Pinning synchronization of delayed neural networks - Semantic Scholar

CHAOS 18, 043111 共2008兲

Pinning synchronization of delayed neural networks Jin Zhou,1,2,a兲 Xiaoqun Wu,2 Wenwu Yu,3 Michael Small,1 and Jun-an Lu2 1

Department of Electronic Information Engineering, Hong Kong Polytechnic University, Hong Kong School of Mathematics Statistics, Wuhan University, Wuhan 430072, China 3 Department of Electronic Engineering, City University of Hong Kong, Hong Kong 2

共Received 31 May 2008; accepted 16 September 2008; published online 12 November 2008兲 This paper investigates adaptive pinning synchronization of a general weighted neural network with coupling delay. Unlike recent works on pinning synchronization which proposed the possibility that synchronization can be reached by controlling only a small fraction of neurons, this paper aims to answer the following question: Which neurons should be controlled to synchronize a neural network? By using Schur complement and Lyapunov function methods, it is proved that under a mild topology-based condition, some simple adaptive feedback controllers are sufficient to globally synchronize a general delayed neural network. Moreover, for a concrete neurobiological network consisting of identical Hindmarsh–Rose neurons, a specific pinning control technique is introduced and some numerical examples are presented to verify our theoretical results. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2995852兴 It is well known that there are many useful network synchronization phenomena in our daily life, such as, the synchronous transfer of digital or analog signals in communication networks. In particular, adaptive synchronization in networks or coupled oscillators has received increasing attention. In this paper, we study adaptive synchronization of a general delayed neural network with pinning mechanism, which has been introduced recently to avoid the impracticality of controlling all the vertices in a large-scale network. Unlike recent works on pinning synchronization which proposed the possibility that synchronization can be reached by controlling a small fraction of neurons, this paper aims to answer the following question: Which neurons should be controlled to synchronize a neural network? We present a novel synchronization algorithm based on Schur complement and Lyapunov stability theory, and also prove that under a mild topology-based condition, an adaptive feedback control is sufficient to globally synchronize a general delayed neural network. In particular, for a concrete neurobiological network consisting of identical Hindmarsh–Rose neurons, we introduce a specific pinning control technique. We exhibit numerical results verifying the validity of our pinning control method. I. INTRODUCTION AND MODEL DEPICTION

The most important component of the central nervous system is a complicated interconnected network of neurons that are responsible for enabling every function of our body. Neurobiological networks have been extensively studied since the 1980s for their potential applications in modelling complex dynamic systems, and various neural networks have been successfully applied to signal processing, linear and nonlinear programming, image processing, pattern recognition,1,2 and so on. a兲

Electronic mail: [email protected].

1054-1500/2008/18共4兲/043111/9/$23.00

As an important phenomenon in neural networks, usually the dynamics of each neuron is derived not only by its own dynamical property, but also by the evolution of its neighbors. A widely studied neural network model is formulated as x˙ i共t兲 = f共xi共t兲,t兲 + c

兺

aijg共x j共t兲兲,

j苸Ni

where 1 艋 i 艋 N, Ni represents the neighborhood of neuron i, the state vector of the ith neuron xi共t兲 = 共xi1共t兲 , xi2共t兲 , . . . , xin共t兲兲T 苸 Rn is a continuous function, f : Rn ⫻ R+ → Rn is a smooth nonlinear vector function, individual neuron dynamics is x˙ 共t兲 = f共x共t兲 , t兲, g : Rn → Rn is the inner-coupling vector function. The outer-coupling weight configuration matrix A = 共aij兲 苸 RN⫻N共aij 苸 兵0 , 1其兲 is symmetric and diffusive satisfying 兺Nj=1aij = 0. This formulation assumes a common outer-coupling strength for all connections and instantaneous information from their neighborhood for all neurons. In a real neural network, however, this is not always the case. Couplings between neurons are not the same in most circumstances even if the diffusive condition is still satisfied. Apart from instantaneous information, neurons will also usually receive delayed information from their neighbors.3–7 One method to solve these problems is to add weights and delayed couplings to this model. Thus the following neural network model is introduced and will be considered throughout the paper: N

