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 Syed.ahmedbukhari83@ee.ceme.edu.pk , Sikandar.javed83@ee.ceme,edu.pk, Sajjad.hussain83@ee.ceme.edu.pk 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 978-1-5090-4090-2/16/$31.00 ©2016 IEEE +∝ 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. 978-1-5090-4090-2/16/$31.00 ©2016 IEEE 177 ISBN 978-1-5090-4089-6 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 978-1-5090-4090-2/16/$31.00 ©2016 IEEE -3 100 110 120 130 140 150 160 170 180 190 200 Figure 7:Time arrangement of the mVdP oscillator by Eq. (3) at e=3.5 178 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 -2.5 100 110 120 130 140 150 160 170 180 190 200 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 -2.5 100 110 120 130 140 150 160 170 180 190 200 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 -2.5 100 110 120 130 140 150 160 170 180 190 200 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] 978-1-5090-4090-2/16/$31.00 ©2016 IEEE 179 ISBN 978-1-5090-4089-6 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 0.8 170 175 180 185 190 195 200 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=0.2 978-1-5090-4090-2/16/$31.00 ©2016 IEEE 165 180 ISBN 978-1-5090-4089-6 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 -3 160 0.5 165 170 175 180 185 190 195 0 200 Figure 16:Action potential generated by classical van der pol oscillator, similar to figure 8 -0.5 -1 III. -1.5 160 DISCUSSION 165 170 175 180 185 190 195 200 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. -1.5 160 165 170 175 180 185 190 195 200 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 978-1-5090-4090-2/16/$31.00 ©2016 IEEE 181 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 [1] L. Sӧrnmo and P. Laguna, Bioelectrical signal processing in cardiac and neurological applications, vol. 1.Academic Press, 2005. [2] S. R. Gois and M. A. Savi, “An analysis of heart rhythm dynamics using a threecoupled oscillator model,” Chaos, Solitons& Fractals, vol. 41, no. 5, pp. 2553–2565, 2009 [3] A. Babloyantz and A. Destexhe, “Is the normal heart a periodic oscillator ,” Biological cybernetics, vol. 58, no. 3, pp. 203–211, 1988. [4] P. V. E. McClintock and A. Stefanovska, “Noise and determinism in cardiovascular dynamics,” Physica A: Statistical Mechanics and its Applications, vol. 314, no. 1, pp. 69–76, 2002. [5]A. Stefanovska, D. G. Luchinsky, and P. V. E. McClintock, “Modelling couplingsamong the oscillators of the cardiovascular system,” Physiological measurement, vol.22, no. 3, pp. 551–564, 2001. [6]A. Stefanovska, M. BracicLotric, S. Strle, and H. Haken, “The cardiovascular systemas coupled oscillators,” Physiological measurement, vol. 22, no. 3, pp. 535–550, 2001. [7]J. P. Zbilut, M. Zak, and R. E. Meyers, “A terminal dynamics model of the heartbeat,” Biological cybernetics, vol. 75, no. 3, pp. 277–280, 1996. [8]Van der pol, b., on relaxation oscillations,philosophical magazine, 7, 978-992, 1926. [9]Van der Pol, B., and van der Mark, J., The HeartbeatConsidered as a Relaxation Oscillation, and an ElectricalModel of the Heart, Philosophical Magazine, 7, 763-775,1928. [10]Van der Pol, B., Forced Oscillations in a Circuit withNonlinear Resistance (Receptance with Reactive Triode),Selected Papers on Mathematical Trends in Control Theory,Bellman and Kalaba, Editors, 1927. [11]Z. Pan, R. Yamaguchi, and S. Doi, "Analysis of Bifurcationsand Variabilities of Rhythm in a Cardiac Pacemaker CellModel," in Proc. of International Symposium on NonlinearTheory and its Applications (NOLTA'09), pp. 360-363, 2009. [12]M. R. Boyett, H. Honjo, and I. Kodama, "The SinoatrialNode, a Heterogeneous Pacemaker Structure," Cardiovasc.Res., vol. 7, pp. 658687, Sep. 2000. [13]Krzysztof Grudzinski and Jan J. Zebrowski, "ModelingCardiac Pacemakers with Relaxation Oscillators", Physica A336, 153-162, 2004. [14]Andersen CM, Geer JF. Power Series Expansions for theFrequency and Period of the Limit Cycle of the van der Pol’sEquation. SIAM Journal on Applied Mathematics; 42:678–693, 1982. [15]Rand RH, Holmes PJ. Bifurcation of Periodic Motions inTwo Weakly Coupled van der Pol’s Oscillators. InternationalJournal of Non-Linear Mechanics; 15:387–399, 1980. [16]Storti DW, Rand RH. Dynamics of Two Strongly Coupledvan der Pol’s Oscillators. International Journal of Non-LinearMechanics; 17(3):143–152, 1982. [17]Cohen AH, Holmes PJ, Rand RH. The Nature of theCoupling between Segmental Oscillators of the LampreySpinal Generator for Locomotion: a Mathematical Model.Journal of Mathematical Biology; 13:345–369, 1982. [18]Di Bernardo D, Signorini MG, Cerutti S. A model of twononlinear coupled oscillators for the study of heartbeatdynamics. International Journal of Bifurcation and Chaos;8(10):1975–1985, 1998. [19]Linkens D A. Analytical Solution of Large Numbers ofMutually Coupled Nearly Sinusoidal Oscillators. IEEETransactions on Circuits and Systems; CAS-21(2):294–300,1974. [20]K. Grudzinski and J. J. Zebrowski, “Modeling cardiac pacemakers with relaxationoscillators,” Physica A: Statistical Mechanics and its Applications, vol. 336, no. 1, pp.153–162, 2004. [21]J.M.T. Thompson, H.B. Steward, Nonlinear Dynamics and Chaos, 2nd Edition, Wiley, New York, 2002. [22]C.R. Katholi, F. Urthaler, J. Macy Jr., T.N. James, A mathematical model of automaticity inthe sinusnode and the AV junction based on weakly coupled relaxation oscillators, Comput. Biomed.Res. 10,529– 543,1977. -0.5 -1 -1.5 160 165 170 175 180 185 190 195 200 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 -2 160 165 170 175 180 185 190 195 200 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. 978-1-5090-4090-2/16/$31.00 ©2016 IEEE 182 ISBN 978-1-5090-4089-6 19th International Conference on Computer and Information Technology, December 18-20, 2016, North South University, Dhaka, Bangladesh [23]B.J. West, A.L. Goldberger, G. Rovner, V. Bhargava, Nonlinear dynamics of the heartbeat, The AVjunction: passive conduict or active oscillator Physica D 17,198–206 ,1985. [24]D. di Bernardo, M.G. Signorini, S. Cerutti, A model of two nonlinear coupled oscillators for the studyof heartbeat dynamics, Int. J. Bifurcations Chaos 8 (10) (1998) 1975–1985. [25]D. Postnov, H. SeungKee, K. Hyungtae, Synchronization of di<usively coupled oscillators near thehomoclinic bifurcation, Phys. Rev. E 60 (3) (1999) 2799–2807. [26]J.M.T. Thompson, H.B. Steward, Nonlinear Dynamics and Chaos, 2nd Edition, Wiley, New York, 2002. [27]L. Glass, P. Hunter, A. McCulloch (Eds.), Theory of Heart, Springer, New York, 1990. 978-1-5090-4090-2/16/$31.00 ©2016 IEEE 183 ISBN 978-1-5090-4089-6