Craig G. Rusin et al- Synchronization engineering: tuning the phase relationship between dissimilar oscillators using nonlinear feedback

8/3/2019 Craig G. Rusin et al- Synchronization engineering: tuning the phase relationship between dissimilar oscillators using

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Phil. Trans. R. Soc. A (2010) 368, 21892204doi:10.1098/rsta.2010.0032

Synchronization engineering: tuning the phaserelationship between dissimilar oscillators using

nonlinear feedback

BY CRAIG G. RUSIN1,*, HIROSHI KORI2, ISTVN Z. KISS3AND JOHN L. HUDSON1

1Department of Chemical Engineering, University of Virginia, Charlottesville,VA 22902, USA

2Division of Advanced Sciences, Ochadai Academic Production,Ochanomizu University, Tokyo 112-8610, Japan

3Department of Chemistry, Saint Louis University, St Louis, MO 63103, USA

A mild, nonlinear, time-delayed feedback signal was applied to two heterogeneousoscillators in order to synchronize their frequencies with an arbitrary and controllablephase difference. The feedback was designed using phase models constructed fromexperimental measurements of the intrinsic dynamical properties of the oscillators. Thefeedback signal produced an interaction function that corresponds to the desired collectivebehaviour. The synchronized phase difference between the elements can be tuned to anyvalue on the interval 0 and 2p by shifting the phase of the interaction function using thefeedback delay. Numerical simulations were conducted and experiments carried out with

electrochemical oscillators.Keywords: synchronization; phase models; nonlinear feedback

1. Introduction

The collective dynamical behaviour of a rhythmic population is dependent onthe interactions between individual elements. Coupling among rhythmic elementscan lead to synchronization, where a number of rhythmic elements organize into asingle group with a uniform phase and frequency. Such synchronization can lead tocoherent light emission from laser systems (Petrov et al. 1997; Oliva & Strogatz2001), neuronal clustering in the suprachiasmatic nucleus (Iglesia et al. 2000;Yamaguchi et al. 2003), oscillations in chemical systems (Epstein & Pojman 1998),epileptic events (Traub & Wong 1982) and Parkinsonian tremors (Tass 1999).

By manipulating the interactions between individual elements, it is possibleto steer the collective behaviour of the system to a desired state. A number ofmethods have been developed to control the collective behaviour of rhythmicsystems (Battogtokh & Mikhailov 1996; Popovych et al. 2006). Proportionalintegralderivative controller-based feedback has been used to create phase-lockedstates (Di Donato et al. 2007), feedback based on mutual information has been

*Author for correspondence ([email protected]).

One contribution of 10 to a Theme Issue Experiments in complex and excitable dynamical systems.

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2190 C. G. Rusin et al.

shown effective in controlling the characteristic time scales of oscillators (Belykhet al. 2005) and tunable heterogeneities have been used to steer the behaviour ofexcitable media (Mikhailov & Showalter 2006).

We have developed a general engineering methodology based on phase modelsthat allows precise control over the collective dynamical behaviour of a rhythmic

system using global nonlinear time-delayed feedback. A key advantage of thismethod is that it can be tailored to the unique dynamical behaviour of a physicalsystem. This is accomplished by using experimental measurements to calibratethe phase model to the target system, eliminating the need for physico-chemicalmodels. Mild feedback is used to gently steer the collective behaviour of thetarget system towards the desired state, ensuring that the rhythmic behaviourof the individual elements of the system is preserved. This method has beenused to generate dynamical behaviours such as phase synchronization, phasedesynchronization, clustering and itinerant clustering within a population ofrhythmic electrochemical elements (Kiss et al. 2007; Kori et al. 2008).

In this paper, we use our methodology to address the general question ofhow to tune the stationary phase difference between two dissimilar oscillatorswithout prior knowledge of their underlying dynamic behaviour. This two-oscillator system represents a canonical model for a large class of rhythmicbiological systems (Winfree 1980; Iglesia et al. 2000) and incorporates manyof the non-idealities that are present in typical experimental systems, yethas a simple analytical solution. Heterogeneities are common in experimentaland natural rhythmic systems where the intrinsic oscillator frequency is notunder experimental control. We demonstrate how to choose and obtain aninteraction function that produces an arbitrary phase difference between two

rhythmic elements, and we demonstrate the application of the method using anexperimental chemical system.