N

j=1

j=1

x˙ i共t兲 = f共xi共t兲,t兲 + 兺 aijg共x j共t兲兲 + 兺 bijh共x j共t − ␶共t兲兲兲, 共1兲 where 1 艋 i 艋 N, xi共t兲 = 共xi1共t兲 , xi2共t兲 , . . . , xin共t兲兲T is the state vector of the ith node, g : Rn → Rn and h : Rn → Rn are the inner-coupling and delayed inner-coupling vector functions, respectively. The initial conditions of Eq. 共1兲 are given by x共t兲 = ⌼共t兲 苸 C共关−¯␶ , 0兴 , Rn兲, where ¯␶ = supt苸R+兵␶共t兲其 , ␶共t兲 is

18, 043111-1

© 2008 American Institute of Physics

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the time delay in the couplings, C共关−¯␶ , 0兴 , Rn兲 represents the set of all continuous functions from 关−¯␶ , 0兴 to Rn. The outercoupling and delayed outer-coupling weight configuration matrices A = 共aij兲 苸 RN⫻N and B = 共bij兲 苸 RN⫻N are diffusive and symmetric. Take A, for example, if there is a link between node i and node j共j ⫽ i兲, then aij = a ji ⬎ 0 and aij is the coupling weight; otherwise, aij = 0. In addition, aii = −兺 j=1,j⫽iaij. For a neurobiological network, control and synchronization have been one of the focal points in many research and application fields.5–12 In particular, adaptive techniques, have emerged as an exciting research topic in nonlinear system control, and have been demonstrated to be an effective way to synchronize a complex network.8–10 However, it is assumed that all the nodes need to be controlled. In view of impracticality of controlling all the nodes in a large-scale network, pinning control has been introduced in recent years.13–15 Many existing works on pinning control presented the possibility of controlling a small fraction of nodes in a network to reach synchronization. However, how to carry out pinning and which nodes should be pinned are still challenging problems. In this paper, a novel criterion for synchronizing a general delayed neural network 共1兲 by pinning control is proposed. In view of the Schur complement and Lyapunov function methods, rigorous theoretical analysis with topologybased conditions is obtained. For a concrete neural network with coupled Hindmarsh–Rose 共HR兲 neurons, a specific pinning control technique is presented by only controlling the membrane potential of each neuron. The paper is organized as follows: In Sec. II, preliminaries are given. The mechanism of globally adaptive pinning synchronization of a general neural network 共1兲 is discussed in Sec. III. A specific pinning control technique for a neural network consisting of HR neurons is then presented in Sec. IV. In Sec. V, numerical examples are provided to verify the theoretical results. Finally, the conclusion is drawn in Sec. VI. II. PRELIMINARIES A. Notations

To begin with, some necessary notations are introduced which will be used throughout the paper. The matrix Iii 苸 Ri⫻i is the identity matrix with dimension i. T denotes the transpose of a matrix or a vector. 储␰储 represents the 2-norm of a vector which is defined as 储␰储 = 冑␰T␰. 丢 denotes the Kronecker product of two matrices.16 ␭m共·兲 represents the maximum eigenvalue of a square matrix. The symmetric matrix G ⬍ 0 means that G is negative definite. The matrix Gl denotes the minor matrix of a matrix G by removing all the ikth 共1 艋 ik 艋 N , 1 艋 k 艋 l , 1 艋 l 艋 N兲 row-column pairs of G. B. Lemmas

In order to derive the main results, the following two lemmas are needed. Lemma 1 共Schur complement17,18兲: The linear matrix inequality 共LMI兲

冉

A共x兲 B共x兲




B共x兲 C共x兲

冊

⬍ 0,

where A共x兲T = A共x兲, C共x兲T = C共x兲, is equivalent to either of the following conditions: 共a兲 A共x兲 ⬍ 0 and C共x兲 − B共x兲TA共x兲−1B共x兲 ⬍ 0; 共b兲 C共x兲 ⬍ 0 and A共x兲 − B共x兲C共x兲−1B共x兲T ⬍ 0. Lemma 2: Assume that M is a diagonal matrix whose ikth 共1 艋 ik 艋 N , 1 艋 k 艋 l , 1 艋 l 艋 N兲 diagonal elements are m and the others are 0, where m ⬎ 0 is a constant. Then for a symmetric matrix G which has the same dimension with M, G − M ⬍ 0 is equivalent to Gl ⬍ 0 when m is large enough. Proof: After an elementary matrix transformation, G − M can be changed into

冉

G1 − mIll G* G *T G l

冊

,

where G1, G* are the corresponding matrices with compatible dimension. If G − M ⬍ 0, i.e.,

