Karima Djabella

Michel Sorine

Crossref

HAL Id: inria-00001197
https://hal.inria.fr/inria-00001197v2
Submitted on 4 Apr 2006
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A Reduced Differential Model of the Electrical Activity
of Cardiac Purkinje Fibres
Karima Djabella, Michel Sorine
To cite this version:
Karima Djabella, Michel Sorine. A Reduced Differential Model of the Electrical Activity of Cardiac
Purkinje Fibres. 28 th Annual International Conference of the IEEE Engineering in Medicine and
Biology Society - EMBS’06, Aug 2006, New York, pp.4167-4170, ￿10.1109/IEMBS.2006.259468￿. ￿inria-
00001197v2￿
A Reduced Differential Model of the Electrical Activity of Cardiac
Purkinje Fibres
Karima Djabella and Michel Sorine
Abstract— A reduced order differential model of cardiac
Purkinje fibres action potential, with only eight state variables,
is presented. Its structure, derived from basic physical princi-
ples can be used for the main other cardiac cell types, a useful
property for some model-based signal or image processing
applications. The electrical activity of cardiac Purkinje fibres is
reconstructed using particular values of the parameters. This
model of the membrane excitation mechanism and intracel-
lular calcium dynamics describes the principal ionic current
underlying autorhythmicity; calcium uptake and release from
the sarcoplasmic reticulum; effects of the binding of calcium
on myoplasmic proteins which affect the Nernst potential
of calcium, and then the membrane potential. The model
allows realistic modelling of cardiac Purkinje fibres action
potential, total ionic current, CICR dependence on intracellular
calcium concentrations. Simulations illustrate the role of the
inward sodium current as the dominant mechanism underlying
pacemaker depolarization during spontaneous activity.
I. INTRODUCTION
The cardiac Purkinje fibres constitute the conduction sys-
tem in the ventricles and, being capable of self-excitation,
they act as an accessory pacemaker. They spontaneously
and rhythmically generate action potentials (AP) resulting
in muscle contraction. Accurate modelling of their AP and
pacemaker activity is difficult due to the complexity of the
excitation mechanisms involving a large number of different
ionic currents. The first models of AP in cardiac Purkinje
fibres (Noble in 1962 [1], updated by McAllister et al [2])
used the Hodgkin & Huxley’s formulation of the properties
of excitable membranes to describe the long lasting AP
and pacemaker activity. DiFrancesco and Noble developed
a new model of Purkinje fibres electrical activity [3] that
was the first cardiac cell model incorporating ionic exchang-
ers, pumps and concentration changes. These models have
their respective advantages and have proven to be useful in
investigating a variety of heart diseases. They differ in many
details from models of atrial or ventricular cardiac cell.
As for other biochemical systems, an apparent paradox in
cardiac cell modelling is that the knowledge on cell structure
imbedded in models is rather rich but it is quite impossible to
experimentally validate the forms of the different nonlinear-
ities used in detailed models and their parameters are poorly
known. In fact, mass and electrical charge conservation laws
are direct applications of fundamental physical principles,
but many model details are Ad-hoc descriptions of observed
behaviors, differing with the authors even for the same
cell. Nevertheless, apparently different models, after some
The authors are with INRIA-Rocquencourt, B.P. 105,
78153 Le Chesnay Cedex, France. Karima.Djabella,
Michel.Sorine@inria.fr
parameter tuning, agree very well with experimental data.
Identifying model structures leading to both these parameter
sensitivity and robustness of model properties is then of par-
ticular interest. This may allow to retain only the necessary
details in a model intended for a given task. This is what
is attempted here to facilitate the development of model-
based signal or image processing applications where 0D or
3D models of the heart and associated inverse problems are
considered [4]. No general technique is available to deter-
mine such structure. Only in very simple situations generic
models are known such as the ubiquitous FitzHugh Nagumo
equations for two-state variable models. Our approach is then
to reduce the state space dimension of different cardiac cell
models trying to reorganize them with a common structure.
In this paper, a reduced differential model for cardiac
Purkinje fibres AP is proposed. It shares the same struc-
ture than the reduced model of ventricular cell in [5]. It
incorporates some myoplasmic calcium proteins which are
involved in the process of excitation–contraction coupling in
ventricular cells and are useful here, as they affect the Nernst
potential of calcium, and then the membrane potential. It
has only eight state variables. Three variables are used to
describe the membrane (membrane potential and two gate
variables of ionic channels), taking into consideration the
dynamics of the main ionic currents (inward sodium, L-
type calcium and outward potassium), Na+/Ca2+ exchang-
ers and Na+/K+ pumps, derived using physical principles
and conservation laws [6]. The remaining five variables are
associated with the fluid compartment model accounting for
intracellular changes in concentrations of Na+, K+ [7] and
the main processes that regulate intracellular calcium con-
centrations: release and uptake by the sarcoplasmic reticulum
(SR), buffering in the SR [8] and in the bulk cytosol [9].
