Comprehensive Procedure for Fast and Accurate Coupled Oscillator Network Simulation

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1 Comprehensive Proceure for Fast an Accurate Couple Oscillator Network Simulation Prateek Bhansali, Shweta Srivastava, Xiaolue Lai an Jaijeet Roychowhury Department of Electrical an Computer Engineering University of Minnesota {bhansali, shwetas, laixl, Abstract Couple oscillator networks occur in various omains such as biology, astrophysics an electronics. In this paper, we present a comprehensive proceure for rapi an accurate simulation of large couple oscillator networks using wiely accepte, fully-nonlinear Perturbation Projection Vector PPV phase macromoels. We valiate our metho against full simulation of x couple network of Brusselator biochemical oscillator an obtain computational speeups of 7x over full simulation. Furthermore, we apply the metho to stuy self-organization phenomenon of Brusselator uner asymmetric coupling an time perio variations. I. INTRODUCTION Networks of couple oscillators occur in multitue of physical an natural systems. For example, the ynamics of auto catalytic an oscillating biochemical Brusselator reaction is moele by couple oscillators []. Natural pacemaker of the heart consists of tens of thousans of cells calle sinoatrial noe. Each pacemaker cell is an oscillator that prouces perioic electrical signals. The reliable an perioic pumping of the heart is riven by rhythmic electrical activity of the network of pacemaker cells []. The synchronous flashing of fireflies is yet another example of couple oscillators in the nature []. Each firefly prouces flash perioically; but, when fireflies gather together, these insects emit a flash upon the sight of lights prouce by neighbouring fireflies. This collaborative behaviour results in a spectacular synchronize flashing. Recently, utility power gri has been stuie using a network of couple oscillators []. Each iniviual oscillator is mathematically moele by a set of nonlinear, couple partial ifferential equations PDEs. Solving such PDEs analytically, even for a single oscillator, is almost always a formiable task. Thus, analysis an simulation of couple oscillator network is of great practical importance. The irect technique is to solve PDEs that governs the ynamics of the oscillator in the time omain. This time-course integration metho is not suitable for simulation of oscillators as it inherently accumulates phase errors without limit uring simulation [5]. So, hunres of timesteps per oscillation cycle are require for acceptable accuracy see Fig. 7, which leas to high computational cost. The situation becomes even worse for couple oscillator systems with numerous oscillators. Another technique is to use phase omain moels, which was pioneere by Winfree [6] an later simplifie by Kuramoto [7]. Winfree constructe the moel for nearly ientical oscillators with the assumption that the coupling among the oscillators is small. This assumption allow the separation of phase an amplitue variations of the oscillator. The amplitue variations ie quickly an oscillator relaxes to a steay state limit cycle, since the perturbation from other oscillators is small. Thus, only phase variations nee to be tracke to etermine the approximate state of the oscillator. In his moel, the rate of change of the phase of an oscillator is etermine by oscillator s interaction function calle phase sensitivity. However, no algorithm was propose to obtain these interaction functions. Kuramoto later simplifie Winfree s moel by approximating the interaction functions as sinusois. But these sinusoial interaction functions accumulate significant errors for non-harmonic oscillators [8], [9]. However, nonlinear PPV macromoel, alreay establishe for iniviual oscillator overcome these errors. Moreover, they can be algorithmically extracte in an computationally efficient manner [5], []. The simulation base on PPV macromoel show large simulation speeups ue to system size reuction an use of larger timesteps, without affecting accuracy. In this paper, we provie the proceure an valiation for a fast metho evelope for simulating large networks of couple oscillators [9]. We start from a PDE escription of a specific reactioniffusion system, the Brusselator, an show how such oscillatory PDE systems can be iscretize in space to obtain a network of oscillators couple spatially. We inicate how perturbation terms require for the application of PPV metho can be obtaine. This metho can be generalize for any spatially couple electronic or biological oscillators. Finally, using the proceure, we simulate a system of a x couple oscillator network uner parametric variations. The remainer