# A New Three-Oscillator Model for the Heart System in the Case of Time Delay and Designing Appropriate Controller for Its Synchronization

Authors: Siroos Nazari; Aghileh Heydari; Mahboobeh Tavakoli; Javad Khaligh
Journal: ISRN Applied Mathematics
 (2014)

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## Abstract

If theSA and AV oscillators are not synchronized, it may arise some kinds of blocking arrhythmias in the system of heart. In this paper, in order to examine the heart system more precisely, we apply the three-oscillator model of the heart system, and to prevent arrhythmias, perform the following steps. Firstly, we add a voltage with rang a1 and ω frequency to SA node. Then, we use delay time factor in oscillators and finally the appropriate control is designed. In this paper, we have explained how simulating and curing these arrhythmias are possible by designing a three-oscillator system for heart in the state of delay and without delay and by applying an appropriate control. In the end, we present the simulation results.


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## Body

## 1. Introduction

As we know, If theSA and AV oscillators are not synchronized, it may cause some kinds of blocking arrhythmias in the system of the heart. At present, there are devices called pacemaker using electric shocks to synchronize if there is an arrhythmia. [1–3]. The main defect of these devices is the error of diagnosis and not having accurate control. For example, they sometimes apply shocking of 600 to 700 voltages to the heart wrongly even when there is no blocking at all, which is too dangerous. Applied shock to the heart muscle constricts all fibers concurrently and consequently all fibers return to recovery state and hope to return to their normal rhythm after that and it may sometimes occur that the patient involves blocking and the device have not diagnosed it so the patient dies. Two popular types of arrhythmias blocking and ectopic are interesting for researchers [4]. In order to simulate ectopic arrhythmias, a voltage ranging a1 and ω frequency is added to SA node.


## 2. Three-Oscillator Modelling of the Heart without Time Delay

As we know, cardiac normal rhythm is firstly produced bySA node (normal pacemaker) and then causes stimulating of AV node. Nevertheless, it has been observed that these two oscillators are not so accurate for producing ECG signal. This is because, the signal of first oscillator is related to the activation of SA node and the right atrium and signal of second oscillator is only related to the depolarization of the left ventricular. Based on this hypothesis, it is possible to produce P curve, but QRS complex may not be produced, because this interval is mainly due to ventricular repolarization [5]. These observations and also the existence of blocking arrhythmias force us to put the third oscillator which represents the spread of the pulse through ventricles, which physiologically represents His-Purkinje complex. To create a general model, we assume that there is a mutual coupling asymmetry among all oscillators [6]. In addition, external stimulation is entered into the system, considering periodic stimulus sentence to each oscillator. This developed model can be shown by a system of differential equations as shown below, in which a Vander pol equation has been considered to model SA oscillator and another one to model the AV oscillator and the third one to model HP oscillator [7]:
(1)(SA){x˙1=x2x˙2=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωtx˙2=+R13(x1-x3)+R15(x1-x5),(AV){x˙3=x4x˙4=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωtx˙4=+R31(x3-x1)+R35(x3-x5),(HP){x˙5=x6x˙6=-d3(x52-1)x6-c3x5+a3cos⁡⁡ωtx˙6=+R51(x5-x1)+R53(x5-x3),