2. General methodology

The phase behaviour of a population of oscillating elements can be approximatedin the limit of weak global interactions as (Kuramoto 1984)

dfidt

=ui +K

N

N

j=1H(fj fi), (2.1)

where fi and ui are the phase and inherent frequency of element i, K is theinteraction strength, N is the number of elements in the population and H(Df)is an interaction function that describes the effect one element has on all otherelements in the system. The interaction function can be derived from measurablequantities of the experimental system as

H(Df)=1

2p

2p0

Z(f)h

x(f+ Df)

df, (2.2)

where Z(f) is the response function, x(f) is the waveform and h(x) is the coupling

function between the elements. The response function, Z(f), represents thesensitivity of a single element to perturbations along its limit cycle. This functionis an intrinsic property of the oscillator and can be experimentally measured

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Synchronization engineering 2191

(Galan et al. 2005; Kiss et al. 2005; Tsubo et al. 2007). The coupling function,h(x), represents the form of the physical connection between the elements inthe system. For the simulations and experiments described in this work, theinteractions among elements are produced by global nonlinear time-delayedfeedback of the form

h(x(t))=S

n=0

knx(t tn Dt)n, (2.3)

where t is the time, S is the polynomial feedback order, kn is the nth orderpolynomial feedback coefficient, tn is the nth order polynomial feedback delay andDt is a feedback delay that is common to all polynomial terms. By manipulatingthe shape of the interaction function, it is possible to control the collectivebehaviour of the system. In general, this can be accomplished using a three-stepprocess: determine a target interaction function that will produce the desired

global behaviour, measure the response function of the rhythmic elements to beused and numerically optimize the shape of the coupling function to achieve thetarget interaction function.

3. Tuning the phase difference between two rhythmic elements

(a) Determining the required interaction function

Applying the Kuramoto phase approximation (equation (2.1)) to a system of

two non-identical elements yields equations for the two phase variables{f

1,f

2}

which when subtracted give a relationship for the phase difference, Df= f1 f2,between the elements

dDfdt

=Du+K

2[H(Df) H(Df)], (3.1)

where Du is the difference between the natural frequencies of the two oscillators.Equation (3.1) can be simplified to

dDfdt

=Du KH(Df), (3.2)

where H is the odd part of the interaction function, defined as

2H(Df)=H(Df) H(Df). (3.3)

From equation (3.2), it is seen that stationary solutions will occur at phasedifferences that satisfy

Du

K=H(Df). (3.4)

For a homogeneous system (identical elements, Du= 0), the stationary solutions

correspond to the roots of H

(Df). The behaviour of a slightly heterogeneoussystem (Du = 0) can be brought close to that of a homogeneous system byincreasing the interaction strength between the elements. In the limit as K,




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the fixed point solutions approach the roots of H(Df). However, caution mustbe used with this approach since an excessively large interaction strength willdisrupt the waveform and invalidate the phase approximation.

For a given interaction function H(Df), the stationary phase difference, Df,is obtained from the roots of its odd part, H(Df)

2H(Df)=H(Df) H(Df)= 0. (3.5)

By shifting the phase of the interaction function by an amount Dt, Df can betuned to take on values between 0 and 1 rad/2p. (As shall be seen below, the phaseshift, Dt, of H(Df) is equal to the common feedback delay in the experimentsto be discussed.) The stationary phase differences between the two oscillatingelements are then given by the solutions of

HDf + Dt

H

Df + Dt

= 0. (3.6)

By definition, H(Df) is a 2p periodic function; therefore equation (3.6) mustalways have roots at 0 and 0.5rad/2p. These roots represent the trivial in-phase and anti-phase synchronization states, respectively. The non-trivial rootsof equation (3.6) are governed by the higher order terms ofH(Df). An interactionfunction with second-order harmonics is sufficient to obtain a non-trivial,stationary phase difference between the two oscillators. A convenient form forthis purpose is the function

H(Df)= sin(Df+ Dt) + R sin(2(Df+ Dt)). (3.7)

Figure 1a illustrates the interaction function defined in equation (3.7) settingR = 0.1 and Dt= 0, while figure 1b illustrates its associated synchronized statesas a function ofDt. The non-trivial bifurcation branches illustrated in figure 1bare relatively narrow, indicating that small changes in the parameter Dt will causelarge changes in the synchronized state. To reduce this sensitivity, the parameterR was selected to maximize the width of the non-trivial bifurcation branches. Tosimplify the analysis, the parameters of equation (3.7) were constrained such thatonly supercritical bifurcations occur.