冉

G1 − mIll G* G *T G l

冊

⬍ 0,

one has Gl ⬍ 0 according to Lemma 1. On the other side, since Gl ⬍ 0, Gl is invertible, and then G1 − mIll − G*Gl−1G*T ⬍ 0 when m is large enough. According to Lemma 1, Gl ⬍ 0 leads to G − M ⬍ 0. The proof is thus completed. C. Hypotheses

For the purpose of obtaining adaptive pinning synchronization criterion for the neural network 共1兲, the hypotheses which will be used in the main results are outlined below. Hypothesis 1 (H1): Suppose that the delay function ␶共t兲 is differentiable and satisfies −␦ 艋 ␶˙ 共t兲 艋 ␦, where 0 艋 ␦ ⬍ 1 is a constant. This hypothesis is practical in real experiments and engineering since ␶共t兲 change slowly. Hypothesis 2 (H2): Assume that there exists a positive constant ␮ satisfying 共␰2 − ␰1兲T共f共␰2 , t兲 − f共␰1 , t兲兲 艋 ␮储␰2 − ␰1储2 for any two vectors ␰1 , ␰2 苸 Rn. Hypothesis 3 (H3): Suppose that the inner-coupling function g共␰兲, where ␰ 苸 Rn, satisfies d1储␰2 − ␰1储2 艋 共␰2 − ␰1兲T共g共␰2兲 − g共␰1兲兲 艋 储␰2 − ␰1储储g共␰2兲 − g共␰1兲储 艋 d2储␰2 − ␰1储2 for any two vectors ␰1 , ␰2 苸 Rn, where d1, d2 are two positive constants. Hypothesis 4 (H4): Assume that the delayed inner-coupling function h共␰兲, where ␰ 苸 Rn, satisfies e1储␰2 − ␰1储2 艋 共␰2 − ␰1兲T共h共␰2兲 − h共␰1兲兲 艋 储␰2 − ␰1储储h共␰2兲 − h共␰1兲储 艋 e2储␰2 − ␰1储2 for any two vectors ␰1 , ␰2 苸 Rn, where e1, e2 are two positive constants. III. PINNING ADAPTIVE SYNCHRONIZATION

In this section, a novel criterion for globally synchronizing the delayed neural network 共1兲 with adaptive controllers is proposed. Suppose that the synchronous state x0共t兲 is a trajectory of the uncoupled system, i.e., x˙ 0共t兲 = f共x0共t兲 , t兲, which can be an equilibrium point, a periodic orbit, an aperiodic orbit, or

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varying coupling delay is derived according to Lyapunov ¯ be a modified matrix of A in which stability theory.19 Let A the diagonal elements aii are replaced by d1aii and other aij ¯ be a changed matrix of B whose are replaced by d2aij, and B diagonal elements and others are e1bii and e2bij, respectively. Theorem 1: Suppose that H1–H4 hold. Then the synchronous solution of the controlled neural network 共2兲 is globally asymptotically stable with adaptive pinning control¯ + 1 / 4␥共1 − ␦兲B ¯ 2兲 兲 ⬍ −共␮ + ␥兲, lers 共3兲, provided that ␭m共共A l where ␥ can be any positive constant. Proof: Consider a Lyapunov candidate as

even a chaotic orbit in the phase space. The objective of synchronization is to control neural network 共1兲 to the given trajectory x0共t兲. By pinning a small fraction of neurons, we apply some adaptive controllers to the network. Assuming that the i1th, i2th, . . . , ilth neurons are controlled, the following controlled network is considered: N

x˙ i共t兲 = f共xi共t兲,t兲 + 兺 aijg共x j共t兲兲 j=1

N

+ 兺 bijh共x j共t − ␶共t兲兲兲 + ui共t兲,

共2兲

1 V共t兲 = 兺 储xi共t兲 − x0共t兲储2 + ␥ 兺 2 i=1 i=1

where 1 艋 i 艋 N, ui are the adaptive controllers designed by uik共t兲 = − ␣ik共t兲共xik共t兲 − x0共t兲兲 1 艋 k 艋 l

␣˙ ik共t兲 = ␤ik储xik共t兲 − x0共t兲储

2

l

1艋k艋l

uik共t兲 = 0

− x0共␴兲储 d␴ + 兺 2

共3兲

k=1

otherwise.

