Computers in Biology and Medicine 37 (2007) 8 – 20
www.intl.elsevierhealth.com/journals/cobm
Phase response characteristics of sinoatrial node cells
D.G. Tsalikakisa, d , H.G. Zhangb , D.I. Fotiadisa, c, d,∗ , G.P. Kremmydasa , Ł.K. Michalisd, e
a Unit of Medical Technology and Intelligent Information Systems, Department of Computer Science, University of Ioannina, GR 45110 Ioannina, Greece
b University of Manchester, Manchester, UK
c Biomedical Research Institute-FORTH, GR 45110 Ioannina, Greece
d Michaelidion Cardiology Center, GR 45110 Ioannina, Greece
e Department of Cardiology, Medical School, University of Ioannina, GR 45110 Ioannina, Greece
Received 29 December 2004; accepted 20 September 2005
Abstract
In this work, the dynamic response of the sinoatrial node (SAN), the natural pacemaker of the heart, to short external stimuli is investigated
using the Zhang et al. model. The model equations are solved twice for the central cell and for the peripheral cell. A short current pulse is
applied to reset the spontaneous rhythmic activity of the single sinoatrial node cell. Depending on the stimulus timing either a delay or an
advance in the occurrence of next action potential is produced. This resetting behavior is quantified in terms of phase transition curves (PTCs)
for short electrical current pulses of varying amplitude which span the whole period. For low stimulus amplitudes the transition from advance
to delay is smooth, while at higher amplitudes abrupt changes and discontinuities are observed in PTCs. Such discontinuities reveal critical
stimuli, the application of which can result in annihilation of activity in central SAN cells. The detailed analysis of the ionic mechanisms
involved in its resetting behavior of sinoatrial node cell models provides new insight into the dynamics and physiology of excitation of the
sinoatrial node of the heart.
䉷 2005 Elsevier Ltd. All rights reserved.
Keywords: Heart; Sinoatrial node; Mathematical models; Cardiac models; Phase resetting; Three-dimensional phase transition curves; Regional differences
1. Introduction
Auto-rhythmic excitation in the cardiac muscle is driven by
the sinoatrial node (SAN), the natural pacemaker of the heart.
Under physiological or pathological conditions during clinical
interventions, such us defibrillation of cardiac muscle or accidental situations like electrocution, the pacemaker of the heart
is subjected to external stimulation [1].
Experimental [2,3] and numerical [4–6] studies indicate that
the application of short current pulses to sinoatrial pacemaker
cells result in changes in the cell’s cycle length which depend
on both the phase and the amplitude of the stimulus [3,7–31].
In most cases, the applied perturbation does not affect the amplitude or the configuration of the pacemaker cell action potential. The magnitude and direction of the phase shift resulting
∗ Corresponding author. Unit of Medical Technology and Intelligent Information Systems, Department of Computer Science, University of Ioannina,
GR 45110 Ioannina, Greece. Tel.: +30 6510 98893; fax: +30 6510 98889.
E-mail address: [email protected] (D.I. Fotiadis).
0010-4825/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compbiomed.2005.09.011
from the alteration of the beat-to-beat interval depends on the
timing as well as on the strength of the stimulus. In terms of the
resetting effect, the total charge injected into the cell seems to
be the crucial factor. Thus, any combinations of various pulse
amplitudes and durations, which result in the injection of the
same amount of current, are equivalent [5].
The response of biological oscillators to external perturbations has been extensively studied using both theoretical and
experimental approaches (reviewed in [32]). Using topological
arguments, Winfree [33] greatly contributed to the qualitative
understanding of phase resetting responses in biological oscillators and pointed out that, in a spatially extended system like
cardiac tissue, such responses play a crucial role in the initiation of arrhythmias. Stimuli of critical amplitude and phase
might lead to annihilation of normal rhythmic activity and the
initiation of complex spiral wave arrhythmias.
Mathematical models which describe the electrical activity of
SAN have been presented by various research groups [1]. The
first models produced by Yanagihara et al. [34] and Noble and
Noble [35], where the basis for models presented later [36,37].
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
The analysis of phase response characteristics provides new
insight into the dynamics of the mathematical models, of the
electrical activities of SAN, and an experimentally verifiable
test for the accurate reconstruction of SAN tissue dynamics.
Simulation of the spatial variation in the electrical properties
of SAN cells and their phase response profiles is necessary
for reconstruction of the complex dynamics of the SAN tissue.
Zhang et al. [38] recently presented one-dimensional model
which describes all of the above.
In this work, we investigate and characterize the effects of
external stimulation on central and peripheral SAN cells using the Zhang et al. model [38]. The equation of the model
along with the appropriate initial conditions are solved using
the Runge–Kutta method. Short (0.5 ms) depolarizing and hyperpolarizing electrical current pulses of varying amplitude and
of timing spanning the whole period are applied and the phase
transition curves (PTC) are obtained [22]. Three-dimensional
PTC [4], relating old phase and stimulus amplitude to the new
phase after stimulation, are generated in order to locate critical
stimuli and singularities in the models. Numerical simulations
for the electrical activity of the transitional cells lying between
the center and the periphery of the SAN [38] are carried out.
