Feedback controller for destroying synchrony in an array of the FitzHugh–NagumooscillatorsArūnas Tamaševičius, Elena Tamaševičiūtė, and Gytis Mykolaitis Citation: Applied Physics Letters 101, 223703 (2012); doi: 10.1063/1.4768938 View online: http://dx.doi.org/10.1063/1.4768938 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/101/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback Chaos 23, 033122 (2013); 10.1063/1.4817393 Harmonics and intermodulation in subthreshold FitzHugh–Nagumo neuron Chaos 19, 033144 (2009); 10.1063/1.3234239 Highly synchronized noisedriven oscillatory behavior of a FitzHugh—Nagumo ring with phaserepulsive coupling AIP Conf. Proc. 887, 89 (2007); 10.1063/1.2709590 Synchronized firing of FitzHugh–Nagumo neurons by noise Chaos 15, 023704 (2005); 10.1063/1.1929687 Parameter dependence of stochastic resonance in the stochastic FitzHugh-Nagumo neuron AIP Conf. Proc. 501, 250 (2000); 10.1063/1.59940

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Feedback controller for destroying synchrony in an array of theFitzHugh–Nagumo oscillators

Ar�unas Tama�sevicius,1 Elena Tama�sevici�ut _e,1 and Gytis Mykolaitis2

1Center for Physical Sciences and Technology, LT-01108 Vilnius, Lithuania2Vilnius Gediminas Technical University, LT-10223 Vilnius, Lithuania

(Received 8 October 2012; accepted 12 November 2012; published online 27 November 2012)

We describe an implementation of an electronic feedback controller, destroying synchrony and/or

suppressing the mean field in arrays of globally coupled nonidentical oscillators. We demonstrate

that the mean field, either artificially nullified or fed back into the array with a negative sign can

break up the phase synchronization. The experiments have been carried out with an array of thirty

electronic oscillators, imitating dynamical behavior of the spiking neurons. We have found that the

negative mean-field technique, depending on the control parameter, can either desynchronize or

synchronize the oscillators, whereas in the both cases, it ensures low mean-field voltage. VC 2012American Institute of Physics. [http://dx.doi.org/10.1063/1.4768938]

Synchronization is a universal and very common phe-

nomenon, widely observed in nature, science, engineering,

and social life.1 Coupled oscillators and their arrays, exhibit-

ing synchrony, range from pendulum clocks to electronic

oscillators, chaotic lasers, chemical systems, and various bio-

logical populations.1–3 Though in the most cases synchroni-

zation plays a positive role, sometimes it has a negative

impact. Strong synchronization in the human brain is an

example. It is widely believed that synchrony of spiking neu-

rons in a large neuronal population causes the symptoms of

the Parkinson’s disease.4

Therefore, the development of the methods and practical

techniques for controlling, more specifically, for suppressing

synchrony of coupled oscillators, in general, and particularly

with possible application to large neuronal arrays, is of great

importance.4–6 Seeking for the frustration of synchrony, the

local feedback methods, based on the inversion of the mean

field,7,8 might be promising, despite some criticism,

expressed by Pyragas et al.6

In this letter, we describe a practical electronic control-

ler for destroying synchrony in an array of globally coupled

oscillators. To achieve the goal, the mean field is either artifi-

cially nullified or fed back into the array with a negative

sign. To be specific, we investigate an array of the mean-

field coupled FitzHugh–Nagumo (FHN) oscillators, which

imitate the dynamics of spiking neurons. However, the

method requires neither the knowledge of dynamics of the

individual oscillators, nor the access to their individual varia-

bles and parameters. Therefore, other systems like the Hind-

marsh–Rose neurons9 are expected to be controlled in a

similar way.

