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9PHYSCON 2009, Catania, Italy, September, 1–September, 4 2009

FEEDBACK-DEPENDENT CONTROL OF STOCHASTICSYNCHRONIZATION IN COUPLED NEURAL SYSTEMS

Philipp H ovel, Sarang A. Shah, Markus A. Dahlem, and Eckehard Scholl∗Institut fur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstraße 38, 10623 Berlin, Germany

∗schoell@physik.tu-berlin.de

AbstractWe investigate the synchronization dynamics of two

coupled noise-driven FitzHugh-Nagumo systems, rep-resenting two neural populations. For certain choicesof the noise intensities and coupling strength, we findcooperative stochastic dynamics such as frequencysynchronization and phase synchronization, where thedegree of synchronization can be quantified by the ra-tio of the interspike interval of the two excitable neu-ral populations and the phase synchronization index,respectively. The stochastic synchronization can beeither enhanced or suppressed by local time-delayedfeedback control, depending upon the delay time andthe coupling strength. The control depends cruciallyupon the coupling scheme of the control force, i.e.,whether the control force is generated from the activa-tor or inhibitor signal, and applied to either component.For inhibitor self-coupling, synchronization is moststrongly enhanced, whereas for activator self-couplingthere exist distinct values of the delay time wherethe synchronization is strongly suppressed even in thestrong synchronization regime. For cross-couplingstrongly modulated behavior is found.

Key wordsSynchronization, noise, coupling, time-delayed feed-

back

1 IntroductionThe control of unstable or irregular states of nonlin-

ear dynamic systems has many applications in differ-ent fields of physics, chemistry, biology, and medicine(Scholl and Schuster, 2008). A particularly simpleand efficient control scheme is time-delayed feedback(Pyragas, 1992) which occurs naturally in a numberof biological systems including neural networks whereboth propagation delays and local neurovascular cou-plings lead to time delays (Haken, 2006; Wilson, 1999;Gerstner and Kistler, 2002). Moreover, time-delayedfeedback loops might be deliberately implemented tocontrol neural disturbances, e.g., to suppress unde-sired synchrony of firing neurons in Parkinson’s dis-

ease or epilepsy (Schiffet al., 1994; Rosenblum andPikovsky, 2004a; Popovychet al., 2005). Here westudy coupled neural systems subject to noise and time-delayed feedback (Hauschildtet al., 2006; Hoveletal., 2009; Schollet al., 2009a; Scholl et al., 2009b).In particular we focus upon the question how stochas-tic synchronization of noise-induced oscillations of twocoupled neural populations can be controlled by time-delayed feedback, and how robust this is with respectto different coupling schemes of the control force.Time-delayed feedback control of noise-induced os-

cillations was demonstrated in a single excitable sys-tem (Jansonet al., 2004; Balanovet al., 2004; Prageret al., 2007; Pototsky and Janson, 2008). The simplestnetwork configuration displaying features of neural in-teraction consists of two coupled excitable systems.In order to grasp the complicated interaction between

billions of neurons in large neural networks, thoseare often lumped into groups of neural populationseach of which can be represented as an effective ex-citable element that is mutually coupled to the otherelements (Rosenblum and Pikovsky, 2004b; Popovychet al., 2005). In this sense the simplest model whichmay reveal features of interacting neurons consists oftwo coupled neural oscillators. Each of these will berepresented by a simplified FitzHugh-Nagumo (FHN)system (FitzHugh, 1960; Nagumoet al., 1962), whichis often used as a generic model for neurons, or moregenerally, excitable systems (Lindneret al., 2004).This paper is organized as follow: We introduce the

model equations and the feedback scheme in Sec. 2.Sec. 3 is devoted to two measures of the stochastic syn-chronization. These are investigated for different cou-pling schemes of the feedback in Sec. 4. Finally, weconlcude in Sec. 5.

2 Model EquationsNeurons are excitable units which can emit spikes or

bursts of electrical signals,i.e., the system rests in a sta-ble steady state, but after it is excited beyond a thresh-old, it emits a pulse. In the following, we consider elec-trically coupled neurons modelled by the FitzHugh-

Figure 1. Schematic diagram of two coupled FitzHugh-Nagumo

systems with time-delayed feedback applied to the first subsystem.