To determine the optimum value of R, a linear stability analysis of equation(3.2) was performed, indicating that stable stationary states occur whendH(Df)/dDf> 0. Therefore, the phase-locked solution, Df*, is stable if

H

(Df

+Dt

) H

(Df

+Dt

)< 0. (3.8)From this, it can be seen that the in-phase synchronization state (Df= 0) isstable if

H(Dt)> 0, (3.9)

while the anti-phase state is stable if

H(Dt+ p)> 0. (3.10)

Equations (3.9) and (3.10) indicate that the trivial bifurcation branchescorrespond to the region where H(Df)> 0. Since the trivial and non-trivial

bifurcation branches are mutually exclusive, the non-trivial branch mustcorrespond to the region where H(Df)< 0. Therefore, a wide non-trivialbifurcation branch will occur when H(Df) has a large domain of negative slope.




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0

0.25

0.50

0.75

1.00

0.1 0.2 0.3

Dt(rad/2p)

Df*(ra

d/2p)

0.4 0.5

0

0.25

0.50

0.75

1.00

H(rads1)

Df*(rad/2p)

1.5

1.0

0.5

0

0.5

1.0

1.5(a) (b)

(c) (d)

Df(rad/2p)

H(rad

s1)

01.5

1.0

0.5

0

0.5

1.0

1.5

0.2 0.4 0.6 0.8 1.0

Figure 1. Relationship between H(Df) and the time asymptotic phase behaviour of a two-oscillator system. (a) Interaction function H(Df)= sin(Df+ Dt) + R sin(2(Df+ Dt)), R = 0.1,Dt= 0rad/2p, and (b) the calculated oscillator phase difference. Solid lines denote stable states,

and dotted lines denote unstable states. (c) Optimized interaction function (R = 0.5, Dt=0rad/2p) and (d) the calculated oscillator phase difference.

The value for the parameter R of equation (3.7) was selected tomaximize the width of the non-trivial branch of the bifurcation diagram,that is where H(Df)< 0. The value of R was constrained to the interval0 R 0.5 owing to the presence of subcritical bifurcations outside thisinterval. Numerical analysis determined that the width of the regionwhere H(Df)< 0 monotonically increased with the value of R in this

interval. As a result, the widest non-trivial bifurcation branch occurredwhen R = 0.5. The target interaction function can be seen in figure 1cfor Dt= 0. With these parameters, equation (3.7) can be analyticallysolved for H(Df)= 0, yielding Df= 0,p, arccos(cos(Dt)/ cos(2Dt)). Thestable and unstable stationary states are shown as a function of Dtin figure 1d.

(b) Numerical simulation of phase difference tuning

Before conducting experiments on the experimental system, the methodology

was verified with numerical simulations using the Brusselator oscillator, a simpletwo-variable ODE system that exhibits self-sustained oscillations (Glansdorff &Prigogine 1971). The phase difference between two interacting Brusselator




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Z(f)

02

0

2

4

0.5

f(rad/2p)

1.0

x(f)

00.5

0

0.5

1.0(a) (b)

0.5

f(rad/2p)

1.0

Figure 2. (a) Waveform and (b) phase response function of the Brusselator oscillator. The parametervalues are A= 1.0 and B= 2.3, resulting in T= 6.43s and u= 0.977rads1.

oscillators is tuned to values between 0 and 2p. The dynamical equations foran interacting homogeneous Brusselator population under global feedback are

dxi

dt = (B 1)xi + A2

xi + f(xi, yi) +K

N

Nj=1 h(xj)

dyidt

=Bxi A2yi f(xi, yi) for i= 1, . . . , N,

f(x, y)=B

Ax2 + 2Axy+ x2y

(3.11)

where h(x) is the feedback function defined in equation (2.3) that is constructedfrom and applied to the variables xi. The variables xi and yi are transformedsuch that the fixed point is shifted to (x, y)= (0,0). For a single uncoupledoscillator, a Hopf bifurcation occurs at B=Bc 1 + A2. The parameters of

equation (3.11) were chosen to be A= 1.0 (so that Bc= 2.0) and B= 2.3.The waveform x(f) and the response functions Z(f) along the x-direction aredisplayed in figure 2. The response function was calculated using the XPPAUTprogram (Ermentrout 2002).