The inherent ability of Purkinje fibres to generate a depo-
larizing unstable resting potential leading to automaticity can
be caused either by decreasing outward current (the time–
dependant decrease in delayed outward potassium current
IK) or the development of a net inward ion flux immediately
following AP repolarization [10]. The precise contribution
of underlying membrane currents to the net inward flow
still remains the subject of controversy (three main hy-
potheses of pacemaker generation have emerged, the inward
hyperpolarization activated current (i f ) which is normally
carried by Na+ and K+, the sustained inward current ist ,
where the major charge carrier is believed to be Na+ and
a Ca2+ "window" current). Despite its moderate number
of state variables, the model includes the main processes
thought to be important in pacemaker activity and it incor-
porates the most important currents underlying pacemaker
depolarization in Purkinje cells. After model tuning, the
mechanisms underlying pacemaker depolarization have been
investigated, and the inward sodium current appears to be
the most dominant one. The model successfully reproduces
experimentally observed cardiac Purkinje fibres pacemaker
depolarization and AP behavior.
The paper is organized as follows: the model is presented
in Section II. Section III shows some simulation results. A
discussion and conclusions are presented in Sections VI and
V respectively.
II. MODEL DESCRIPTION
The proposed model for the cardiac Purkinje fibres AP
is the system (1) of eight first order ordinary differential
equations (ODE) (notations are defined in table I):

dV
dt = −
IK,t+INa,t+ICa,t+INaK+INaCa
Cm
dKi
dt =
2INaK−IK,t
FVC
dNai
dt = −
INa,t+3INaK+3INaCa
FVC
dCai
dt =
2INaCa−ICa,t
2FVC +Jleak+Jrel−Jup
−∑b∈IBBb dθbdt , IB ={T n,Cal} ,
dgX
dt =
gX∞−gX
τgX
, X ∈ {Na,K}
dθb
dt = k
on
b |Cai|(1−θb)− k
o f f
T n θb, b ∈ IB
(1)
The gate dynamics are defined by
gX∞ = 12
[
1+tanh
(
V−VgX
RT/2F
)]
,
1
τgX
= 1τX cosh
(
V−VgX
RT/2F
)
, X ∈{Na,K,d,m}
(2)
where X = d,m represent fast Ca, Na activation gating,
denoted as usual d∞ = gd∞, m∞ = gm∞. Setting
VX =
RT
zX F
log
∣∣∣∣XeXi
∣∣∣∣ , X ∈{Ca,Na,K} , (3)
the currents through the membrane are then:
IK,t = ¯IKgK sinh
(
V−VK
2RT/F
)
+ ¯Ib,K (4)
INa,t = ¯INagNam∞ sinh
(V−VNa
2RT/F
) (5)
ICa,t = [ ¯ICa(1−gK)d∞+¯Ib,Ca]sinh
(V−VCa
RT/F
) (6)
INaK = ¯INaK tanh
(V+2VK−3VNa−VATP
2RT/F
) (7)
INaCa = ¯INaCa sinh
(V+2VCa−3VNa
2RT/F
) (8)
Finally, the CICR mechanism is described using
Jup = QupJmax
∣∣∣ CaiKm f
∣∣∣H − ∣∣∣CaSRKmr
∣∣∣H
1+
∣∣∣ CaiKm f
∣∣∣H + ∣∣∣CaSRKmr
∣∣∣H (9)
Jrel = Kreld∞(CaSR −CaiJCT ) (10)
Jleak = Kleak(CaSR −CaiJCT ) (11)
CaiJCT =
BJCT Cai
|Cai|+KJCT
(12)
CaSR +
BSR CaSR
|CaSR|+KSR
=
VC
VSR
(
CaT−Cai−∑
b∈IB
Bbθb−CaiJCT
)
(13)
TABLE I
NOTATIONS USED IN THE TEXT
Notations Definitions
V Action potential
Vext External stimulus voltage
IX ,t Total X current through all channels
INaCa Na+ −Ca2+ exchanger current
INaK Na+ −K+ pump current
gK Potassium activation gating
gNa Sodium activation gating
d∞ Fast Calcium activation gating
m∞ Fast Sodium activation gating
Jrel Calcium-Induced Calcium Release current
Jup Pump current taking up calcium in the SR
Jleak Leakage current from SR to the cytoplasm
Xi Intracellular concentration of the free ion X
Xe External concentration of the ion free X
CaT Total calcium concentration in the cell
CaSR Free calcium concentration in the SR
θT n Fraction of Troponin–C sites bound with Ca
θCal Fraction of Calmodulin sites bound with Ca
CaiJCT SR Calcium buffered in the junction
CaT =
1
2
(
Cm
FVC
(V −Vext)−Nai −Ki
)
(14)
Vext = −
FVC
Cm
(Nae +Ke +2Cae) (15)
III. RESULTS
The system of nonlinear ODE is solved with a second
order modified Rosenbrock method with variable steplength.