in which pairs (x1,x2), (x3,x4), and (x5,x6) show SA, AV, and HP oscillators, respectively.To choose appropriate parameters from the Vander-pol system which are close to the system of the heart, we use trying and error test [8]. Regarding this test, appropriate parameters have been mentioned in Table 1. By choosing parameters as shown in Table 1, any oscillator system is synchronized with another system that has different parameters. In the case that two systems are not synchronized, with the help of changing parameters and coupling coefficients, arrhythmias will occur [9, 10].Table 1
Appropriate parameters to synchronize three-oscillator system.
Parameter
Definition
Three-oscillators system
c
1
SA frequency
1
c
2
AV frequency
2
c
3
HP frequency
1
a
1
SA voltage range
5
a
2
AV voltage range
6
a
3
HP voltage range
4
d
1
SA damping Coefficient
5
d
2
AV damping Coefficient
6
d
3
HP damping Coefficient
7
ω
Frequency
2.001
≤
ω
<
2.05
R
i
j
Coupling coefficients betweenxi and xj
R
13
1
R
15
1
R
35
1
R
51
2
R
31
5
R
53
3
x
1
SA membrance flow
x
2
SA membrance voltage
x
3
AV membrance flow
x
4
AV membrance voltage
x
5
HP membrance flow
x
6
HP membrance voltageBy changingd1, d2, and d3 coefficients, we can produce different kinds of alternative responses. These coefficients influence the nonlinear sentence of the equation and cause stability of limit cycle in the phase plane (x1,x2). Having limit cycle with the heart behavior is adapted physiologically to the behavior of the heart [11].In the discussion of synchronization, synchronizing of the slow oscillators with the fast oscillators is done more easily. In the case of heart, slower pacemakers such asAV and HP, should be synchronized with faster pacemakers such as SA. If it is not synchronized by changing the coupling coefficients, we try to synchronize oscillators by applying appropriate input controller to slow oscillators as shown below
(2)(SA){x˙1=x2x˙2=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωtx˙2=+R13(x1-x3)+R15(x1-x5),(AV){x˙3=x4x˙4=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωtx˙4=+R31(x3-x1)+R35(x3-x5)+u1,(HP){x˙5=x6x˙6=-d3(x52-1)x6-c3x5+a3cos⁡⁡ωtx˙6=+R51(x5-x1)+R53(x5-x3)+u2.
The state variables of slow systems will be converged to state variables of fast system after a transient time. In fact, the second and third oscillators should follow the first oscillator which contains dominant frequency, so synchronization error is considered as follows [12]:
(3)e1=x1-x3,e2=x2-x4,e3=x3-x5,e4=x4-x6.
The purpose of the synchronization is to annihilate errors. So, selected control functions to vanish errors as e˙i=0, i=1,2,3,4, are measured as follows:
(4)e˙1=x˙1-x˙3,e˙2=x˙2-x˙4=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωt+R13(x1-x3)+R15(x1-x5)+d2(x32-1)x4+c2x3-a2cos⁡⁡ωt-R31(x3-x1)-R35(x3-x5)-u1,e˙3=x˙3-x˙5,e˙4=x˙4-x˙6=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωt+R31(x3-x1)+R35(x3-x5)+u1+d3(x52-1)x6+c3x5-a3cos⁡⁡ωt-R51(x5-x1)-R53(x5-x3)-u2.
So, by choosing control rules as follows:
(5)u1=-d1x12x2+a1cos⁡⁡ωt+d2x32x4-c1x3-a2cos⁡⁡ωt+d1x4+c2x5-d2x2+k1e1+k2e2+k3e3,u2=-d2x32x4+a2cos⁡⁡ωt+d3x52x6+c3x5-a3cos⁡⁡ωt+d^2x6-c2x5-d3x4+k4e1+k5e3+k6e4+u1+k7,
matrix form of error equations is
(6)e˙=Ae.
By applying controllers u1 and u2, the results of simulations show that diagrams related to synchronization are tracking each other and
(7)lim⁡t→∞⁡∥e∥=0.
Simulation diagrams related to synchronization in the case of three oscillators without time delay have been shown below. See Figures 1, 2, and 3 (the horizontal axis time (s) and the vertical axis millivolts).Synchronization of statesx2, x4 without time delay befor control (a) and after control (b).
(a)
(b)Synchronization of statesx4, x6 without time delay befor control (a) and after control (b).
(a)
(b)Error statesx2, x4 and x4, x6 without time delay after control ((a), (b)).
(a)
(b)


