Neurocomputing 74 (2011) 3296–3304

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

n Corr

Pusan N

Republi

E-m

kshong@1 Te2 Te

journal homepage: www.elsevier.com/locate/neucom

Synchronization of multiple chaotic FitzHugh–Nagumo neurons with gapjunctions under external electrical stimulation

Muhammad Rehan 1,a, Keum-Shik Hong a,b,n, Muhammad Aqil 2,a

a Department of Cogno-Mechatronics Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Koreab School of Mechanical Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Korea

a r t i c l e i n f o

Article history:

Received 15 October 2010

Received in revised form

14 February 2011

Accepted 13 May 2011

Communicated by D. Barberneurons in addition to the conventionally considered synchronization errors between the master and

Available online 24 June 2011

Keywords:

Chaos synchronization

L2 gain reduction

Nonlinear, robust and adaptive control

FitzHugh–Nagumo neurons

12/$ - see front matter & 2011 Elsevier B.V. A

016/j.neucom.2011.05.015

esponding author at: Department of Cogn

ational University, 30 Jangjeon-dong, Geu

c of Korea. Tel.: þ82 51 510 2973; fax: þ82 5

ail addresses: rehan@pusan.ac.kr (M. Rehan),

pusan.ac.kr (K.-S. Hong), aqil@pusan.ac.kr (M

l.: þ82 51 510 2454; fax: þ82 51 514 0685.

l.: þ82 51 510 2454; fax: þ82 51 514 0685.

a b s t r a c t

This paper discusses the synchronization of three coupled chaotic FitzHugh–Nagumo (FHN) neurons

with different gap junctions under external electrical stimulation. A nonlinear control law that

guarantees the asymptotic synchronization of coupled neurons (with reduced computations) is

proposed. The developed control law incorporates the synchronization error between two slave

the slave neurons, which make the proposed scheme computationally more efficient. Further, a novel L2

gain reduction criterion has been developed for multi-input multi-output systems with non-zero initial

conditions, and is applied to robust synchronization of FHN neurons under L2 norm bounded distur-

bance and uncertainties. Furthermore, a robust adaptive nonlinear control law is developed, which is

capable of handling variations in nonlinear part of synchronization error dynamics, without using any

neural-network-based training-oriented adaptive scheme. The proposed control schemes ensure global

synchronization with computational simplicity, easy way of design and implementation and avoiding

extra measurements. The results obtained with the proposed control laws are verified through

numerical simulations.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Dynamic behavior of neuron is widely studied to explore thechief role of neuronal spiking for effective neurotransmission andbrain signal processing [1–5]. External electrical stimulation(EES), like deep brain stimulation, is a therapy for cognitivedisorders such as Parkinson’s disease, epilepsy and dystonia [6].Synchronization of chaotic neurons under external stimulationplays a major role in the transmission of neural signals andenables efficient communication between the brain and themuscles [7,8]. Investigation of neuronal synchronization, for keyunderstanding of the neural circuit functioning and the braininformation processing, has become one of the widely studiedproblems in the field of neuroscience [9–12]. It has attractedmany brain researchers over the past decade in order to under-stand the underlying mechanism of external stimulation andhence to improve the stimulation therapy based treatments for

ll rights reserved.

o-Mechatronics Engineering,

mjeong-gu, Busan 609-735,

1 514 0685.

. Aqil).

cognitive diseases. FitzHugh–Nagumo (FHN) neural model, undersinusoidal electrical stimulation, is widely utilized for the syn-chronization study due to its potential ability of representingdynamical aspects of neurons [13–15].

Researchers have used various control strategies for synchro-nization of different models of neural networks without gapjunctions [14–16]. Gap junctions are protein channels by whichneurons communicate with each other [10,17–19]. Incorporationof the strength of gap junctions into the dynamics of coupledchaotic FHN neurons renders derivation of a control law for theirsynchronization difficult [20–22]. Some researchers have incor-porated gap junctions for studying the behavior of two coupledchaotic neurons [23], and, in fact, various control strategies havebeen used for synchronization of two chaotic neurons with gapjunctions. For instance in [20,21], synchronization has beenachieved by canceling the gap junction terms present in FHNneuron dynamics, which cannot be possible due to variation inthe strength of gap junctions. Moreover, the control laws reportedrecently for two neurons with gap junctions [20,21] are compu-tationally complex. Two control inputs have been utilized for asingle slave neuron, which is not applicable in reality, making thetechnique [20] still more conservative.

Dynamical behavior, network architecture and synchroniza-tion of multiple interacting chaotic FHN neurons with gap junc-tions have been although addressed in the literature [18,24], but

M. Rehan et al. / Neurocomputing 74 (2011) 3296–3304 3297

still, the development of control strategies for synchronization ofmultiple coupled chaotic FHN neurons with different and uncer-tain gap junctions remained elusively rare up to this date.Synchronization of multiple interlinked chaotic neurons withdifferent and uncertain gap junctions is challenging due to theirinherently complex coupling. Nonetheless, development of acontrol law for synchronization of a network of FHN neurons istantalizingly attractive owing to its potential utility for restora-tion of effective neuro-system communication under EES. Mostrecently, synchronization of multiple oscillators has been dis-cussed in [25], but much detailed work remains to be done.