The parameters values are listed in table II. Some simulations
of the autorhythmic activity are presented.
Figure 1 shows the simulated Purkinje fibres AP. It
presents the characteristic spike, followed by the rapid de-
polarization, then the potential moves again in a positive
direction, thus generating the ‘notch’ characteristic of many
experimental records [2]. The plateau phase after the ‘notch’
follows a somewhat rounded (concave downwards) time
course similar to experimental records [2]. Repolarization
terminates followed by the slow spontaneous depolarization
(pacemaker potential) until the threshold is reached and
another action potential is initiated.
0 0.5 1 1.5 2 2.5 3 3.5
−0.09
−0.08
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
Time (s)
V 
(V
)
Fig. 1. Simulated Purkinje fibres action potential
0 0.5 1 1.5 2  2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Sl
ow
 g
at
es
gK
gCa=1−gK
gNa
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fa
st
 g
at
es
Time (s)
m
∞
d
∞
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
To
ta
l N
a 
an
d 
Ca
 g
at
in
g
GNa=gNam∞
GCa=gCad∞
Fig. 2. Top: Simulated slow gating variables. Middle: Simulated fast gating
variables. Bottom: Simulated total Na and Ca gating variables.
In Fig. 2, the time course of the computed gating variables
changes is shown. It can be seen that the slow gates gCa and
gNa rise to a high value during the pacemaker depolarization
phase (100% and 85% are the fractions of Ca and Na
channels in the open state respectively). However, the fast
gating variables corresponding to the sodium and calcium
channels fall (the probability of calcium channels being open
is zero and is approximately 2% for sodium channels). Thus,
only the inward sodium current contributes to pacemaker
activity. This result is clearly shown in the bottom of the
figure 2 and in the top where the slow potassium gating
variable falls to zero. Only Ca and Na have a quantitative role
in the spike action potential phase. By contrast, potassium
and calcium play an important role in the plateau and in the
repolarization phases (calcium is more important during the
plateau than during the repolarization phase).
In Fig. 3, the time courses of the five ionic currents and
the intracellular calcium concentration changes are shown.
TABLE II
MODEL PARAMETERS
Parameter Value Unit
VC 16.404 pL
VSR 1.094 pL
F 96.486 Cmmol−1
Vm −56.8610−3 V
Vd −510−3 V
VgK −2610−3 V
VgNa −71.5510−3 V
τNa = τK 0.2 s
VAT P −45010−3 V
Cm 85 pF
R 8.31410−3 Jmmol−1K−1
T 310 K
zNa = zK 1
zCa 2
¯INa 112.7 pA
¯IK 32.9 pA
¯Ib,K 70 pA
¯ICa 26.2 pA
¯Ib,Ca 0.01645 pA
¯INaK 11.46 pA
¯INaCa 1400 pA
Ke 5.4 mM
Nae 140 mM
Cae 2 mM
Kleak 510−3 s−1
H 1.787
Km f 0.24610−3 mM
Kmr 1.7 mM
Jmax 28610−3 mMs−1
Qup 2.6
Krel 25103 s−1
konT n 32.7103 mM−1s−1
ko f fT n 19.6 s−1
BT n 7010−3 mM
konCal 34103 mM−1s−1
ko f fCal 238 s
−1
BCal 2410−3 mM
KJCT 1310−3 mM
BJCT 4.610−3 mM
KSR 65010−5 mM
BSR 0.14 mM
The intracellular Ca2+ transient rises to a peak within a
few milliseconds and decays as expected in a biphasic
response before the faster phase of repolarization begins.
This corresponds well to the experimental results on Purkinje
fibres [11]. In principle, two peaks are possible since the
Ca2+ is made up of two components: the smaller one due to
the calcium current and the larger one due to internal release.
While INa,t is the main time–dependent gated current
that contributes to pacemaker activity, other currents also
contribute substantially. The net increase in INa,t during
the pacemaker depolarization in figure 3 is -15 pA. By
comparison, Ib,Ca shows a fall of -15 pA which produces the
fall of the ICa,t , IK,t and INaCa remains equal to zero during
the pacemaker potential, and INaK carries about 1.5 pA.