An array of the mean-field coupled FHN non-identical

oscillators is given by

_xi ¼ xi � x3i =3� yi þ ci þ kðxm � xiÞ;

_yi ¼ eðxi � byiÞ:(1)

Here i¼ 1,2,…,N. The xi and the yi correspond to the mem-

brane potential and the recovery variable,10 respectively (in

the original paper by FitzHugh,10 the equations have slightly

different form), k is the coupling coefficient. Note that the

parameter ci is different for each individual oscillator, thus

making them non-identical units. In Eq. (1), the xm is the

mean value of the xi

xm ¼1

N

XN

i¼1

xi: (2)

Therefore, the set of oscillators given by Eq. (1) is called ei-

ther globally or mean-field coupled array. This type of cou-

pling is widely known to give the synchronization effect.

Following the paper by Tsimring et al.7 we call the non-

identical oscillators synchronized or phase-locked, if they

have fixed (not necessarily zero) phase differences. There

are many other coupling possibilities, described in literature,

yielding synchronous behavior of periodic and chaotic oscil-

lators, including the FHN systems. We just mention here

synchronization of two weakly coupled chaotic FHN oscilla-

tors11 (chaotic oscillations can appear in the FHN system

due to the periodic driving force), where synchrony is

achieved by means of applying appropriate external input(s).

The purpose of this work is to demonstrate that using

the feedback control of the mean field it is possible either to

destroy the synchronized state of the individual oscillators,

and/or to diminish essentially their mean field. To achieve

the goal, we replace in Eq. (1) the xm with the controlled

mean x�m, which is found from an algebraic equation (the

physical meaning of this equation is illustrated in the experi-

mental part with a specific example of the Kirchhoff’s law)

kXN

i¼1

ðxi � x�mÞ � Cx�m ¼ 0; (3)

where C is a parameter of the external control (C is either

positive or negative). It follows from Eq. (3) that

x�m ¼kN

kN þ Cxm: (4)

Without the control (C¼ 0), the x�m ¼ xm, as expected.

0003-6951/2012/101(22)/223703/5/$30.00 VC 2012 American Institute of Physics101, 223703-1

APPLIED PHYSICS LETTERS 101, 223703 (2012)

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There are two special and important cases of control.

Case 1: if jCj � kN, then the controlled mean x�m � 0, i.e.,

the attractive coupling is nullified. We note the difference

between the uncoupled oscillators (k¼ 0) and the nullified

coupling (x�m¼ 0): the presence of terms –kxi in the latter

case. However, for k < 1, these terms involve only small

local damping and do not cause any synchronization effect

of the array. Case 2: if C < –kN, then the controlled mean x�mbecomes negative. Specifically, at C ¼ �2kN, the x�m ¼ �xm.

The special case with x�m ¼ �xm, called the repulsive cou-

pling, has been considered analytically and numerically for

the array of simple one-dimensional phase oscillators (the

Kuramoto model).7,8 Below, we present the results, both nu-

merical and experimental, for a more complicated model,

namely, the two-dimensional FHN system.

Simulation results, obtained from Eq. (1), are shown in

Fig. 1. The following parameter values have been used:

e¼ 0.3, b¼ 0.1, k¼ 0.1, N¼ 30, the individual parameters ci

range from �5.0 to �4.5 with the increment ciþ1¼ ci þ 0.05

and c1 ¼ �5. Phase portrait (x1, x30) in Fig. 1(a) and all the

other phase portraits xi, xj 6¼i (not plotted in Fig. 1) demon-

strate synchronization effect of the all oscillators in the

uncontrolled array (k 6¼ 0, xm > 0). The complicated phase

portrait shown in Fig. 1(b), in contrast, evidences that the

controlled oscillators (k 6¼ 0, x�m¼�xm) are not phase

synchronized. The high amplitude of the mean-field variable

xm (Fig. 1(c)), observed in the case of the synchronized oscil-

lators (C¼ 0), decreases drastically, by a factor of more than

10 (Fig. 1(d)), when the control is applied (C¼�2kN).

In addition, we have carefully examined the time series

xi(t) of all the thirty individual oscillators in the controlled

array. An interesting observation is that the oscillations are

amplitude-modulated. The modulation depth is about 5%,

the envelopes of the waveforms seem to be chaotic.