In this paper, we address the issue of synchronization of threecoupled chaotic FHN neurons with gap junctions, by applyingnonlinear, robust and robust adaptive control strategies. A novelL2 gain reduction criterion for systems with non-zero initialconditions has been developed and applied to synchronize FHNneurons under L2 norm bounded disturbance. As gap junctionsrepresent the properties of the medium between any twoconnected neurons, which differ for each of the connection, wetake different and uncertain gap junctions between any twointerlinked FHN neurons and ensure robust synchronization ofuncertain chaotic neurons. The proposed control schemes arecomputationally more efficient than the results reported in [20–22],and utilize a single control input for each of the slave neurons.Moreover, consideration of controller design without cancelingthe gap junction terms makes the proposed strategies less con-servative than the conventional methods. Our schemes considersynchronization of three FHN neurons, which is helpful forsynchronization of a network of FHN neurons with known neuralconnections.

In order to deal with model variations associated with non-linear part of neural dynamics, a computationally efficient robustadaptive control law has also been developed (see also [26–30]).This robust adaptive control strategy guarantees asymptoticsynchronization of three interconnected FHN neurons andensures the L2 gain reduction from disturbance to synchroniza-tion error. The proposed control schemes are utilizing linearmatrix inequalities (LMI’s) for computing large matrices in orderto reduce parameter tuning efforts [31–34]. The main contribu-tion of this paper is described below:

(1)

This paper proposes a novel L2 gain reduction criterion frominput to output of a multi-input multi-output (MIMO)continuous-time system with non-zero initial condition andprovides its application for synchronization of chaotic FHNneurons under disturbance. The alternative l2 gain reductioncriterion for discrete-time systems is also reported.

(2)

This paper describes nonlinear, robust and robust adaptivecontrol schemes for synchronization of three coupled chaoticFHN neurons with uncertain and different strengths of gapjunctions. This work provides an understanding of issues forsynchronization of multiple FHN neurons.

(3)

The proposed control strategies are computationally efficient,global, easy to design and implement, avoid extra measure-ments and reduce parameter tuning efforts, hence suitable forimplementation.

Simulation results that validate the proposed methodology arealso presented. This work can be a step towards the synchroniza-tion control of a network of coupled FHN neurons with uncertainand different strengths of gap junctions.

This paper is organized as follows. Section 2 provides theL2 and l2 gain reduction criteria for MIMO systems with non-zeroinitial conditions. Section 3 presents the FHN model of threeneurons coupled with different and uncertain gap junctions.Section 4 demonstrates the computationally efficient nonlinear

and robust nonlinear control laws for synchronization of cou-pled neurons. Section 5 discusses the proposed robust adap-tive nonlinear control law for neuronal synchronization alongwith the pertinent simulation results. Section 6 draws theconclusions.

Notations: We use standard notations. The L2 norm of a vectord(t) is represented by

:dðtÞ:2 ¼

Z 10

:dðtÞ:2dt

� �1=2

,

where :d(t): represents the Euclidean norm of d(t) and t repre-sents time. The L2 gain from a vector d(t) to another vector z(t) isrepresented by sup:d:2 a0ð:zðtÞ:=:dðtÞ:2Þ. For discrete-time sys-tems, the l2 norm of a vector d(n) is represented by

:dðnÞ:l2 ¼Xt ¼ 1t ¼ 0

:dðnÞ:2

!1=2

,

where n denotes the nth sample; and the l2 gain from a vectord(n) to another vector z(n) is represented by sup:d:l2 a0ð:zðnÞ:l2=

:dðnÞ:l2Þ. Identity and null matrices of appropriate dimensions arerepresented by I and O, respectively.

2. L2 and l2 gain reduction criteria

Many physical systems are affected by disturbances and noisesthat can be bounded in L2 norm sense. The main objective, forstabilization, regulation, tracking and synchronization of physicalsystems is to minimize the effect of L2 norm bounded distur-bances and noises at output. However, the traditional L2 gainreduction criteria [35–37], from input to output of a system,are based on the supposition of zero initial condition. The aim ofthis section is to provide a novel L2 gain reduction criterion frominput to output of general MIMO continuous-time control sys-tems having non-zero initial conditions. Additionally, due tosignificance of discrete-time control systems, the counterpartof this criterion in terms of l2 gain reduction has also beenpresented.

Consider a general class of continuous-time MIMO systemsgiven by

_s ¼ f ðs,w,t,tÞ, z¼ gðs,w,t,tÞ ð1Þ

where s(t)ARm, w(t)ARp and z(t)ARq represent the vectors forstate, input and output, respectively; f(s,w,t,t)ARm and g(s,w,t,t)ARq. Here, t and t represent time and time-delay, respectively. Notethat the delay t can also be zero for systems without time-delay.We take the following assumption:

Assumption 1. The L2 norm of input w is bounded by wm, that is:w:2rwm, where wm is a positive scalar.

To address the L2 gain reduction from input to output of aMIMO system (1) with any initial condition, we introduce thefollowing important lemma.

Lemma 1. Consider the system (1) satisfying Assumption 1. Suppose

that there exists a Lyapunov function E(s,t)40 (or E(s,t,t)40) such

that

_Eðs,tÞþzT z�g2wT wo0, ð2Þ

where g40 is a scalar. Then, the following are ensured.

(i)

The system (1) is asymptotically stable if w(t)¼0.

(ii)

The output z satisfies :zðtÞ:2

2o ðg2þðEðs,0Þ=w2

mÞÞ:wðtÞ:2

2 , if

w(t)a0. In other words, the L2 gain from w(t) to z(t) is less

thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðEðs,0Þ=w2

mÞp

.

M. Rehan et al. / Neurocomputing 74 (2011) 3296–33043298

Proof. If _Eðs,tÞþzT z�g2wT wo0, then the following two caseshold:

(a)

If w(t)¼0, then _Eðs,tÞþzT zo0, which implies that _Eðs,tÞo0.Hence the system (1) is asymptotically stable.