IV. DISCUSSION
In this paper, a model for cardiac Purkinje fibres action
potential is proposed. With only eight state variables, it is
realistic enough to exhibit many of the characteristics of
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
Ca
i (µ
M)
0 0.5 1 1.5 2 2.5 3 3.5
−600
−400
−200
0
I C
a,
t (p
A)
0 0.5 1 1.5 2 2.5 3  3.5
0
10
20
30
40
50
I K
,t 
(pA
)
0  0.5 1 1.5 2 2.5 3 3.5
−300
−200
−100
0
I N
a,
t (p
A)
0 0.5 1 1.5 2 2.5 3 3.5
−5
0
5
10
15
I N
aK
 
(pA
)
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5
−1500
−1000
−500
0
500
I N
aC
a 
(pA
)
Time (s)
Fig. 3. Simulated intracellular calcium transients and ionic currents
larger pacemaker cell models. Its structure, common with
ventricular cardiac cells, involves Na+, K+ and Ca2+ ions,
their respective channels, the Na+/Ca2+ exchanger, and
the Na+/K+ pump. It also includes a description of the
dynamics of the main calcium buffers in the bulk cytosol
and in the SR, taking into account the calcium uptake and
release from SR.
The model can be tuned to reproduce Purkinje fibres
action potential, the behavior of the main currents and the
intracellular calcium dynamics involved during pacemaking.
The mechanisms underlying pacemaker depolarization of
Purkinje fibres model were examined by ascertaining the
dominant membrane currents during each phase of the action
potential. Pacemaker depolarization was found to arise from
the net contribution of a number of inward currents, the
largest of which was the Na+ current. In contrast, the delayed
rectifier K+ does not play a role in pacemaker depolarization,
remaining equal to zero throughout most of this phase.
Due to its medium complexity, this model can be used in
multi–beat simulations from the cell to the heart scales. It
can also be useful in direct computations to gain insights and
for its predictive capabilities, and is also a good candidate
in inverse problems to estimate state and parameters from
measurements.
V. CONCLUSION
We have presented a reduced differential model of the
membrane potential, ionic currents and calcium dynamics
in a Purkinje fibre cell. It is in good agreement with more
complete models. The model represents a good trade-off
between descriptive power and complexity. It is a good
candidate to be imbedded in heart simulators for applications
where both direct and inverse problems are considered.
REFERENCES
[1] D. Noble, “A modification of the Hodgkin–Huxley equations applica-
ble to Purkinje fibre action and pace-maker potentials,” J Physiol, vol.
160, pp. 317–52, 1962.
[2] R. E. McAllister, D. Noble, and R. W. Tsien, “Reconstruction of the
electrical activity of cardiac Purkinje fibres,” J Physiol, vol. 251, no. 1,
pp. 1–59, 1975.
[3] D. DiFrancesco and D. Noble, “A model of cardiac electrical activity
incorporating ionic pumps and concentration changes,” Philos Trans
R Soc Lond B Biol Sci, vol. 307, no. 1133, pp. 353–98, 1985.
[4] CardioSense3D, “An INRIA Large Initiative Action,” //www-
sop.inria.fr/CardioSense3D, 2006.
[5] K. Djabella and M. Sorine, “A differential model of excitation-
contraction coupling in a cardiac cell for multicycle simulations,” in
Proceedings of the 3rd European Medical and Biological Engineering
Conference, Prague, Czech Republic, November 20-25 2005.
[6] L. P. Endresen, K. Hall, J. S. Hoye, and J. Myrheim, “A theory for
the membrane potential of living cells,” Eur Biophys J, vol. 29, pp.
90–103, 2000.
[7] T. J. Hund, J. P. Kucera, N. F. Otani, and Y. Rudy, “Ionic charge
conservation and long-term steady state in the Luo-Rudy dynamic cell
model,” Biophys Journal, vol. 81, pp. 3324–3331, 2001.
[8] K. H. W. J. T. Tusscher, D. Noble, P. J. Noble, and A. V. Panfilov, “A
model for human ventricular tissue,” Am J Physiol Heart Circ Physiol,
vol. 286, pp. 1573–1589, 2004.
[9] T. R. Shannon, F. Wang, J. Puglisi, C. Weber, and D. M. Bers, “A
mathematical treatment of integrated Ca dynamics within the ventriclar
myocite,” Biophys Journal, vol. 87, pp. 3351–3371, 2004.
[10] D. M. Bers, “Calcium and cardiac rhythms: Physiological and patho-
physiological,” Circ Res, vol. 90, pp. 14–17, 2002.