## 3. Three-Oscillator Modelling of the Heart with Time Delay

Since even small delays may change the dynamics of the system, including of time delay in differential equations can cause drastic changes and chaos in a system that has regular behavior [7, 13]. Thus, system (1) is changed as below in which τ represents time delay and xiτ=xi(t-τ):
(8)(SA){x˙1=x2x˙2=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωtx˙2=+R13(x1-x3τ13)+R15(x1-x5τ15),(AV){x˙3=x4x˙4=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωtx˙4=+R31(x3-x1τ31)+R35(x3-x5τ35),(HP){x˙5=x6x˙6=-d3(x52-1)x6-c3x5+a3cos⁡⁡ωtx˙6=+R51(x5-x1τ51)+R53(x5-x3τ53).
By substitution, we will have
(9)(SA){x˙1=x2x˙2=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωtx˙2=+R13(x1-x3(t-τ13))x˙2=+R15(x1-x5(t-τ15)),(AV){x˙3=x4x˙4=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωtx˙4=+R31(x3-x1(t-τ31))x˙4=+R35(x3-x5(t-τ35)),(HP){x˙5=x6x˙6=-d3(x52-1)x6-c3x5+a3cos⁡⁡ωtx˙2=+R51(x5-x1(t-τ51))x˙2=+R53(x5-x3(t-τ53)).
Now, by applying the appropriate controller on it which is as follows [14]:
(10)(SA){x˙1=x2x˙2=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωtx˙2=+R13(x1-x3(t-τ13))x˙2=+R15(x1-x5(t-τ15)),(AV){x˙3=x4x˙4=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωtx˙4=+R31(x3-x1(t-τ31))x˙4=+R35(x3-x5(t-τ35))+u1,(HP){x˙5=x6x˙6=-d3(x52-1)x6-c3x5+a3cos⁡⁡ωtx˙6=+R51(x5-x1(t-τ51))x˙6=+R53(x5-x3(t-τ53))+u2,
the state variables of slow systems will be converged to state variables of fast system, after a transient time. In fact, the second and third oscillators should follow the first oscillator containing dominant frequency, so synchronization error is considered as follows [11]:
(11)e1(t)=x1(t)-x3(t-τ13),e2(t)=x2(t)-x4(t-τ24),e3(t)=x3(t)-x5(t-τ35),e4(t)=x4(t)-x6(t-τ46).
The purpose of the synchronization is to annihilate errors. So, chose control functions to vanish errors as e˙i=0, i=1,2,3,4, that are measured as follows:
(12)e˙1=x˙1-x˙3(t-τ13),e˙2=x˙2-x˙4(t-τ24)=-d1(x12-1)x2-c1x1+a1cos⁡⁡ωt+R13(x1-x3(t-τ13))+R15(x1-x5(t-τ15))+d2(x32-1)x4+c2x3-a2cos⁡⁡ωt-R31(x3-x1(t-τ31))-R35(x3-x5(t-τ35))-u1,e˙3=x˙3-x˙5(t-τ35),e˙4=x˙4-x˙6=-d2(x32-1)x4-c2x3+a2cos⁡⁡ωt+R31(x3-x1(t-τ31))+R35(x3-x5(t-τ35))+u1+d3(x52-1)x6+c3x5-a3cos⁡⁡ωt-R51×(x5-x1(t-τ51))-R53(x5-x3(t-τ53))-u2.
In this case,
(13)e˙2=x˙2-x˙4(t-τ24)=-d1x12x2+d1x2+d1x4-d1x4-c1x1+c1x3-c1x3+a1cos⁡⁡ωt+R13(x1-x3(t-τ13))+R15(x1-x5(t-τ15))+d2x32x4-d2x4+d2x2-d2x2+c2x3+c2x5-c2x5-a2cos⁡⁡ωt-R31(x3-x1(t-τ31))-R35(x3-x5(t-τ35))-u1.
By substitution, we will have
(14)e˙2=-d1x12x2+d2x32x4+a1cos⁡⁡ωt-a2cos⁡⁡ωt+R13e1+R31e1-R35e3+R15e1+R15e3+d1e2+d1x4-c1e1-c1x3+d2e2-d2x2+c2e3+c2x5-u1.
Also,
(15)e˙2=-d1x12x2+d2x32x4+a1cos⁡⁡ωt-a2cos⁡⁡ωt+d1x4-c1x3-d2x2+c2x5+(R13+R31+R15-c1)e1+(d^2+d1)e2+(-R35+R15+c2)e3-u1.
And control functions are as follows:
(16)u1=-d1x12x2+d2x32x4+a1cos⁡⁡ωt-a2cos⁡⁡ωt+d1x4-c1x3-d2x2+c2x5+k1e2+k2e2+k3e3.
Also,
(17)e˙4=x˙4-x˙6(t-τ46)=-d2x32x4+d2x4+d2x6-d2x6-c2x3+c2x5-c2x5+a2cos⁡⁡ωt-R31e1+R35e3+u1+d3x52x6-d3x6+d3x4-d3x4+c3x5-a3cos⁡⁡ωt-u2.
By substitution, we will have
(18)e˙4=x˙4-x˙6(t-τ46)=-d2x32x4+d3x52x6+a2cos⁡⁡ωt-a3cos⁡⁡ωt-R31e1+R35e3+u1+c3x5+R51e1+R51e3+R53e3+d2e4+d2x6-c2e3-c2x5+d3e4-d3x4-u2.
And control functions is as follows:
(19)u2=-d2x32x4+d3x52x6+a2cos⁡⁡ωt-a3cos⁡⁡ωt+u1+c3x5+d2x6-c2x5-d3x4+k4e1+k5e3+k6e4+k7
and matrix form of error equations is
(20)e˙=Ae.
In this case, the third oscillator acts as a pacemaker and its advantage is that instead of a sudden shock and a voltage of DC, it applies sinusoidal voltage for treatment.Simulation diagrams related to synchronization in the case of three oscillators with time delay have been shown below. See Figures4, 5, and 6 (the horizontal axis time (s) and the vertical axis millivolts).Synchronization of statesx2, x4 with time delay befor control (a) and after control (b).
(a)
(b)Synchronization of statesx4, x6 with time delay befor control (a) and after control (b).
(a)
(b)Error statesx2, x4 and x4, x6 with time delay after control ((a), (b)).
(a)
(b)


## 4. Conclusion

In this paper, it has been explained that how simulating and curing some arrhythmias are possible, that one example of curing method, is modelling three-oscillator heart system in the state of delay and without delay and by applying an appropriate control.In this paper after applying controlu to the model of three-oscillator heart system in case of time delay and without time delay and also observing of simulated diagrams corresponding to synchronization of this model, we observe that, the time of synchronization x2, x4 and x4, x6 and also the time when the error converges to zero in x2, x4 and x4, x6 after applying designed controller; in case of without time delay is less than 2(s), and in case of time delay less than 1.5(s).These results indicate that the model of three-oscillator of the heart system in case of time delay and designed controller corresponding to synchronization of this model, is more appropriate and accurate than the state of without time delay.



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*Source: 168247-2014-06-05.xml*