(b)

If w(t)a0, then, integrating (2) from 0 to To-N, we get

limTo-1

ðEðs,ToÞ�Eðs,0ÞÞþ

Z To

0zT z dt�g2

Z To

0wT w dt

� �o0: ð3Þ

As E(s,To)40, (3) implies

:z:2

2�g2:w:2

2�Eðs,0Þo0, ð4Þ

which can be written as

ð:z:2

2=:w:2

2Þ�g2�ðEðs,0Þ=:w:2

2Þo0 ð5Þ

Using :w:2rwm, we get

ð:z:2

2=:w:2

2Þ�g2�ðEðs,0Þ=w2

mÞo0, ð6Þ

which implies that :zðtÞ:2

2oðg2þðEðs,0Þ=w2

mÞÞ:wðtÞ:2

2. Hence, the

L2 gain from input w(t) to output z(t) is less thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðEðs,0Þ=w2

mÞp

, which completes the proof of Lemma 1. Note

that (6) also implies

ð:z:2

2=:w:2

2Þ�g2�ðEðs,0Þ=w2

mÞ�2gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðs,0Þ=w2

m

qo0, ð7Þ

which states that the L2 gain from w(t) to z(t) is also less than

ðgþffiffiffiffiffiffiffiffiffiffiffiffiEðs,0Þ

p=wmÞ. &

Remark 1. In the past (for instance, see [35–38]), a large number

of research papers used inequalities like _Eðs,tÞþzT z�g2wT wo0for stabilization, regulation, tracking and synchronization oflinear, nonlinear, continuous-time, discrete-time and time-delaysystems, assuming s(0)¼0 (or s(t�t)¼0, 8trt, for time-delaysystems). It ensures that the L2 gain from w(t) to z(t) is less than g.

However, Lemma 1 shows that the inequality _Eðs,tÞþzT z�g2wT

wo0, providing the L2 gain from w(t) to z(t) less thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðEðs,0Þ=w2

mÞp

, is also well-applicable for MIMO systems (1)

with non-zero initial conditions. The results proposed by Lemma1 are more general, because if E(s,0)¼0 (corresponding tos(0)¼0), the upper bound on L2 gain from w(t) to z(t) becomesg. It is interesting to note that the L2 gain also depends on theinitial condition of a system because E(s,0) depends on s(0).

Remark 2. The minimization of the effect of input w(t) at output

z(t) can be achieved by minimizing the L2 gainffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðEðs,0Þ=w2

mÞp

or ðgþffiffiffiffiffiffiffiffiffiffiffiffiEðs,0Þ

p=wmÞ, which requires the minimization of g and

E(s,0). Here, the minimization of E(s,0) depends on the selection ofa Lyapunov function. For most frequently used quadratic Lyapu-nov function E(s,t)¼sT(t)Ps(t) with P¼PT40, we can introduce a

new inequality sT ð0ÞPsð0ÞrmsT ð0Þsð0Þ with scalar m40 and mini-mize E(s,0) by minimizing m. If the bound wm and s(0) are known,

thenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðmsT ð0Þsð0Þ=w2

mÞp

can be minimized to obtain the

optimal value offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðsT ð0ÞPsð0Þ=w2

mÞp

, otherwise a weighted

combination of g (or g¼ g2) and m can be used for optimization.We suggest to use more weight for g (or g) in optimizationbecause m is associated with the Lyapunov function that has tosatisfy other constrains like stability, convergence rate, perfor-mance and robustness. It is also worth mentioning that if initialcondition s(0) is not exactly known, then we can introduce abound on initial condition. For instance, the upper bound on the

L2 gain becomesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þZ=w2

m

pif we confine the initial condition

within an ellipsoid s(0)AsT(0)Ps(0)rZ for a positive scalar Z.

In discrete-time systems, the l2 gain reduction from input tooutput is used for minimizing the effects of unwanted noisesand disturbances (for example, see [38]). Now, we provide thecounterpart of Lemma 1 due to wide applicability of the l2 gainreduction in discrete-time control systems. Consider the MIMOdiscrete-time system given by

sðnþ1Þ ¼ f ðs,w,nÞ, zðnÞ ¼ gðs,w,nÞ: ð8Þ

Assumption 2. The l2 norm of input w(n) is bounded by a positivescalar wm, that is :wðnÞ:l2rwm.

Lemma 2. Consider the system (8) satisfying Assumption 2. Suppose

that there exists a Lyapunov function E(s,n)40 such that

Eðsðnþ1Þ,nþ1Þ�EðsðnÞ,nÞþzT z�g2wT wo0, ð9Þ

where g40 is a scalar, then the following are ensured:

(i)

The system (8) is asymptotically stable, if w(n)¼0. (ii) The output z satisfies :zðnÞ:2

l2o ðg2þðEðs,0Þ=w2

mÞÞ:wðnÞ:2

l2 , if

w(n)a0. That is, the l2 gain from w(n) to z(n) is less thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðEðs,0Þ=w2

mÞp

.

Proof. If (9) holds, then the following two cases hold:

(a)

If w(n)¼0, then E(s(nþ1),nþ1)�E(s(n),n)o0. It implies thatthe system (8) is asymptotically stable.