K andτ denote the feedback gain and time delay, respectively, and

C is the coupling strength.

Nagumo system in the excitable regime:

ε1du1

dt= f (u1, v1) + C (u2 − u1) (1a)

dv1dt

= g (u1, v1) +D1ξ1 (1b)

ε2du2

dt= f (u2, v2) + C (u1 − u2) (2a)

dv2dt

= g (u2, v2) +D2ξ2 (2b)

with f (ui, vi) = ui−u3

i /3−vi andg (ui, vi) = ui+a(i = 1, 2). The fast activator variablesui (i = 1, 2)refer to the transmembrane voltage, and the slow in-hibitor variablesvi are related to the electrical conduc-tance of the relevant ion currents. The parameterais the excitability parameter. For the purposes of thispaper,a is fixed at1.05, such that there are no au-tonomous oscillations (excitable regime).C is the dif-fusive coupling strength betweenu1 andu2. To intro-duce different time scales for both systems,ε1 is set to0.005 andε2 is set to0.1. Both systems, when uncou-pled, are driven entirely by independent noise sources,which in the above equations are represented byξi(i = 1, 2, Gaussian white noise with zero mean andunity variance).Di is the noise intensity, and for thepurposes of this paper,D2 will be held fixed at0.09(Hauschildtet al., 2006).The control force which we apply only to the first

of the neural populations as schematically depictedin Fig. 1 is known as time-delay autosynchroniza-tion (TDAS) or time-delayed feedback control. Thismethod was initially introduced by Pyragas (Pyragas,1992) for controlling periodic orbits in chaotic systems.It has been effective in a variety of experimental appli-cations at controlling oscillatory behavior and can beeasily implemented in many analog devices (Scholl andSchuster, 2008). TDAS constructs a feedbackF fromthe difference between the current value of a controlsignalw and the value for that quantity at timet − τ .

The difference is then multiplied by the gain coefficientK

F (t) = K[w(t− τ) − w(t)], (3)

wherew determines which components of the systementer the feedback as will be discussed in the following.The variablew in the control force can be either the

activatoru1 or the inhibitorv1. Also, the control forcecan either be applied to the activator or the inhibitordifferential equation. These possibilities lead to twoself-coupling schemes (uu andvv) where either the ac-tivator is coupled to the activator equation or the in-hibitor is coupled to the inhibitor equation, and twocross-coupling schemes (uv andvu). Thus, Eqs. (1)of the first subsystem can be rewritten including time-delayed feedback as

(

ε1du1

dtdv1dt

)

=

(

f (u1, v1) + C (u2 − u1)g (u1, v1) +D1ξ1

)

(4)

+K

(

Auu Auv

Avu Avv

)(

u1(t− τ) − u1(t)v1(t− τ) − v1(t)

)

,

where the coupling matrix elementsAij with i, j ∈{u, v} define the specific coupling scheme.Next, we will discuss cooperative stochastic dynamics

resulting in frequency synchronization and phase syn-chronization in the following Sections.

3 Measures of SynchronizationA measure of frequency synchronization is the ratio of

the interspike intervals (ISI) of the two neural popula-tions (Hauschildtet al., 2006; Hovelet al., 2009). Therespective average ISI of each neural population is de-noted by〈T1〉 and〈T2〉. The ratio〈T1〉/〈T2〉 comparesthe average time scales of both systems, where unity ra-tio describes two systems spiking at the same averagefrequency. It is for this reason that the ISI ratio is oftenconsidered as a measure of frequency synchronization.It does not contain information about the phase of syn-chronization, and a given ISI ratio can also result fromdifferent ISI distributions.In order to account for the phase difference between

two systems, one can define a phase (Pikovskyet al.,1996; Pikovskyet al., 2001; Hauschildtet al., 2006)

ϕ(t) = 2πt− ti−1

ti − ti−1

+ 2π(i− 1) (5)

wherei = 1, 2, . . . . ti denotes the time of theith spike.The phase difference between two consecutive spikes is2π. The phase difference of 1:1 synchronization is

∆ϕ(t) = |ϕ1(t)− ϕ2(t)| , (6)

whereϕ1(t) andϕ2(t) are the phases of the first andsecond system, respectively. Two systems that are

