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Phil. Trans. R. Soc. A (2009) 367, 2225–2255
doi:10.1098/rsta.2008.0283
Mathematical models of the electrical action
potential of Purkinje fibre cells
B Y P HILIP S TEWART 1 , O LEG V. A SLANIDI 1 , D ENIS N OBLE 2 , P ENELOPE
J. N OBLE 2 , M ARK R. B OYETT 3 AND H ENGGUI Z HANG 1, *
1
School of Physics and Astronomy, and 3Faculty of Medical and Human
Sciences, University of Manchester, Manchester M13 9PL, UK
2
Department of Physiology, Anatomy and Genetics, University of Oxford,
Oxford OX1 3PT, UK
Early development of ionic models for cardiac myocytes, from the pioneering
modification of the Hodgkin–Huxley giant squid axon model by Noble to the iconic
DiFrancesco–Noble model integrating voltage-gated ionic currents, ion pumps and
exchangers, Ca2C sequestration and Ca2C-induced Ca2C release, provided a general
description for a mammalian Purkinje fibre (PF) and the framework for modern cardiac
models. In the past two decades, development has focused on tissue-specific models with
an emphasis on the sino-atrial (SA) node, atria and ventricles, while the PFs have largely
been neglected. However, achieving the ultimate goal of creating a virtual human heart
will require detailed models of all distinctive regions of the cardiac conduction system,
including the PFs, which play an important role in conducting cardiac excitation and
ensuring the synchronized timing and sequencing of ventricular contraction. In this
paper, we present details of our newly developed model for the human PF cell including
validation against experimental data. Ionic mechanisms underlying the heterogeneity
between the PF and ventricular action potentials in humans and other species are
analysed. The newly developed PF cell model adds a new member to the family of human
cardiac cell models developed previously for the SA node, atrial and ventricular cells,
which can be incorporated into an anatomical model of the human heart with details of
its electrophysiological heterogeneity and anatomical complexity.
Keywords: model; Purkinje; cardiac; conduction; electrophysiology
1. Introduction
Purkinje fibre (PF) cells are tertiary pacemakers of the heart, normally
suppressed by the primary pacemaker, the sino-atrial (SA) node (Vassalle
1970, 1977). The PF network is an important part of the cardiac conduction
system, responsible for ensuring the synchronized timing and sequencing of
ventricular contraction (Fozzard et al. 1991). It can also be a major source for
generating life-threatening ventricular arrhythmias (Nattel & Quantz 1988;
Nibley & Wharton 1995; Asano et al. 1997; Berenfeld & Jalife 1998). Under
* Author for correspondence ([email protected]).
One contribution of 15 to a Theme Issue ‘The virtual physiological human: tools and applications II’.
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P. Stewart et al.
normal conditions, the PF network serves as a fast conduction pathway to
conduct electrical excitation waves, originating from the SA node, into the
ventricles. In some abnormal conditions, the PF network may produce a series of
ectopic focal activities that rapidly drive the surrounding ventricular tissue,
leading to ventricular tachycardia and fibrillation (Pogwizd & Corr 1992; Arnar
et al. 1997, 2001; Chung et al. 1997; Pogwizd et al. 1998; Arnar & Martins 2002).
Conduction block in either the left or right bundle branches of the PFs can lead
to uncoordinated ventricular excitation and contraction (Fantoni et al. 2005;
Imanish et al. 2006; Niu et al. 2006). Additionally, under some circumstances, a
localized temporal functional conduction block in the PF network can generate
re-entrant excitation waves giving rise to ventricular fibrillation (Arnar et al.
1997, 2001; Xing & Martins 2004).
The PF network is a distinctive tissue of the heart with intrinsic electrical
properties remarkably different from other cardiac tissues including that of the
ventricle (Tseng & Boyden 1989; Yu et al. 1995; Cordeiro et al. 1998; Han et al.
2001a, 2002; Dumaine & Cordeiro 2007). In many species, the PF cell action
potentials (APs) have unique features, including a larger upstroke velocity, lower
plateau and longer AP duration (APD; Baláti et al. 1998; Burashnikov &
Antzelevitch 1999; Schram et al. 2002; Lu et al. 2005). Importantly, PF cells can
present automaticity due to slow spontaneous diastolic depolarization (Yu et al.
1995; Baláti et al. 1998). Such differences in electrical APs are associated with
different kinetics and current densities in a number of major ion channels (Han
et al. 2001a, 2002; Dumaine & Cordeiro 2007). Considering both the important
role of the PF system in ensuring normal ventricular excitation and generating
life-threatening ventricular arrhythmias, and their distinctive properties in ion
channel kinetics, it is necessary to develop a biophysically detailed model for the
electrical APs of PF cells that can be incorporated into a realistic model of
the human heart.
(a ) Half a century of progress
The seminal work by Hodgkin & Huxley (1952), in which they derived a
quantitative description of the ionic currents and hence the AP of the giant squid
axon, began what is now more than half a century of development in mathematical
models describing the electrophysiology of biologically excitable cells. A decade
later, Noble (1962) pioneered the application of the work of Hodgkin & Huxley
(1952) to cardiac myocytes, proposing modifications that would result in a simple
ionic model of a mammalian PF cell, reproducing the much longer APD and
pacemaker potential of the PF. The model retained a single fast sodium current,
but split the potassium current into two components, IK1 (note IK1 here differs
from the inward rectifier potassium current referred to later) and IK2, with the
conductance of IK1, gK1, instantaneously dependent on membrane potential, and
gK2, the conductance of IK2, rising slowly as the membrane depolarizes.
The advent of the first successful voltage-clamp measurements by Deck &
Trautwein (1964), discoveries of the cardiac calcium current by Reuter (1967)
and multiple components of the potassium current, IK, by Noble & Tsien (1969)
resulted in a need for a successor to the Noble (1962) model. In response to the
growing wealth of experimental data (Noble & Tsien 1968, 1969) and knowledge
at the time, McAllister et al. (1975) developed a model to reproduce the AP of
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Models of Purkinje fibre cells
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the cardiac PF using nine ionic currents. Their model added a new secondary
inward calcium current, ICa, a transient outward chloride current, ICl, and fast
and slow potassium currents, Ix1 and Ix2. The resultant model could reproduce a
much wider range of experimental observations known at the time, some with
a high degree of accuracy.
The model of McAllister et al. (1975) was superseded by the DiFrancesco &
Noble (1985) PF cell model. DiFrancesco (1981) had already shown that what
was previously called IK2 was actually an inward, hyperpolarization-activated
pacemaker current, If, as opposed to an outward, depolarization-activated
current, as described by McAllister et al. (1975) in their model. In addition to the
incorporation of If, the model of DiFrancesco & Noble (1985) included dynamic
changes of intra- and extracellular ion concentrations, ionic pumps and
exchangers that are necessary to restore and maintain the transmembrane ion
concentration gradients, and a description of the Ca2C handling in the
sarcoplasmic reticulum (SR). The depletion of potassium ions in the extracellular
spaces made the inclusion of the sodium–potassium (INaK) pump a necessity,
otherwise If would not resemble a potassium current. As a consequence of
introducing the NaC –KC pump and hence changes in the potassium
concentration, it also became necessary to include concentration changes for
sodium and calcium, leading to the introduction of the sodium–calcium
exchanger (INaCa) and a description of calcium release from the SR, including
calcium-induced calcium release, described by Fabiato & Fabiato (1975). The
DiFrancesco & Noble (1985) model had thus become the first electrophysiologically detailed model capable of describing both ionic currents and
concentration changes.