Simulations are conducted when specific ionic currents vary.
The application of a critical depolarizing stimulus (about
0.4 nA) during the late repolarization phase of action potential
resulted in annihilation of the activity in central SAN cells, revealing the existence of a stable singularity in the corresponding model configuration [22]. Detailed analysis of the phase
response characteristics of the peripheral cells failed to show
a similar singularity and annihilation of normal activity. This
difference in phase response behavior of central and peripheral
cells is portrayed in the structure of the corresponding PTC.
9
where V is the membrane potential, t is the time, C is the
membrane capacitance of SAN and
Itot = (INa + ICa,L + ICa,T + Ito + Isus + IK,r + IK,s
+ If + Ib,Na + Ib,Ca + Ib,K + INaCa + Ip ).
The transition from central to peripheral SAN activity was
modelled as in [38], using a scaling factor which depends on
the distance of each cell from the center of the SAN. The set of
differential Eqs. (1)–(16) given in Tables 2–9 is solved using a
fourth order Runge–Kutta method with time step 0.1 ms. Further details for the formulation of the equations described in
Tables 2–9 and the experimental background behind the equations are sited in [38]. The set of differential Eqs. (1)–(16) are
solved twice: first the set of equations is solved for the central
cell with initial conditions shown in the left column of Table 9
and second for the peripheral cell with initial conditions shown
in the right column in Table 9. The model equations for central,
peripheral and transitional cell configurations are integrated until a stable limit cycle is obtained. The minimum diastolic potential is determined on the stable limit cycle and the vector
of dynamic variables at this point is recorded and used as the
initial condition in the subsequent simulations. For each of the
configurations examined, a single 0.5 ms depolarizing or hyperpolarizing current stimulus of a given amplitude is applied at
several time steps during the whole period of integration. The
starting reference point is selected at the membrane potential
level of (−10 mV) during the depolarization phase of the action
potential and at each subsequent run the stimulus timing increases by 1 ms. The stimulation current amplitude varies from
0 to 7.5 nA. For a given stimulus amplitude, the corresponding
PTC are obtained [22], for the whole range of stimulus amplitudes, three-dimensional PTC are also obtained [4].
2. Methods
3. Results
We assume that we apply a short pulse at the SAN and we
obtain the PTC using the model proposed by Zhang et al. [38].
The spatial effects are ignored and the problem is reduced to
a time dependent problem which is described by the set of
ordinary differential equations.
2.1. Model equations
The electrical activity of central and peripheral SAN cells is
simulated using the model proposed by Zhang et al. [38] which
was developed using experimental data. The model includes
formulations for the ionic currents (INa , ICa,L , ICa,T , Ito ),
sensitive sustained current (Isus , IK,r , IK,s , If ) and background currents (Ib,Na , Ib,Ca , Ib,K , Ip , INaCa ). Table 1 shows
the parameters entering the model. The total ionic current
Itot appearing in the differential equation which models
the membrane potential V is comprised of the currents:
INa , ICa,L , Ia,T , Ito , Isus , IK,r , IK,s , If , Ib,Na , Ib,Ca , Ib,K , INaCa
and Ip . The total membrane potential is modelled as
dV
−1
=
Itot ,
dt
C
(1)
The normal electrical activity of central, peripheral, and transitional SAN cells is shown in Fig. 1. The membrane potential V of central (dark line—in Fig. 1(a)) and peripheral (dark
line—in Fig. 1(c)) SAN cells is plotted vs time. The gray lines
indicate the electrical activity of the transitional cells. The duration of the simulation is 1 s. The transition between central
and peripheral activity is modelled using a scaling factor (ranging from 0 to 1) which depends on the distance of the cell from
the center of the SAN (see Eqs. (80) and (81) in Ref. [38] and
Ref. [39]). The scaling factor is used in order to calculate the
capacitance and the ionic conductances of the transitional cells
given their corresponding values at the center and the periphery
of the SAN.
For the transitional cell activity shown in Fig. 1 the scaling
factor values (0.04 in Fig. 1(a) and 0.13 in plot Fig. 1(c)) were
selected to indicate the point where a sharp change in membrane potential configuration is evident. Thus, in Fig. 1(c) the
transitional cell appears to have the characteristic spike-anddome configuration of peripheral cells.
The total ionic current Itot is plotted vs membrane potential
V for each of the corresponding simulations Fig. 1(b) and (d).