(b)

If w(n)a0, then, summing (9) from 0 to N, we get

limN-1

EðsðNÞ,NÞ�Eðsð0Þ,0ÞþXN�1

n ¼ 0

zT z�g2XN�1

n ¼ 0

wT w

!o0: ð10Þ

Using a similar procedure as for the proof of Lemma 1, we get

ð:z:2

l2=:w:2

l2Þ�g2�ðEðs,0Þ=w2

mÞo0: ð11Þ

Hence the l2 gain from w(n) to z(n) is less thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðEðs,0Þ=w2

mÞp

,

which completes the proof of Lemma 2. For l2 gain optimization,the same method described by Remark 2 can be used. &

3. FHN model description

Consider the following model of three coupled chaotic FHNneurons [18,39] with different gap junctions:

dx1

dt¼ x1ðx1�1Þð1�rx1Þ�y1� ~g12ðx1�x2Þ� ~g13ðx1�x3Þ

þða=oÞcosotþd1,

dy1

dt¼ bx1, ð12Þ

dx2

dt¼ x2ðx2�1Þð1�rx2Þ�y2� ~g12ðx2�x1Þ� ~g23ðx2�x3Þ

þða=oÞcosotþd2,

dy2

dt¼ bx2, ð13Þ

dx3

dt¼ x3ðx3�1Þð1�rx3Þ�y3� ~g13ðx3�x1Þ� ~g23ðx3�x2Þ

þða=oÞcosotþd3,

dy3

dt¼ bx3, ð14Þ

where x and y represent the state variables of a neuron represent-ing the activation potential and the recovery voltage, respec-tively; x1 and y1 are the states of the master FHN neuron, x2 and y2

are the states of the first slave FHN neuron, and x3 and y3 are thestates of the second slave FHN neuron. Here ~g12, ~g13 and ~g23

M. Rehan et al. / Neurocomputing 74 (2011) 3296–3304 3299

represent the strengths of gap junctions between the masterand the first slave neurons, between the master and the secondslave neurons, and between the two slave neurons, respectively.Disturbances at the master, the first slave and the second slaveneurons are represented by d1, d2 and d3, respectively. The termða=oÞcosot represents the external stimulation current with timet and angular frequency o. In the present study, we use theangular frequency o and the amplitude a as dimensionlessquantities as specified for FHN model [21,30,39].

Traditionally, researchers have considered the synchronizationof identical neurons (for example, see [14–20,22–24]); however,these coupled neurons can in no way be identical. If slightdifferences in the strengths of gap junctions ( ~g12, ~g13 and ~g23)are accounted, the coupled neurons (12)–(14) are not completelyidentical. Additionally, the parameters representing the strengthsof gap junctions are uncertain due to the property variations ofthe medium between interlinked neurons. As synchronization ofneurons is affected by the properties of gap junctions, we considerrobust synchronization of FHN neurons under different anduncertain parameters ( ~g12, ~g13 and ~g23). Due to these facts, themodel (12)–(14), in contrast to [18], has different and uncertainparameters ( ~g12, ~g13 and ~g23) for each of the linkages among the

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2-0.5

0

0.5

1

1.5

2

y 1x 2

x1

-0.4 -0.2 0 0.2-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x 3

y 2

y1

-0.5 0 0.5-0.5

0

0.5

1

1.5

2

y 3

x1-0.4 -0.2 0 0.2

-0.5

0

0.5

1

1.5

2

y 2

Fig. 1. Chaotic behavior of coupled FHN neurons with gap junctions in EES. All three neu

(c) state space of x3 and y3, (d) state space of x1 and x2, (e) state space of x1 and x3, (f) s

(i) state space of y2 and y3.

three neurons. We take

~g12 ¼ g12þDg12, ~g13 ¼ g13þDg13, ~g23 ¼ g23þDg23, ð15Þ

where g12, g13 and g23 represent the nominal values and Dg12,Dg13 and Dg23 represent the uncertainties in strengths of gapjunctions. We fix the parameters of the model as

r¼ 10, g12 ¼ 0:011, g13 ¼ 0:012, g23 ¼ 0:013,

o¼ 0:254p, b¼ 1 and a¼ 0:1, ð16Þ

and the initial condition as

x1ð0Þ ¼ 0, y1ð0Þ ¼ 0, x2ð0Þ ¼ 0:3, y2ð0Þ ¼ 0:3,

x3ð0Þ ¼�0:3 and y3ð0Þ ¼ �0:3: ð17Þ

Fig. 1 shows state-space plots of the three coupled chaotic FHNneurons under zero uncertainties and disturbances (see also[18,39]). Note that all three neurons are non-synchronous sincethe plots in Fig. 1(d)–(i) do not make straight lines of slope 1passing through the origin.

To synchronize three FHN neurons, we use two control inputsu1 and u2 for the first and second slave neurons, respectively. Thecoupled nonlinear model (12)–(14) with two control inputs is,

0.4 0.6 0.8 1x1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x 3

x2

1 1.5 2y1

-0.5 0 0.5 1 1.5 2-0.5

0

0.5

1

1.5

2

y 3

y2

0.4 0.6 0.8 1x2

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

y 3

x3

rons are non-synchronous: (a) state space of x1 and y1, (b) state space of x2 and y2,

tate space of x2 and x3, (g) state space of y1 and y2, (h) state space of y1 and y3 and

M. Rehan et al. / Neurocomputing 74 (2011) 3296–33043300

then, given by

dx1

dt¼ x1ðx1�1Þð1�rx1Þ�y1� ~g12ðx1�x2Þ� ~g13ðx1�x3Þ

þða=oÞcosotþd1,

dy1

dt¼ bx1, ð18Þ

dx2

dt¼ x2ðx2�1Þð1�rx2Þ�y2� ~g12ðx2�x1Þ� ~g23ðx2�x3Þ

þða=oÞcosotþu1þd2,

dy2

dt¼ bx2, ð19Þ

dx3

dt¼ x3ðx3�1Þð1�rx3Þ�y3� ~g13ðx3�x1Þ� ~g23ðx3�x2Þ

þða=oÞcosotþu2þd3,

dy3

dt¼ bx3: ð20Þ

The goal of the present study is to design proper controlsignals u1 and u2 for synchronization of both the slave neuronswith the master neuron.