An individual FHN type electronic cell, used as a build-

ing block of the array, is presented in Fig. 2. The nominal

values of the circuit elements, employed to build the hard-

ware oscillators, are the following: R1¼R2¼ 1 kX,

R3¼R5¼ 510 X, R4¼ 30 X, R6¼ 220 X, R7 has been set dif-

ferent for each individual oscillator, R7i ¼ [25 þ (i � 1)] kX,

i¼ 1,…,30, the inductor coil L¼ 10 mH, C¼ 3.3 nF. The

OA are the NE5534 type operational amplifiers; D1 and D2

are the BAV99 type p–n junction diodes. The dc bias voltage

Vdc ¼ �15 V.

A similar electronic FHN type oscillator, having some-

what different biasing circuitry and operating at 100 times

lower frequencies, i.e., with the inductance element L¼ 1 H

(implemented in the form of an active operational amplifier

based gyrator) and the capacitor C¼ 330 nF, has been

described by Tama�sevicius et al.12

The coupled array of the electronic oscillators and the

electronic feedback controller are sketched in Fig. 3 (the

uncontrolled array of the coupled oscillators and its proper-

ties have been described in details elsewhere13).

The input resistance R0 of the feedback controller is neg-

ative, specifically R0¼�R03 for R01¼R02. An important

note is that the coupling node M, in general, cannot be

accessed directly, but via a buffer resistor R**, which should

be assumed of the same order like the coupling resistors R*

Therefore, the node M cannot be simply “grounded” to nul-

lify the mean field.

FIG. 1. Array of coupled FitzHugh�Nagumo oscillators, N¼ 30. (a) and (b)

Phase portraits, x30 vs. x1. (c) and (d) Waveforms xm(t). (a) and (c) Uncon-

trolled array, C¼ 0. (b) and (d) controlled array, C¼�6.

FIG. 2. The FitzHugh�Nagumo type electronic oscillator.

FIG. 3. Array of the mean-field coupled oscillators 1,2,…,N (left) and feed-

back controller (right). R* ¼ R** ¼ 5.1 kX, R01 ¼ R02 ¼ 510 X, R03 ¼ 10 kX(adjustable), switch S is shown in the OFF position.

223703-2 Tama�sevicius, Tama�sevici�ut _e, and Mykolaitis Appl. Phys. Lett. 101, 223703 (2012)

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Using the first and the second Kirchhoff’s circuit laws

(the current and the voltage laws), the array of coupled oscil-

lators can be described by the following set of ordinary dif-

ferential equations

CdVi

dt¼ Vi

R3

� IDi � Ii þVdc � Vi

R7iþ V�m � Vi

R�;

LdIi

dt¼ Vi � R6Ii:

(5)

Here, i¼ 1,2,…,30, IDi¼ ID1i þ ID2i. The current-voltage

characteristics of the diode-resistor composites D1�R4 and

D2�R5 are given by the transcendental expressions

ID1i ¼ �Is expID1iR4 � Vi

VT� 1

� �;

ID2i ¼ Is expVi � ID2iR5

VT� 1

� �:

(6)

In Eq. (6), Is is the saturation current of the diodes

(�5� 10�9 A for the BAV99 devices), the VT¼ kBT/q is the

thermal potential (�25 mV at 300 K).

The voltage V�m at the coupling node M, taking into

account the current leakage via the buffer resistor R** and

the input resistance R0 of the controller, is found from the

first Kirchhoff’s law

XN

i¼1

Vi � V�mR�

� I�m ¼ 0: (7)

We emphasize that the controller is an essentially

two-terminal feedback device. It senses the voltage V�m at the

coupling node M and feeds back into the array the current I�m¼ V�m/(R** þ R0).