of paper is organize as follows. In Section II, a brief review of the nonlinear PPV phase macromoel is presente. Section III eals with Brusselator biochemical oscillator moel escription an etails to evelop ifferential-algebraic equation DAE of the oscillator. We show how the PPV phase macromoel of an inepenent oscillator can be abstracte from the PDE of the Brusselator by observing perturbation terms. In Section IV, etails to solve the oscillator network for phase of each oscillator an using them to reconstruct the oscillator states across the network are presente. In Section V, we valiate our metho against a full simulation for a couple oscillator network, an show phase error accumulation in time-course transient simulation of a simple two oscillator network. Next, in Section VI, collective behaviour of the Brusselator system is stuie uner asymmetric coupling strength an time perio variations. II. NONLINEAR PHASE MACROMODEL In this section, we review the nonlinear phase macromoelling technique using the PPV phase macromoel [5]. The PPV phase macromoels are well suite for large systems involving couple oscillators as they ramatically improves efficiency an accuracy. A starting point is the ifferential equation which escribes oscillator uner external perturbations x t + f x = bt where xt is a vector of oscillator states an bt represents the vector of perturbation signals applie to corresponing states of free running oscillator. Assuming bt an the amplitue variations are small, the solution to this oscillator subject to perturbation can be shown [5] to take the following form x p t = x s t + αt where x p t represents the waveform of the perturbe oscillator, x s t is the steay state perioic solution of unperturbe oscillator an αt is the phase eviation cause by the external perturbation bt. In [5], Demir et al prove that the phase eviation αt is governe by a Authorize license use limite to: Univ of Calif Berkeley. Downloae on December 7, 8 at :56 from IEEE Xplore. Restrictions apply.

2 simple scalar equation x αt t = V T t + αt bt where V T is the PPV of system size. Each component in PPV represents the oscillator s phase sensitivity to the applie perturbation signal. In the next section, we apply PPV macromoelling technique to single Brusselator biochemical oscillator an show how this technique of the phase macromoelling can be extene to couple oscillator networks. The PPV macromoels can be obtaine both in frequency an time omain algorithmically [], unlike Winfree s interaction functions. y y i,j i,j+ i,j i,j i+,j III. DESCRIPTION OF THE NUMERICAL PROCEDURE FOR BRUSSELATOR EQUATION It is well known that many types of biochemical reaction-iffusion RD systems [] prouce intricate an beautiful spatial patterns phase waves. For example, the patterns in Fig. b result from a simple type of biochemical RD system calle the Brusselator. The PDEs of a RD system have the form cx, y,t t = R c,t+d c c where x, y are spatial imensions an c is a vector of s of the chemical species taking part in RD reaction. The generation spee of chemical species is given by R c,t an its functional form epens on the specific system being stuie. The iffusion coefficients are represente by matrix D c an secon term in the RHS of, D c c, represents change in the c by iffusion. For example, the Brusselator system is moele by the following nonlinearly couple PDEs [] u t = A B+u+u v+d u u v t = Bu u v+d v v where A an B are constants specific to reaction, u an v are the chemical s of two species, an their temporal erivatives represent the rate of change of their s with time. The iffusion coefficients are represente as D u an D v. We use 5 to present the couple oscillator simulation metho. In orer to solve 5 numerically, we choose rectangular gri as shown in Fig. to iscretize PDEs spatially. Let the gri points be enote by i, j, where i {,N }, j {,N } an thus the network size is N N. Each gri point represents an oscillator, as we will verify later, escribe by the nonlinearly couple PDE 5. The line joining the gri points correspons to coupling of an oscillator with its neighbour. To iscretize the PDE 5, we use secon-orer central ifference scheme for the Laplacian operator,, obtaine using Taylor series expansion as u u i, jt+u i+, j t u i, j t i, j x + u i, j t+u i, j+ t u i, j t y v v 6 i, jt+v i+, j t v i, j t i, j x + v i, j t+v i, j+ t v i, j t y where x an y are space iscretization steps in x an y imensions respectively. For bounary points, we use non-flux bounary 5 Fig.. x Rectangular gri for iscretizing the Brusselator PDEs. conitions given as u n v n = i=[,n ]; j=[,n ] = i=[,n ]; j=[,n ] where n is normal to the bounary. The bounary conitions play a irect role in etermining the perturbations across the gri. Then, 5 is evaluate at each gri point using 6 an 7. For example, at any internal point i, j we have t u i, jt = A B+u i, j t+u i, j tv i, jt+c u t i, j 8 t v i, jt = Bu i, j t u i, j tv i, jt+c v t i, j where, C u t i, j = D u [ ui, j t+u i+, j t u i, j t x + u i, j t+u i, j+ t u i, j t y C v t i, j = D v [ vi, j t+v i+, j t v i, j t x + v i, j t+v i, j+ t v i, j t y For simplicity, we efine c ux, c uy, c vx an c vy as Du c ux = x c vx = Dv x Du, c uy = y, c vy = Dv y,. ], ]. 