4. Robust nonlinear control

This section describes a nonlinear state feedback control lawfor synchronization of three FHN neurons (18)–(20), by selectingcontrol signals u1 and u2 to achieve x2,x3-x1 and y2,y3-y1.A robust nonlinear synchronization control law has also beendeveloped against model uncertainties and disturbances. Theproposed control law for synchronization of three coupled chaoticFHN neurons is given by

u1 ¼ Coðx1�x2Þ�ðð1þrÞx22�rx3

2Þþðð1þrÞx21�rx3

1Þ, ð21Þ

u2 ¼ Coðx1�x3Þ�ðð1þrÞx23�rx3

3Þþðð1þrÞx21�rx3

1Þ, ð22Þ

where Co is a constant to be determined. Similar to [40], twolinear feedback terms Co(x1�x2) and Co(x1�x3) are used to ensurethe convergence of the states of two slave neurons to the state ofthe master neuron; while nonlinear feedback terms ðð1þrÞx2

1�

rx31Þ, ðð1þrÞx2

2�rx32Þ and ðð1þrÞx2

3�rx33Þ are used to cancel the

nonlinear terms in synchronization error dynamics, which willbe discussed later. Now we provide a sufficient condition forsynchronization of the three FHN neurons (18)–(20) using controllaws (21) and (22) in the absence of uncertainties anddisturbances.

Theorem 1. Consider the coupled FHN neurons under d1¼d2¼d3¼

0 and Dg12¼Dg13¼Dg23¼0. The nonlinear control laws ensure the

asymptotic synchronization of three neurons if the following matrix

inequalities are satisfied:

P40, AT PþPAo0, ð23Þ

where

A¼

�ð1þCoþ2g12Þ �g13 g23 �1 0 0

�g12 �ð1þCoþ2g13Þ �g23 0 �1 0

g12 �g13 �ð1þCoþ2g23Þ 0 0 �1

b 0 0 0 0 0

0 b 0 0 0 0

0 0 b 0 0 0

2666666664

3777777775

,

ð24Þ

and PAR6�6 is a symmetric matrix.

Proof. Incorporating (21) and (22) into (19) and (20), theoverall closed-loop system becomes

dx1

dt¼ x1ðx1�1Þð1�rx1Þ�y1� ~g12ðx1�x2Þ� ~g13ðx1�x3Þ

þða=oÞcosotþd1,

dy1

dt¼ bx1, ð25Þ

dx2

dt¼�rx3

1þð1þrÞx21�x2�y2� ~g12ðx2�x1Þ

� ~g23ðx2�x3Þþða=oÞcosotþCoðx1�x2Þþd2

( ),

dy2

dt¼ bx2, ð26Þ

dx3

dt¼�rx3

1þð1þrÞx21�x3�y3� ~g13ðx3�x1Þ

� ~g23ðx3�x2Þþða=oÞcosotþCoðx1�x3Þþd3

( ),

dy3

dt¼ bx3: ð27Þ

The following synchronization errors are taken into account.

e1 ¼ x1�x2, e2 ¼ x1�x3, e3 ¼ x2�x3, ð28Þ

e4 ¼ y1�y2, e4 ¼ y1�y2 and e6 ¼ y2�y3: ð29Þ

The purpose of control laws (21) and (22) is to ensure theconvergence of synchronization errors (28) and (29) to zero for aproper selection of Co. The derivatives of the error terms in (28) and(29) along (25)–(27) yield the synchronization error dynamics:

de1

dt¼�ð1þCoþ2 ~g12Þe1� ~g13e2þ ~g23e3�e4þd1�d2, ð30Þ

de2

dt¼� ~g12e1�ð1þCoþ2 ~g13Þe2� ~g23e3�e5þd1�d3, ð31Þ

de3

dt¼ ~g12e1� ~g13e2�ð1þCoþ2 ~g23Þe3�e6þd2�d3, ð32Þ

de4

dt¼ be1, ð33Þ

de5

dt¼ be2, ð34Þ

de6

dt¼ be3, ð35Þ

which can be rewritten in the state space form as

de

dt¼ AeþDAeþB ~d, ð36Þ

where

e¼ e1 e2 e3 e4 e5 e5� �T

, ð37Þ

DA¼DA11 O

O O

� �, ð38Þ

DA11 ¼

�2Dg12 �Dg13 Dg23

�Dg12 �2Dg13 �Dg23

Dg12 �Dg13 �2Dg23

264

375, ð39Þ

B¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

264

375

T

, ð40Þ

~d ¼ d1�d2 d1�d3 d2�d3� �T

, ð41Þ

and A has been already defined in (24). From (36), it is clear that thesynchronization error dynamics is linear, due to the cancelation of

(ii)

M. Rehan et al. / Neurocomputing 74 (2011) 3296–3304 3301

the nonlinear terms in the model with those in the control law. Nowconsider the following quadratic Lyapunov function candidate:

Eðe,tÞ ¼ eTgPe, ð42Þ

where P40, and we take g¼1 for the present case. The derivative of(42) along (36) becomes

_Eðe,tÞ ¼ eTgðAT PþPAÞeþeTgðDAT PþPDAÞeþgeT PB ~dþg ~dTBT Pe:

ð43Þ

Using ~d ¼ 0, DA¼0 and g¼1, and ATPþPAo0, _Eðe,tÞo0 is con-cluded, which completes the proof of Theorem 1. &

Remark 3. The proposed control laws (21) and (22) are muchsimpler than that in [20–22], as it does not include the gapjunctions and terms like y1�y2 and y1�y3, among others. More-over, only a single control input is used for each slave neuron toachieve a computationally efficient control law. Furthermore, thiscontrol law does not require extra measurements of recoveryvoltages. Such a control law is advantageous for real-timesynchronization of multiple chaotic coupled FHN neurons.

