1), can take any real or complex value [50, 51].
FC has been successfully applied in various fields such as modeling of dynamical systems in control
theory, electrical circuits, transmission lines, and viscolelasticicy [53,54]. Recent studies have shown that FC,
when applied to problems in physics and engineering, provides significant advantage over conventional integer
order calculus in the modeling and control of systems [53-57]. The importance of fractional order models is that
they yield a more accurate description and give a deeper insight into the physical processes underlying a long
range memory behavior [44].
Capacitors are one of the most indispensable elements in integrated circuits and are used extensively in
many analog and digital electronic systems [58]. As capacitors occupy significant amount of die area, it is
essential that they are area-efficient [58]. However, various studies have demonstrated that the ideal capacitor
cannot exist in nature [46-48]. Westerlund and Ekstam have shown that the dielectric materials exhibit a
fractional behavior yielding electrical impedances of the form 1/(jωCF)q
, with q ϵ R+
[49].
Proposed fractional HRd model
The HRd model has been formulated using ODE. This model fails to capture the accurate refractory
period of the canine ventricular action potential. In fact, even in the steady state, ion exchange continues across
the membrane. This phenomenon cannot be modeled accurately through a static (or ideal) capacitor. Intuitively,
it should be possible to better capture the dynamic ion-exchange mechanism across the membrane of cardiac
myocyte through a leaky (or fractional) capacitor. Hence, we propose a modified form of the HRd model that
utilizes fractional capacitor. Later in this section, it will be shown that our proposed modification leads to
accurate modeling of absolute refractory period of canine ventricular cardiac action potential. Use of fractional
capacitor facilitates the modeling of the ion exchange mechanism of cardiac myocytes using FDE. The order of
an FDE, however, cannot exceed the number of available initial conditions [59]. Hence, the minimum number
of initial conditions required is the smallest integer greater than the order of the FDE. The HRd model has only
one set of initial conditions. This implies that the order of the FDE cannot exceed 1. The modified HRd model is
given by
f
tot
q
m
q
C
I
dt
Vd 
 (3)
where,
q Order of fractional derivative,
Vm Voltage difference across membrane,
Cf Fractional capacitance,
Itot Total transmembrane current,
Itot = ICa,t + INa,t + IK,t + ICl,t + Istim (4)
ICa,t Total transmembrane Ca2+
current,
ICa,t = ICa(L) + ICa,b + Ip,Ca – 2INaCa (5)
INa,t Total transmembrane Na+
current,
INa,t = INa + 3INaK + ICa,Na + 3INaCa + INa,L (6)
IK,t Total transmembrane K+
current,
IK,t = IKs + IKr + IK1 + ICa,K + Ito1 + IKp – 2INaK (7)
ICl,t Total transmembrane Cl-
current,
ICl,t = Ito2 + ICl,b (8)

Istim Stimulus current, μA/μF
The initial value of the membrane potential Vm was -84.4 mV and the current stimulus Itot was 1 µA.
The current stimulus is applied for a period of 0.01 ms. Once the spike reaches a threshold value of -70 mV,
action potential is fired. The value of the fractional capacitance was 0.0015903 µF/cm2
. The detailed list of ion
concentrations and other parameters used in the model is described in table 1 [14].
Table 1: Ionic concentrations at rest*
IV. Results
Determination of fractional order
The modified HRd model with fractional capacitance was simulated for different values of order of
derivative q, and corresponding values of absolute refractory period Tr, were obtained. A mathematical relation
was found between q and Tr using curve fitting method. A polynomial equation of third order was obtained as
given by Eq. 9.
43
2
2
3
1 pTpTpTpq rrr  (9)
where,
q = fractional order
Tr = absolute refractory period
p1 = -7.476 * 10-6
p2 =0.004595
p3 = -0.9419
p4 = 65.377
Fig. 2 shows the plot of fractional order corresponding to the given refractory period. Along the x-axis
is the value of the absolute refractory period in milliseconds (ms) and the y-axis shows the corresponding order
of fractional derivative (q). Thus, given a refractory period value, its corresponding order of derivative can be
found from the graph. The scatter plot in Fig. 2 shows the values obtained through simulations of the modified
HRd model for various values of q. The solid line in the same figure shows the values obtained from Eq. 9. The
graph shows the absolute refractory period corresponding to fractional order from 0.982 to 1. It is well known
that Dq
has an m-dimensional kernel, and therefore we certainly need to specify m initial conditions in order to
obtain a unique solution of the straightforward form of a fractional differential equation, viz.
Dq
y(x) = f(x,y(x)) (10)
with some given function f. But, using Eq. 9 the graph can be extrapolated in both directions. This
overcomes the limitation imposed by the availability of a single initial condition which restricts the maximum
value of order of derivative to 1.
The order of fractional derivative for a given absolute refractory period can be obtained from the above
equation. The error between the data obtained from simulation and that from the equation was calculated. The
accuracy turned out to be 99.8 percent.
The value of absolute refractory period obtained from the simulation of the HRd model was 186.4 ms,
whereas the actual recorded value is 200.45 ms [60]. It can be noted that there is significant difference between
the refractory period obtained from the HRd model and that from the actual data. We obtained the exact value of