0 0.2 0.4 0.6 0.8D1

0

0.2

0.4

0.6

C

(a)

0 0.2 0.4 0.6 0.8 1D1

0.5 0.6 0.7 0.8 0.9 1

(b)

Figure 2. (Color online) Panels (a) and (b) show the ratio of inter-

spike intervals〈T1〉/〈T2〉 and the phase synchronization indexγof the two subsystems as color code in dependence on the coupling

strengthC and noise intensityD1, respectively. No control is ap-

plied to the system. The dots mark the parameter choice for different

synchronization regimes used in the following. Other parameters:

ε1 = 0.005, ε2 = 0.1, a = 1.05, andD2 = 0.09.

phase synchronized at a given time satisfy∆ϕ = 0.Finally, the overall time-averaged phase synchroniza-tion of two systems can be quantified using the syn-chronization index

γ =√

〈cos∆ϕ(t)〉2 + 〈sin∆ϕ(t)〉2. (7)

A value of 0 indicates no synchronization, while avalue of unity indicates perfect synchronization.Figure 2 depicts both measures for stochastic synchro-

nization in the(D1, C) plane, both exhibiting very sim-ilar behavior. Panel (a) refers to the frequency synchro-nization characterized by the ratio of the average ISIs〈T1〉/〈T2〉 and panel (b) shows the phase synchroniza-tion indexγ. The green dots mark parameter valuesused in Sec. 4. Note that both panels share the samecolor code. For a small value ofD1 and large couplingstrength, the two subsystems display well synchronizedbehavior,〈T1〉/〈T2〉 ≈ 1 andγ ≈ 1. The timescales inthe interacting systems adjust themselves to1 : 1 syn-chronization. On average, they show the same numberof spikes and the two subsystems are in-phase which isindicated by yellow color. The two subsystems are lesssynchronized in the dark blue and black regions.In the following we show the ratio of the average

interspike interval〈T1〉/〈T2〉 and the phase synchro-nization indexγ which are color coded in the(τ,K)plane for fixed combinations ofD1 andC. For eachcoupling scheme of time-delayed feedback control(cross-coupling schemesuv andvu and self-couplingschemesuu andvv) we present a selection of(D1, C)values. In all cases, only one element of the couplingmatrix A is equal to unity and all other elements arezero.

4 Coupling SchemesAfter the introduction of the system and the cou-

pling schemes, we will present results on frequency andphase synchronization in the following. We consider16

different combinations of the noise intensityD1 and thecoupling strengthC which are marked as green dots inFig. 2. The ordering of panels in Figs. 3 to 10 is the fol-lowing: The rows correspond to fixed coupling strengthchosen asC = 0.01, 0.21, 0.41, and0.61 from bottomto top. The columns in each figure are calculated forconstant noise intensityD1 = 0.01, 0.34, 0.67, and1.0from left to right.

4.1 Frequency SynchronizationFigures 3 to 6 show frequency synchronization mea-

sured by the ratio of average interspike intervals〈T1〉/〈T2〉 calculated from the summarized activatorvariableuΣ = u1 + u2 as color code in dependenceon the feedback gainK and the time delayτ . Thesystem’s parameters are fixed in each panel as de-scribed above. Figures 3 and 6 correspond to self-coupling (uu- andvv-coupling) and Figs. 4 and 5 de-pict the cross-coupling schemes (uv- andvu-coupling).The dynamics in the white regions is outside the ex-citable regime and does not show noise-induced spik-ing, but rather the system exhibits large-amplitude self-sustained oscillations.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 3. uu-coupling: Ratio of average interspike intervals

〈T1〉/〈T2〉 as color code. Rows and columns correspond to con-

stant coupling strengthC and noise intensityD1, respectively, as

marked in Fig. 2 as green dots, and specified in the text. Otherpa-

rameters as in Fig. 2.

One can see that appropriate tuning of the controlparameters leads to enhanced or deteriorated synchro-nization displayed by bright yellow and dark blue ar-eas, respectively. In each figure, all panels show qual-itatively similar features like a modulation of the ratio〈T1〉/〈T2〉 whose range between maximum and min-imum depends onD1 andC. Comparing the rows,the systems are less (more strongly) synchronized for

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 4. uv-coupling: Ratio of average interspike intervals

〈T1〉/〈T2〉 as color code. Rows and columns correspond to con-

stant coupling strengthC and noise intensityD1, respectively.