It is convenient to introduce the following dimensionless

variables and the dimensionless parameters:

xi ¼Vi

VT; yi ¼

qIi

VT; t! t

R3C; x�m ¼

V�mVT

;

a ¼ R3Is

VT; b ¼ R6

q; ci ¼

R3

R7i

Vdc

VT; gi ¼

R3

R7i;

q ¼ffiffiffiffiL

C

r; e ¼ R3

q; k ¼ R3

R�; C ¼ R3

R�� þ R0

:

(8)

Using the notations (8) and neglecting small terms �gixi (gi

� 0.01), we come to a set of differential equations

_xi ¼ xi � f ðxiÞ � eyi þ ci þ kðx�m � xiÞ;_yi ¼ eðxi � byiÞ:

(9)

Here, the nonlinear function f(xi) for small correction terms

ID1R4 and ID2R5 in Eq. (6) can be presented analytically as

f ðxiÞ ¼ a½expðxiÞ � expð�xiÞ� ¼ 2a sinhðxiÞ: (10)

Though the hyperbolic sine function sinh(x) formally differs

from the cubic parabola x3 in Eq. (1), these two nonlinear

functions are qualitatively similar. Whereas, the controlled

mean x�m, derived from Eq. (7), exactly coincides with for-

mula (4).

Experiments with the hardware array (Fig. 4) of coupled

oscillators confirm the main features demonstrated by means

of numerical simulations. Single loop in the phase portrait

(Fig. 4(a)) shows that the oscillators are synchronized in the

uncontrolled array (C ¼ 0). The multi-loop phase portrait

(Fig. 4(b)), in contrast, indicates that the oscillators are not

in synchrony in the controlled array (R**þR0 ¼ �85 X,

C ¼ �6). Also, the high/low amplitudes of the mean-field

voltage Vm are typical characteristics of the uncontrolled/

controlled arrays, (Fig. 4(c))/(Fig. 4(d)) arrays, respectively.

For the circuit element values, used in the experiment, the

coupling coefficient k � 0.1; for N ¼ 30, the kN � 3. Thus,

the control parameter C ¼ �6, introduced by the controller,

corresponds to C ¼ �2kN and provides x�m ¼ �xm, i.e.,

implements experimentally the repulsive coupling technique,

considered theoretically by Tsimring et al.7 We have not pre-

sented here the experimental results for the case of the nulli-

fied mean (jCj � kN, x�m ! 0), because qualitatively they

look the same as for C ¼ �6 (Figs. 4(b) and 4(d)).

Using an analog spectrum analyzer, we have taken the

power spectra of the mean-field voltage Vm at four different

values of the control parameter (Fig. 5).

FIG. 4. Experimental results from the circuit in Fig. 3. (a) and (b) Phase por-

traits, V30 vs. V1. (c) and (d) Waveforms of the mean-field voltage Vm. (a)

and (c) Uncontrolled array, C¼ 0. (b) and (d) Controlled array, C¼�6 (R**

þ R0 ¼ �85 X).

223703-3 Tama�sevicius, Tama�sevici�ut _e, and Mykolaitis Appl. Phys. Lett. 101, 223703 (2012)

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Depending on the value of the control parameter C, sev-

eral different situations are observed.

Situation 1: at C ¼ 0 (no control), all the cells are

synchronized due to the positive (attractive) mean field Vm

and oscillate, as expected, at the same frequency fm, which is

indicated by a single discrete line in Fig. 5(a). Here, the 2nd

harmonic 2 fm ¼ 24 kHz and the higher harmonics of the fmare out of the spectral range 11 to 16 kHz.

Situation 2: at jCj � kN (nullified mean field, V�m � 0),

all the cells are desynchronized and oscillate at their natural

frequencies, discretely distributed from 12.3 to 13.8 kHz in

the narrow band of approximately 1.5 kHz. In Fig. 5(b), not

all the 30 lines are distinguishable, because some cells oscil-

late at frequencies separated by only 1 or 2 Hz, i.e., less than

the spectral resolution of the analyzer (3 Hz). The mean-field

voltage Vm is low, like in Fig. 4(d).

Situation 3: at C < �kN, e.g., at C ¼ �6, the individual

oscillators are also desynchronized, however, their frequen-

cies are continuously spread in the “broadband” spectrum

(Fig. 5(c)), typical to chaotic signals. The Vm is low (Fig.

