Synchronization of noise-induced oscillations bytime-delayed feedback

Philipp Hövel, Markus A. Dahlem, Eckehard Schöll

Institut für Theoretische Physik, Technische UniversitätBerlin, 10623 Berlin, Germany

Abstract. We consider two mutually coupled neural populations represented by two FitzHugh-Nagumo systems prepared at parameter values at which no autonomous oscillations occur. Eachsystem is driven by its own source of random fluctuations realized by Gaussian white noise. Weshow that an extended time-delayed feedback scheme is able to influence the global cooperativedynamics of the ensemble of neural populations by local application of the stimulus to a singlesystem. We discuss effects on the stochastic synchronization of coupled neural oscillators in de-pendence upon the control parameters given by the time delay, feedback strength, and memoryparameter. We show that increasing the memory parameter enhances the cooperative dynamics ofthe two subsystems.

Keywords: Synchronization, noise, coupling, time-delayed feedbackPACS: 05.45.-a, 05.40.Ca, 05.45.Tp, 87.19.La

The network of neurons in the brain exhibits a subtle balanceof dynamic chaos andselforganized order. A number of neurological diseases like Parkinson or epilepsy arecharacterized by a disturbance of this balance, e.g. synchronized firing of electricalpulses of the neurons [1, 2]. Modern concepts of time-delayed feedback control haverecently been applied to suppress this undesired synchrony[3, 4].

Various models have been introduced to describe the neural systems. Here, we con-sider two mutually coupled FitzHugh-Nagumo systems which are subject to independentrandom fluctuations [See Fig. 1(a)]:

ε1ẋ1(t) = x1(t)−x31(t)

3−y1(t)+C[x2(t)−x1(t)] (1a)

ẏ1(t) = x1(t)+a+K∞

∑n=0

Rn [y1(t − (n+1)τ)−y1(t−nτ)]+D1ξ1(t)

ε2ẋ2(t) = x2(t)−x32(t)

3−y2(t)+C[x1(t)−x2(t)] (1b)

ẏ2(t) = x2(t)+a+D2ξ2(t),

wherex1,y1 and x2,y2 represent two different neurons or neuron populations, whichare diffusively coupled with coupling strengthsC. The variablesx1,x2 are related tothe transmembrane voltage andy1,y2 correspond to various quantities connected to theelectrical conductance of the relevant ion currents. Throughout this paper, we fix thesystem’s parameters asa = 1.05, ε1 = 0.005, andε2 = 0.01. Each neuron is driven byGaussian white noiseξi(t)(i = 1,2) with zero mean and unity variance. The noise inten-sities are denoted by parametersD1 andD2, respectively. We keepD2 fixed atD2 = 0.09

-2 -1 0 1 2x

-1

0

1

y

(a)

x 1, x

2, x

Σ

D1=

0

x 1, x

2, x

Σ

D1=

0.05

0 20 40 60 80 100

x 1, x

2, x

Σ

t

D1=

1

(b)

FIGURE 1. Panel (a): Dynamics of a single FitzHugh-Nagumo system without control according toEq. (1a) for a noise intensityD = 0.6. Panel (b): Time series of two coupled FitzHugh-Nagumo systems,x1, x2, and their sumxΣ = x1 + x2 for different noise intensitiesD1 = 0,0.05, and 1 in the absence ofcontrol, i.e. K = 0. Other parameters:C = 0.07 andD2 = 0.09. The red dashed and dash-dotted linesdenote the null-isoclines.

in the following.The parameters of the time-delayed feedback scheme are the feedback gainK, the

time delayτ, and the memory parameterR. With this method, a control force is con-structed from the differences of the states of the system which are one time unitτ apart.The memory parameterR∈ (−1,1) can be seen as a weight of states that are further inthe past. One could also consider application of the feedback scheme to both subsystemsand effects of different values of the control parameters for each subsystem, but theseinvestigations are out of the scope of this work.

The caseR= 0, also known as Pyragas control [5], was investigated in Ref. [6]. Thus,we extend the work of Hauschildtet al.[6] by application of a different external stimula-tion with multiple time delays. This method, also known as extended time-delayed feed-back, was initially proposed by Socolaret al.in order to stabilize unstable periodic orbits[7]. It generalizes the Pyragas scheme by introducing an additional memory parameter.Previously, time-delayed feedback has also been used to influence noise-induced oscil-lations of a single excitable system [8, 9]. The effect of extended time-delayed feedbackcontrol upon noise-induced oscillations below a Hopf bifurcation has beeen studied in[10]. For further application of this control scheme see Ref. [11] and the referencestherein.

In the absence of control (K = 0) and for proper choices of the noise intensities andcoupling strength, the two subsystems exhibit cooperativedynamics as can be seen fromthe time series in Fig. 1(b). There are various measures of the synchronization of cou-pled systems. For instance, one can consider the average interspike intervals of eachsubsystem,i.e. 〈T1〉 and〈T2〉, calculated from thex variable of the respective subsystemand their ratio〈T1〉/〈T2〉 as depicted in Fig. 2. The bright regions in Fig. 2(c) indicate astrongly synchronized behaviour of the two subsystems.