25
0.05
0
0.00
-25
-0.05
-50
-0.10
-75
-0.15
0.0
0.2
0.4
(a)
0.6
0.8
1.0
t
-75
(b)
-50
-25
V
0
25
30
0.35
0
0.00
-30
-0.35
-60
-0.70
Itot
V
Itot
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
V
10
-1.05
-90
0.0
0.2
0.4
0.6
0.8
1.0
t
(c)
-90
-60
(d)
-30
0
30
V
Fig. 1. Normal electrical activity in central (a,b), peripheral (c,d) and transitional sinoatrial node cells (gray lines in a–d). In figures (b) and (d) the total ionic
current Itot is plotted vs the membrane potential V shown in figures (a) and (c). Figures (a) and (c) indicate the variations of membrane potential V with time t.
0.05
40
T0
T
0.00
V
Itot
0
-0.05
-40
-0.10
-80
0.0
(a)
0.4
0.8
t
1.2
1.6
-75
(b)
-50
-25
V
0
25
Fig. 2. (a) Single pulse perturbation protocol for the central sinoatrial node cell. A depolarizing stimulus of 0.45 nA is applied with a time delay after the
detection (at V = −10 mV during the depolarization phase) of the second action potential. The gray trace corresponds to the unperturbed activity. (b) The total
ionic current Itot vs the membrane potential V .
The difference in Itot amplitude between central and peripheral
cells is mainly due to the contribution of INa which is not
present in central cells. The Itot –V plots provide a means to
visualize the effects of external perturbation on the SAN cell
activity (see Fig. 2) and can be considered as a representation
of the complete multi-dimensional phase space of the model.
The single pulse perturbation protocol used is shown in
Fig. 2. The gray solid line represents the normal unperturbed
electrical activity of central SAN cell with a period T0 =334 ms.
The onset of the action potential is marked at V = −10 mV (depolarization phase). At a time interval after the onset of the
action potential a short (0.5 ms) depolarizing current pulse is
applied resulting in early depolarization of the cell membrane
(dark line). The phase of the stimulus is /T0 while the new
phase of the perturbed oscillator is = + 1 − T /T0 where
T is the period of the perturbed cycle [22]. The electrical activity resumes immediately after the first perturbed cycle as is
evident from the Itot –V plot (Fig. 2(b)) in which both simulations are shown (gray line corresponds to the unperturbed limit
cycle oscillation of the central SAN cell and the black line corresponds to the limit cycle oscillation of the SAN).
The individual ionic currents comprising Itot in both the central and peripheral SAN cell models are shown in Fig. 3(a),
(b). Current traces of 1 s of spontaneous activity are plotted.
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
1
Itot
Itot
0.04
-5
0.0
ICa.L
ICa.L
-0.10
0.02
-0.10
-1.4
0.00
ICa.T
-0.004
-0.08
0.001
0.012
IK.s
IK.s
ICa.T
0.000
-0.002
0.004
0.004
INaCa
INaCa
0.000
-0.002
-0.010
INaK
0.026
0.005
0.012
0.00
0.00
Ib
Ib
INaCa
0.008
-0.04
0.001
0.03
If
If
-0.01
-0.002
0.0
(a)
11
0.2
0.4
0.6
0.8
-0.05
0.0
1.0
t
(b)
0.2
0.4
0.6
0.8
1.0
t
Fig. 3. (a) Ionic currents for central sinoatrial node cell model; (b) Ionic currents for peripheral sinoatrial node cell model.
A comparison of the two plots reveals the contribution of each
ionic current in Itot . The sodium current INa is the principal
component of Itot in peripheral while it is absent in central
SAN cells.
In the central cell model, the maximum upstroke velocity of
the action potential is 6 V s−1 , the minimum diastolic potential
is −68 mV and the maximum systolic potential is 22 mV, while
in the peripheral cell model the maximum upstroke velocity of
the action potential is 86 V s−1 , the minimum diastolic potential
is −81 mV and the maximum systolic potential is 25 mV.
For each one of the experiments shown in Fig. 1(a) and
(c), we repeat those with from 0 to T0 (correspondingly
0 < < 1) and current stimulus amplitudes from 0 to 7.5 nA.
The results are shown in Fig. 4 where representative PTC are
depicted. Fig. 4(a)–(d) correspond to low amplitude stimuli
(−0.3 nA) while Fig. 4(e)–(g),(j) traces correspond to high amplitude stimuli. Fig. 4(a)–(d), (j) correspond to central and peripheral cells while the Fig. 4(b), (c), (f), (g) correspond to the
transitional cell configurations shown in Fig. 1 (gray lines in
Fig. 1(a) and (c), respectively).
The advance or delay in the occurrence of the first action
potential after perturbation for a set of experiments for which
the stimulus phase varies from 0 to 1 can be visualized by
plotting the ratio of the corresponding period T1 (during which
the stimulus was applied) to the period T0 of the unperturbed
cycle over the stimulus phase . Such plots were obtained for
the central and peripheral SAN (Fig. 5).