[11] W. G. Wier and G. Isenberg, “Intracellular [Ca] transients in voltage-
clamped cardiac Purkinje fibres,” Pflugers Arch. Eur. J. Physiol, vol.
392, pp. 284–290, 1982.


A Reduced Differential Model of the Electrical Activity of Cardiac Purkinje Fibres

English

HAL Id: inria-00001197
https://hal.inria.fr/inria-00001197v2
Submitted on 4 Apr 2006
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A Reduced Differential Model of the Electrical Activity
of Cardiac Purkinje Fibres
Karima Djabella, Michel Sorine
To cite this version:
Karima Djabella, Michel Sorine. A Reduced Differential Model of the Electrical Activity of Cardiac
Purkinje Fibres. 28 th Annual International Conference of the IEEE Engineering in Medicine and
Biology Society - EMBS’06, Aug 2006, New York, pp.4167-4170, ￿10.1109/IEMBS.2006.259468￿. ￿inria-
00001197v2￿
A Reduced Differential Model of the Electrical Activity of Cardiac
Purkinje Fibres
Karima Djabella and Michel Sorine
Abstract— A reduced order differential model of cardiac
Purkinje fibres action potential, with only eight state variables,
is presented. Its structure, derived from basic physical princi-
ples can be used for the main other cardiac cell types, a useful
property for some model-based signal or image processing
applications. The electrical activity of cardiac Purkinje fibres is
reconstructed using particular values of the parameters. This
model of the membrane excitation mechanism and intracel-
lular calcium dynamics describes the principal ionic current
underlying autorhythmicity; calcium uptake and release from
the sarcoplasmic reticulum; effects of the binding of calcium
on myoplasmic proteins which affect the Nernst potential
of calcium, and then the membrane potential. The model
allows realistic modelling of cardiac Purkinje fibres action
potential, total ionic current, CICR dependence on intracellular
calcium concentrations. Simulations illustrate the role of the
inward sodium current as the dominant mechanism underlying
pacemaker depolarization during spontaneous activity.
I. INTRODUCTION
The cardiac Purkinje fibres constitute the conduction sys-
tem in the ventricles and, being capable of self-excitation,
they act as an accessory pacemaker. They spontaneously
and rhythmically generate action potentials (AP) resulting
in muscle contraction. Accurate modelling of their AP and
pacemaker activity is difficult due to the complexity of the
excitation mechanisms involving a large number of different
ionic currents. The first models of AP in cardiac Purkinje
fibres (Noble in 1962 [1], updated by McAllistert al [2])
used the Hodgkin & Huxley’s formulation of the properties
of excitable membranes to describe the long lasting AP
and pacemaker activity. DiFrancesco and Noble developed
a new model of Purkinje fibres electrical activity [3] that
was the first cardiac cell model incorporating ionic exchang-
ers, pumps and concentration changes. These models have
their respective advantages and have proven to be useful in
investigating a variety of heart diseases. They differ in many
details from models of atrial or ventricular cardiac cell.
As for other biochemical systems, an apparent paradox in
cardiac cell modelling is that the knowledge on cell structure
imbedded in models is rather rich but it is quite impossible to
experimentally validate the forms of the different nonlinear-
ities used in detailed models and their parameters are poorly
known. In fact, mass and electrical charge conservation laws
are direct applications of fundamental physical principles,
but many model details are Ad-hoc descriptions of observed
behaviors, differing with the authors even for the same
cell. Nevertheless, apparently different models, after some
The authors are with INRIA-Rocquencourt, B.P. 105,
78153 Le Chesnay Cedex, France.Karima.Djabella,
Michel.Sorine@inria.fr
parameter tuning, agree very well with experimental data.
Identifying model structures leading to both these parameter
sensitivity and robustness of model properties is then of par-
ticular interest. This may allow to retain only the necessary
details in a model intended for a given task. This is what
is attempted here to facilitate the development of model-
based signal or image processing applications where 0D or
3D models of the heart and associated inverse problems are
considered [4]. No general technique is available to deter-
mine such structure. Only in very simple situations generic
models are known such as the ubiquitous FitzHugh Nagumo
equations for two-state variable models. Our approach is then
to reduce the state space dimension of different cardiac cell
models trying to reorganize them with a common structure.