Building on the successes of these models, development shifted towards
different cardiac tissues across a variety of species, notably including the
mammalian ventricular models of Luo & Rudy (1991, 1994a,b) and the human
models for atrial (Courtemanche et al. 1998; Nygren et al. 1998) and ventricular
cells (Iyer et al. 2004; ten Tusscher et al. 2004; ten Tusscher & Panfilov 2006).
During this time, the development of further PF models has largely been
neglected (Boyett et al. 2005).
(b ) The virtual human heart
We are fast approaching the ultimate goal of constructing a virtual human
heart with detailed ionic models of all distinctive regions of the cardiac
conduction system already available, including the atria (Courtemanche et al.
1998; Nygren et al. 1998) and the ventricles (Iyer et al. 2004; ten Tusscher
et al. 2004; ten Tusscher & Panfilov 2006). A simple caricature model for the
human SA node has been developed (Seemann et al. 2006) as has been a detailed
anatomical geometry of the whole human heart (Sachse et al. 2000). The human
PF cell is one of the remaining missing models for the human cardiac conduction
system. Minor modifications to the maximum conductance of IKs and INa in the
ten Tusscher et al. (2004) and ten Tusscher & Panfilov (2006) human ventricle
model were proposed by ten Tusscher & Panfilov (2008), which resulted in a
simple human PF model and allowed them to simulate the cardiac conduction
system in the ventricles. However, their resultant AP lacks many of the
characteristics observed experimentally in human PF cells (Dangman et al. 1982;
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P. Stewart et al.
Lee et al. 2004). We therefore aim to develop a biophysically detailed model of
the human PF cell AP to fulfil the impending requirement to build a virtual
human heart.
2. Methods
The dynamics of the membrane potential in a cardiac cell are described by the
following differential equation:
dV
ZKðIion C Istim Þ;
ð2:1Þ
dt
where Cm is the membrane capacitance; V is the membrane potential; t is the
time; Iion is the sum of the transmembrane ionic currents; and Istim is an
externally applied stimulus current (Hodgkin & Huxley 1952). Numerous
biophysically detailed descriptions of Iion have already been developed for
many different cardiac tissue types, in a variety of species.
Cm
(a ) Human Purkinje fibre model
We developed a description of Iion for the human PF cell based on the model of
the human endocardial cell by ten Tusscher et al. (2004) and ten Tusscher &
Panfilov (2006). We modified their model based on the experimental data of
Han et al. (2002) describing the properties of potassium currents in human
PF cells. Our description of Iion required the addition of two currents: a
hyperpolarization-activated current, If, and a sustained potassium current, Isus,
resulting in a total of 14 ionic currents, as given in equation (2.2). In addition to
the introduction of the new currents, the descriptions for the inward rectifier
current, IK1, and the transient outward current, Ito, were reformulated, and the
maximum conductance of the rapid and slow delayed rectifier potassium
currents, IKr and IKs, and the fast sodium current, INa, were altered because
these channels are distinctively different in channel kinetics and current densities
between PF and ventricular cells:
Iion Z IKr C IKs C IK1 C Ito C Isus C INa C Ib;Na C ICa ;L C Ib;Ca C INaK
C INaCa C Ip;Ca C Ip;K C If :
ð2:2Þ
A detailed listing of all equations and parameters for the developed human PF
cell AP can be found in appendices A and B, respectively. The model is available
online from the CellML repository (http://www.cellml.org/) and was developed
with CELLULAR OPEN RESOURCE (Garny et al. 2003). Below we describe details of
modifications made to the ten Tusscher et al. (2004) and ten Tusscher & Panfilov
(2006) model of endocardial cells for each individual current.
(i) Transient outward current, Ito, and sustained current, Isus
The transient outward potassium current, Ito, and the sustained potassium
current, Isus, are both present in human PF and ventricular cells; however, their
channel properties (i.e. channel kinetics and current densities) are different
between the two cell types (Han et al. 2002). In the PF cells, it was observed that
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Ito is significantly smaller in current density and slower in inactivation and
recovery, but Isus current density is substantially larger (Han et al. 2002). The
clear difference between the PF and ventricular cells in their sensitivities to
potassium channel blocks (e.g. 4AP and tetraethylammonium) may be due to a
different molecular basis forming the Ito and Isus channels in the two cell types as
seen in canines (Han et al. 2000, 2001b).
To reflect the fundamental differences in the Ito and Isus channel properties, the
equations of the ten Tusscher et al. (2004) model for the steady-state activation
variable, rN, and inactivation variable, sN, of Ito were reformulated based on the
experimental data of Han et al. (2002) on human PF cells. This resulted in an
increase in slope factor of rN and sN from 6 and 5 mV, respectively, to 13 mV, and
an additional shift in the half-inactivation (figure 1a) by K1 mV. Equations for the
activation and inactivation time constants, tr and ts, were also reformulated to fit
the experimental data of Han et al. (2002). The resultant tr and ts are significantly
larger than in the ventricle model (a maximal increase by 72 and 1300% for tr and
ts, respectively; figure 1b). Maximum conductance, Gto, was determined by fitting
the current–voltage (I–V ) relationship (figure 1c) to the experimental data of Han
et al. (2002), resulting in an increase of 12 per cent with respect to the endocardial
model of ten Tusscher et al. (2004).
In the original ten Tusscher et al. (2004) and ten Tusscher & Panfilov (2006)
model, there is no formulation for the sustained potassium current, Isus.
Experimental data from both animal and human studies suggested the presence of
the current in PF cells, which is distinctively different in molecular basis from its
ventricular counterpart (Han et al. 2000, 2001a,b). Based on the experimental data of
Han et al. (2002), Isus was introduced with a single instantaneous activation variable,
a, described by a single exponential sigmoid function. To determine the maximum
conductance, Gsus, the simulated I–V relationship (figure 1d ) obtained using the
same voltage-clamp protocol used experimentally was fitted to the experimental
data of Han et al. (2002). Model parameters and equations were validated by
the consistency of the resulting simulated current traces of ItoCIsus (figure 1e) and
the I–V relationship with those observed experimentally (Han et al. 2002).
(ii) Hyperpolarization-activated current, I f
The hyperpolarization-activated current, If, is believed to play an important
role in producing spontaneous diastolic depolarization leading to automaticity in
some cardiac tissues, such as the SA node and PF cells (DiFrancesco 2006). If has
been recorded from both animal and human PF cells (Callewaert et al. 1984;
Cerbai et al. 1997; Shi et al. 1999; Han et al. 2002). In the absence of a description
of If in the ten Tusscher et al. (2004) and ten Tusscher & Panfilov (2006) model,
we introduced If based on the model of Zhang et al. (2000) for the rabbit SA
node. The model equations for the steady-state activation variable, yN, and the
voltage-dependent time constant of activation, ty, were reformulated based on
the experimental data on the channel kinetics of human PF If (Han et al. 2002).