7 9 The iffusion terms, C u t an C v t, contribute to perturbations of an oscillator from other oscillators in the gri. Thus, the characteristics of a single oscillator can be capture by eliminating the iffusion terms, C u t an C v t, from 8 as t ut A+B+ut u tvt = t vt But+u tvt = Note that in the above equation, we have roppe the gri point inex i, j to make it clear that if we put the iffusion terms equal to zero, Authorize license use limite to: Univ of Calif Berkeley. Downloae on December 7, 8 at :56 from IEEE Xplore. Restrictions apply.

3 there will be no coupling effect or perturbations from the neighboring oscillators an the PDE at each noe in the network will be the same an can be represente by. Next we evelop the DAE of. It will be same for every oscillator in the gri as we have eliminate the iffusion terms, C u t an C v t. These terms will be incorporate later while solving for the couple oscillator network in Section IV as external perturbations to an oscillator. To represent the PDEs of the Brusselator in a convenient circuit s DAE form, we start with t q x+ f x+ bt = where x are the unknowns. For we can write, q x = x an bt is the vector of perturbations from neighbouring oscillators. Thus, for an inepenent oscillator, bt =. Next, we consier x = [ut vt] T, then f x, f q x an x for can be written as ut x = vt A+B+ut u f x tvt = But+u tvt f B+ utvt u x = t B+utvt u t q x = Using the above DAE of the Brusselator, we extract the steay state waveform an PPV [] of u an v for A =.5, B =.7 using the harmonic balance metho. The waveforms obtaine are shown in Fig. a an Fig. b u v time a Plot of steay state. Phase sensitivity ppv_u ppv_v time s b Plot of PPVs. Fig.. Plot of the steay state an the PPV of u an v. The PPV waveforms nee to be extracte only once an can be reuse for a couple oscillator network simulation as we show next. IV. SIMULATION DETAILS OF COUPLED OSCILLATOR NETWORK USING PPV MACROMODELS In this section, we present the etails for simulating large couple oscillator networks. The PPV phase macromoelling technique for a single oscillator can be extene for couple oscillator networks, by moeling bt in to account for coupling among oscillators. For example, the oscillators at gri points i, j +, i, j, i+, j an i, j cause perturbations in oscillator states at gri point i, j through coupling as shown in Fig.. The vector of perturbations, b i, j t, at the gri point i, j can be obtaine by comparing 8 an. The components of b i, j t are C u t i, j an C v t i, j, for the u an v state of the oscillator respectively. Therefore, for any gri point i, j in the network, perturbation vector b i, j t can be written Fig.. as where, j i,j i i,j+ i,j i,j i+,j Contribution of perturbation from the neighboring gri points. b i, j t = b_ui, j t b_v i, j t b_u i, j t = C u t i, j 5 b_v i, j t = C v t i, j 6 Next, we solve for the phase eviation of oscillators in the network using PPV, V T t an the perturbation signal, bt. At any point i, j, the phase eviation α i, j t can be calculate using as the solution of following equation α i, j t = V T t + α i, jt b i, j t α i, j t = ppv_ut + α i, j t ppv_vt + α i, j t b_ui, j t b_v i, j t = ppv_ut + α i, j tb_u i, j t+ ppv_vt + α i, j tb_v i, j t = ft,α i, j t 7 where ppv_ut an ppv_vt are PPVs of for u an v, respectively. The secon orer Runge-Kutta RK numerical metho can be use to solve for αt. Accoring to RK algorithm, the phase eviation for point i, j at time t n+ = t n + h, where h is timestep use in the simulation, is solve as shown α n+ i, j = α n i, j + k i, j k i, j = h f t n + h, α n i, j + k i, j = h[ppv_u t n + h + α n i, j + k i, j + ppv_v an k is calculate as t n + h + α n i, j + k i, j k i, j = h ft n, α n i, j = h[ppv_ut n + α n i, j b_u i, j t n + ppv_vt n + α n i, j b_v i, j t n ] b_u i, j t n + h b_v i, j t n + h ] 8 9 Assuming the amplitue variations are small, the perturbe solutions for ut an vt uner the effect of bt are given by. Thus, for gri point i, j at time t = t n u i, j t n = u ss t n + α i, j t v i, j t n = v ss t n + α i, j t Authorize license use limite to: Univ of Calif Berkeley. Downloae on December 7, 8 at :56 from IEEE Xplore. Restrictions apply.