In this paper, a reduced differential model for cardiac
Purkinje fibres AP is proposed. It shares the same struc-
ture than the reduced model of ventricular cell in [5]. It
incorporates some myoplasmic calcium proteins which are
involved in the process of excitation–contraction coupling i
ventricular cells and are useful here, as they affect the Nernst
potential of calcium, and then the membrane potential. It
has only eight state variables. Three variables are used to
describe the membrane (membrane potential and two gate
variables of ionic channels), taking into consideration the
dynamics of the main ionic currents (inward sodium, L-
type calcium and outward potassium),Na+/Ca2+ exchang-
ers andNa+/K+ pumps, derived using physical principles
and conservation laws [6]. The remaining five variables are
associated with the fluid compartment model accounting for
intracellular changes in concentrations ofNa+, K+ [7] and
the main processes that regulate intracellular calcium con-
centrations: release and uptake by the sarcoplasmic reticulum
(SR), buffering in the SR [8] and in the bulk cytosol [9].
The inherent ability of Purkinje fibres to generate a depo-
larizing unstable resting potential leading to automaticity an
be caused either by decreasing outward current (the time–
dependant decrease in delayed outward potassium current
IK) or the development of a net inward ion flux immediately
following AP repolarization [10]. The precise contribution
of underlying membrane currents to the net inward flow
still remains the subject of controversy (three main hy-
potheses of pacemaker generation have emerged, the inward
hyperpolarization activated current (i f ) which is normally
carried byNa+ and K+, the sustained inward currentist,
where the major charge carrier is believed to beNa+ and
a Ca2+ "window" current). Despite its moderate number
of state variables, the model includes the main processes
thought to be important in pacemaker activity and it incor-
porates the most important currents underlying pacemaker
depolarization in Purkinje cells. After model tuning, the
mechanisms underlying pacemaker depolarization have been
investigated, and the inward sodium current appears to be
the most dominant one. The model successfully reproduces
experimentally observed cardiac Purkinje fibres pacemaker
depolarization and AP behavior.
The paper is organized as follows: the model is presented
in Section II. Section III shows some simulation results. A
discussion and conclusions are presented in Sections VI and
V respectively.
II. MODEL DESCRIPTION
The proposed model for the cardiac Purkinje fibres AP
is the system (1) of eight first order ordinary differential
equations (ODE) (notations are defined in table I):

























dV
dt = −
IK,t+INa,t+ICa,t+INaK+INaCa
Cm
dKi
dt =
2INaK−IK,t
FVC
dNai
dt = −
INa,t+3INaK+3INaCa
FVC
dCai
dt =
2INaCa−ICa,t
2FVC
+Jleak+Jrel−Jup
−∑b∈IBBb
dθb
dt , IB={Tn,Cal} ,
dgX
dt =
gX∞−gX
τgX
, X ∈ {Na,K}
dθb
dt = k
on
b |Cai |(1−θb)−k
o f f
Tn θb, b∈ IB
(1)
The gate dynamics are defined by
gX∞ = 12
[
1+tanh
(
V−VgX
RT/2F
)]
,
1
τgX
= 1τX cosh
(
V−VgX
RT/2F
)
, X∈{Na,K,d,m}
(2)
where X = d,m represent fastCa, Na activation gating,
denoted as usuald∞ = gd∞, m∞ = gm∞. Setting
VX =
RT
zXF
log
∣
∣
∣
∣
Xe
Xi
∣
∣
∣
∣
, X∈{Ca,Na,K} , (3)
the currents through the membrane are then:
IK,t = ĪKgK sinh
(
V−VK
2RT/F
)
+ Īb,K (4)
INa,t = ĪNagNam∞ sinh
(
V−VNa
2RT/F
)
(5)
ICa,t = [ĪCa(1−gK)d∞+Īb,Ca]sinh
(
V−VCa
RT/F
)
(6)
INaK = ĪNaK tanh
(
V+2VK−3VNa−VATP
2RT/F
)
(7)
INaCa = ĪNaCasinh
(
V+2VCa−3VNa
2RT/F
)
(8)
Finally, the CICR mechanism is described using
Jup = QupJmax
∣
∣
∣
Cai
Km f
∣
∣
∣
H
−
∣
∣
∣
CaSR
Kmr
∣
∣
∣
H
1+
∣
∣
∣
Cai
Km f
∣
∣
∣
H
+
∣
∣
∣
CaSR
Kmr
∣
∣
∣
H (9)
Jrel = Kreld∞(CaSR−CaiJCT) (10)
Jleak = Kleak(CaSR−CaiJCT) (11)
CaiJCT =
BJCT Cai
|Cai |+KJCT
(12)
CaSR +
BSRCaSR
|CaSR|+KSR
=
VC
VSR
(
CaT−Cai−∑
b∈IB
Bbθb−CaiJCT
)
(13)
TABLE I
NOTATIONS USED IN THE TEXT
Notations Definitions
V Action potential
Vext External stimulus voltage
IX,t Total X current through all channels
INaCa Na+ −Ca2+ exchanger current
INaK Na+ −K+ pump current
gK Potassium activation gating
gNa Sodium activation gating
d∞ Fast Calcium activation gating
m∞ Fast Sodium activation gating
Jrel Calcium-Induced Calcium Release current
Jup Pump current taking up calcium in the SR
Jleak Leakage current from SR to the cytoplasm
Xi Intracellular concentration of the free ionX
Xe External concentration of the ion freeX
CaT Total calcium concentration in the cell
CaSR Free calcium concentration in the SR
θTn Fraction of Troponin–C sites bound with Ca
θCal Fraction of Calmodulin sites bound with Ca
CaiJCT SR Calcium buffered in the junction
CaT =
1
2
(
Cm
FVC
(V −Vext)−Nai −Ki
)
(14)
Vext = −
FVC
Cm
(Nae+Ke+2Cae) (15)
III. RESULTS
The system of nonlinear ODE is solved with a second
order modified Rosenbrock method with variable steplength.