Low amplitude stimulation shown in Fig. 5(a), (b), in both
the central and peripheral SAN cells, produces only minimal
advances (T1 /T0 < 1) or delays (T1 /T0 > 1) for most of the cycle excluding a small region around = 0.4 where a significant
advance is observed. At this region the membrane is in the absolute refractory period and the stimulus amplitude is sufficient
to initiate an action potential and thus to shorten T1 . Increasing
the stimulation amplitude extends the region where an advance is observed. In central cells the delay in the occurrence
of the action potential at intermediate stimulus levels observed
close to = 0.6 is associated with the approximation of a singularity in the phase space. Around this singularity T1 becomes
close to 2 × T0 (Fig. 5(c)) while for 0.4 < < 0.6 approaching T1 /T0 = 2, there are missing points which correspond to
T1 /T0 → ∞.
The PTC shown in Fig. 4 indicate that, in central, peripheral, and transitional SAN cells, both Type 0 (Fig. 4(a)–(d)) and
Type 1 (Fig. 4(e)–(j)) phase resetting is obtained. Low amplitude stimuli produce continuous PTC since the effect of the
external perturbation on the phase of the limit cycle oscillator
is less pronounced. The discontinuities of the PTC for stronger
12
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
φ'
1
φ'
1
0
0
0
(a)
1
φ
0
(b)
1
1
φ'
φ'
1
0
0
1
0
φ
(c)
1
0
φ
(d)
1
φ'
1
φ'
0
0
0
1
φ
(e)
0
1
φ
(f)
1
φ'
φ'
1
0
(g)
φ
0
φ
0
1
(h)
0
φ
1
Fig. 4. Phase transition curves for central ((a), (e)), peripheral ((d), (j)), and transitional cells ((b), (c), (f), (g)).
stimuli (Type 1 phase resetting) are indicative of the existence
of a singularity in the limit cycle.
Each of the PTC in Fig. 4 represents a set of experiments like
the one shown in Fig. 2 where the stimulus amplitude is constant while the interval varies. A set of such curves for various stimulus amplitudes produces a three-dimensional phase
transition surface [4] relating both and stimulus amplitude
to the new phase of the perturbed oscillator. Such three-
dimensional phase transition plots allow further investigation
of the phase resetting behavior of an oscillator. Moreover, such
PTC can help in finding the exact stimulus phase and amplitude
combination which brings the oscillator close to the singularity
range.
The three-dimensional phase transition plots for the central, transitional and peripheral SAN cells are shown in
Fig. 6 as contour plots with the new phase is color-coded
T1
T0
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
2.0
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.2
0.4
T1
T0
0.6
0.8
1.0
(a)
0.0
0.2
0.4
2.0
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.6
0.8
1.0
(b)
0.2
0.2
0.0
0.2
0.4
0.6
0.8
1.0
(c)
T1
T0
13
0.0
0.2
0.4
(d)
2.0
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.2
0.4
0.6
0.8
1.0
(e)
0.0
0.2
0.4
(f)
2.0
1.8
1.6
1.4
T1
T0
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
(g)
0.4
0.6
0.8
1.0
Fig. 5. Alteration of the timing of the first action potential after stimulation is plotted T1 /T0 vs for various stimulation amplitudes. (a) Peripheral cell at
−0.5 nA, (c) peripheral cell at −2.5 nA, (e) peripheral cell at −5 nA, (g) peripheral cell at −6.5 nA and (b) central cell at −0.2 nA, (d) central cell at −0.4 nA,
(f) central cell at −0.6 nA (T0,central = 320 ms, T0,peripheral = 160 ms).
on a gray scale from 0 to 1. The ordering of the plots is similar to that of Fig. 4 and the corresponding PTC of Fig. 4 can
be considered as horizontal “slices” of the data at the corresponding current stimulus amplitude (Istim ). The difference
in the stimulus amplitude for the corresponding plots is due
to the difference in the amplitude of the corresponding total
membrane current (a stronger stimulus is required to produce
Type 1 phase resetting behavior of Fig. 1).
In the three-dimensional phase transition plots shown in
Fig. 6, there exist regions where the new phase is very dense
14
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
Central
Peripheral
0.0
0.1
-0.3
-1.5
0.2
0.3
-0.6
-3.0
Istim
0.4
0.5
-0.9
-4.5
-1.2
-6.0
-1.5
-7.5
0.6
0.7
0.8
0.9
0.0
0.2
0.4
0.6
0.8
1.0
(a)
0.0
0.2
0.4
0.6
0.8
1.0
1.0
(b)
Transitional
Transitional
0.0
0.1
-0.4
-1
0.2
-0.8
0.3
Istim
-2
0.4
-1.2
0.5
0.6
-3
-1.6
0.7
-2.0
0.8
-4
0.9
-2.4
1.0
-5
0.0
(c)
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(d)
Fig. 6. Three-dimensional contour plots of the new phase vs old phase and amplitude: for (a) central, (b) peripheral, and (c), (d) transitional SAN cell.
60
V
30
0
-30
-60
0.0
0.5
1.0
t
1.5
2.0
Fig. 7. A critical stimulus of −0.4 nA applied at t = 0.53 s resulting in permanent annihilation of sinoatrial node repetitive activity.