冋

N

兺 i=1

N

共xi共t兲 − x0共t兲兲
f共xi共t兲兲 − f共x0共t兲兲 +

兺 j=1

N

aijg共x j共t兲兲 +

兺 j=1

bijh共x j共t − ␶共t兲兲兲

N

+ ␥ 兺 关储xi共t兲 − x0共t兲储 − 共1 − ␶˙ 共t兲兲储xi共t − ␶共t兲兲 − x0共t − ␶共t兲兲储 兴 + 2

2

i=1

N

兺 i=1

=

兺 兺 i=1 j=1

t

t−␶共t兲

储xi共␴兲

1 共␣i 共t兲 − ␣兲2 , 2 ␤ ik k

册

l

−

␣i 共t兲储xi 共t兲 − x0共t兲储2 兺 k=1 k

k

l

共␣i 共t兲 − ␣兲储xi 共t兲 − x0共t兲储2 兺 k=1 k

k

N

N

共xi共t兲 − x0共t兲兲
共f共xi共t兲兲 − f共x0共t兲兲兲 +

冕

where ␣ is a sufficiently large positive constant to be determined. Taking the derivative of V共t兲 along the trajectories of Eqs. 共2兲 and 共3兲, one obtains

Using the above controllers, a criterion for globally synchronizing a general weighted neural network with time-

V˙共t兲 =

N

N

j=1

N

aij共xi共t兲 − x0共t兲兲
共g共x j共t兲兲 − g共x0共t兲兲兲 +

N

兺 bij共xi共t兲 兺 i=1 j=1

N

− x0共t兲兲 共h共x j共t − ␶共t兲兲兲 − h共x0共t − ␶共t兲兲兲兲 + ␥ 兺 关储xi共t兲 − x0共t兲储2 − 共1 − ␶˙ 共t兲兲储xi共t − ␶共t兲兲 − x0共t − ␶共t兲兲储2兴


i=1

l

␣储xi 共t兲 − x0共t兲储2 兺 k=1

−

k

N

艋

N

N

N

N

␮储xi共t兲 − x0共t兲储 + 兺 aiid1储xi共t兲 − x0共t兲储 + 兺 biie1储xi共t − ␶共t兲兲 − x0共t − ␶共t兲兲储 + 兺 兺 aijd2储xi共t兲 − x0共t兲储 兺 i=1 i=1 i=1 i=1 j=1,j⫽i 2

N

⫻储x j共t兲 − x0共t兲储 +

2

N

兺 兺 i=1 j=1,j⫽i

2

N

bije2储xi共t兲 − x0共t兲储储x j共t − ␶共t兲兲 − x0共t − ␶共t兲兲储 + ␥ 兺 关储xi共t兲 − x0共t兲储2 i=1

l

− 共1 − ␶˙ 共t兲兲储xi共t − ␶共t兲兲 − x0共t − ␶共t兲兲储2兴 −

␣储xi 共t兲 − x0共t兲储2 兺 k=1 k

¯ ⌬共t兲 + ⌬
共t兲B ¯ ⌬共t − ␶共t兲兲 + ␥⌬
共t兲⌬共t兲 − ␥共1 − ␦兲⌬
共t − ␶共t兲兲⌬共t − ␶共t兲兲 − ⌬
共t兲⌳⌬共t兲 艋 ␮⌬
共t兲⌬共t兲 + ⌬
共t兲A

=

冉

⌬共t兲 ⌬共t − ␶兲

冊




冢

¯ −⌳ 共␮ + ␥兲INN + A

1¯ B 2

1¯ B 2

− ␥共1 − ␦兲INN

冣

冉

冊

⌬共t兲 , ⌬共t − ␶兲

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where ⌬共t兲 = 共储x1共t兲 − x0共t兲储 , 储x2共t兲 − x0共t兲储 , . . . , 储xN共t兲 − x0共t兲储兲T, and ⌳ 苸 RN⫻N is a diagonal matrix whose ikth 共1 艋 k 艋 l兲 elements are ␣ and the others are 0. ¯ + 1 / 4␥共1 − ␦兲B ¯ 2兲 兲 ⬍ −共␮ + ␥兲, Due to the fact that ␭m共共A l ¯ + 1 / 4␥共1 − ␦兲B ¯ 2 − ⌳ ⬍ 0 when ␣ is one has 共␮ + ␥兲INN + A large enough according to Lemma 2. Furthermore,