The parameters values are listed in table II. Some simulations
of the autorhythmic activity are presented.
Figure 1 shows the simulated Purkinje fibres AP. It
presents the characteristic spike, followed by the rapid de-
polarization, then the potential moves again in a positive
direction, thus generating the ‘notch’ characteristic of many
experimental records [2]. The plateau phase after the ‘notch’
follows a somewhat rounded (concave downwards) time
course similar to experimental records [2]. Repolarization
terminates followed by the slow spontaneous depolarization
(pacemaker potential) until the threshold is reached and
another action potential is initiated.
0 0.5 1 1.5 2 2.5 3 3.5
−0.09
−0.08
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
Time (s)
V
 (
V
)
Fig. 1. Simulated Purkinje fibres action potential
0 0.5 1 1.5 2  2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
S
lo
w
 g
at
es
g
K
g
Ca
=1−g
K
g
Na
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F
as
t g
at
es
Time (s)
m∞
d∞
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
T
ot
al
 N
a 
an
d 
C
a 
ga
tin
g
G
Na
=g
Na
m∞
G
Ca
=g
Ca
d∞
Fig. 2. Top: Simulated slow gating variables. Middle: Simulated fast gating
variables. Bottom: Simulated total Na and Ca gating variables.
In Fig. 2, the time course of the computed gating variables
changes is shown. It can be seen that the slow gatesgCa and
gNa rise to a high value during the pacemaker depolarization
phase (100% and 85% are the fractions of Ca and Na
channels in the open state respectively). However, the fast
gating variables corresponding to the sodium and calcium
channels fall (the probability of calcium channels being open
is zero and is approximately 2% for sodium channels). Thus,
only the inward sodium current contributes to pacemaker
activity. This result is clearly shown in the bottom of the
figure 2 and in the top where the slow potassium gating
variable falls to zero. Only Ca and Na have a quantitative rol
in the spike action potential phase. By contrast, potassium
and calcium play an important role in the plateau and in the
repolarization phases (calcium is more important during the
plateau than during the repolarization phase).
In Fig. 3, the time courses of the five ionic currents and
the intracellular calcium concentration changes are shown.
TABLE II
MODEL PARAMETERS
Parameter Value Unit
VC 16.404 pL
VSR 1.094 pL
F 96.486 Cmmol−1
Vm −56.8610−3 V
Vd −510−3 V
VgK −2610
−3 V
VgNa −71.5510
−3 V
τNa = τK 0.2 s
VATP −45010−3 V
Cm 85 pF
R 8.31410−3 Jmmol−1K−1
T 310 K
zNa = zK 1
zCa 2
ĪNa 112.7 pA
ĪK 32.9 pA
Īb,K 70 pA
ĪCa 26.2 pA
Īb,Ca 0.01645 pA
ĪNaK 11.46 pA
ĪNaCa 1400 pA
Ke 5.4 mM
Nae 140 mM
Cae 2 mM
Kleak 510−3 s−1
H 1.787
Km f 0.24610−3 mM
Kmr 1.7 mM
Jmax 28610−3 mMs−1
Qup 2.6
Krel 25103 s−1
konTn 32.710
3 mM−1s−1
ko f fTn 19.6 s
−1
BTn 7010−3 mM
konCal 3410
3 mM−1s−1
ko f fCal 238 s
−1
BCal 2410−3 mM
KJCT 1310−3 mM
BJCT 4.610−3 mM
KSR 65010−5 mM
BSR 0.14 mM
The intracellularCa2+ transient rises to a peak within a
few milliseconds and decays as expected in a biphasic
response before the faster phase of repolarization begins.
This corresponds well to the experimental results on Purkinje
fibres [11]. In principle, two peaks are possible since the
Ca2+ is made up of two components: the smaller one due to
the calcium current and the larger one due to internal releas.