(e.g. for = 0.55 and Istim = −0.4 nA). Such regions indicate
the existence of a singularity and should be further investigated
by fine-tuning of both interval and Istim amplitude.
Further experimentation with finer values for and Istim
within those regions revealed the existence of a stable singularity for the central SAN cell. In the simulation shown in Fig. 7 a
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
critical stimulus of −0.4 nA is applied at t = 0.53 s resulting in
permanent annihilation of the normal repetitive activity. Transitional and the peripheral cells do not exhibit similar behavior.
The application of stimulus of appropriate amplitude and
phase (e.g. = 0.4 and Istim = −5.4) results in a series of low
amplitude oscillations of the membrane potential with normal
limit cycle behavior resuming after 2–3 cycles.
4. Discussion
Investigation of the dynamic behavior of biological oscillators and their response to external perturbation is of great
importance in biological research since biological oscillations
are involved in many vital processes in living systems [32,33].
Understanding the dynamic response of the SAN to external
perturbations is important in elucidating its behavior under normal and pathological conditions [33]. Studies in the past have
concentrated on the elucidation of the phase resetting dynamics of the SAN using biophysical ionic models [15,29,5] but
did not address the issue of regional differences between central and peripheral node cells (the corresponding ionic models
described central cell behavior).
The SAN is a complex structure and regional differences in
the electrical activity of regional cells seem to be important in
the behavior of the natural pacemaker of the heart [39,40,30].
The Zhang et al. model [38] used in this study accounts for
regional differences and can be used to model the electrical
activity of central, peripheral, and transitional cells.
In this work, the analysis of phase resetting characteristics
of each type of SAN cell by means of three-dimensional phase
transition plots, revealed important differences in their response
to external electrical stimuli (see Fig. 6). Although all types of
cells display both Type 0 and Type 1 phase resetting behavior,
the stimulus amplitude at which Type 1 phase resetting behavior
first appears varies. This is a result of the regional differences
in Itot amplitude (see Fig. 1): Itot is higher in the periphery of
SAN (mainly due to the dominance of INa ) and thus a higher
amplitude of external stimulus is required in order to significantly affect the normal limit cycle behavior. We have produced
Itot –V plots to provide a means to visualize the effects of external perturbation on the SAN cell electrical activity (Fig. 2).
Our simulations show the existence of a critical stimulus
amplitude-phase combination which is capable of annihilating
the electrical activity of the central SAN cells (Fig. 7). So far,
there is no reference in the literature describing and proofing
the existence of such a combination in Zhang model. It is believed that such critical stimuli are involved in the generation
of certain cardiac arrhythmias [33] and gives new insights into
the investigation of cardiac models.
In a complex structure like the SAN, regional differences
and spatiotemporal interactions could play an important role in
its pathophysiological response to external perturbations and
should be further investigated. This must be included in our next
set of experiments targeting one-dimensional cells. However,
the use of the finite element method might help in the analysis of
two- and three-dimensional structures. The comparison of phase
resetting characteristics of other models will be included too.
15
Investigation of the effects of increases of various concentration
on annihilation of SAN activity must be further studied.
5. Summary
In the present work, we use the model proposed from Zhang
et al. [38] and we examine the dynamic response of the SAN to
external stimulus. We apply a short current to reset the spontaneous rhythmic activity of the single SAN. This application of
the external stimulus results in the prolongation of the normal
activity and a new phase is obtained. This resetting behavior is
quantified in terms of phase transition curves (PTCs) for short
electrical current pulses of varying amplitude which span the
whole period. Discontinuities, as we show in the figures, reveal critical stimulus. The application of this critical stimulus
can result in annihilation of activity on SAN cells. Detailed
analysis of the ionic mechanisms which involved in the resetting behavior could be useful and provide new insight into the
physiology of the excitation in SAN cell. Our work, is a step
forward to a future work on higher dimensionality model were
we believe the resetting technic can be more significant.
Acknowledgements
This work is partially funded by the Greek Secretariat for Research and Technology PENED-01E511 to D.G. Tsalikakis.
Appendix
The parameters used in the model are given in Table 1. The
set of differential equations (1)–(16) given in Tables 2–9 was
solved using a fourth order Runge–Kutta method with time step
0.1 ms.