Remark 4. In the literature [14–16,19–22,25,30,31], synchroni-zation errors only between the master and slave neurons wereconsidered. In the present scenario, however, the synchronizationerrors between two slave neurons (e3, e6) are considered inaddition to those between the master and the slave neurons (e1,e2, e4 and e5). Note that the convergence of e1, e2, e4 and e5 to zeroensures the convergence of e3 and e6. This implies that e3 and e6

should not be considered in (28) and (29) as synchronizationerrors. However, if in the control law derivation, state differencesin slave neurons (e3, e6) are not used, one has to incorporateredundant terms in control signals (21) and (22) for cancelation ofterms g23e3, �g23e3 and �e6 present in (30)–(32). Hence, for thesynchronization of multiple neurons, it is vital to consider thesynchronization errors between the states of any two slaveneurons for obtaining a simple controller with less computationalcomplexity.

The main problem for synchronization of chaotic systemsunder L2 disturbance is that the initial condition cannot be zero(that is, e(t)a0 at t¼0), while the traditional controller designtechniques are based on the assumption e(0)¼0. This problemcan be resolved by applying Lemma 1. To address the robustsynchronization of FHN neurons under disturbance and uncer-tainty, we take the following assumptions:

Assumption 3. The L2 norm of disturbance ~d is bounded as: ~d:2r

~dm, where ~dm is a positive scalar.

Assumption 4. The uncertain terms Dg12, Dg13 and Dg23 arebounded as 9Dg129ogm, 9Dg139ogm and 9Dg239ogm, where gm

is a positive scalar.

Now, by virtue of Lemma 1, we provide a sufficient condition

for robust synchronization of three FHN neurons (18)–(20) under

bounded disturbance and parametric uncertainties.

Theorem 2. Consider the three coupled FHN neurons satisfying

Assumption 3-4. Suppose the optimization problem:

minc1gþc2msuch that

P¼ PT 40, PomI, g40, e40, ð44Þ

c¼

AT PþPAþeM PB I P

n �g 0 0

n n �g 0

n n 0 �e

26664

37775o0, ð45Þ

with

M¼M11 O

O O

� �, ð46Þ

M11 ¼ g2m

6 3 �3

3 6 3

�3 3 6

264

375, ð47Þ

where m, g and e are scalars, P is a symmetric matrix and scalars c1

and c2 are the optimization weights. The nonlinear control laws

ensure the following:

(i)

Asymptotic synchronization of the FHN neurons, if ~d ¼ 0 .

The L2 gain from ~d to e is less than

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðgeð0ÞT Peð0Þ= ~d

2

mÞ

q.

Proof. Consider the following inequalities:

Eðe,tÞ ¼ eTgPeogmeT e, ð48Þ

_Eðe,tÞþeT e�g2 ~dT ~do0: ð49Þ

Here (48) ensures PomI. By application of Lemma 1 for

Lyapunov function (42), the inequality (49) ensures asymptotic

synchronization of FHN neurons if ~d ¼ 0 and the L2 gain from ~d to

e is less than

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2þðgeð0ÞT Peð0Þ= ~d

2

mÞ

q. Using (43) into (49), we get

eT ðAT PþPAÞeþeT ðDAT PþPDAÞeþeT PB ~d

þ ~dTBT Peþg�1eT e�g ~d

T ~do0: ð50Þ

Using (38), (39), (46), (47) and Assumption 4, it is trivial to obtain

DATDA¼DAT

11DA11 O

O O

" #rM, ð51Þ

due to the specific structure of DA in (38) and (39). For any scalar

e40 and using (51), we obtain

DAT PþPDAreDATDAþe�1P2reMþe�1P2: ð52Þ

Incorporating (52) into (50), we get

eT ðAT PþPAþeMþe�1P2þg�1IÞeþeT PB ~d

þ ~dTBT Pe�g ~d

T ~do0, ð53Þ

which is equivalent to

AT PþPAþeMþe�1P2þg�1I PB

n �g

" #o0: ð54Þ

By applying the Schur complement [41,42], we obtain the

matrix inequalities (44) and (45), which complete the proof of

Theorem 2 &.