The maximal channel conductance, Gf, was determined by fitting the simulated
I–V relationship to the experimental data (Han et al. 2002). The model equations
and parameters were validated by the agreement of the simulated If current
traces (figure 2a) and I–V relationship (figure 2b) during voltage clamp with
experimental data (Han et al. 2002).
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(b)
1.0
time constants ( ) (ms)
steady-state
activation ( ) / inactivation ( )
(a)
P. Stewart et al.
0.5
Ito (pA pF–1)
(c)
– 80
– 40
0
V (mV)
40
80
25
0
20
40
60
V (mV)
6
(d) 4
3
Isus (pA pF–1)
0
–120
50
0
2
0
– 30
0
30
V (mV)
60
– 40
–20
0
20
V (mV)
40
60
Ito+Isus (pA pF–1)
(e) 10
5
0
100 ms
Figure 1. Modelling Ito and Isus. (a) Steady-state activation (filled circles) and inactivation (open
circles) curves for Ito. (b) Time constants tr (filled circles) and ts (open circles) for Ito. (c) Current–
voltage relationship for Ito. (d ) Current–voltage relationship for Isus. (e) Resultant current traces
for ItoCIsus during voltage clamp simulations. In all cases, solid lines represent the simulated values
and circles represent the experimental data.
(iii) Inward rectifier current, IK1
Experimental data suggested a different IK1 current density between the PF
and ventricular cells (Han et al. 2002). In rabbit hearts, it was shown that
the measured IK1 density was much smaller than in ventricular myocytes
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Models of Purkinje fibre cells
(b)
–120
–100
V (mV)
–80
If (pA pF–1)
0
– 60
– 40
0
–1
–1
–2
–2
–3
–3
–4
–4
0.5
IK1 (pA pF–1)
1000 ms
(c)
0
If (pA pF–1)
(a)
– 0.5
–1.0
–100
–80
–60
– 40
V (mV)
–20
0
Figure 2. Modelling If and IK1. (a) Resultant current traces and (b) current–voltage relationship for
If obtained during voltage clamp for the model (solid line) and experimentally (open circles).
(c) Simulated (solid line) and experimental (open circles) current–voltage relationship for IK1.
(Cordeiro et al. 1998). Based on the experimental data of human PF cells (Han
et al. 2002), the endocardial description (ten Tusscher et al. 2004) of the timeindependent inward rectification factor, x K1N, of IK1 was reformulated. The
maximum conductance, GK1, was determined by fitting the simulated I–V
relationship (figure 2c) to the experimental data (Han et al. 2002).
(b ) All-or-nothing repolarization
Following the methods of Vassalle (1966) and McAllister et al. (1975), the
phenomenon of all-or-nothing repolarization (Weidmann 1951) can be used as a
method of validating qualitatively the behaviour of the AP model. APs are
elicited by a suprathreshold stimulus at a time interval of 1 s and remain
unperturbed during the resultant AP. During the 10th AP, at times of 40, 60 and
80 ms after the AP has been elicited, the membrane potential, V, is clamped for
20 ms to a holding potential, Vhold, and then released. The response to varying
Vhold is determined by whether the AP repolarizes earlier than a normal AP.
If Vhold is above a threshold value, the membrane will depolarize and the AP will
repolarize later than normal. If Vhold is at or below the threshold, the membrane
will repolarize earlier than usual.
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(c ) Overdrive suppression
Rapid stimulation of PF cells can result in a phenomenon called overdrive
suppression (Vassalle 1970; Valenzuela & Vassalle 1983; Boyett & Fedida 1984),
where the pacemaker activity of the PF cells is suppressed for a period of time
following high-frequency stimulation after which it restarts again. Under normal
sinus rhythm in the heart, the pacemaking of the PF cells is usually suppressed
by the SA node. Overdrive suppression was simulated using the following
protocol: no external stimulus was applied for the first 100 s, after which
the cell was paced by a series of periodic suprathreshold stimuli (with an
amplitude of K52 pA pFK1 and a duration of 1 ms) at frequencies of 1.5, 2.0 or
2.5 Hz, before the external stimulation was stopped 10 min later.
3. Results
(a ) Simulated action potential of human Purkinje fibre cells
The simulated time course of the autorhythmic human PF APs (figure 3a), major
underlying ionic channel currents (figure 3b–j ) and the transient of intracellular
Ca2C concentration (figure 3k) are shown in figure 3. The simulated AP begins with
a rapid phase-0 depolarization upstroke, accompanied by the activation of INa
(figure 3b). Following the rapid depolarization, there is a rapid phase-1
repolarization caused by the activated Ito (figure 3d ), producing a sharp spike and
notch. The phase-2 plateau is maintained by the activation of ICaL (figure 3c), which
is followed by the phase-3 repolarization as a consequence of an integral action of I Kr,
I Ks and I K1 (figure 3e–g). Activation of If (figure 3h) produces a phase-4 diastolic
depolarization leading to automaticity. During the time course of APs, activation of
INaK (figure 3i ) and INaCa (figure 3j ) contributes to dynamic changes of ion
concentrations and also to the morphology of the APs. The reconstructed sharp
spike/notch and the phase-4 diastolic depolarization leading to automaticity are
features of PF cells that are distinctive compared with ventricular myocytes.
The simulated AP has characteristics comparable to the experimental data of
human PF cells. Experimentally measured maximal diastolic potential (MDP)
of PF APs is between K79 and K85 mV (Dangman et al. 1982; Lee et al. 2004). In
simulations, the computed MDP is K75.53 mV. The experimentally measured
amplitude of APs (APA, measured from the MDP to the overshoot of the AP) for
human PF cells is between 107 and 114 mV (Dangman et al. 1982; Lee et al. 2004).
In the model, the computed APA is 123.13 mV. PF cells have a larger upstroke
velocity than ventricular myocytes. The measured maximal upstroke velocity
from human PF cells is from 207 to 387 V sK1 (Dangman et al. 1982; Lee et al.
2004). In the model, the computed maximal upstroke velocity is 327.42 V sK1.
The experimentally measured APD90 is 319G23 ms (Lee et al. 2004), while the
computed value is approximately 293 ms. The measured slope of the diastolic
potential from paced PF cells is 3.7G1.0 mV sK1 (Lee et al. 2004). In automatic
cells, it is expected that there will be a larger slope of the diastolic potential. In
the model, the simulated APs are automatic with a computed slope of diastolic
potential of 10 mV sK1. The computed cycle length for spontaneous APs
is approximately 1.1 s, which is close to the experimentally observed range of
1.3–3.0 s (Schmidt & Thews 1993; Lee et al. 2004).