Table 1
Parameters of the model
4-AP
AM
APD
CL
Cm
a (x), C s (x)
Cm
m
dL , dT
dNa,Ca
dV /dtmax
Da , Ds
EK,s
ENa , ECa , EK
ECa,L , ECa,T
F
FK,r
FNa
4-Aminopyridine
Atrial muscle
Action potential duration
Spontaneous cycle length
Cell capacitance
Capacitance of atrial muscle cell or SA node
cell in one-dimensional model of intact
SA node at distance x from center of SA node
Activation variables for ICa,L and ICa,T
Denominator constant for INa,Ca
Maximum upstroke velocity of action potential
Diffusion coefficient between atrial muscle cells
or SA node cells in one-dimensional
model of the intact SA node
Reversal potential for IK,s
Equilibrium potentials for Na+ , Ca2+ , and K+
Reversal potentials for ICa,L and ICa,T
Faraday’s constant
Fraction of activation of IK,r that occurs slowly
Fraction of inactivation of INa that occurs slowly
16
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
Table 1 (continued)
Table 1 (continued)
fL , fT
gp , gc
g a (x), g s (x)
gNa
gCa,L , gCa,T
gto , gsus
gK,r , gK,s
gf,Na , gf,K
gb,Na , gb,Ca , gb,K
h1 , h2
h
INa
ICa,L , iCa,T
Ito , Isus
IK,r , IK,s
IK
If
If,Na , If,K
Ib,Na , Ib,Ca , Ib,K
INaCa
Ip
Ip
Itot
a (x), I s (x)
Itot
tot
Ist
IK,ACh
IK,AP
kNa,Ca
Km,Na , Km,K
L
Ls
m
MDP
n∞
pa
pa,f , pa,s
pi
Q10
r
R
q
SA node, SAN
t
T
TOP
V
V a, V s
V a (x), V s (x)
Inactivation variables for ICa,L and ICa,T
Conductance of a current in peripheral or central
SA node cell models
Conductance of a current in atrial muscle cell
or SA node cell in one-dimensional model
of intact SA node at distance x from center of SA
node
Conductance of INa
Conductance of ICa,L and gCa,T
Conductance of Ito and Isus
Conductance of IK,r and IK,s
Conductance of Na+ and K+
components of if
Conductance of Ib,Na , Ib,Ca and Ib,K
Fast and slow inactivation variables for INa
Net fractional availability of INa
TTX-sensitive Na+ current
L- and T-type Ca2+ currents
Transient and sustained components of 4-APsensitive current
Rapid and slow delayed rectifying K+ currents
Sum of IK,r and IK,s
Hyperpolarization-activated current
Na+ and K+ components of If
Background Na+ , Ca2+ , and K+ currents
Na+ /Ca2+ exchanger current
Na+ –K+ pump current
Maximum Ip
Total ionic current in a cell
Total ionic current in atrial
muscle cell or SA node cell in one-dimensional
model of intact SA node at distance x
from center of SA node
Sustained current
ACh-activated K+ current
ATP-sensitive K+ current
Scaling factor for INa,Ca
Dissociation constants for Na+ and K+ activation
of Ip
Length of string of SA node and atrial
tissue in one-dimensional
model of intact SA node
Length of string of SA node tissue
in one-dimensional model of intact SA node
Activation variable for iNa
Maximum diastolic potential
Steady-state value of n
General activation variable for IK,r
Fast and slow activation variables forIK,r
Inactivation variable for IK,r
Fractional change in a variable
with a 10 ◦ C increase in temperature
Activation variable for Ito
Universal gas constant
Inactivation variable for Ito
Sinoatrial node
Time
Absolute temperature
Takeoff potential
Membrane potential
Membrane potential of atrial muscle
cell or SA node cell in one-dimensional model of
intact SA node
Membrane potential of atrial muscle
cell or SA node cell in one-dimensional model of
intact SA node at distance x from center of SA node
x
xs
y
z
[Na+ ]i , [Ca2+ ]i
an
bn
NaCa
n
Distance from center of SA node
in one-dimensional model of intact SA node
Activation variable for IK,s
Activation variable of If
Valency of ion
Intracellular Na+ , Ca2+ , concentrations
Voltage-dependent opening rate constant of n
Voltage-dependent closing rate constant of n
Position of Erying rate theory energy
barrier controlling voltage dependence of
INaCa
Time constant of n
Space constant
Table 2
Formulation of the equations of Na+ current (INa )