4(d)).

Situation 4: at�5� C<�3, the cells become synchron-

ized again as evidenced by a single spectral line in Fig. 5(d).

The spectral line is shifted towards slightly higher frequen-

cies, compared with the case of the uncontrolled array (Fig.

5(a)). However, in contrast to the uncontrolled array, the

mean-field voltage Vm remains low, like in Fig. 4(d). This

indicates that the oscillators are in the antiphase states. More

precisely, the phases are distributed on the interval between

0 and 2p.

To summarize, we have investigated the dynamics of

the mean-field coupled non-identical FHN type oscillators.

We have proposed an electronic feedback controller, which

enables to avoid synchrony and/or essentially reduces the

mean field of the interacting oscillators.

From a methodological point of view, the electrical cir-

cuits (Fig. 2 and Fig. 3, left), used in this work and in the pre-

vious paper by Tama�sevici�ut_e et al.13 to mimic the dynamical

behavior of coupled neurons, as well as many other designs,

e.g., an electronic analog of a mammalian cochlea,14 can be

treated as simple and extremely fast analog modeling devices

for investigating complex biological systems. Whereas, the

described electronic feedback controller (Fig. 3, right) can be

considered as a prototype of a practical device for suppressing

undesirable synchrony of coupled oscillators.

An important feature of the controller is that it is a two-

terminal device. Thus it differs from more complicated four-

terminal feedback designs, developed so far for suppressing

neural synchrony, e.g., described by Tukhlina et al.15 and in

the related papers.16–18 The feedback controller, proposed in

this letter, uses the same single node to register (to measure)

the mean-field voltage V�m and to feed back the stimulation

by injecting the current I�m ¼ V�m/(R** þ R0) into the array. In

the paper by Pyragas et al.,6 an assumption is made that the

simultaneous registration and stimulation of the whole array

is impossible and, therefore, the repulsive coupling7 and

some other advanced feedback techniques fail to suppress

the synchrony of the coupled oscillators. In contrast to this,

we demonstrate the effectiveness of the local feedback tech-

nique either to destroy phase synchronization of the oscilla-

tors or at least to suppress the mean field in the array, using a

single node for registration and stimulation. The possibility

to control a dynamical system via a single node arises due to

the fact that the controller senses the electric voltage, but

feeds back the electric current. Therefore, there is no inter-

ference between the two different electrical measures.

Concerning practical application of the proposed method

to real neural systems, we point out that the same electrode

setup, used in the conventional deep brain stimulation19,20

(DBS), namely, the probes implanted in either globus pallidus

or subthalamic nucleus can be exploited. The implanted pulse

generator (IPG) used for the DBS should be replaced with the

implanted feedback controller (IFC). An important advantage

of the proposed technique over the DBS is that the signals

sent into the brain from the IFC are estimated to be nearly 100

times smaller than those sent from the IPG.

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FIG. 5. Experimental power spectra of the mean-field voltage Vm in the

range of 11–16 kHz with spectral resolution of 3 Hz at different control pa-

rameters C. (a) Uncontrolled, C¼ 0, fm¼ 12.0 kHz (since the mean-field

voltage Vm is high, the signal has been reduced by 30 dB), (b)–(d) controlled

array; (b) nullified mean-field voltage Vm*, jCj ! 1 (jR** þ R0j � 0), fm ¼

12.3…13.8 kHz, (c) and (d) repulsive control; (c) C¼ �6 (R** þ R0 ¼ �85

X), (d) C¼ �5 (R** þ R0 ¼ �103 X), fm¼ 13.3 kHz.

223703-4 Tama�sevicius, Tama�sevici�ut _e, and Mykolaitis Appl. Phys. Lett. 101, 223703 (2012)

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223703-5 Tama�sevicius, Tama�sevici�ut _e, and Mykolaitis Appl. Phys. Lett. 101, 223703 (2012)

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