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Models of Purkinje fibre cells
(a)
40
60
–80
100 ms
V (mV)
0
–40
(b)
(c)
(d )
(e)
(f)
(g)
(h)
(i)
( j)
(k)
[Ca2+]i
INaCa
INaK
If
IK1
IKs
IKr
Ito
ICaL
INa
(µM) (pA pF –1) (pA pF –1) (pA pF –1) (pA pF –1) (pA pF –1) (pA pF –1) (pA pF –1) (pA pF –1) (pA pF –1)
–80
0
–180
0
–12
3
0
0.45
0
0.4
0
0.25
0
0.1
– 0.1
0.44
0.22
0.45
– 0.45
0.9
0
250 ms
Figure 3. (a) Simulated autorhythmic APs. Inset: comparison between APs evoked by an external
stimulus from uncorrected (black) and corrected (grey) parameters for heart failure-induced
electrical remodelling. (b–j ) Time traces for major ionic currents and (k) calcium transient.
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(b ) All-or-nothing repolarization and overdrive suppression
Figure 4a,b shows the simulated all-or-nothing repolarization results obtained
using the model. In figure 4a, time courses of repolarized APs clamped 80 ms
after the AP was elicited to various potentials for 20 ms are shown. When the
clamp potential is above K25 mV, the model response is a secondary
depolarization that extends the AP repolarization duration and increases the
APD. However, when the clamp potential is at or below K25 mV, the model
response results in successive repolarization that shortens the AP repolarization
duration. Thus, the model presents the existence of a threshold (K25 mV) at
which repolarization can be accelerated. Figure 4b shows the computed threshold
for all-or-nothing repolarization at 40, 60 and 80 ms after the AP was elicited.
The computed threshold is dynamical, which shifts towards the plateau potential
with time. The simulated all-or-nothing repolarization and dynamical shift of the
determined threshold for forcing repolarization with varied refractory timing are
consistent with experimental observations on PF tissues (Weidmann 1951;
Vassalle 1966).
Rapid stimulation of PF cells can result in overdrive suppression (Vassalle
1970; Valenzuela & Vassalle 1983; Boyett & Fedida 1984), a phenomenon that is
reproduced by the model (figure 4c). During the first 100 s period, the PF cell
model is stably autorhythmic. In the following 10 min period, the PF model is
stimulated by a series of rapid stimuli at 2.5 Hz, each of which evokes an AP.
When the external stimulus is switched off, there is a period of quiescence before
stable automaticity resumes (figure 4c(i)). The characteristics of the simulated
overdrive suppression are similar to the experimental observations by Boyett
et al. (1987).
To investigate possible mechanisms underlying the genesis of overdrive
suppression, time courses of intracellular NaC concentration and the NaC–KC
pump current are considered (figure 4c(ii)(iii)). During the period of rapid
stimulation, the intracellular NaC concentration rises slowly towards an
asymptotic level, approximately 4 mM higher than the initial value. Associated
with the increased intracellular NaC concentration is a monotonic increase in the
NaC–KC pump current. This increase in NaC–KC pump current suppresses the
spontaneous pacemaking activity when the external stimulus is switched off,
leaving the cell model in a quiescent state. During the quiescent period,
intracellular NaC falls comparatively quickly to the initial value (figure 4c(ii)),
resulting in decreased NaC–KC pump current. When the NaC–KC pump
current decreases to a critical amplitude, comparable with the amplitude of INaK
during the diastolic phase of the AP before rapid stimulation, the spontaneous
pacemaking activity of the PF cell model resumes.
An increase in intracellular NaC during rapid pacing has been observed
experimentally (Boyett et al. 1987) and is believed to be responsible for
suppression of automaticity following prolonged periods of rapid stimulation
(Valenzuela & Vassalle 1983), as it produces a rate-dependent increase in
the NaC–KC pump activity (Kline & Kupersmith 1982; Boyett & Fedida 1984).
The simulations presented support this hypothesis (figure 4c). The link between
overdrive suppression and the rate-dependent increase in the NaC–KC pump
current due to intracellular NaC overload can be further studied by removing
the contribution of INaK to cell membrane potential (i.e. it is removed from
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Models of Purkinje fibre cells
(a)
60
(b)
V (mV)
30
0
–25
–30
–30
–36
– 60
–90
V (mV)
(c)
40
50 ms
(i)
(d)
0
– 40
INaK (pA pF–1) [Na+]i (mM)
–80
14
(ii)
11
8
0.6
(iii)
0.4
0.2
2 min
Tq (s)
(e) 90
45
0
1.5
2.0
f (Hz)
2.5
Figure 4. Reproduction of experimental phenomena. All-or-nothing repolarization: (a) 80 ms after
the AP is elicited, the membrane is clamped to a holding potential for 20 ms and then released,
resulting in either successive depolarization and a prolongation of the APD or repolarization and a
shortening of the APD. (b) The threshold for repolarization approaches the plateau potential as
the time after the AP is elicited, at which the membrane is clamped, increases. (c) Overdrive
suppression of the pacemaker after a period of rapid stimulation. (i) The cell is autorhythmic before
rapid stimulation at 90, 120 or 150 beats per minute for 10 min, after which a period of quiescence
occurs, the pacemaker recovers and eventually resumes spontaneous activity. Slow rises in both
(ii) [NaC]i and (iii) INaK occur during rapid pacing, returning to normal levels shortly after
automaticity resumes. (d ) Removal of INaK from equation (2.2) while retaining the NaC–KC pump
function resulted in no period of quiescence after rapid pacing. (e) Effect of pacing frequency, f, on
period of quiescence, Tq.
equation (2.2)), while retaining the NaC–KC pump function in sustaining the
homoeostasis of NaC and KC ions. The removal of the NaC–KC pump current
from equation (2.2) results in a relatively small intracellular NaC overload and
negligible increase in NaC–KC pump current, and as a result abolishes the period
Phil. Trans. R. Soc. A (2009)
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2236
P. Stewart et al.
of quiescence after the rapid stimulation (figure 4d ). This provides further
support for the hypothesis that the overloading of intracellular NaC concentration is responsible for the genesis of overdrive suppression (Vassalle 1970).
Boyett & Fedida (1984) demonstrated that overdrive suppression is rate
dependent and the period of quiescence, Tq, is longer at higher stimulation
frequencies. Such a rate-dependent prolongation of the suppression period is
reproduced by the model for stimulation frequencies greater than 1.5 Hz.
The model was paced at 1.5, 2 and 2.5 Hz (corresponding to pacing rates of 90,
120 and 150 beats per minute, respectively) and with the increasing frequency,
the measured Tq increased from 50 to 83 s (figure 4e).
(c ) Ionic mechanisms underlying human Purkinje fibre cells
Simulations were performed to elucidate the role of each major individual ionic
current in generating autorhythmic human PF APs (figure 5), especially the
genesis of diastolic depolarization leading to automaticity.
(i) Effect of INa on the action potential
The role of INa was investigated by blocking INa either partially or completely
(figure 5b). Blocking INa by 50 per cent slows down the automatic activity.
Compared with the control condition, the measured cycle length increases from
1.1 to 2.6 s, while the overshoot is decreased from 30 to 18 mV, as is the maximal
upstroke velocity, which decreases to 24 V sK1. There is no noticeable change in
the MDP or the APD. Blocking INa by 100 per cent results in the abolition of
automaticity with the membrane potential resting at K66.5 mV.
(ii) Effect of ICaL on the action potential
The role of ICaL was determined by blocking ICaL either partially or
completely (figure 5a). Blocking ICaL by 50 per cent resulted in a slowing of the
automaticity, increasing the cycle length to 1.4 s. Additionally, it decreases the
overshoot, shortens APD and also decreases the plateau potential of the APs.