A feedback parameter set {kn, tn} is chosen to create a tunable non-trivialsolution such that an arbitrary stationary phase difference Df can be achieved.The target interaction function was selected to be equation (3.7), where R = 0.5.The phase shift of the interaction function can be directly controlled usingthe common feedback delay parameter Dt. For the purpose of determining thefeedback parameters, the common feedback delay can be set to zero. Second-order

feedback is used in order to produce the required first and second harmonics in thetarget interaction function. The constant term in equation (2.3) is arbitrary andcan be safely neglected (k0 = 0). Although both the linear and quadratic termsof the feedback function contain weak third (and higher) order harmonics, theireffect on the interaction function is negligible since the third (and higher) orderharmonics of the response function are small. The feedback parameters are foundby numerically solving for h(x) (Kori et al. 2008), with one solution being

(k1, k2, t1, t2) (2.56, 4.68, 5.34, 1.98). (3.12)

As expected, the magnitudes of the higher harmonics are very small,

|h3Z3| 0.01 and |h4Z4| 0.004, (3.13)

for the parameter set (equation 3.12).




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Df*

0

p/2

p

0.8 1.2 1.6 2.0

a(rad)2.4

Figure 3. Numerical simulation using a system of two Brusselator oscillators. Stationary phasedifference Df as a function ofa, where auDt. The lines show the theoretical prediction (stable,solid; unstable, dotted) from the phase model. Data points are numerically obtained, where Df

for p


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counter electrode

reference

potentiostat

ZRA

computerfeedback (dV)I(t)

V(t) =V0 + dV

Rp

working electrode

3M H2SO4

Figure 4. Schematic diagram of the experimental apparatus. Rp represents a set of resistors, onefor each element in the population. The computer is a real-time data acquisition computer, whichcalculates the feedback signal, dV.

to the mean amplitude of the population (Amean):

Im(t)=Amean

Am(Im(t) IOffsetm ). (4.1)

The host machine was used to continuously determine the offset and amplitudeof each rhythmic element in the population over time. The feedback signal, dV,fed back to the potentiostat was calculated using the equation

dV=K

N

Nm=1

h(xm(t)), (4.2)

where K is the overall feedback gain (i.e. interaction strength), h(x) is equation(2.3) and xm(t) is the potential drop across the double layer for the mth electrode,calculated as

xm(t)= V(t) Im(t)Rp, (4.3)

where V(t) is the applied voltage and Rp is the channel resistance.Before the nickel electrode array was placed into the system, it was polished

with a wet rotary polisher to remove any initial oxide layer that may have beenpresent. Six polishing discs were used, decreasing in roughness from 180 to 4000grit. This ensured that the electrodes started from roughly identical conditionsand that these initial conditions were approximately identical for each experiment.After polishing, the electrodes were placed in the acid solution and the potentialwas ramped from 0.68 to 1.25 V and back to 0 V without any resistors present.

This caused a thin passive oxide layer to form on each electrode. After this cyclewas complete, the resistors were reconnected and the system was brought up tothe desired operating voltage. The system was allowed to line out at this voltage




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1.15(a) (b)

1.10

1.05

1.00

0.95

0.900 2 4 6

time (s)

E

(V)

8 10

1.4

1.2

0.8

1.0

0.60 126 18 24

time (s)

Df

(rad/2p)

30 36

Figure 5. (a) Time series of electrode potential, E, during electrochemical dissolution of twonickel wires in 3M sulphuric acid without feedback (V= 1.165V, Rp = 650U, u1 = 0.4796Hz,u2 = 0.5105 Hz). (b) Unbounded phase difference between the two waveforms seen in ( a).

for approximately 1.52.0 h, as this was found to reduce the amount of drift in theinherent frequency of the oscillators. The electrode potential waveform of bothelements can be seen in figure 5a. In the absence of feedback, the phase differencebetween the two elements increases over time (figure 5b).

(b) Determining the response function

The response function can be determined from equation (2.2), providedthat the interaction function and feedback function are both known. Work byMiyazaki & Kinoshita (2006) was extended to provide a method for measuringthe interaction function of a rhythmic system composed of two non-identicaloscillators under weak global feedback. Physically, the interaction functionrepresents the change in the frequency of a rhythmic element as a function ofits phase relative to all other interacting elements in the system. Therefore,the interaction function can be determined by recording the frequency (orperiod) of two interacting oscillators as a function of their phase difference.Two electrochemical oscillators were created, with different inherent frequencies,and a weak feedback signal was applied to the system. The magnitude ofthe feedback was selected such that the system was unable to synchronize,allowing the phase difference between the elements to grow over time (figure 6a).

The weak interaction caused the period of the two oscillators to fluctuate asthe phase difference between the elements changed (figure 6b). By integratingthe instantaneous frequency of one of the oscillators over the observed periodof a single cycle and equating this to 2p rad (Miyazaki & Kinoshita 2006), theinteraction function can be determined using the equation

H(Df)=2p

KP2base[P(Df) Pbase], (4.4)

where Pbase is the intrinsic period of the oscillator and K is the overallfeedback gain.