While INa,t is the main time–dependent gated current
that contributes to pacemaker activity, other currents also
contribute substantially. The net increase inINa,t during
the pacemaker depolarization in figure 3 is -15 pA. By
comparison,Ib,Ca shows a fall of -15 pA which produces the
fall of the ICa,t , IK,t and INaCa remains equal to zero during
the pacemaker potential, andINaK carries about 1.5 pA.
IV. D ISCUSSION
In this paper, a model for cardiac Purkinje fibres action
potential is proposed. With only eight state variables, it is
realistic enough to exhibit many of the characteristics of
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
C
a i
 (
µM
)
0 0.5 1 1.5 2 2.5 3 3.5
−600
−400
−200
0
I C
a,
t (
pA
)
0 0.5 1 1.5 2 2.5 3  3.5
0
10
20
30
40
50
I K
,t 
(p
A
)
0  0.5 1 1.5 2 2.5 3 3.5
−300
−200
−100
0
I N
a,
t (
pA
)
0 0.5 1 1.5 2 2.5 3 3.5
−5
0
5
10
15
I N
aK
 (
pA
)
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5
−1500
−1000
−500
0
500
I N
aC
a 
(p
A
)
Time (s)
Fig. 3. Simulated intracellular calcium transients and ionic currents
larger pacemaker cell models. Its structure, common with
ventricular cardiac cells, involvesNa+, K+ andCa2+ ions,
their respective channels, theNa+/Ca2+ exchanger, and
the Na+/K+ pump. It also includes a description of the
dynamics of the main calcium buffers in the bulk cytosol
and in the SR, taking into account the calcium uptake and
release from SR.
The model can be tuned to reproduce Purkinje fibres
action potential, the behavior of the main currents and the
intracellular calcium dynamics involved during pacemaking.
The mechanisms underlying pacemaker depolarization of
Purkinje fibres model were examined by ascertaining the
dominant membrane currents during each phase of the action
potential. Pacemaker depolarization was found to arise from
the net contribution of a number of inward currents, the
largest of which was theNa+ current. In contrast, the delayed
rectifierK+ does not play a role in pacemaker depolarization,
remaining equal to zero throughout most of this phase.
Due to its medium complexity, this model can be used in
multi–beat simulations from the cell to the heart scales. It
can also be useful in direct computations to gain insights and
for its predictive capabilities, and is also a good candidate
in inverse problems to estimate state and parameters from
measurements.
V. CONCLUSION
We have presented a reduced differential model of the
membrane potential, ionic currents and calcium dynamics
in a Purkinje fibre cell. It is in good agreement with more
complete models. The model represents a good trade-off
between descriptive power and complexity. It is a good
candidate to be imbedded in heart simulators for applications
where both direct and inverse problems are considered.
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160, pp. 317–52, 1962.
[2] R. E. McAllister, D. Noble, and R. W. Tsien, “Reconstruction of the
electrical activity of cardiac Purkinje fibres,”J Physiol, vol. 251, no. 1,
pp. 1–59, 1975.
[3] D. DiFrancesco and D. Noble, “A model of cardiac electrical activity
incorporating ionic pumps and concentration changes,”Philos Trans
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[4] CardioSense3D, “An INRIA Large Initiative Action,” //ww-
sop.inria.fr/CardioSense3D, 2006.
[5] K. Djabella and M. Sorine, “A differential model of excitation-
contraction coupling in a cardiac cell for multicycle simulations,” in
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Conference, Prague, Czech Republic, November 20-25 2005.
[6] L. P. Endresen, K. Hall, J. S. Hoye, and J. Myrheim, “A theory for
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90–103, 2000.
[7] T. J. Hund, J. P. Kucera, N. F. Otani, and Y. Rudy, “Ionic charge
conservation and long-term steady state in the Luo-Rudy dynamic cell
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[8] K. H. W. J. T. Tusscher, D. Noble, P. J. Noble, and A. V. Panfilov, “A
model for human ventricular tissue,”Am J Physiol Heart Circ Physiol,
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[9] T. R. Shannon, F. Wang, J. Puglisi, C. Weber, and D. M. Bers, “A
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myocite,” Biophys Journal, vol. 87, pp. 3351–3371, 2004.
[10] D. M. Bers, “Calcium and cardiac rhythms: Physiologicaland patho-
physiological,”Circ Res, vol. 90, pp. 14–17, 2002.
[11] W. G. Wier and G. Isenberg, “Intracellular [Ca] transients in voltage-
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A Reduced Differential Model of the Electrical Activity of Cardiac Purkinje Fibres

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