冉

冊

¯ − ⌳ 1/2B ¯ 共␮ + ␥兲INN + A ⬍0 ¯ − ␥共1 − ␦兲I 1/2B NN

in view of Lemma 1. Based on LaSalle’s invariance principle,19 every solution of the system converges to the largest invariant set ⌫ = 兵V˙ = 0其, in which ⌬共t兲 = 0. Thus limt→⬁ ⌬共t兲 = 0, in other words, limt→⬁ xir共t兲 − x0r共t兲 = 0共1 艋 i 艋 N , 1 艋 r 艋 n兲. As a conclusion, the synchronous solution of the controlled weighted delayed neural network 共2兲 is globally asymptotically stable with the adaptive pinning controllers 共3兲. This completes the proof. Remark 1: From this theorem, we find that for an appropriate positive constant ␥, if the topology-based condition ¯ + 1 / 4␥共1 − ␦兲B ¯ 2兲 兲 ⬍ −共␮ + ␥兲 is satisfied, the i th, ␭m共共A l 1 i2th, . . . , ilth neurons can be controlled to reach globally asymptotic synchronization for this type of neurobiological network. Remark 2: It should be noted that since the condition in the theorem is just sufficient, it does not mean that synchronization cannot be reached if the condition is not satisfied. Remark 3: At the first stage of studying pinning synchronization on delayed neural networks, constant and delayed couplings are both considered. The constant ␮ in Hypothesis 2 can be considered as the passivity degree,22 where ␮ ⬎ 0

N

共xi − x0兲 兺 i=1

N




冢 冣 兩xi1 − x01兩

共f共xi兲 − f共x0兲兲 艋 兺 兩xi2 − x02兩 i=1 兩xi3 − x03兩




冢

N

j=1

j=1

The motivation for the Hindmarsh–Rose 共HR兲 model was to isolate the essentially mathematical properties of excitation and propagation from the electrochemical properties of sodium and potassium ion flow. The HR neuron model is described by2 x˙ 共t兲 = f共x共t兲兲, where x共t兲 = 共x1共t兲 , x2共t兲 , x3共t兲兲T,

冢 冣冢

Here, x1共t兲 is the membrane potential, x2共t兲 and x3共t兲 are the recovery variables, and ␹1 , ␹2 , ␹3 , ␬1 , ␬2 are dimensionless parameters. For a HR model, the following inequality can be attained 共a proof is given in the Appendix兲:

共␹1 + ␬1兲M +

1 ␬ 2␹ 2 + 1 2 2

−1

0

␬ 2␹ 2 + 1 2

0

− ␬2

冢

g1共x j1共t兲兲 0 g共x j共t兲兲 = 0

h共x j共t − ␶共t兲兲兲 =

共4兲

冣

␹1x21共t兲 − x31共t兲 − x2共t兲 − x3共t兲 f 1共x共t兲兲 . f共x共t兲兲 = f 2共x共t兲兲 = 共␹1 + ␬1兲x21共t兲 − x2共t兲 f 3共x共t兲兲 ␬2共␹2x1共t兲 − x3共t兲 + ␹3兲

1 共␹1 + ␬1兲M + 2

x˙ i共t兲 = f共xi共t兲,t兲 + 兺 aijg共x j共t兲兲 + 兺 bijh共x j共t − ␶共t兲兲兲,

where g共x j共t兲兲 and h共x j共t − ␶共t兲兲兲 are specified as

IV. A SPECIFIC NEUROBIOLOGICAL NETWORK AND THE CORRESPONDING CONTROL

2␹1 M

Usually, in the realistic neurobiological network, neurons are coupled with the first variable, namely, membrane potential. Moreover, only the membrane potentials can be observed by electrodes in practical experiments. Taking that into consideration, we describe this kind of neurobiological network model by N

means that each node needs energy from outside to stabilize the network, while ␮ ⬍ 0 means that each node itself is already stable. The derived conditions in the theorems in this paper imply that ␮ ⬍ 0 when A = 0. Here, only ␮ ⬎ 0 is considered throughout the paper for simplicity; otherwise the network can achieve self-synchronization even without control. There are some results about synchronization in networks with solely delayed couplings where A = 0 and ␮ ⬎ 0 in the literature, where the problem can be investigated by a different scenario and will be our future works.

冢

冣

冣

冢 冣 兩xi1 − x01兩

兩xi2 − x02兩 . 兩xi3 − x03兩

and

h1共x j1共t − ␶兲兲 0 0

冣

.