Measuring interaction functions under different feedback conditions allowedthe response function (figure 6d) to be calculated from equation (2.2). Thesesets of data were used in a multi-objective optimization to find the single best




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2.10

2.05

pe

riod(s)

2.00

1.950 50 100 150

time (s)

E(V)

200

(a) (b)

(c) (d)

4

3

Df

(rad/2p)

1

2

0 50 100 150

time (s)

200

1.0

0.5

H(Df)

(rads

1)

0.5

0

1.00 0.25 0.50 0.75

Df(rad/2p)

1.00

401.15

0.850 2p

f

20

Z(rad/2p)

0

20

400 0.25 0.50 0.75

f(rad/2p)

1.00

Figure 6. Measuring a response function. (a) Unbounded phase difference of the heterogeneoussmooth oscillator seen for the experimental system under first-order feedback. K= 0.07, k0 = 0 V,k1 = 1, t1 = 0.012 rad/2p. (b) Fluctuations in the period of a single oscillator owing to feedback.(c) Measured interaction function obtained using equation (4.4). (d) Optimized response function

Z(f) and waveform (inset) of a single oscillator.

response function that could reproduce all the measured interaction functionssimultaneously. This fitting was necessary to reduce the effects of experimentalnoise in the interaction function measurements. The Fourier coefficients of theresponse function were used as the optimization parameters. The number ofFourier terms to be optimized was determined by the number of higher harmonicspresent in the waveform and the measured interaction functions. Close to the Hopfbifurcation point (approx. 1.105 V), only one to two terms were necessary sincethe system was largely sinusoidal. At higher circuit potentials (approx. 1.20 V),the system became relaxational and the response function required approximatelyseven terms. At the experimental circuit potential of 1.165 V, four Fourier termswere used since the system was at a slightly higher voltage than the Hopfbifurcation point, but still exhibited relatively smooth oscillations. Additionally,it was found that increasing the number of coefficients did not significantly alterthe shape of the response function. A simplex optimization algorithm was usedto determine the value of the Fourier coefficients by minimizing an objectivefunction of the form

error=

Nn=1

Ptsi=1

(|Tni Dni | )

21/2

, (4.5)




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(b) 1.5

1.0

0.5

0

0.5

1.0

1.50 0.25 0.50 0.75

Df(rad/2p)

1.00

(a) 2

1

0

H

(rads

1)

1

20 0.25 0.50 0.75

Df(rad/2p)

1.00

Figure 7. (a) The target interaction function (dotted line) and the optimized interaction function(solid line) determined by feedback parameter optimization. (b) Interaction function measuredusing the experimental system (dots) and phase model prediction (line) using the responsefunction in figure 6 and feedback parameters (K= 0.03, k0 = 0.03V, k1 = 1.72, k2 =4.6816V1,t1 = 0.012rad/2p, t2 = 0.143rad/2p).

where Tni is the ith data point of the nth measured interaction function, D isthe interaction function calculated using the optimized response function, N isthe number of measured interaction function datasets and Pts is the number ofmeasured points in the function. The initial conditions were taken to be theFourier coefficients of a sine wave, since the response function approaches asine wave as the system approaches the Hopf bifurcation point. On average, theoptimization required approximately 1200 iterations to converge, correspondingto approximately 20 min.

(c) Calculating the feedback signal

The target interaction function (figure 7a) was engineered into theexperimental system using the nonlinear time-delayed feedback defined inequation (2.3). The feedback parameters kn and tn can be calculated usingequation (2.2), provided that the response function of the system and the targetinteraction function are known. The common feedback delay, Dt, was set to zerofor this calculation. A simplex optimization algorithm was used to minimize theerror between the calculated and target interaction functions by manipulatingthe feedback parameters. The objective function for this optimization was

error=

Ptsi=1

(|Ti Hi|)21/2

, (4.6)

where Ti is the ith data point of the target interaction function, H is theoptimized interaction function and Pts is the number of data points in thefunctions T and H. Second-order feedback was used, since this is the highestorder harmonic present in the target interaction function. The initial conditions

for the optimization were taken to be kn= 10n

1 and tn= 0.01rad/2p. This allowseach polynomial feedback term to be of the same order of magnitude, since |x|< 1.The optimization successfully found a feedback parameter set that produced the




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target interaction function with sufficient accuracy:

(K, k0, k1, k2, t1, t2)= (0.03,0.03V,1.72,4.6816V1,0.012rad/2p,0.143rad/2p).