the refractory period, 200.45 ms [60] (refer to figure 5), for order of FDE taken as 0.985, the set of parameters
and initial conditions being the same. Fig. 3 shows the change in refractory period as the order of differentiation
is changed from 1 to 0.985. Fig. 4 shows the comparison of the proposed model with HRd model and
experimental data. The experimental data was obtained from the paper by Sicouri and Antzelevitch [60].
Figure 2: Fractional order corresponding to given refractory period. The solid line shows the plot obtained from
curve fitted eq. 9 and the dotted line shows the plot obtained from the simulations of the modified HRd model
for different values of order of derivative versus absolute refractory period.
Figure 3: Action potential for order of derivative q = 1 and q = 0.985.

Figure 4: Comparison of the proposed model with HRd model and experimental data.
Figure 5: Electrical abstraction of the cardiac cell membrane. Cf is the fractional capacitor. GNa, GK, GKs, GCl,
GCaL, and GCab are conductances of Na+
ions, K+
ions, slow delayed rectifier K+
ion channels, Cl-
ions, L-type
Ca2+
ion channels, and background Ca2+
ion channels, respectively. INa, IK, IKs, ICl, ICaL, and ICab are the
corresponding currents through these channels.
Circuit model
The modified HRd model was translated into an electrical network comprising fractional capacitor,
resistors, and voltage and current sources as shown in Fig. 5. This circuit describes the changes in Na+
, K+
, and
Ca2+
ions in the myoplasm, and the Ca2+
ions in the sarcoplasmic reticulum. The semipermeable membrane of
the myocyte acts as a fractional capacitor by creating a charge separation between the interior of the cell and the
extracellular environment. The fractional capacitor accounts for the dielectric losses in the membrane. The
Nernst potential is created by the difference of ion concentrations across the cell membrane in resting phase, and
is represented by a battery. Conductances of the voltage dependent ion channels are represented by variable
resistors. Ion channels which do not depend on voltage are represented as current sources.
V. Discussion
The current across a capacitor is proportional to the first order derivative of the potential difference
between two points. Whereas, the current across a resistor is linearly proportional to the potential difference
between the two points. A leaky capacitor exhibits a behavior similar to a resistor, to a certain degree. The order
of the fractional derivative 'q' is determined by this behavior. Thus, the fractional capacitance implies that the
property of the capacitor lies between an ideal capacitor and an ideal resistor. The closer the order 'q' lies to 1,

the more it resembles a capacitor. As the order decreases, the behavior of the capacitor tends to resemble a
resistor. Hence, Equation (2) models the behavior of a capacitor more accurately. Equation (3) shows the HRd
model generalized for fractional order.
In order to verify graphically whether the proposed methodology leads to high accuracy, we evaluate
the modified HRd model and compare the results with both the actual recorded signal and the HRd model graph.
Figure 4 shows the results of the comparison. It can be observed that the value of the absolute refractory period
obtained from the fractional HRd model is closer to actual recorded data as compared to that of the original HRd
model. This validates our proposed modification. The results obtained for canine action potential can be
extended to the human cardiac action potential as well.
Animal research has been instrumental in developing most of the medical treatments for humans [61-
65]. Hence, the above model applied to canine cardiac action potential, can also be extended to the modeling of
cardiac action potential in various regions of the heart and subsequently ECG signals in human beings. For
example, the cardiac action potential can be mapped to the PR interval. Estimation of PR interval is critical to
cardiac diagnosis [66, 67].
VI. Conclusions
In this paper we have proposed a modified form of the HRd model using fractional capacitance. This
model captures the refractory period of the canine ventricular action potential precisely. Though several studies
have been conducted for modeling the cardiac action potential, none of them were able to model the refractory
period accurately. The refractory period is significant from the point of view of early detection of harmful
cardiac arrhythmia which may, otherwise, lead to fatal conditions like myocardial infarction or heart attack.
We have also found a mathematical relation between refractory period and order of fractional
derivative. This relation allows us to estimate the refractory period for order of derivative greater than 1 which
was not possible otherwise.
An electrical circuit of the proposed model has also been designed.
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