2016. Ahmed Modified Van der Pol (Vdp) Oscillator based Cardiac Pacemakers

19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
Modified Van der Pol (Vdp) Oscillator based
Cardiac Pacemakers
Syed Hassaan Ahmed,Sikandar Javed ,Syed KashifAbbas,Sajjad Hussain
Department of Electrical Engineering
National University of Sciences and Technology (NUST), Islamabad, Pakistan
[email protected] , [email protected],edu.pk, [email protected]
1
model ECG waveform and the action potentials, generated
in different nodes of human heart [2]. Various arguments
are reported in the literature, for example by Babloyantz
and Destexhe [3], about the fact whether the heart perfectly
behaves like a periodic oscillator or not. It is arguable that
in an actually recorded ECG signal, there might be slight
morphological difference between the consecutive
heartbeats (due to noise or other external stochastic
disturbances like artefacts etc.) which produces chaos-like
random wandering of the states in the phase space [3]. But
there is no doubt that human physiological systems exhibit
some kind of complex periodic waveforms like other
different cardiovascular signals, e.g. respiration, ECG, heart
rate variability (HRV), blood pressure, blood flow etc. [4],
[5], [6], which motivates modeling of such systems using
equivalent coupled oscillators. Representing each of the
physiological oscillations with the corresponding
characteristic frequency, a coupled oscillator model is
developed in Stefanovska et al. [5], [6] for cardiovascular
system in human body. Three ordinary differential equation
based model was proposed by Zbilut et al. [7] to represent
variation in the inter-beat length, although the obtained
signal is not similar to the P-QRS-T complex of an ECG
signal.Balthazar van der Pol’s introduced natural
philosophy in the research lab during the 1920's and
1930's[8-10].The Vdp’s oscillator has been taken into
consideration by different analysts [11-13].Its result has
been computed to high-order[14]and in literature,
synchronization tests have been studied for coupled
oscillators[15-16]. Coupled Vdp's oscillators have been
considered to show numerous complex organic frameworks
[17-19].
The action potentials generated from sino-atrial (SA) and
atrio-ventricular (AV) nodes have been modeled using
mixed VdP/Duffing type relaxation oscillators by
Grudzinski and Zebrowski[20].
For cardiac pacemaker modeling we use Van der Pol
oscillators (Vdp) because the Vdposcillator, given by Eq.(1)
adapts its intrinsic frequency to the frequency of its external
driving signal without changing of its amplitude[21].
Abstract—The paper involves the study of nonlinear
dynamics oscillator and numerical methodology to analyze
and resolute the nonlinear dynamics model. Based on either
the classical Vander Pol oscillator or other nonlinear
oscillators, these models were interesting rather because of the
physical phenomena that could be obtained (chaos and
synchronization).Here, we can simulate many important
physiological features of true physiological action potentials in
practical systems by adjusting the parameter. We also show
different ways to change pacemaker actions. As van der pol
oscillator can model heart beat phenomenon so here these
oscillator models are modified so that they can match well
with the results obtained by actual pacemakers. We model
electrical activity of cardiac electric system including atrial
and ventricular muscles solving a set of coupled nonlinear
oscillator equations. A new mathematical model for the
electrical activity of the heart is proposed. In this paper a
modified Vander Pol oscillator model was designed in order to
reproduce the time series of the action potential generated by
a natural pacemaker of the heart (i.e., the SA or the AV node).
The model represents a special singularly perturbed threedimensional system of ordinary differential equations with one
fast and two slow variables.
Keywords—heartbeat, cardiac pacemaker, van der pol
oscillator
I.
INTRODUCTION
Mathematical modeling of biological signals is quite
challenging and is an emerging field of research. This
provides better understanding of the underlying physical
phenomena which results in different physiological signals
in human body like Electrocardiogram (ECG),
Electroencephalogram (EEG), Electromiogram (EMG) etc.
[1]. These physiological signals are widely used, over the
years by the clinicians to diagnose irregular behaviour of
human organs. Therefore, mathematical modeling for the
generation of such signals from a system theoretic point of
view may lead us to the root of the complicated
physiological processes, responsible for their generation, to
detect healthy and unhealthy behaviour of human organs.
The fundamental principle behind generation of any
complex biological waveform is governed by few set of
nonlinear differential equations. Several dynamical system
theoretic approaches have been proposed to mathematically
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+∝
176
− 1) +
=0
(1)
ISBN 978-1-5090-4089-6
19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
A. Modification of the van der pol oscillator:
This model is based on modified Van der Pol oscillator
(mvP)[22].In this oscillator harmonic force is replaced by
cubic Duffing term:
)
)
− ) +
+ =0
(2)
+
The phase space of Eq. (2) resembles the phase space of
physiological oscillators[23]of the Morris-Lecar model of
the neuron membrane[24].