(4.7)

Figure 7a shows the optimized interaction function (solid line) compared withthe target interaction function (dashed line). There is good agreement betweenthe target interaction function and the optimized interaction function. Since theoptimized function has a large domain of negative slope, it will produce a largenon-trivial synchronization region.

The feedback parameters were experimentally validated by measuring theinteraction function produced by the optimized feedback conditions. Figure 7billustrates the agreement between the predicted interaction function and themeasured interaction function, indicating that the optimized parameters weresuccessful in producing the desired interaction function.

(d) Experimental results

The feedback parameters (equation (4.7)) were applied to the experimentalsystem and the common feedback delay, Dt, was adjusted to achieve the desiredphase difference between the two rhythmic elements. The phase of an elementwas linearly interpolated between adjacent peaks in the observed waveform;the phases of the peaks were defined as 0 and 2p rad, respectively. WhenDt= 0rad/2p, the elements phase-synchronized with Df= 0rad/2p (figure8a,d). Under these conditions, the system exhibited only one stable stationary

state, corresponding to the root ofH

(Df

) located atDf=

0 (figure 8g

).Increasing Dt to 0.23rad/2p caused the interaction function to shift, changingthe synchronized phase difference to Df= 0.73 rad/2p (figure 8b,e). In thiscase, the system exhibited bi-stability, in which two stable stationary states(Df= 0.73 and Df= 0.28rad/2p) coexist simultaneously (figure 8h). Furtherincreasing Dt to 0.5rad/2p caused the elements to synchronize in an anti-phaseconfiguration, where Df= 0.5 rad/2p (figure 8c,f). This anti-phase configurationcorresponded to the root ofH(Df) located at Df= 0.5rad/2p (figure 8i), whichwas previously unstable.

A more quantitative comparison between the phase model predictions andexperimental results can be seen in figure 9. The bifurcation diagram in figure 9a

was generated by tracking the roots ofH(Df) (and their associated derivatives)as a function ofDt. Therefore, the diagram represents all the possible stationarystates (both stable and unstable) for the experimental system at different valuesof Dt. To determine the accuracy of these predictions, the stationary phasedifference between two electrochemical elements was experimentally measuredwith different values ofDt. These measurements were made by first synchronizingthe system on the upper non-trivial branch of the bifurcation diagram usingthe appropriate delay and initial conditions. When the initial phase differencebetween the elements was 0.5


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1.15(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

1.10

1.05

E(V)

1.00

0.95

0.90

1.00

0.75

0.50

Df(rad/2p)

0.25

0 2 4time (s)

6

1.5

1.00.5

0

H(rads

1)

0.5

1.0

1.50 0.25 0.50 0.75

Df(rad/2p)

1.00

0 2 4time (s)

6 0 2 4time (s)

6

0.6

0.40.2

0

0.2

0.4

0 0.25 0.50 0.75

Df(rad/2p)

1.00

1.5

1.00.5

0

0.5

1.0

1.50 0.25 0.50 0.75

Df(rad/2p)

1.00

Figure 8. (ac) Time series of the electrode potential, (df) phase difference and (gi) H(Df)of a system of two elements with second-order global feedback (K= 0.03, k0 = 0.03V, k1 = 1.72,k2 =4.6816V1, t1 = 0.012 rad/2p, t2 = 0.143rad/2p). (a,d,g) In-phase synchronization (Dt=

0rad/2p). (b,e,h) Representative out-of-phase synchronization (Dt= 0.23rad/2p). (c,f,i) Anti-phase synchronization (Dt= 0.5rad/2p). The phase loop diagrams indicate the relative position of

the elements in the system and the direction of rotation. The roots of H(Df) are indicated withthe associated stationary state stability (black circles denote stable states, grey squares denoteunstable states).

(a)

0.1 0.2 0.3 0.4 0.5 1.000.750.50

experimental data (rad/2p)

0.25

1.00

0.75

0.50

0.25Df*(rad

/2p)

Dt(rad/2p)

0

(b) 1.00

0.75

0.50

0.25

phasemode

l(rad/2p)

0

Figure 9. (a) Stationary phase difference values of a system of two rhythmic elements undersecond-order feedback (K= 0.03, k0 = 0.03V, k1 = 1.72, k2 =4.6816V1, t1 = 0.012 rad/2p, t2 =0.143rad/2p). Lines represent phase model predictions of the stable stationary states (solid) and

unstable stationary states (dotted). Circles represent experimental measurements with positivefeedback, and triangles represent measurements with negative feedback. (b) Parity plot ofexperimental measurements versus phase model predictions.