There was no noticeable change in the MDP. However, blocking ICaL further to
100 per cent accelerates, rather than decelerates, the automaticity. In this
condition, the measured CL decreases to 0.9 s. The accelerated automaticity is
due to a shortening of the APD as a consequence of the loss of the AP plateau,
similar to previous experimental observations in peripheral rabbit SA node
cells, in which the application of nifedipine, a inhibiter of ICaL, shortened
the rabbit SA node APD and accelerated its pacemaking activity (Kodama
et al. 1997).
(iii) Effect of IKr on the action potential
Blocking IKr by 50 or 100 per cent produces a prolonged APD and a reduction
in the rate of automaticity (figure 5c). However, it has negligible effect on the
overshoot and MDP. The slowing down in the automaticity can be attributed to
the prolonged APD as observed in rabbit SA node cells (Kodama et al. 1997).
Phil. Trans. R. Soc. A (2009)
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Models of Purkinje fibre cells
(i)
40
V (mV)
(a)
0
(b)
(c)
(i)
(i)
(ii)
(ii)
(e)
(f)
(i)
(i)
(ii)
(ii)
– 40
–80
V (mV)
(ii) 40
0
– 40
–80
(i)
40
V (mV)
(d)
0
– 40
–80
V (mV)
(ii) 40
0
– 40
–80
500 ms
Figure 5. Effects of blocked ionic current on autorhythmic APs. (a) ICaL blocked by (i) 50% and
(ii) 100%, (b) INa blocked by (i) 50% and (ii) 100%, (c) IKr blocked by (i) 50% and (ii) 100%, (d ) IKs
blocked by (i) 50% and (ii) 100%, (e) Ito blocked by (i) 50% and (ii) 100% and (f ) If blocked by (i) 30%
and (ii) 100%. In all cases, the grey trace is the control AP and the black trace is the effect of the block.
(iv) Effect of IKs on the action potential
Blocking IKs by 50 or 100 per cent produces negligible effects on the overshoot,
MDP, maximal upstroke velocity and the automaticity of the APs (figure 5d ).
It does, however, prolong the measured APD50 from 231 to 309 ms with a
100 per cent block.
(v) Effect of Ito on the action potential
Blocking Ito by 50 or 100 per cent produces negligible effects on the overshoot,
MDP, maximal upstroke velocity and the automaticity of APs (figure 5e).
However, it has remarkable effects on the phase-1 repolarization. Blocking Ito
Phil. Trans. R. Soc. A (2009)
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2238
P. Stewart et al.
slows down the phase-1 repolarization producing an elevated plateau potential
and a less marked phase-1 spike/notch. By blocking Ito by 100 per cent, the
unique feature of the phase-1 notch of the PF cell AP disappears.
(vi) Effect of If on the action potential
Blocking If has the most dramatic effect on the automaticity of the PF
cell model, though its effect on the overshoot, MDP, maximal upstroke velocity
and APD is negligible (figure 5f ). Blocking If by 30 per cent increased the
measured cycle length of APs from 1.1 to 1.7 s under the control condition.
Blocking If by 100 per cent abolished the automaticity with the membrane
potential resting at K75.5 mV, close to the MDP. However, the cell model remains
excitable and with an external suprathreshold stimulus, a full AP can be evoked.
(d ) Comparison between human and canine Purkinje fibre cell
action potential models
The electrical properties of PF cells are species dependent (Lu et al. 2001).
Experimental data indicate dramatic differences in the morphology of human and
canine PF APs. Primarily, canine PF cells have a much lower AP plateau and
more predominant notch (Dumaine & Cordeiro 2007) than human PF cells (Lee
et al. 2004). It is possible that such differences in their APs are due to different
properties of ion channels in the two species. Using the model we developed for
canine PF cells (O. V. Aslanidi et al. unpublished data) and the present model,
we have identified Ito as the main factor contributing to differences between
human and canine PF APs. There are experimental data suggesting that the
current density of Ito measured at 20 mV from canine PF cells is significantly
larger than from human PF cells (figure 6a). Blocking Ito in the canine PF cell
model produces APs with substantially changed morphology (figure 6b): an
elevated plateau and less marked notch as seen in human PF cells (Han et al.
2002). Therefore, we conclude that the large differences in the AP plateau
potential and notch between the canine and human PF cells seen experimentally
(Lee et al. 2004; Dumaine & Cordeiro 2007) can be explained by significant
differences in the transient outward current between the two species (Han et al.
2001a, 2002).
4. Discussion
In this study, we have developed a biophysically detailed model for the electrical
AP of the human PF cell based on modifications to the ten Tusscher et al.
(2004) and ten Tusscher & Panfilov (2006) model of human ventricular
myocytes, which incorporate extant voltage-clamp data recorded from human
PF cells (Han et al. 2002). Conductance, steady-state activation and
inactivation curves and time constants for Ito, IKr, IKs and IK1 were updated,
and two additional currents were introduced: a hyperpolarization-activated
pacemaking current, If, and a sustained potassium current, Isus, absent in the
ventricular cell models but present in PF cells, which we fitted to the
experimental data of Han et al. (2002). The resultant model reproduces the PF
Phil. Trans. R. Soc. A (2009)
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Models of Purkinje fibre cells
(a) 10
(b)
60
V (mV)
Ito (pA pF–1)
30
5
0
– 30
– 60
0
– 90
canine
human
100 ms
Figure 6. (a) Ito current density measured at 20 mV in human (Han et al. 2002) and canine PF
(Han et al. 2001a) cells. (b) Effect of blocking Ito in the canine PF cell model (black) produced
human PF cell-like AP (grey).
cell AP with characteristics consistent with experimental recordings (Dangman
et al. 1982; Lee et al. 2004). Inclusion of I f in the model produces autorhythmic
APs with a cycle length of approximately 1.1 s, which is consistent with
experimental data (Schmidt & Thews 1993; Lee et al. 2004). The model is also
validated by its ability to reproduce the all-or-nothing repolarization
phenomenon observed in PF tissues (Wiedmann 1951; Vassalle 1966), and the
well-known physiological phenomenon of overdrive suppression (Vassalle 1970;
Valenzuela & Vassalle 1983; Boyett & Fedida 1984). Using the model, we
compute the functional role of several major ionic currents (INa, ICaL, Ito, IKr, IKs
and If) in producing the unique features of human PF APs, especially the fast
phase-1 repolarization, the phase-4 diastolic depolarization and the automaticity. It is shown that while Ito plays an important role in producing the phase-1
notch, INa, ICaL and I f all play an important role in controlling the automaticity
of PF cells.
(a ) Comparison to other species
In contrast to many other species including canine (Dumaine & Cordeiro
2007), rabbit (Dumaine & Cordeiro 2007) and sheep (Boyett 1981), the human
PF AP lacks a number of characteristics, such as a longer APD, lower plateau
potential and less marked phase-1 notch than its ventricular counterpart. The
human PF AP is, in fact, more ventricular-like than other species (Dangman
et al. 1982; Lee et al. 2004). The presented human PF model reproduces this
observation. It is of scientific interest to investigate the ionic mechanisms
underlying such differences in the AP characteristics between human and animal
models. Using the present model and the model we developed for canine PF cells
(O. V. Aslanidi et al. unpublished data), we have shown that the marked
differences in the morphology of PF APs between the human and canine hearts
can be explained by the different Ito densities of PF cells measured in the two
species, as observed experimentally.