Eq. (2) shows that the distance between the node and the
saddle cannot be changed so for changing it we modify Eq.
(2) by introducing a new parameter e:
+
)
− ) +
II.
)
=0
(3)
NUMERICAL METHODOLOGY
A. Method of Multiple Scale:
Here method of multiple scales (MMS) is used for
analytical solution of Van der pol oscillators. Ordinary
differential equation changed into a system of partial
differential equations which allows generality to obtain an
perfect approximation with the help of Method of Multiple
scales.
Figure 1: Flow Chart diagram of modified Vdp oscillator
based cardiac pacemaker
Numerical simulations were done using Runge–Kutta
method for solving nonlinear equations on Matlab.Here
Van der pol oscillator equation are implemented in Matlab
and required results are obtain using Runge-Kutta
techniques using Matlab.
Figure 1 describes the flow chart diagram of modified Van
der pol oscillator based cardiac pacemaker. Figure 1 shows
that Van der pol oscillator is used and which is modified in
first part and classical Van der pol oscillator is used in
second part. Modified Van der pol oscillator is analyzed
using Runge Kutta method then its action potential and time
series analysis is generated and observed. Afterwards action
potential is generated and analyzed using classical van der
pol oscillator.
II. BLOCK DIAGRAM
III. RESULTS
B. Runge–Kutta method
A. By applying Modified Vdp (mVdp) oscillator model
Figures [2-4], show the phase portrait of the modified van
der pol oscillator (mVdp) as mention in Eq. (2). For high
values of µ, systems undergoes a global homoclinic
bifurcation which results in the trajectory ending in the
stable node [25].Figures [2-4] are very similar to blue sky
catastrophe[26],where trajectories goes to infinity(∞) while
here it is attracted to the node.
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19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
action potential become shorter when nodal and saddle are
moved away from each other’s. Trajectories tend to spend
more time around the saddle when node point approaches to
saddle point.
Phase portrait
3
mu=1.4
2
1
Time History of x
2
e=6
1.5
0
1
-1
0.5
0
-2
-0.5
-3
-7
-6
-5
-4
-3
-2
-1
0
1
-1
2
-1.5
Figure 2:Phase portrait of the mVdP oscillator given by Eq.
(2) atµ=1.4
-2
-2.5
100
110
120
130
140
150
160
170
180
190
200
Phase portrait
1.5
Figure 5:Time arrangement of the mVdPoscillator by Eq. (3)
at e=6
mu=0.2
1
0.5
Time History of x
2
e=4.5
0
1.5
1
-0.5
0.5
0
-1
-0.5
-1.5
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-1
0.8
-1.5
Figure 3: Phase portrait of the mVdP oscillator given by Eq.
(2) atµ=0.2
-2
-2.5
Phase portrait
-3
100
2.5
110
120
130
140
150
160
170
180
190
200
mu=1
Figure 6:Time arrangement of the mVdPoscillator by Eq. (3)
at e=4.5
2
1.5
1
Time History of x
2
0.5
e=3.5
1.5
0
1
-0.5
0.5
-1
0
-1.5
-0.5
-2
-2.5
-3
-1
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
Figure 4: Phase portrait of the mVdP oscillator given by Eq.
(2) at µ=1
-2
-2.5
Figures [5-7], show the time series effect of the modified
van der pol (mVdP) oscillator given by Eq. (3).Here
parameter e change the depolarization period. Figures [4-6]
demonstrates the reaction for different estimations of e
which demonstrates that parameter e specifically influences
the depolarization time of pacemaker. Intervals of the
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-3
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110
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140
150
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Figure 7:Time arrangement of the mVdP oscillator by Eq. (3)
at e=3.5
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ISBN 978-1-5090-4089-6
19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
As a further modification, we replace the damping term ∝
− ) by a term: ∝
− 1) − 2)
Time History of x
2
1.5
Our model now has the form:
+∝
− 1)
1
)
− 2) +
)
=0
0.5
(4)
0
v1and v2must have inverse signs (v1v2<0) for preservation
of self-oscillatory behavior of the system.
-0.5
-1
Figures [8-10], show the action potential effect generated
by Eq. (4).By varying the values of parameters v1 and v2,
we can lower and raise the values of the resting potential
(RT) which results in the decreasing and increasing of the
values of action potential (AC).
-1.5
-2
-2.5
100
110
120
130
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150
160
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190
200
Figure 10: Action potential generated by Eq. (4) at v1=0.95;
v2=-1.1 ,keeping e=6;d=3;α=3
Time History of x
2
v1=1;v2=-1
1.5
Figure 11 shows a comparison between real cardiac
pacemaker [27] and the response generated by our model.
Here it is cleared that our model is exactly same the original
model of cardiac pacemakers but here frequency of cardiac
pacemakers are too low as compared with original model
[25].