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the feedback was removed and the system was allowed to drift into the range ofinitial conditions required to reach the lower branch (0


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systems are considered; it is usually a formidable task to construct an appropriatedetailed mathematical model of a biological system; however, the investigation ofthe phase response function is often possible.

This work was supported in part by the National Science Foundation through grant CBET-0730597.I.Z.K. thanks the organizers for financial aid to attend ECC10. H.K. acknowledges financialsupport from the Grants-in-Aid for Young Scientists (no. 19800001) and the Sumitomo Foundation(no. 071019).

References

Battogtokh, D. & Mikhailov, A. 1996 Controlling turbulence in the complex GinzburgLandauequation. Physica D 90, 8495. (doi:10.1016/0167-2789(95)00232-4)

Belykh, V. N., Osipov, G. V., Kucklander, N., Blasius, B. & Kurts, J. 2005 Automatic controlof phase synchronization in coupled complex oscillators. Physica D 200, 81104. (doi:10.1016/j.physd.2004.10.008)

Di Donato, P. F. A., Macau, E. E. N. & Grebogi, C. 2007 Phase locking control in the circle map.Nonlinear Dyn. 47, 7582. (doi:10.1007/s11071-006-9055-7)

Epstein, I. R. & Pojman, J. A. 1998 Chemical dynamics: oscillations, waves, patterns, and chaos.Oxford, UK: Oxford University Press.

Ermentrout, G. B. 2002 Simulating, analyzing, and animating dynamical systems: a guide toXPPAUT for researchers and students. Philadelphia, PA: SIAM.

Galan, R. F., Ermentrout, G. B. & Urban, N. N. 2005 Efficient estimation of phase-resettingcurves in real neurons and its significance for neural-network modeling. Phys. Rev. Lett. 94,158101. (doi:10.1103/PhysRevLett.94.158101)

Glansdorff, P. & Prigogine, I. 1971 Thermodynamic theory of structure, stability, and fluctuations.London, UK: Wiley.

Iglesia, H. O., Meyer, J., Carpino, A. & Schwartz, W. J. 2000 Antiphase oscillation of the left and

right suprachiasmatic nuclei. Science 290, 799801. (doi:10.1126/science.290.5492.799)Kiss, I. Z., Zhai, Y. M. & Hudson, J. L. 2005 Predicting mutual entrainment of oscillatorswith experiment-based phase models. Phys. Rev. Lett. 94, 248301. (doi:10.1103/PhysRevLett.94.248301)

Kiss, I. Z., Rusin, C. G., Kori, H. & Hudson, J. L. 2007 Engineering complex dynamicalstructures: sequential patterns and desynchronization. Science 316, 18861889. (doi:10.1126/science.1140858)

Kori, H., Rusin, C. G., Kiss, I. Z. & Hudson, J. L. 2008 Synchronization engineering: theoreticalframework and application to dynamical clustering. Chaos 18, 026111. (doi:10.1063/1.2927531)

Kuramoto, Y. 1984 Chemical oscillations, waves and turbulence. New York, NY: Springer.Mikhailov, A. S. & Showalter, K. 2006 Control of waves, patterns and turbulence in chemical

systems.Phys. Rep. 425

, 79194. (doi:10.1016/j.physrep.2005.11.003)Miyazaki, J. & Kinoshita, S. 2006 Method for determining a coupling function in coupledoscillators with application to BelousovZhabotinsky oscillators. Phys. Rev. E 74, 056209.(doi:10.1103/PhysRevE.74.056209)

Oliva, R. A. & Strogatz, S. H. 2001 Dynamics of a large array of globally coupledlasers with distributed frequencies. Int. J. Bifurcation Chaos 11, 23592374. (doi:10.1142/S0218127401003450)

Petrov, V., Ouyang, Q. & Swinney, H. L. 1997 Resonant pattern formation in a chemical system.Nature 388, 655657. (doi:10.1038/41732)

Popovych, O. V., Hauptmann, C. & Tass, P. A. 2006 Control of neuronal synchrony by nonlineardelayed feedback. Biol. Cybern. 95, 6985. (doi:10.1007/s00422-006-0066-8)

Tass, P. A. 1999 Phase resetting in medicine and biology. Stochastic modeling and data analysis.

Berlin, Germany: Springer.Traub, R. D. & Wong, R. K. S. 1982 Cellular mechanism of neuronal synchronization in epilepsy.