4 where u ss t n an v ss t n are the steay state solution at t = t n in the absence of the perturbation signal, bt. Using the above proceure, we calculate the phase eviation, αt an use it to compute the perturbe solution, ut an vt of the network. that nonlinear PPV base couple oscillator simulation metho results in a close match with full system simulation with sufficiently high number of time steps, which valiates our metho. However, the propose metho is computationally cheaper than the full simulation. V. NUMERICAL RESULTS In this section, we apply an valiate the technique presente above using the Brusselator oscillator. A simple, couple oscillator network of two oscillators is represente schematically in Fig.. The line joining the oscillators represent the coupling between u an v noes of each oscillator. We first simulate the two oscillator network using the PPV phase macromoelling technique for couple oscillators. The two oscillators start with ranom initial phase between an π an we take A =.5, B =.7 an c ux = c uy = c vx = c vy = Oscillator Oscillator 6 8 Fig.. Schematic of two couple oscillators. Fig. 6. Phase locke u waveforms of two oscillators using PPV macromoel. The phase eviation an oscillator state waveforms obtaine are α shown in Fig. 5 an Fig. 6. After a few oscillation cycles, t becomes constant, inicating that the oscillators get phase locke. We can see from u waveforms in Fig. 6 that oscillators inee get phase locke. The coupling between the two oscillators forces them to get into phase. Now, we compare the results obtaine with the time-course full simulation of the above system, for ifferent time steps uner same initial conitions. The full simulation results are compute using same system parameters an RK numerical metho. From Fig. 7, we can see the efficiency of propose metho against the full simulation. For example, using timesteps per oscillation cycle in the full simulation an the propose metho, the former metho accumulates significant phase errors. However, when we increase the number of timesteps in the full simulation the phase error reuces. As it can be seen, the result closely match with larger number of time points in the full simulation. Phase shift s Oscillator Oscillator 6 8 Fig. 5. Phase shift in the two oscillator using the nonlinear PPV phase macromoel. Next, we simulate a x couple Brusselator oscillator network using full system simulation an propose metho for similar initial conitions an a non-flux bounary conition for bounary points. The parameters A, B, c ux, c uy, c vx an c vy are chosen to be the same as before. We take timesteps per oscillation cycle for PPV base simulation an 5 timesteps for full simulation, to get comparable accuracies. We plot the simulation results of u state of oscillator at position, an v state of oscillator at, in the couple network in Fig. 8 an Fig. 9. It can be seen from Fig. 8 an Fig Macromoel: steps FS: steps FS: steps FS: 5 steps Fig. 7. Plot of u waveform for oscillator using PPV macromoel an full simulation FS. On an AMD Athlon6 + base workstation, running MAT- LAB on Linux kernel.6.9, our technique using PPV phase macromoels took 7 s for oscillation cycles as compare to s with full simulation for comparable accuracy implemente on the same workstation. This is a 7-fol speeup an is expecte to go higher for large couple oscillator networks, involving a larger number of oscillators. The spee up is ue to the fact that we calculate nonlinear phase macromoel only once an use that across the network for further analysis. VI. EXPLORING SELF-ORGANIZATION UNDER ASYMMETRIC COUPLING AND TIME PERIOD VARIATION In this section, we simulate a larger couple oscillator network, using the proceure evelope in the previous sections to explore the effect of asymmetric coupling an time perio variations in the network for ranom an given initial conitions of α at t =. The Brusselator self-organizes itself forming spatio-temporal patterns an this has been experimentally observe [], []. The network is mae up of ientical oscillators, with free running frequency of each oscillator as f = T an coupling parameters c ux, c uy, c vx an c vy, as efine in Section III. The self-organization an subsequent pattern formation epens on A, B an coupling parameters. For simulation purposes, we chose A =.5, B =.7 an nominal c ux =, c ux =, c ux = an c ux =. Authorize license use limite to: Univ of Calif Berkeley. Downloae on December 7, 8 at :56 from IEEE Xplore. Restrictions apply.