Phil. Trans. R. Soc. A (2009)
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P. Stewart et al.
(b ) Adjustment of model parameters due to heart failure-induced ion
channel remodelling
The experimental data of Han et al. (2002) on properties of potassium currents
of human PF cells were obtained from explanted failing human hearts. It is
well known that heart failure induces changes in channel properties of several
major ion channels (i.e. ion channel remodelling) responsible for electrical APs
in both PF and ventricular cells (Priebe & Beuckelmann 1998; Han et al. 2001a).
Experimental data also suggested that heart failure-induced ion channel
remodelling is comparable between PF and ventricular cells (Han et al.
2001a). In human ventricular cells, it was shown that the AP is prolonged in
patients with heart failure (Priebe & Beuckelmann 1998). Associated with the
changes in the AP is the downregulation of IK1 (Beuckelmann et al. 1993; Koumi
et al. 1995) and Ito (Beuckelmann et al. 1993; Näbauer et al. 1993). There is no
evidence for heart failure-induced remodelling on other potassium channels, such
as IKr and IKs, nor on the fast sodium current INa (Priebe & Beuckelmann 1998;
Han et al. 2001a). However, there is evidence that the current densities and
kinetics of ICa are unaltered (Beuckelmann et al. 1993; Mewes & Ravens 1994;
Ouadid et al. 1995), though Ca2C handling is altered and the activity of the
NaC–Ca2C exchanger is enhanced (Gwathmey et al. 1987; Beuckelmann et al.
1992; Flesch et al. 1996; Reinecke et al. 1996) in the failing hearts. Data obtained
from canine studies suggested that congestive heart failure (CHF) produced
compatible ion channel remodelling between PF and ventricular myocytes, with
the main changes involving downregulation of both IK1 and Ito densities and
slowed inactivation of ICaL, but no change in other currents such as IKs, IKr, INaCa
and ICaT (Priebe & Beuckelmann 1998; Tomaselli & Marbán 1999; Han et al.
2001a, 2002).
The present PF cell model is based on the experimental data of Han et al.
(2002) obtained from failing human hearts. Some major ion channels, including
IK1 and Ito, may be affected by heart failure-induced ion channel remodelling.
Thus, it is necessary to adjust some channel parameters in order to model a
normal human PF cell. We assumed heart failure-induced ion channel
remodelling to be consistent across PF and ventricular myocytes and followed
the approach of Priebe & Beuckelmann (1998). Namely, heart failure would
produce a 36 per cent reduction in Ito and a 20 per cent reduction in IK1 current
densities. Therefore, in the normal PF cell model, Gto and GK1 were increased
by 36 and 20 per cent, respectively. As equations and parameters for INaCa and
Ca2C handling were inherited from the original ten Tusscher et al. (2004)
and ten Tusscher & Panfilov (2006) model, we assumed they are for healthy cells,
and therefore did not require adjustment.
In the model with adjusted parameters to compensate for heart failure-induced
ion channel remodelling of Ito and IK1, the simulated PF APs (the grey line in the
inset of figure 3a) evoked by an external stimulus are similar to those of the
uncorrected model, except for a more marked phase-1 repolarization. This is
consistent with the experimental observation of Han et al. (2001a) on canine PF
cells: the characteristics of canine PF cells are very close between normal hearts
and hearts with CHF, except a less marked phase-1 repolarization and higher
plateau voltage in CHF. There were no significant differences in resting potential,
AP amplitude or APD between control and CHF cells.
Phil. Trans. R. Soc. A (2009)
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Models of Purkinje fibre cells
2241
(c ) Limitations
The model was constructed based on the experimental data of Han et al.
(2002) on properties of potassium currents in human PF cells isolated from
failing hearts, which were treated by a variety of medications. As both disease
and medication can change the kinetics and current density of ion channels
(Han et al. 2001a), it is possible that the data of Han et al. (2002) may not truly
reflect the electrical properties of healthy human PF cells. These are wellrecognized limitations for virtually all electrophysiological studies of human
cardiac cells in the literature, based on which all other models for human cardiac
cells were developed. Though we have adjusted possible electrical remodelling
induced by heart failure for Ito and IK1 based on experimental studies in canine
and humans (Priebe & Beuckelmann 1998; Han et al. 2001a), the adjustment for
parameters may be incomplete, as other channels, such as IKr and IKs, may also
be remodelled by heart failure.
In the absence of detailed experimental data, a number of major currents
including INa and ICaL, and intracellular Ca2C handling, were inherited from the
ten Tusscher et al. (2004) and ten Tusscher & Panfilov (2006) models. It is
possible that these inherited descriptions, notably the Ca2C handling
mechanisms, are different between PF and ventricular cells (e.g. due to a lack
of t-tubules in PF cells; Sommer & Johnson 1968; Boyden et al. 2000). These
limitations must be addressed in the future when more experimental data are
available and can be used to improve the validity of the current model. The
present model is quiescent following rapid stimulation at pacing rates over
90 beats per minute (i.e. stimulus frequency higher than 1.5 Hz), faster than an
average adult human heart rate at normal physiological conditions, potentially
due to the inheritance of ventricle data.
Additionally, recent studies have identified a number of currents absent in the
model which are believed to play an important role in AP morphology of PF cells,
notably IK(ACh), ICaT and INaL (Gaborit et al. 2007; Dun & Boyden 2008). Though
identified in human PF cells, a lack of experimental data on IK(ACh) and ICaT
prevents their inclusion in the current model, while the presence of ICaT in human
PF cells has yet to be observed (Dun & Boyden 2008). The role of INaL and ICaT has
been studied in more detail in canine PF cells (O. V. Aslanidi et al. unpublished
data), which revealed that ICaT and IK(ACh) played a minor role, whereas the effect
of INaL on the APD was much more prominent. Future models of human PF cells
should incorporate effects of the latter, subject to availability of experimental
data. The developed model can however reproduce the typical features of APs in
human PF cells, such as the marked phase-1 notch, automaticity, all-or-nothing
repolarization and overdrive suppression; thus, it can be used to simulate the
conduction system of PF network in the whole heart model.
(d ) Role of If in pacemaking activity of Purkinje fibre cells
Controversy still surrounds the mechanism underlying the genesis of
automaticity in cardiac pacemaking cells including the SA node and PF cells.
Blocking If in the present model abolishes the automaticity of the PF cells
(figure 5f ), providing evidence to support the hypothesis that If is the primary
factor responsible for generating pacemaking activity (DiFrancesco 1981, 2006).