Science 216, 745747. (doi:10.1126/science.7079735)


http://dx.doi.org/doi:10.1016/0167-2789(95)00232-4http://dx.doi.org/doi:10.1016/j.physd.2004.10.008http://dx.doi.org/doi:10.1016/j.physd.2004.10.008http://dx.doi.org/doi:10.1016/j.physd.2004.10.008http://dx.doi.org/doi:10.1007/s11071-006-9055-7http://dx.doi.org/doi:10.1103/PhysRevLett.94.158101http://dx.doi.org/doi:10.1103/PhysRevLett.94.158101http://dx.doi.org/doi:10.1103/PhysRevLett.94.158101http://dx.doi.org/doi:10.1126/science.290.5492.799http://dx.doi.org/doi:10.1126/science.290.5492.799http://dx.doi.org/doi:10.1103/PhysRevLett.94.248301http://dx.doi.org/doi:10.1103/PhysRevLett.94.248301http://dx.doi.org/doi:10.1126/science.1140858http://dx.doi.org/doi:10.1126/science.1140858http://dx.doi.org/doi:10.1126/science.1140858http://dx.doi.org/doi:10.1063/1.2927531http://dx.doi.org/doi:10.1016/j.physrep.2005.11.003http://dx.doi.org/doi:10.1103/PhysRevE.74.056209http://dx.doi.org/doi:10.1142/S0218127401003450http://dx.doi.org/doi:10.1142/S0218127401003450http://dx.doi.org/doi:10.1142/S0218127401003450http://dx.doi.org/doi:10.1142/S0218127401003450http://dx.doi.org/doi:10.1038/41732http://dx.doi.org/doi:10.1007/s00422-006-0066-8http://dx.doi.org/doi:10.1126/science.7079735http://dx.doi.org/doi:10.1126/science.7079735http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://dx.doi.org/doi:10.1126/science.7079735http://dx.doi.org/doi:10.1007/s00422-006-0066-8http://dx.doi.org/doi:10.1038/41732http://dx.doi.org/doi:10.1142/S0218127401003450http://dx.doi.org/doi:10.1142/S0218127401003450http://dx.doi.org/doi:10.1103/PhysRevE.74.056209http://dx.doi.org/doi:10.1016/j.physrep.2005.11.003http://dx.doi.org/doi:10.1063/1.2927531http://dx.doi.org/doi:10.1126/science.1140858http://dx.doi.org/doi:10.1126/science.1140858http://dx.doi.org/doi:10.1103/PhysRevLett.94.248301http://dx.doi.org/doi:10.1103/PhysRevLett.94.248301http://dx.doi.org/doi:10.1126/science.290.5492.799http://dx.doi.org/doi:10.1103/PhysRevLett.94.158101http://dx.doi.org/doi:10.1007/s11071-006-9055-7http://dx.doi.org/doi:10.1016/j.physd.2004.10.008http://dx.doi.org/doi:10.1016/j.physd.2004.10.008http://dx.doi.org/doi:10.1016/0167-2789(95)00232-4


16/16


Tsubo, Y., Takada, M., Reyes, A. & Fukai, T. 2007 Layer and frequency dependencies of phaseresponse properties of pyramidal neurons in rat motor cortex. Eur. J. Neurosci. 25, 34293441.(doi:10.1111/j.1460-9568.2007.05579.x)

Winfree, A. T. 1980 The geometry of biological time. New York, NY: Springer.Yamaguchi, S., Isejima, H., Matsuo, T., Okura, R., Yagita, K., Kobayashi, M. & Okamura, H.

2003 Synchronization of cellular clocks in the suprachiasmatic nucleus. Science 302, 14081412.(doi:10.1126/science.1089287)Zhai, Y., Kiss, I. Z. & Hudson, J. L. 2004 Emerging coherence of oscillating chemical reactions

on arrays: experiments and simulations. Ind. Eng. Chem. Res. 43, 315326. (doi:10.1021/ie030164z)


http://dx.doi.org/doi:10.1111/j.1460-9568.2007.05579.xhttp://dx.doi.org/doi:10.1126/science.1089287http://dx.doi.org/doi:10.1021/ie030164zhttp://dx.doi.org/doi:10.1021/ie030164zhttp://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://dx.doi.org/doi:10.1021/ie030164zhttp://dx.doi.org/doi:10.1021/ie030164zhttp://dx.doi.org/doi:10.1126/science.1089287http://dx.doi.org/doi:10.1111/j.1460-9568.2007.05579.x

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Craig G. Rusin et al- Synchronization engineering: tuning the phase relationship between dissimilar oscillators using nonlinear feedback