5 .5 Full Simulation Macromoel.5 a Phase plot at t= 5T. b Phase plot at t= T..5 Fig.. Phase plot of the network with ranom initial conitions. Fig Plot of u waveform of, oscillator Full Simulation Macromoel Network simulation with variations: We now simulate the network with small variations in the time perio, T an in coupling parameters. In the first simulation, the coupling relate parameters c ux, c uy, c vx an c vy are varie is such a way that they follow a Gaussian istribution, N,9. The plot of the phase of oscillators at t = T is shown in Fig.. It can be seen that spiral patterns, almost similar to Fig., are forme an network is robust against variations in coupling. This variations can be physically mappe to the anisotropy of the physical system uner consieration. For example, heart is not isotropic an homogeneous in all irections []. A practical an useful application of the propose metho can be to simulate auto-oscillatory cells of the mammalian heart Fig. 9. Plot of v waveform of, oscillator. a Phase plot at t= T. A. Ranom initial conition simulations The oscillators in the gri are assigne ranom initial phase between to π as shown in Fig.. Each small ot in the phase plot represents the phase of that oscillator. The oscillators with the same phase have the same color. The network is simulate for oscillation cycles t=t an solve for αt. Fig.. Phase plot with ranom initial phase an variations in the coupling parameters. In the secon simulation, the normalize time perio of oscillators in the network is varie as a Gaussian istribution, N,e-7. The phase plots are shown in Fig.. Again, similar patterns form with small ranom time-perio variations, but the pattern formation in the network is not robust against it. Fig.. Ranom initial phase of oscillators in the network at t=. a Phase plot at t= 5T. b Phase plot at t= T. Network simulation without variation: We simulate the network with ranom initial phase for each oscillator. For simulation without any variation in the coupling parameter or the time perio, Fig. shows the phase plot of oscillators at ifferent time instants. As time evolves, the mutual coupling between the oscillators forces phase lock between them in a localize fashion, which can be seen in the form of spiral patterns or phase waves in Fig. a at t = 5T. These patterns grow an emerges as seen in Fig. b at t=t. We call these spirals as see. Fig.. perio. Phase plot with ranom initial phase an variations in the time B. Given initial conition simulations In this subsection, we analyze the collaborative behaviour of the oscillators in the network for a given initial phase. We start by placing a see at the center of network as shown in Fig., enclose by a ashe rectangle for illustration purpose, an analyze its growth. Authorize license use limite to: Univ of Calif Berkeley. Downloae on December 7, 8 at :56 from IEEE Xplore. Restrictions apply.