However, experimental studies have also suggested an alternative hypothesis
Phil. Trans. R. Soc. A (2009)
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2242
P. Stewart et al.
that reverse excitation–contraction (EC) coupling (Dangman & Miura 1987;
Boyden et al. 2000; Lakatta et al. 2003; ter Keurs & Boyden 2007) may play a
critical role underlying cardiac automaticity. The present model is not sufficient
to investigate the possible role of the major mechanisms of reverse EC coupling—
Ca2C sparks and waves—in initiating PF cell automaticity, since the model lacks
consideration of spatially extended features of Ca2C handling and diffusion (Tao
et al. 2008). Such an approach would involve considering spatio-temporal
dynamics of subcellular variables, which is beyond the scope of the present paper.
(e ) Looking forward
By combining a geometric model with suitable models of the AP in single cells,
it is possible to reconstruct the electrical activity and activation sequence of the
whole heart. The newly developed PF cell model adds a new member to the
family of human cardiac cell models developed in previous studies for the SA
node (Seemann et al. 2006), atrial (Courtemanche et al. 1998; Nygren et al. 1998)
and ventricular (Iyer et al. 2004; ten Tusscher et al. 2004; ten Tusscher &
Panfilov 2006) cells, which can be incorporated into an anatomical model of the
human heart (Sachse et al. 2000) with details of its electrophysiological
heterogeneity and anatomical complexity.
P.S. is supported by a UK EPSRC DTA studentship. O.V.A., M.R.B. and H.Z. are supported by
the UK BBSRC (BBS/B/1678X) project grant.
Appendix A. Model equations
(a ) Inward rectifier current, I K1
I K1 Z G K1 x K1NððV K8ÞK E K Þ;
x K1N Z
1
0:1ðVC75:44Þ
1 Ce
ðA 1Þ
ðA 2Þ
:
(b ) Transient outward current, Ito
Ito Z Gto rsðV K E K Þ;
rN Z
1
1 C eð20KV Þ=13
ðA 3Þ
ðA 4Þ
;
2
tr Z 10:45 e KðVC40Þ =1800 C 7:3;
sN Z
1
1 C eðVC27Þ=13
Phil. Trans. R. Soc. A (2009)
1 Ce
ðA 6Þ
;
5
2
ts Z 85 e KðVC25Þ =320 C
ðVK40Þ=5
ðA 5Þ
C 42:
ðA 7Þ
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Models of Purkinje fibre cells
2243
(c ) Sustained current, Isus
Isus Z Gsus aðV K E K Þ;
aN Z
1
ð5KV Þ=17
1 Ce
ðA 8Þ
ðA 9Þ
:
(d ) Hyperpolarization-activated current, If
If Z i f; K C i f;Na ;
ðA 10Þ
i f;K Z Gf;K yðV K EK Þ;
ðA 11Þ
i f;Na Z Gf;Na yðV K ENa Þ;
ðA 12Þ
yN Z
1
1 C eðVC80:6Þ=6:8
ðA 13Þ
;
ay Z e K2:9Kð0:04V Þ ;
ðA 14Þ
by Z e3:6Cð0:11V Þ ;
ðA 15Þ
ty Z
4000
:
ay C by
ðA 16Þ
(e ) Fast sodium current, INa
INa Z GNa m 3 hjðV K E Na Þ;
ðA 17Þ
1
mN Z ;
ðK56:86KV Þ=9:03 2
1 Ce
ðA 18Þ
am Z
bm Z
1
1 Ce
ðK60KV Þ=5
ðA 19Þ
;
0:1
0:1
C
;
ðVC35Þ=5
ðVK50Þ=200
1 Ce
1 Ce
ðA 21Þ
tm Z am bm ;
hN Z 1
2
1 C eðVC71:55Þ=7:43
;
ah Z 0
)
if V RK40;
ah Z 0:057 e KðVC80Þ=6:8
otherwise;
Phil. Trans. R. Soc. A (2009)
ðA 20Þ
ðA 22Þ
ðA 23Þ
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2244
P. Stewart et al.
9
=
if V RK40;>
0:77
bh Z
0:13 1 C eKðVC10:66Þ=11:1
0:079V
bh Z 2:7 e
5
C 3:1 !10 e
th Z
0:3485V
otherwise;
>
;
1
;
ah C bh
ðA 25Þ
1
jN Z ;
ðVC71:55Þ=7:43 2
1 Ce
aj Z 0
aj Z
ðA 24Þ
if V RK40;
ðK2:5428 !104 e0:2444V K6:948 !10K6 eK0:04391V ÞðV C 37:78Þ
1 C e0:311ðVC79:23Þ
ðA 26Þ
ðA 27Þ
otherwise;
ðA 28Þ
0:6 e0:057V
bj Z
1 C eK0:1ðVC32Þ
0:02424 eK0:01052V
bj Z
1 C eK0:1378ðVC40:14Þ
tj Z
9
>
if V RK40; >
=
>
>
otherwise; ;
1
:
aj C bj
ðA 29Þ
ðA 30Þ
(f ) L-type calcium current, ICaL
ICaL Z GCaL d f f 2 f Cass 4
ðV K15ÞF 2 0:25 ½Ca2Css e2ðV K15ÞF=RT K½Ca2Co
; ðA 31Þ
RT
e2ðV K15ÞF=RT K 1
dN Z
ad Z
1
ðK35KV Þ=13
1 Ce
C 0:25;
ðA 33Þ
1:4
;
1 C eðVC5Þ=5
ðA 34Þ
1:4
;
1 C eð50KV Þ=20
ðA 35Þ
bd Z
Phil. Trans. R. Soc. A (2009)
ðA 32Þ
1 Ce
1:4
gd Z
;
ðK8KV Þ=7:5
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Models of Purkinje fibre cells
ðA 36Þ
td Z ad bd C gd ;
1:4
;
1 C eðVC20Þ=7
fN Z
ðA 37Þ
2
af Z 1102:5 eðKðVC27Þ=15Þ ;
ðA 39Þ
180
C 20;
1 C eðVC30Þ=10
ðA 40Þ
ðA 41Þ
tf Z af C bf C gf ;
f2N Z
ðA 38Þ
200
;
1 C eð13KV Þ=10
bf Z
gf Z
2245
0:67
C 0:33;
1 C eðVC35Þ=7
2
af 2 Z 600 e KððVC25Þ =170Þ ;
31
ðA 42Þ
ðA 43Þ
;
ðA 44Þ
;
ðA 45Þ
tf 2 Z af 2 C bf 2 C gf 2 ;
ðA 46Þ
bf 2 Z
gf 2 Z
fCa ssN Z
1 C eð25KV Þ=10
16
1 C eðVC30Þ=10
0:6
½Ca2Css
1C
0:05
tf Ca ss Z
2 C 0:4;
ðA 47Þ
2 C 2:
ðA 48Þ
80
½Ca2Css
1C
0:05
(g ) Slow delayed rectifier current, IKs
IKs Z G Ks x 2s ðV K E Ks Þ;
x sN Z
Phil. Trans. R. Soc. A (2009)
1
1 Ce
ðK5KV Þ=14
;