5. Robust adaptive control

In the previous section, a nonlinear robust control law, based oncancelation of nonlinear terms in synchronization error dynamics,has been developed. However, due to uncertainty in the nonlinearpart of FHN neurons, exact cancelation cannot be possible. Thissection addresses a robust adaptive control law to ensure synchro-nization of neurons under uncertainty in the nonlinear part of neuraldynamics, in addition to L2 bounded disturbance and uncertaintiesin gap junctions. The classical way of dealing with the nonlinear partuncertainty is to develop a control law by considering wholenonlinear function as an uncertainty. For instance, the adaptationlaw [22], approximating whole nonlinear function using a neural-network-based approach, is however computationally complex due

M. Rehan et al. / Neurocomputing 74 (2011) 3296–33043302

to the use of a number of neural nodes. Moreover, such synchroni-zation techniques, requiring additional software for training of theneural networks, are capable to provide the local synchronizationdue to the use of local approximation of nonlinear functions. Theother less conservative way is to adapt uncertain parameter r

present in the nonlinear part of FHN neurons (see [43–45]). In thissection, we present a computationally simpler continuous-timerobust adaptive control law ensuring global synchronization of threeFHN neurons. The proposed control law is given by

u1 ¼ Coðx1�x2Þ�ðð1þ r̂Þx22�r̂x3

2Þþðð1þ r̂Þx21�r̂x3

1Þ, ð55Þ

u2 ¼ Coðx1�x3Þ�ðð1þ r̂Þx23�r̂x3

3Þþðð1þ r̂Þx21�r̂x3

1Þ, ð56Þ

where r̂ is a time varying adaptive parameter of the proposedcontrol law.

Assumption 5. The parameter r is bounded by a positive value rm.That is

9r9orm: ð57Þ

The adaptation law for r̂ is given by

_̂r¼�0:5ðeT PFðxÞþFT ðxÞPeÞS, ð58Þ

where S is a positive scalar, and

FðxÞ ¼ f1 f2 f3 0 0 0h iT

, ð59Þ

with

f1 ¼ ðx22�x3

2�x21þx3

1Þ, ð60Þ

f2 ¼ ðx23�x3

3�x21þx3

1Þ, ð61Þ

f3 ¼ ðx23�x3

3�x22þx3

2Þ: ð62Þ

Theorem 3. Consider the three coupled FHN neurons satisfying

Assumptions 3–5. Suppose the following optimization problem:

min c1gþc2msuch that

P40, PomI, ð63Þ

g40, e40, co0, ð64Þ

where m, g and e are scalars, P is a symmetric matrix and scalars c1

and c2 are the optimization weights. The nonlinear control law along

with the adaptation law (58) ensures the following:

(i)

(ii)

Asymptotic synchronization of the FHN neurons, if ~d ¼ 0 .ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq

The L2 gain from ~d to e is less than g2þðgeð0ÞT Peð0Þ= ~d

2

mÞ .

Proof. Incorporating the control law (55) and (56) into (19) and(20), the closed-loop system becomes

dx1

dt¼�rx3

1þð1þrÞx21�x1�y1� ~g12ðx1�x2Þ� ~g13ðx1�x3Þ

þða=oÞcosotþd1,

dy1

dt¼ bx1, ð65Þ

dx2

dt¼�r̂x3

1þð1þ r̂Þx21�ðr�r̂Þx3

2þðr�r̂Þx22�x2�y2� ~g12ðx2�x1Þ

� ~g23ðx2�x3Þþða=oÞcosotþCoðx1�x2Þþd2,

dy2

dt¼ bx2, ð66Þ

dx3

dt¼�r̂x3

1þð1þ r̂Þx21�ðr�r̂Þx3

3þðr�r̂Þx23�x3�y3� ~g23ðx3�x1Þ

� ~g23ðx3�x2Þþða=oÞcosotþCoðx1�x3Þþd3,

dy3

dt¼ bx3: ð67Þ

Using the same procedure as in the previous section, synchro-

nization error dynamics for this case becomes

de

dt¼ AeþDAeþFðxÞðr̂�rÞþB ~d: ð68Þ

Consider the following Lyapunov function candidate [32,33]:

E¼ eTgPeþðr̂�rÞ2gS�1, ð69Þ

with g40, P40 and S40. Taking the derivative of (69) along

(68), we obtain

_Eðe,tÞ ¼ gðeT ðAT PþPAÞeþeT ðDAT PþPDAÞeþeT PB ~dþ ~dTBT Pe

þeT PFðxÞðr̂�rÞþFT ðxÞPeðr̂�rÞþ2ðr̂�rÞ_̂rS�1Þ: ð70Þ

Choosing eT PFðxÞþFT ðxÞPeþ2_̂rS�1 ¼ 0 from the adaptation law

(58) and (70) reduces to (43). Further using the same steps as in

the proof of Theorem 2. we obtain the matrix inequalities (63)

and (64), which complete the proof of Theorem 3. &

Remark 5. The proposed adaptation law (58) does not use anykind of neural networks. Hence, there is no training required, incontrast to [22]. Moreover, the proposed control law (55) and (56)and adaptation law (58) are computationally simpler than [22].This makes the proposed control strategy suitable and indeedadvantageous for practical implementation.

Remark 6. The approaches developed in the present study areglobal and based on the solution of LMI’s for a proper selection ofcontroller parameter Co. In contrast to [15,22], the selection oflarge matrices can be made easily using LMI-based tools. Itbecomes necessary to use such LMI tools for selection of largematrices like P when dealing with synchronization of multiplecoupled FHN neurons.

Remark 7. In [22], the robust adaptive technique developed forsynchronization of two neurons separated by gap junctions isbased on the convergence of synchronization errors within asmall region rather than to zero, even for zero disturbance. Theadaptive control proposed in the present work is less conservativebecause it ensures convergence of synchronization errors to zerofor ~d ¼ 0. Moreover, the adaptive-control-based synchronizationof FHN neurons proposed in [14,20] does not provide anyindication of stability. Furthermore, the selection of a largenumber of control parameters, with the aforementioned meth-odologies, is difficult.