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
100
110
120
130
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150
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Figure 8: Action potential generated by Eq. (4) at v1=1; v2=-1
keeping e=6; d=3; α=3
Time History of x
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
100
110
120
130
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150
160
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190
200
Figure 9: Action potential generated by Eq. (4) atv1=1.05;
v2=-0.95 keeping e=6; d=3; α=3
Figure 11: Upper figure- action potentials obtained from
pacemaker [21]
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19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
Phase portrait
6
mu=1
Lower figure—the action potentials generated by Eq. (4) at
v1=0.83; V2=-0.83 keeping e=6; d=3; α=3
4
Figures 9 and 11 depict that how Van der pol oscillator
model waveforms match the actual waveforms generated in
the heart.
Figures [12-14], show the phase portrait generated by
mVdp oscillator model by Eq. (4).so here it is cleared that
modification will not affect the phase space of the system.
However figures [12-14] show the effect of changing µ on
the phase space of pacemaker.
2
0
-2
-4
-6
-2.5
Phase portrait
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
8
mu=1.4
Figure 14: Phase portrait of the system given by Eq. (4) atµ=1
6
4
B. By applying classical van der Pol oscillator model
2
-2
-6
-8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
) =
−
Ωt)
(5)
Here in Eq. (5), F is the amplitude of the forcing function,
Ω is its frequency, is the system frequency,µ and α are
the tuning damping coefficients.
Figures [15-16] show the response generated by applying
classical Van der pol oscillator model as mentioned in Eq.
(5) and it is evaluated that classical model reproduce the
same results obtained in figures [8-11].
-4
-10
-2.5
−
+
0
2.5
Figure 12:Phase portrait of the system given by Eq. (4)
atµ=1.4
Time History of x
1.5
Phase portrait
1
mu=0.2
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
160
-1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
170
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185
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200
Figure 15:Action potential generated by classical van der pol
oscillator, similar to figure 7
Figure 13: Phase portrait of the system given by Eq. (4) At
µ=0.2
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19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
response so for positive values of F, amplitude will increase
and for negative values of F, amplitude will increase but it
become inverted and for F=0 ,its amplitude is reduced as
shown in figures [17-20].
Time History of x
3
2
1
At F=0:
0
Time History of x
1.5
-1
1
-2
-3
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0
200
Figure 16:Action potential generated by classical van der pol
oscillator, similar to figure 8
-0.5
-1
III.
-1.5
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Figure 17: Action potential generated by Eq. (5) at F=0
It is evaluated from above discussion that better model of
cardiac pacemakers are obtained by slight variation in the
Van der Pol oscillators. Followings are the effect of
different parameters which affect the system response.
At F=1;
Time History of x
1.5
A. Effect of frequency on the system response
1
If we set Ω=0 then we get sinusoidal response from the
system. Modulation index of the system is changed when
we increase Ω.
Figure 15 and 16 show the waveforms for different values
of Ω.Ω must be 1.1µ.
0.5
0
-0.5
B. Effect of µ on the response
-1
This parameter affects the amplitude, frequency and shape
of the output. When µ=0, it will give pure sinusoidal
waveform. When value of µ is increased then amplitude is
increased but its output frequency will decrease as well as it
will de-shape the output. To achieve the similar results we
must set µaround 0.3 to 1.
-1.5
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Figure 18: Action potential generated by Eq. (5) at F=1
At F=-1;
C. Effect of α on the response
This parameter directly affects the amplitude of the system.
When α=0 then system become unstable and it can’t be
negative.When value of α is increased then amplitude of the
output is decreased and vice versa.
D. Effect of F on the response
Here F is the amplitude of the forcing function. This
parameter directly affects the amplitude of output of system
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ISBN 978-1-5090-4089-6
19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh
ACKNOWLEDGMENT
Time History of x
1.5
The author would like to thank Prof. Dr Imran Akhtar for
guidance and also thanks to Mr. Krzysztof Grudzinski and
Jan J. Zebrowski for interesting discussion.
1
0.5
BIBLIOGRAPHY
0
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Figure 19:Action potential generated by Eq. (5) at F=-1
At F=3;
Time History of x
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
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165
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Figure 20:Action potential generated by Eq. (5) at F=3
IV.
CONCLUSION
The global structure of the action potential (AC) is analyzed
by model of Grudzinski and Zebrowski.The effects of
changing the amplitude of external forcing on the
pacemaker are examined. In this paper it is cleared that Van
der Pol oscillator can be successfully used to model the
cardiac pacemaker. Here cardiac pacemakers are modeled
through self-excited Van der Pol oscillators and then by
forced Van der Pol oscillators.
The results obtained through modified Van der Pol
oscillator and classical Van der pol oscillators are
implemented and compared.
In future we perform the Fast Fourier Transform (FFT) on
both of the system to analyze the frequency response of the
system for it detailed analysis.
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