6 Further, we emonstrate the collaborative behaviour of oscillators in the network uner parametric variations for given initial conitions. a Phase plot at t=5t. b Phase plot at t= T. Fig. 7. Phase plot with initial see an Gaussian noise in time perios. Fig.. Phase plot of oscillators with given initial conition seee. Simulation without noise: After placing the see, we simulate the network for similar values of A, B an coupling parameters mentione in Section VI-A. This results in a stable spiral pattern growth encompassing the whole network an phase synchronization among oscillators takes place, as illustrate in Fig. 5. Fig. 5. a Phase plot at t= 5T. b Phase plot at t= T. Phase plot with given initial conitions illustrating see growth. Simulation with noise: In a similar manner as in Section VIA, we investigate the effect of parameter variations on the see growth. In the first simulation, we simulate the network with Gaussian noise in the coupling parameters, c ux, c uy, c vx an c vy, of N,9 i.e. with a stanar eviation SD of 7.5%. The phase plot at ifferent time instants is shown in Fig. 6. It can be seen that the network is robust an basic elliptical patterns are still present. a Phase plot at t= 5T. b Phase plot at t= T. Fig. 6. Phase plot with seee network an variations in coupling parameter. In the secon simulation, we simulate the network with normalize time perio varying as a Gaussian istribution, N,e-7, across the network. The phase plots at ifferent time instants are shown in Fig. 7. Thus, the network is affecte by the noise in time perio but the basic see pattern remain the same. couple Brusselator biochemical oscillator emonstrating speeups over the time-course full simulation. In aition, the propose metho provies useful insights into self-organization phenomenon of Brusselator uner parametric variations. Currently, we are working on its application to simulate the conuction system of heart. Acknowlegments We thank the MARCO GSRC program, which supporte this work. Computational an infrastructural resources from the University of Minnesota s Digital Technology Center is gratefully acknowlege. REFERENCES [] P. K. Maini, K. J. Paintera, an H. N. P. Chaub. Spatial pattern formation in chemical an biological systems. J. Chem. Soc., Faraay T rans., pages 6, 997. [] Alexaner V. Panfilov an Arun V. Holen. Computational Biology of the Heart. John Wiley an Sons, Chichester, 997. [] Steven Strogatz. Sync: The Emerging Science of Spontaneous Orer. Hyperion,. [] G. Filatrella, A. H. Nielsen, an N. F. Peersen. Analysis of a power gri using a kuramoto-like moel. The European Physical Journal B - Conense Matter an Complex Systems, 6:85 9, 8. [5] A. Demir, A. Mehrotra, an J. Roychowhury. Phase noise in oscillators: a unifying theory an numerical methos for characterization. IEEE Trans. on Circuits an Systems-I:Funamental Theory an Applications, 75:655 67, May. [6] A. T. Winfree. Biological rhythms an the behavior of populations of couple oscillators. J. Theor. Biol., 967. [7] Y. Kuramoto. Chemical Oscillations, Waves, an turbulence. Springer, Berlin, 98. [8] Steven H. Strogatz. From kuromoto to crawfor: exploring the onset of synchronization in populations of couple oscillators. Physica D,. [9] X. Lai an J. Roychowhury. Fast simulation of large networks of nanotechnological an biochemical oscillators for investigating selforganization phenomena. In Proc. IEEE Asia South-Pacific Design Automation Conference, January 6. [] A. Demir an J. Roychowhury. A reliable an efficient proceure for oscillator ppv computation, with phase noise macromoelling applications. IEEE Trans. on Computer-Aie Design of Integrate Circuits an Systems, :88 97, February. [] Irving R Epstein an John A Pojman. An introuction to nonlinear chemical ynamics : oscillations, waves, patterns, an chaos. New York : Oxfor University Press, 998. [] A. L. Lin, A. Hagberg, A. Arelea, M. Bertram, H. L. Swinney, an E. Meron. Four-phase patterns in force oscillator systems. Physical Review E, 6:79 798,. [] E. Etienne Verheijck, Any Wessels, Antoni C. G. van Ginneken, Jan Bourier, Marry W. M. Markman, Jacqueline L. M. Vermeulen, Jacques M. T. e Bakker, Wouter H. Lamers, Tobias Opthof,, an Lennart N. Bouman. Distribution of atrial an noal cells within the rabbit sinoatrial noe : Moels of sinoatrial transition. Circulation, 97:6 6, 998. VII. CONCLUSIONS We have presente a comprehensive proceure for simulation of couple oscillator networks using the phase-omain nonlinear PPV macromoels. The metho is generally applicable to any spatially couple oscillator network. We have valiate our metho for x Authorize license use limite to: Univ of Calif Berkeley. Downloae on December 7, 8 at :56 from IEEE Xplore. Restrictions apply.