Other parameters as in Fig. 2.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 5. vu-coupling: Ratio of average interspike intervals

〈T1〉/〈T2〉 as color code. Rows and columns correspond to con-

stant coupling strengthC and noise intensityD1, respectively.

Other parameters as in Fig. 2.

small (large) values ofC indicated by dark blue (yel-low) color. As the noise intensityD1 increases, thedynamics of the coupled subsystems is more and morenoise-dominated and the dependence on the time delayτ becomes less pronounced.

Note the symmetry in the cross-coupling schemesshown as Figs. 4 and 5 betweenK and its negativevalue−K for the inverse cross-coupling. The reasonis that enhancing the activator yields a similar effectson the dynamics as diminishing the inhibitor variable.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 6. vv-coupling: Ratio of average interspike intervals

〈T1〉/〈T2〉 as color code. Rows and columns correspond to con-

stant coupling strengthC and noise intensityD1, respectively.

Other parameters as in Fig. 2.

4.2 Phase SynchronizationFigures 7 to 10 depict the phase synchronization indexγ as color code depending on the control parametersK andτ for uu-, uv-, vu-, andvv-coupling, respec-tively. The noise intensityD1 and coupling strengthCare fixed for each panel as described in Sec. 4.1.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 7. uu-coupling: Phase synchronization indexγ. Noise in-

tensityD1 and coupling strengthC chosen as described in Sec. 4.1.

Other parameters as in Fig. 2.

Comparing Figs. 7 to 10 with the respective plots forfrequency synchronization, i.e., Figs. 3 to 6, one cansee that both types of synchronization coincide qual-itatively, but the phase synchronization index is more

sensitive to the modulation features. Similar to the caseof frequency synchronization, time delayed feedbackcan lead to either enhancement or suppression of phasesynchronization depending on the specific choice of thefeedback gainK and time delayτ indicated by yellowand dark blue regions. In general, these effects becomeless sensitive on the time delay asD1 increases. Forlarger values ofC, the two subsystems show enhancedphase synchronization.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 8. uv-coupling: Phase synchronization indexγ. Noise in-

tensityD1 and coupling strengthC chosen as described in Sec. 4.1.

Other parameters as in Fig. 2.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 9. vu-coupling: Phase synchronization indexγ. Noise in-

tensityD1 and coupling strengthC chosen as described in Sec. 4.1.

Other parameters as in Fig. 2.

8

2

0 2 4 6

τ

-2

-1

0

1

K

-2

2

-1

0

1

K

-2

2

-1

0

1

K

-2-1

0

1

2

K

0 82 4 6

τ0 82 4 6

τ0 2 4 6 8

τ

0

0.2

0.4

0.6

0.8

1

Figure 10. vv-coupling: Phase synchronization indexγ. Noise in-

tensityD1 and coupling strengthC chosen as described in Sec. 4.1.

Other parameters as in Fig. 2.

5 ConclusionIn summary, we have shown that stochastic synchro-

nization in two coupled neural populations can be tunedby local time-delayed feedback control of one popula-tion. Synchronization can be either enhanced or sup-pressed, depending upon the delay time and the cou-pling strength. The control dependents crucially uponthe coupling scheme of the control force. For inhibitorself-coupling (vv) synchronization is most strongly en-hanced, whereas for activator self-coupling (uu) thereexist distinct values ofτ where the synchronization isstrongly suppressed even in the strong synchronizationregime. For cross-coupling (uv, vu) there is mixed be-havior, and both schemes exhibit a strong symmetrywith respect to inverting the sign ofK. These obser-vations might be important in the context of the delib-erate application of control with the aim of suppress-ing synchronization, e.g. as therapeutic measures forParkinson’s disease.

AcknowledgementsThis work was supported by DFG in the framework

of Sfb 555 (Complex Nonlinear Processes). S. A. S.acknowledges support of the Deutsche AkademischeAustauschdienst (DAAD) in the framework of the pro-gram Research Internships in Science and Engineering(RISE).

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