The sodium current INa is given by the equation
INa = gNa (m)3 h[Na+ ]o
(F )2.0 (e(V −ENa )F /RT − 1.0)
V
RT
(eVF/RT − 1.0)
Sodiumcurrent m gate
dm
(m∞ − m)
=
dt
m
m∞ =
1.0
(1.0 + e−V /5.46 )
(2)
1.0/3.0
m =
0.0006247
(0.832e−0.335(V +56.7) + 0.627e0.082(V +65.01) )
+0.00004)
Sodium current h gate
h = ((1.0 − FNa )h1 + FNa h2 )
(h1,∞ − h1 )
dh1
=
dt
h1
(3)
(h2,∞ − h2 )
dh2
=
dt
h 2
(4)
h1,∞ =
1.0
(1.0 + e(V +66.1)/6.4 )
h2,∞ = h1,∞
0.000003171e−0.2815(V +17.11)
h1 =
+ 0.0005977
(1.0 + 0.003732e−0.3426(V +37.76) )
h2 =
0.00000003186e−0.6219(V +18.8)
+ 0.003556
(1.0 + 0.00007189e−0.6683(V +34.07) )
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
Table 3
Formulation of the equations of L-type Ca2+ current, ICa,L
Table 4 (continued)
L-type Ca channel ICa,L
0.006
ICa,L = gCa,L fL dL +
(V − ECa,L )
−(V
+14.1)/6.0
(1.0 + e
)
d,T =
17
1.0
(d,T + d,T )
dT,∞ =
1.0
(1.0 + e−(V +37.0)/6.8 )
L-type Ca channel d gate
(dL,∞ − dL )
ddL
=
dt
d,L
d,L = −14.19
(V + 35.0)
42.45V
− −0.208V
(e
− 1.0)
(e−(V +35.0)/2.5 − 1.0)
(5)
(fT,∞ − fT )
dfT
=
dt
f,T
(8)
f,T = 15.3e−(V +71.7)/83.3
5.71(V − 5.0)
d,L = 0.4(V −5.0)
(e
− 1.0)
f,T = 15.0e(V +71.7)/15.38
1.0
d,L =
(d,L + d,L )
dL,∞ =
T-type Ca channel f gate
1.0
(f,T + f,T )
f,T =
1.0
(1.0 + e−(V +23.1)/6.0 )
fT,∞ =
1.0
(1.0 + e(V +71.0)/9.0 )
L-type Ca channel f gate
dfL
(fL,∞ − fL )
=
dt
f,L
f,L =
(6)
Table 5
Formulation of the equations for sensitive currents (Ito and Isus ) and delayed
rectifying current (IK,r )
3.12(V + 28.0)
(e(V +28.0)/4.0 − 1.0)
Ito
f,L =
f,L =
25.0
Ito = gto qr(V − EK )
(1.0 + e−(V +28.0)/4.0 )
Isus
Isus = gsus r(V − EK )
1.0
(f,L + f,L )
fL,∞ =
q gate
(q∞ − q)
dq
=
dt
q
1.0
(1.0 + e(V +45.0)/5.0 )
q∞ =
(9)
1.0
(1.0 + e(V +59.37)/13.1 )
0.06517
+ 0.000024e0.1(V +50.93)
0.57e−0.08(V +49.0)
Table 4
Formulation of the equations of T-type Ca2+ current (ICa,T )
q = 0.0101 +
T-type Ca channel ICa,T
4-AP sensitive currents r gate
ICa,T = gCa,T dT fT (V − ECa,T )
(r∞ − r)
dr
=
dt
r
T-type Ca channel d gate
(dT,∞ − dT )
ddT
=
dt
d,T
d,T = 1068.0e(V +26.3)/30.0
r∞ =
(7)
(10)
1.0
(1.0 + e−(V −10.93)/19.7 )
r = 0.00298 +
d,T = 1068.0e−(V +26.3)/30.0
References
0.01559
+ 0.369e−0.12(V +23.84) )
(1.037e0.09(V +30.61)
18
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
Table 6 (continued)
Table 5 (continued)
Rapid delayed rectifying potassium current IK,r
14.0
(1.0 + e−(V −40.0)/9.0 )
x s =
IK,r = gK,r pa pi (V − EK )
pa = ((1.0 − FK,r )pa,f + FK,r pa,s )
xs = e−V /45.0
Rapid delayed rectifying potassium current pa,f gate
(pa,f,∞ − pa,f )
dpa,f
=
dt
pa,f
pa,f,∞ =
pa,f =
(11)
xs∞ =
1.0
(1.0 + e−(V +14.2)/10.6 )
1.0
(xs + xs )
xs =
1.0
(37.2e(V −9.0)/15.9 + 0.96e−(V −9.0)/22.5 )
xs
(xs + xs )
Hyperpolarization activated current If
If = (If,Na + If,K )
Rapid delayed rectifying potassium current pa,s gate
If,Na
(pa,s,∞ − pa,s )
dpa,s
=
dt
pa,s
(12)
pa,s,∞ = pa,f,∞
(13)
pa,s =
1.0
(4.2e(V −9.0)/17.0 + 0.15e−(V −9.0)/21.6 )
If,Na = gf,Na y(V − ENa )
If,K
If,K = gf,K y(V − EK )
Hyperpolarization activated current y gate
(y∞ − y)
dy
=
dt
y
Rapid delayed rectifying potassium pi gate
dpi
(pi,∞ − pi )
=
dt
p i
pi,∞ =
(14)
1.0
(1.0 + e(V +18.6)/10.1 )
y = e−(V +78.91)/26.62
y = e(V +75.13)/21.25
y∞ =
Table 6
Formulation of the equations for slow delayed rectifying current (IK,s ) and
hyperpolarization current (If )
Slow delayed rectifying potassium current IK,s
IK,s = gK,s (xs )2.0 (V − EK,s )
Slowdelayed rectifying potassium current xs gate
dxs
(xs∞ − xs )
=
dt
xs
(15)
y =
y
(y + y )
1.0
(y + y )
(16)
D.G. Tsalikakis et al. / Computers in Biology and Medicine 37 (2007) 8 – 20
Table 7
Formulation of the equations for background, pump and exchanger currents
Sodium background current Ib,Na
Ib,Na = gb,Na (V − ENa )
Potassium background current Ib,K
Ib,K = gb,K (V − EK )
Calcium background current Ib,Ca
Ib,Ca = gb,Ca (V − ECa )
