Processes 2019, 7,20
Ca 2+ influx. Also chronic cardiac insufficiency may appear with an elongation of the AP by a reduction
of the repolarising K + currents.
1.2. The Mathematical Model
The history of the modelling of APs of excitable biological cells as neurons and cardiac muscle cells
starts with the famous and pioneering Hodgkin-Huxley model in 1952 [19]. In this paper, the authors
established a mathematical approach that can be used to model an AP of excitable biological cells,
i.e., one uses a Hodgkin-Huxley (type) formalism for the description of APs as systems of ordinary
differential equations. The first model of a cardiac cell is the Noble model [20] of a generic Purkinje cell.
In 1991, Luo and Rudy published an ionic model for cardiac action potential in guinea pig ventricular
cells. Moreover, the Ten Tusscher-Noble-Noble-Panfilov model [21] from 2004 describes a model for
human ventricular tissue, cf. also [2]. Such conductance-based models are based on an equivalent
circuit representation of a cell membrane. These models represent a minimal biophysical interpretation
for an excitable biological cell in which current flow across the membrane is due to charging of the
membrane capacitance and movement of ions across ion channels. Ion channels are selective for
particular ionic species, such as calcium (Ca 2+ ) or potassium (K + ), giving rise to currents I Ca 2+ or I K + ,
respectively. Our simplistic model reads as follows:
dV
dt
= −
I K + + I Ca 2+
C m
,
(1)
with the membrane capacity C m = 1
μF
m 2 and ion currents
I K + := G K + · x · (V − E K + ) and I Ca 2+ := G Ca 2+ · d · f · (V − E Ca 2+ ),
(2)
where the different gating variables d, f and x are satisfying the differential equation
dy
dt
=
y ∞ (V) − y
τ y
(3)
and y represents the gating variables d, f and x, while
y ∞ := y ∞ (V)=
1
1 + exp
V−V Ty
ky
(4)
with V Ty ∈ R, k y ∈ R\ {0} denotes the equilibrium of the corresponding gating variable and τ y is the
corresponding relaxation time constant for each of d, f and x. The gating variables d, f and x ∈ [0, 1]
are important for the activation (opening) and inactivation (closing) of the ion channels and therefore
for the ion current interaction, see [15]. Moreover, the Nernst potentials of these ion currents are
denoted by E Ca 2+ and E K + , while the corresponding conductance are represented by G Ca 2+ = 0.025
mS
cm 2
and G K + = 0.05
mS
cm 2 , respectively. Furthermore, the relaxation time constants are given by τ f = 80 ms
and τ x = 300 ms. We have to remark that in [1] it is assumed that the gating variable d is equal to its
steady state. Please note that if τ d tends to zero, we have the situation as in [1], since
τ d
dd
dt
=(d ∞ − d) ⇒ 0 =(d ∞ − d),
as τ d → 0. In this paper, we will use the relaxation time constant of d, i.e., τ d , as further non-zero
parameter. Moreover, the choice τ d = 0.1 ms yields the same trajectory as in [1], but also smaller
values of τ d are conceivable. In Figure 1 some examples of EADs are presented with τ d = 0.1 ms
and G Ca 2+ ∈
0.029
mS
cm 2 ; 0.03
mS
cm 2 ; 0.031
mS
cm 2 ; 0.035
mS
cm 2
(from left to right). Please compare Figure 6a in
[15] with the second trajectory in Figure 1. Here, we see that the four dimensional system behaves
66
Ca 2+ influx. Also chronic cardiac insufficiency may appear with an elongation of the AP by a reduction
of the repolarising K + currents.
1.2. The Mathematical Model
The history of the modelling of APs of excitable biological cells as neurons and cardiac muscle cells
starts with the famous and pioneering Hodgkin-Huxley model in 1952 [19]. In this paper, the authors
established a mathematical approach that can be used to model an AP of excitable biological cells,
i.e., one uses a Hodgkin-Huxley (type) formalism for the description of APs as systems of ordinary
differential equations. The first model of a cardiac cell is the Noble model [20] of a generic Purkinje cell.
In 1991, Luo and Rudy published an ionic model for cardiac action potential in guinea pig ventricular
cells. Moreover, the Ten Tusscher-Noble-Noble-Panfilov model [21] from 2004 describes a model for
human ventricular tissue, cf. also [2]. Such conductance-based models are based on an equivalent
circuit representation of a cell membrane. These models represent a minimal biophysical interpretation
for an excitable biological cell in which current flow across the membrane is due to charging of the
membrane capacitance and movement of ions across ion channels. Ion channels are selective for
particular ionic species, such as calcium (Ca 2+ ) or potassium (K + ), giving rise to currents I Ca 2+ or I K + ,
respectively. Our simplistic model reads as follows:
dV
dt
= −
I K + + I Ca 2+
C m
,
(1)
with the membrane capacity C m = 1
μF
m 2 and ion currents
I K + := G K + · x · (V − E K + ) and I Ca 2+ := G Ca 2+ · d · f · (V − E Ca 2+ ),
(2)
where the different gating variables d, f and x are satisfying the differential equation
dy
dt
=
y ∞ (V) − y
τ y
(3)
and y represents the gating variables d, f and x, while
y ∞ := y ∞ (V)=
1
1 + exp
V−V Ty
ky
(4)
with V Ty ∈ R, k y ∈ R\ {0} denotes the equilibrium of the corresponding gating variable and τ y is the
corresponding relaxation time constant for each of d, f and x. The gating variables d, f and x ∈ [0, 1]
are important for the activation (opening) and inactivation (closing) of the ion channels and therefore
for the ion current interaction, see [15]. Moreover, the Nernst potentials of these ion currents are
denoted by E Ca 2+ and E K + , while the corresponding conductance are represented by G Ca 2+ = 0.025
mS
cm 2
and G K + = 0.05
mS
cm 2 , respectively. Furthermore, the relaxation time constants are given by τ f = 80 ms
and τ x = 300 ms. We have to remark that in [1] it is assumed that the gating variable d is equal to its
steady state. Please note that if τ d tends to zero, we have the situation as in [1], since
τ d
dd
dt
=(d ∞ − d) ⇒ 0 =(d ∞ − d),
as τ d → 0. In this paper, we will use the relaxation time constant of d, i.e., τ d , as further non-zero
parameter. Moreover, the choice τ d = 0.1 ms yields the same trajectory as in [1], but also smaller
values of τ d are conceivable. In Figure 1 some examples of EADs are presented with τ d = 0.1 ms
and G Ca 2+ ∈
0.029
mS
cm 2 ; 0.03
mS
cm 2 ; 0.031
mS
cm 2 ; 0.035
mS
cm 2
(from left to right). Please compare Figure 6a in
[15] with the second trajectory in Figure 1. Here, we see that the four dimensional system behaves
66