To confirm the validity of the proposed control methodology,we select Co¼5. By solving the matrix inequalities of Theorem 2or 3 for gm¼0.2, c1¼1 and c2¼0.1, we obtain

P¼

4:6333 0:1665 �0:1639 0:9945 �0:0246 0:0246

0:1665 4:6298 0:1614 �0:0248 0:9946 �0:0247

�0:1639 0:1614 4:6262 0:0250 �0:0250 0:9948

0:9945 �0:0248 0:0250 6:2878 0:7104 �0:7109

�0:0246 0:9946 �0:0250 0:7104 6:2890 0:7115

0:0246 �0:0247 0:9948 �0:7109 0:7115 6:2902

2666666664

3777777775

,

ð71Þ

m¼ 7:41, g¼ 1:34, e¼ 103:2: ð72Þ

y 1x 2

x1

x1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

y 2

x2

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

y 3

x3

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x 3

x1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x 3

x2

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2-0.5

0

0.5

1

1.5

2

y 2

y1

-0.5 0 0.5 1 1.5 2-0.5

0

0.5

1

1.5

2

y 3

y1

-0.5 0 0.5 1 1.5 2-0.5

0

0.5

1

1.5

2

y 3

y2

Fig. 2. Synchronization of three coupled chaotic FHN neurons with uncertain and different gap junctions under disturbance using the proposed adaptive nonlinear control:

(a) state space of x1 and y1, (b) state space of x2 and y2, (c) state space of x3 and y3, (d) state space of x1 and x2, (e) state space of x1 and x3, (f) state space of x2 and x3, (g) state

space of y1 and y2, (h) state space of y1 and y3 and (i) state space of y2 and y3.

M. Rehan et al. / Neurocomputing 74 (2011) 3296–3304 3303

Fig. 2 shows the results, so obtained, by utilizing the proposedadaptive control. The initial condition of r̂ is taken to be zero,and Dg12¼0.1, Dg13¼0.14 and Dg23¼0.18. The disturbances aretaken as

d1 ¼ 0:02sinðtÞ,

d2 ¼ 0:02sinð1:1tÞ,

d3 ¼ 0:02sinð1:2tÞ: ð73Þ

It is evident that all three of the coupled chaotic FHN neuronsare synchronized for identical behaviors because the plots inFig. 2(d)–(i) make straight lines having slope 1 passing throughthe origin, which shows the robustness in the presence ofdisturbances and parametric uncertainties.

6. Conclusions

This paper discussed the synchronization of three coupledchaotic FHN neurons, in which one of them was considered as themaster and the other two as slave neurons. The strengths of gapjunctions between the neurons of each connection were assumeddifferent and uncertain. A new L2 gain reduction criterion from inputto output of a MIMO system with non-zero initial condition was

established and was effectively used for synchronization of FHNneurons under L2 norm bounded disturbance. Robust nonlinear/adaptive control laws for synchronization of FHN neurons weredeveloped. The developed control laws are simple in implementa-tion, avoiding additional measurements for recovery voltages andreducing parameter tuning efforts. The simulation results demon-strated the success and effectiveness of the overall scheme. Thepresent work can be extended to the synchronization of a networkof coupled FHN neurons with uncertain and different gap junctions.

Acknowledgement

This research was supported by the World Class Universityprogram through the National Research Foundation of Koreafunded by the Ministry of Education, Science and Technology,Republic of Korea (grant no. R31-20004).

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Muhammad Rehan received his M.Sc. degree in Elec-tronics from Quaid-e-Azam University (QAU) in 2005and M.S. degree in Systems Engineering from PakistanInstitute of Engineering and Applied Sciences (PIEAS),Islamabad, Pakistan. He is currently a faculty memberin the Department of Electrical Engineering, PIEAS, andis also a Ph.D. candidate in the Department of Cogno-Mechatronics Engineering (under the World ClassUniversity program, MEST, Korea), Pusan NationalUniversity, Busan, Republic of Korea. His researchinterests include robust, nonlinear and adaptive con-trol, anti-windup design and implementation, and

chaotic systems control.

Keum-Shik Hong received B.S. degree in MechanicalDesign and Production Engineering from SeoulNational University in 1979, M.S. degree in MechanicalEngineering from Columbia University in 1987, andboth M.S. degree in Applied Mathematics and Ph.D. inMechanical Engineering from the University of Illinoisat Urbana-Champaign in 1991. Dr. Hong is currentlythe Editor-in-Chief of the Journal of Mechanical Scienceand Technology. He served as an Associate Editor forAutomatica (2000–2006) and as an Editor for theInternational Journal of Control, Automation, and Sys-tems (2003–2005). Dr. Hong received Fumio Hara-

shima Mechatronics Award in 2003 and the Korean

Government Presidential Award in 2007. Dr. Hong’s research interests includenonlinear systems theory, adaptive control, distributed parameter system control,robotics, autonomous vehicles and brain-computer interfaces.

Muhammad Aqil received his M.Sc. degree in AppliedPhysics with specialization in Electronics from Univer-sity of Karachi in 2001 and M.S. degree in SystemsEngineering from Pakistan Institute of Engineering andApplied Sciences (PIEAS) in 2004. Since then, he isa faculty member in the Department of ElectricalEngineering, PIEAS. Currently, he is a Ph.D. candidatein the Department of Cogno-Mechatronics Engineering(under the World Class University program, MEST,Korea), Pusan National University, Busan, Republic ofKorea. His research interests include applied intelligentcontrol of embedded systems, brain signal processing

and design and instrumentation of brain computerinterface and brain machine interface systems.