Sodium calcium exchanger INaCa
INaCa = KNaCa
(([Na+ ]i )3.0 [Ca2+ ]o e0.03743V NaCa −([Na+ ]o )3.0 [Ca2+ ]i e0.0374V (NaCa −1.0) )
(1.0+dNaCa ([Ca2+ ]i ([Na+ ]o )3.0 +[Ca2+ ]o ([Na+ ]i )3.0 ))
Sodium potassium pump Ip
2.0
3.0 [Na+ ]i
[K+ ]o
Ip =Ip,max
(Km,K + [K+ ]o )
(Km,Na + [Na+ ]i )
1.6
×
(1.5 + e−(V +60.0)/40.0 )
Table 8
Formulation of the equations for reversal and equilibrium potentials
Reversal and equilibrium potentials
RT [Na+ ]o
ln
F
[Na+ ]i
ENa =
EK =
RT [K+ ]o
ln +
F
[K ]i
ECa =
[Ca2+ ]o
RT
ln
2.0F
[Ca2+ ]i
EK,s =
RT ([K+ ]o + 0.12[Na+ ]o )
ln
F
([K+ ]i + 0.12[Na+ ]i )
Table 9
Initial conditions for central and peripheral cell
Variable
V
m
h1
h2
dL
fL
dT
fT
y
r
q
xs
pa,f
pa,s
pii
Value
Central
Peripheral
−55.4434
0.158763
0.0365616
0.0515223
0.00328396
0.853578
0.0622561
0.129637
0.0230307
0.0334583
0.288777
0.137793
0.29045
0.561665
0.973981
−78.4021
0.0531058
0.341077
0.0207796
0.000109311
0.97761
0.00226369
0.278358
0.0215392
0.0110545
0.435506
0.0950833
0.313397
0.3783
0.991546
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Dimitrios G. Tsalikakis was born in Thessaloniki, Greece, in 1977. He
received the Diploma degree in Mathematics from the University of Ioannina
and he is a PhD candidate in the Department of Cardiology since 2002.
His research interests include cardiac modeling, monophasic action potential
analysis, automated systems analysis, virtual instrumentation, and scientific
computing. He is member of the Unit of Medical Technology and Intelligent
Information Systems.
Henggui Zhang, born in 1964 in China, educated in Physics (BSc, 1985),
Computer Science (MSc, 1985–1988), Non-linear Science (PhD, 1988–1990),
and Biomedical Science (PhD, 1991–1994). Since 1991, he has been working
as a research assistant (1991–1994, Leeds, funding from MRC), postdoctoral
research fellow (1994–1995, JHU, funding from NSF, USA; 1995–1996,
Leeds; funding from Welcome Trust; 1996–2000, Leeds; funding from the
British Heart Foundation), and then a senior research fellow (2000–2001,
Leeds, funding from British Heart Foundation). In October 2001, he moved
to UMIST to take up a lectureship. Currently he is a senior lecturer in
Biological Physics Group, School of Physics and Astronomy, the University
of Manchester.
Dimitrios I. Fotiadis was born in Ioannina, Greece, in 1961. He received
the Diploma degree in Chemical Engineering from the National Technical
University of Athens and the PhD degree in Chemical Engineering from the
University of Minnesota, USA. Since 1995, he has been with the Department
of Computer Science, University of Ioannina, Greece, where he is currently
an Associate Professor. He is the director of the Unit of Medical Technology
and Intelligent Information Systems. His research interests include biomedical
technology, biomechanics, scientific computing and intelligent information
systems.
Georgios P. Kremmydas, born in 1972 in Argostoli, Kefalonia Island, Greece.
He received his BSc (Hon) Degree in Physiology in 1994, University of Leeds,
UK and a PhD in 2000, Department of Biomedical Sciences, University
of Leeds, UK. He is currently the Director of Research Technology and
Development Department of L.U.M.C. of Kefalonia and Ithaki and a visiting
researcher of the Computer Science Department, University of Ioannina,
Greece.
Lampros K. Michalis was born in Arta, Greece, in 1960. He received the
MD degree with distinction from the Medical School, University of Athens,
Greece, in 1984. Since 1995, he has been with the Medical School, University
of Ioannina, Greece, where he is currently a Professor in Cardiology. He is in
charge of the coronary care unit and the catheter laboratory of the University
Hospital of Medical School, Ioannina, Greece. His research interests focus
on bioengineering, interventional cardiology, and electrophysiology.