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

If the SA 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 a 1 andω 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.


Introduction
As we know, If the  and  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][2][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  1 and  frequency is added to  node.

Three-Oscillator Modelling of the Heart without Time Delay
As we know, cardiac normal rhythm is firstly produced by  node (normal pacemaker) and then causes stimulating of  node.Nevertheless, it has been observed that these two oscillators are not so accurate for producing  signal.This is because, the signal of first oscillator is related to the activation of  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  curve, but  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  oscillator and another one to model the  oscillator and the third one to model  oscillator [7]: in which pairs ( 1 ,  2 ), ( 3 ,  4 ), and ( 5 ,  6 ) show , , and  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].
By changing  1 ,  2 , and  3 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 ( 1 ,  2 ).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 as  and , should be synchronized with faster pacemakers such as .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) 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]: The purpose of the synchronization is to annihilate errors.So, selected control functions to vanish errors as ė  = 0,  = 1, 2, 3, 4, are measured as follows: So, by choosing control rules as follows: matrix form of error equations is By applying controllers  1 and  2 , the results of simulations show that diagrams related to synchronization are tracking each other and lim 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).

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  represents time delay and    =   ( − ):  By substitution, we will have   Now, by applying the appropriate controller on it which is as follows [14]:

ISRN Applied Mathematics
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]: The purpose of the synchronization is to annihilate errors.So, chose control functions to vanish errors as ė  = 0,  = 1, 2, 3, 4, that are measured as follows: In this case, By substitution, we will have Also, And control functions are as follows: By substitution, we will have (18) And control functions is as follows: and matrix form of error equations is ė = .
(20) In this case, the third oscillator acts as a pacemaker and its advantage is that instead of a sudden shock and a voltage of , it applies sinusoidal voltage for treatment.Simulation diagrams related to synchronization in the case three oscillators with time delay have been shown below.See Figures 4, 5, and 6 (the horizontal axis time (s) and the vertical axis millivolts).

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 control  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  2 ,  4 and  4 ,  6 and also the time when the error converges to zero in  2 ,  4 and  4 ,  6 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.

Figure 2 :
Figure 2: Synchronization of states  4 ,  6 without time delay befor control (a) and after control (b).

Figure 5 :
Figure 5: Synchronization of states  4 ,  6 with time delay befor control (a) and after control (b).

Table 1 :
Appropriate parameters to synchronize three-oscillator system.