It was shown in Ref. [6] that application of time-delayed feedback of Pyragas typeis able to change coherence, time-scales, and synchronization of noise-induced oscil-lations. In the following, we consider effects of the additional control parameterR forthe cases of moderately, weakly, and strongly synchronizedrealizations of the neuralsystem by chosing parameters as (i)C = 0.2,D1 = 0.6, (ii) C = 0.1,D1 = 0.6, and (iii)

0.1 1D1

2

4

6

<T

1>, <

T2>

0.4

0.6

0.8

1

<T

1>/<

T2>

0 0.1 0.2 0.3 0.4 0.5C

4

6

8

<T

1>, <

T2>

0.4

0.6

0.8

1

<T

1>/<

T2>

(a)

(b)

1.0

0.8

0.6

0.4

D1

C

(c)0.80.60.40.20

0.4

0.2

0

FIGURE 2. Absence of control (K = 0): Average interspike intervals of each subsystem (〈T1〉 as blacksolid curve and〈T2〉 as red dashed curve) and their ratio (〈T1〉/〈T2〉 as green dotted curve) on dependenceon the noise intensityD1, the coupling strengthC in panels (a) and (b), respectively. Panel (c): Ratio ofthe average interspike intervals in the (D1,C) plane.

FIGURE 3. Ratio of the average interspike intervals of moderately, weakly, and strongly synchronizedsystems in the left, middle, and right plot, respectively. In each subfigure, panel (a), (b), (c), and (d)corresponds to a memory parameter ofR= 0,0.35,0.7, and 0.9, respectively.

C = 0.2,D1 = 0.15, respectively.The results are shown in Fig. 3. The ratio of the average interspike intervals is de-

picted in the plane parameterized by the time delayτ and feedback gainK, where theleft, middle, and right plot corresponds to the case of moderately, weakly, and stronglysynchronized systems, respectively. The memory parameteris chosen asR= 0,0.35,0.7,and 0.9 in panel (a), (b), (c), and (d) of each subfigure, respectively. In all three cases,one can see that an increase of the memory parameterRyields a more uniform behaviourover a large range of control parameters in comparison to thecase ofR= 0. For instance,the minima atτ = 2 are less pronounced. The extended time-delayed feedback methodis less likely to desynchronize the two subsystems. Therefore, this control method canbe considered more robust in terms of a variation of the feedback gainK and time delayτ. However, not only the cases of strong desynchronization are weakened, but the re-alizations of enhanced cooperative dynamics loose some synchronization as well. Thiscan be seen, for instance, in the right plot of Fig. 3, where the bright regions becomedarker.

These effects can also be seen in Fig. 4, which shows the ratioof the average inter-spike intervals in dependence on the time delayτ. The feedback gain is fixed atK = 1.5in all plots and the memory parameterR is chosen asR= 0 and 0.9 in panels (a) and

3

4

5

6

<T

1>, <

T2>

0.7

0.75

0.8

0.85

<T

1>/<

T2>

0 2 4 6 8 10τ

3

4

5

6

<T

1>, <

T2>

0.7

0.75

0.8

0.85

<T

1>/<

T2>

(a)

(b)

2

3

4

5

6

<T

1>, <

T2>

0.55

0.6

0.65

0.7

0.75

<T

1>/<

T2>

0 2 4 6 8 10τ

3

4

5

6

<T

1>, <

T2>

0.55

0.6

0.65

0.7

0.75

<T

1>/<

T2>

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

4

6

8

10

<T

1>, <

T2>

0.92

0.96

1

<T

1>/<

T2>

0 2 4 6 8 10τ

4

6

8

10

<T

1>, <

T2>

0.92

0.96

1

<T

1>/<

T2>

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

FIGURE 4. Ratio of the average interspike intervals of moderately, weakly, and strongly synchronizedsystems in the left, middle, and right plot, respectively. In each subfigure, panel (a) and (b) corresponds toa memory parameter ofR= 0 and 0.9, respectively. The color code is as in Fig. 2(a) The feedback gainKis fixed atK = 1.5.

(b) of each subfigure, respectively. Thus, Fig. 4 can be understood as a horizontal cutthrough Fig. 3. For a larger memory parameter, both the maxima and minima are lesspronounced.

In conclusion, we have investigated the effect of extended time-delayed feedback onthe cooperative dynamics of two coupled excitable neural systems in the presence ofnoise. We have shown that incorporating states into the control force, which are fur-ther in the past, the feedback scheme is able to enhance the synchronization of the twosubsystems. An increase of the additional memory parameterleads to an enhanced syn-chronization of the two subsystems.

This work was supported by DFG in the framework of Sfb 555 (Complex NonlinearProcesses). P.H. acknowledges support of the Deutsche Akademische Austauschdienst(DAAD) and thanks Kazuyuki Aihara and his group for stimulating discussions.

REFERENCES

1. P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky,J. Volkmann, A. Schnitzler, and H. J.Freund,Phys. Rev. Lett.81, 3291 (1998).

2. P. Grosse, M. J. Cassidy, and H. J. Freund,Clin. Neurophysiol.113, 1523 (2002).3. M. G. Rosenblum, and A. S. Pikovsky,Phys. Rev. Lett.92, 114102 (2004).4. O. V. Popovych, C. Hauptmann, and P. A. Tass,Phys. Rev. Lett.94, 164102 (2005).5. K. Pyragas,Phys. Lett. A170, 421 (1992).6. B. Hauschildt, N. B. Janson, A. G. Balanov, and E. Schöll,Phys. Rev. E74, 051906 (2006).7. J. E. S. Socolar, D. W. Sukow, and D. J. Gauthier,Phys. Rev. E50, 3245 (1994).8. N. B. Janson, A. G. Balanov, and E. Schöll,Phys. Rev. Lett.93, 010601 (2004).9. A. G. Balanov, N. B. Janson, and E. Schöll,Physica D199, 1 (2004).10. J. Pomplun, A. G. Balanov, and E. Schöll,Phys. Rev. E75, 040101 (2007).11. E. Schöll, and H. G. Schuster, editors,Handbook of Chaos Control, Wiley-VCH, Weinheim, 2007,

second completely revised and enlarged edition.