ðA 49Þ
ðA 50Þ
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2246
P. Stewart et al.
1400
axs Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
1 C eð5KV Þ=6
1
bxs Z
1 Ce
ðVK35Þ=15
ðA 51Þ
ðA 52Þ
;
ðA 53Þ
txs Z axs bxs C 80:
(h ) Rapid delayed rectifier current, IKr
rffiffiffiffiffiffiffiffiffiffiffiffi
½KCo
x x ðV K E K Þ;
IKr Z GKr
5:4 r1 r2
x r1N Z
1 Ce
450
axr1 Z
bxr1 Z
1
ðK26KV Þ=7
ðK45KV Þ=10
1 Ce
;
ðA 55Þ
;
ðA 56Þ
;
ðA 57Þ
6
ðVC30Þ=11:5
1 Ce
ðA 58Þ
txr1 Z axr1 bxr1 ;
x r2N Z
axr2 Z
bxr2 Z
1
ðA 54Þ
;
ðA 59Þ
;
ðA 60Þ
1:12
;
1 C eðVK60Þ=20
ðA 61Þ
1 C eðVC88Þ=24
3
ðK60KV Þ=20
1 Ce
txr2 Z axr2 bxr2 :
ðA 62Þ
(i ) NaC/Ca2C exchange current, INaCa
INaCa Z kNaCa
egVF=RT ½NaC3i ½Ca2Co KeðgK1ÞVF=RT ½NaC3o ½Ca2Ci a
: ðA 63Þ
ðK3mNa i C ½NaC3o ÞðKmCa C ½Ca2Co Þð1 C k sat eðgK1ÞVF=RT Þ
Phil. Trans. R. Soc. A (2009)
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Models of Purkinje fibre cells
2247
(j ) NaC/KC pump current, INaK
INaK Z
PNaK ½KCo ½NaCi
;
ð½KCo C KmK Þð½NaCi C KmNa Þð1 C 0:1245 e K0:1VF=RT C 0:0353 e KVF=RT Þ
ðA 64Þ
IpCa Z GpCa
IpK Z GpK
½Ca2Ci
;
KpCa C ½Ca2Ci
ðA 65Þ
VKEK
:
1 C eð25KV Þ=5:98
ðA 66Þ
(k ) Background current, Ib
IbNa Z GbNa ðV K ENa Þ;
ðA 67Þ
IbCa Z GbCa ðV K ECa Þ:
ðA 68Þ
(l ) Calcium dynamics
Ileak Z Vleak ð½Ca2Csr K½Ca2Ci Þ;
Iup Z
Vmaxup
;
1 C K2up =½Ca2Ci
ðA 70Þ
Irel Z Vrel Oð½Ca2Csr K½Ca2Css Þ;
ðA 71Þ
Ixfer Z Vxfer ð½Ca2Css K½Ca2Ci Þ;
ðA 72Þ
OZ
k 1 ½Ca2C2ss R
;
k 3 C k 1 ½Ca2C2ss
dR
C k 4 ð1 C RÞ;
ZKk 2 ½Ca2Css R
dt
Phil. Trans. R. Soc. A (2009)
ðA 69Þ
ðA 73Þ
ðA 74Þ
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2248
P. Stewart et al.
k1 Z
k10
;
k Ca sr
k 2 Z k 20 k Ca sr ;
k Ca sr Z maxsr K
ðA 75Þ
ðA 76Þ
max sr Kmin sr
;
1 C ðEC=½Ca2Csr Þ2
ðA 77Þ
½Ca2Ci !Buf c
;
½Ca2Ci !KBufc
ðA 78Þ
½Ca2Ci Bufc Z
IbCa C IpCa K2INaCa Vsr
d½Ca2Ci total
ðI K Iup Þ C Ixfer ;
ZK
C
dt
2Vc F
Vc leak
ðA 79Þ
½Ca2Csr !Buf sr
;
½Ca2Csr C KBufsr
ðA 80Þ
d½Ca2Csr total
ZKIleak C Iup K Irel ;
dt
ðA 81Þ
½Ca2Csr Bufsr Z
½Ca2CssBufsr Z
½Ca2Css !Buf ss
;
½Ca2Css C KBufss
d½Ca2Css total
I
V
V
ZK CaL C sr Irel K c Ixfer :
dt
2Vss F Vss
Vss
ðA 82Þ
ðA 83Þ
(m ) Sodium and potassium dynamics
INa C IbNa C i f;Na C 3INaK C 3INaCa
d½NaCi
ZK
;
dt
Vc F
IK1 C Ito C IKr C IKs C i f;K C Isus K2INaK C IpK C Istim
d½KCi
ZK
:
dt
Vc F
Phil. Trans. R. Soc. A (2009)
ðA 84Þ
ðA 85Þ
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Models of Purkinje fibre cells
2249
Appendix B. Model parameters
(a ) Parameters
parameter
value
GK1
Gto
Gsus
Gf,K
Gf,Na
GKr
GKs
GNa
R
T
F
Cm
S
r
Vc
Vsr
Vss
[KC]o
[NaC]o
[Ca2C]o
pKNa
GCaL
k NaCa
g
KmCa
KmNai
k sat
a
PNaK
KmK
KmNa
GpK
GpCa
KpCa
GbNa
GbCa
Vmaxup
Kup
Vrel
k 10
k 20
0.065 nS pFK1
0.08184 nS pFK1
0.0227 nS pFK1
0.0234346 nS pFK1
0.0145654 nS pFK1
0.0918 nS pFK1
0.2352 nS pFK1
130.5744 nS pFK1
8.3143 J KK1 molK1
310 K
96.4867 C mmolK1
2.0 mF cmK2
0.2 mmK1
162 U cm
16.404 mm3
1.094 mm3
0.05468 mm3
5.4 mM
140 mM
2 mM
0.03 (dimensionless)
3.980K5 cm msK1 mFK1
1000 pA pFK1
0.35 (dimensionless)
1.38 mM
87.5 mM
0.1 (dimensionless)
2.5 (dimensionless)
2.724 pA pFK1
1 mM
40 mM
0.0146 nS pFK1
0.1238 nS pFK1
0.0005 mM
0.000290 nS pFK1
0.000592 nS pFK1
0.006375 mM msK1
0.00025 mM
40.8 mM msK1
0.15 mMK2 msK1
0.045 mMK1 msK1
(Continued.)
Phil. Trans. R. Soc. A (2009)
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2250
P. Stewart et al.
parameter
value
k3
k4
EC
maxsr
minsr
Vleak
Vxfer
Bufc
KBufc
Bufsr
KBufsr
Bufss
KBufss
0.060 msK1
0.000015 msK1
1.5 mM
2.5 (dimensionless)
1 (dimensionless)
0.00036 mM msK1
0.0038 mM msK1
0.2 mM
0.001 mM
10 mM
0.3 mM
0.4 mM
0.00025 mM
(b ) Initial conditions
parameter
value
V
[NaC]i
[KC]i
½Ca2Ci
[Ca2C]sr
[Ca2C]ss
m
h
j
x r,1
x r,2
xs
r
s
d
f1
f2
fCass
R
O
y
K74.7890522727
8.5447311020
136.9896086978
0.0001720623
3.2830723338
0.0006146554
0.0145766758
0.2979720207
0.0692509548
0.4663168269
0.3657472179
0.0486609588
0.0006830833
0.9717098312
0.0001356656
0.5943228461
0.8265709174
0.9767040566
0.8199969443
0.0000006152
0.0184308075
Phil. Trans. R. Soc. A (2009)
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Models of Purkinje fibre cells
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