In what follows, we introduce two useful Hypotheses to reach our main results in this section. Hypothesis 3⬘ 共H3⬘兲. Suppose that g1共␨兲, where ␨ 苸 Rn, satisfies d1⬘ 艋 共g1共␨2兲 − g1共␨1兲兲 / 共␨2 − ␨1兲 艋 d2⬘ for any two sca-

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lar variables ␨1 , ␨2 苸 R, where d1⬘, d2⬘ are two positive constants. Hypothesis 4⬘ 共H4⬘兲. Assume that h1共␨兲, where ␨ 苸 Rn, satisfies e1⬘ 艋 共h1共␨2兲 − h1共␨1兲兲 / 共␨2 − ␨1兲 艋 e2⬘ for any two scalar variables ␨1 , ␨2 苸 R, where e1⬘, e2⬘ are two positive constants. To achieve synchronization in this kind of neural network consisting of HR neurons, we can design a pinning controlled network as follows: x˙ i共t兲 = f共xi共t兲,t兲 N

N

j=1

j=1

¯ ⬘ as a similarly modified matrix of B. placed by d2⬘aij, and B Then the globally adaptive pinning synchronization theorem for neurobiological network 共4兲 is attained. Theorem 2: Suppose that H1, H3⬘, H4⬘ hold. ¯ ⬘ + 1 / 4␥共1 − ␦兲B ¯ ⬘2兲 ⬍ −共共共␹ + ␬ 兲M + 1 兲2 + 1 / ␬ 共␬ ␹ If ␭m共A l 1 1 2 2 2 2 + 1 / 2兲2 + 2␹1M + ␥兲, the synchronous solution of the controlled network 共5兲 will be globally asymptotically stable using the adaptive pinning controllers 共6兲. Proof: Consider a Lyapunov candidate as

+ 兺 aijg共x j共t兲兲 + 兺 bijh共x j共t − ␶共t兲兲兲 + ui共t兲, 共5兲

N

1 V共t兲 = 兺 储xi共t兲 − x0共t兲储2 2 i=1

where the controllers are uik共t兲 =

冢

− ␣ik共t兲共xi1共t兲 − x01共t兲兲 0 0

␣˙ ik共t兲 = ␤ik共xi1共t兲 − x01共t兲兲

2

冣

N

+ ␥兺

1艋k艋l

i=1

+兺

1艋k艋l

uik共t兲 = 0

k=1

otherwise.

¯ ⬘ as a modified matrix of A in which the diagDenote A onal elements aii are replaced by d1⬘aii and other aij are re-

冋

N

l

共6兲

冕

t

t−␶共t兲

共xi1共␴兲 − x01共␴兲兲2d␴

1 共␣i 共t兲 − ␣兲2 , 2 ␤ ik k

where ␥ is a positive constant, ␣ is a sufficiently large positive constant to be determined. The derivative of V共t兲 along the trajectory 共5兲 and 共6兲 is

N

N

j=1

j=1

册

l

V˙共t兲 = 兺 共xi共t兲 − x0共t兲兲
f共xi共t兲兲 − f共x0共t兲兲 + 兺 aijg共x j共t兲兲 + 兺 bijh共x j共t − ␶共t兲兲兲 − 兺 ␣i共t兲共xi1共t兲 − x01共t兲兲2 i=1

N

l

i=1

k=1

i=1

+ ␥ 兺 关共xi1共t兲 − x01共t兲兲2 − 共1 − ␶˙ 共t兲兲共xi1共t − ␶共t兲兲 − x01共t − ␶共t兲兲兲2兴 + 兺 共␣ik共t兲 − ␣兲共xik1共t兲 − x01共t兲兲2 N

N

i=1

i=1

N

N

兺

艋 兺 共xi共t兲 − x0共t兲兲
共f共xi共t兲兲 − f共x0共t兲兲兲 + 兺 aiid1共xi1共t兲 − x01共t兲兲2 + 兺 N

N

+ 兺 biie1共xi1共t − ␶共t兲兲 − x01共t − ␶共t兲兲兲 + 兺 2

aijd2共xi1共t兲 − x01共t兲兲共x j1共t兲 − x01共t兲兲

i=1 j=1,j⫽i

N

兺

bije2共xi1共t兲 − x01共t兲兲共x j1共t − ␶共t兲兲 − x01共t − ␶共t兲兲兲

i=1 j=1,j⫽i

i=1

N

l

+ ␥ 兺 关共xi1共t兲 − x01共t兲兲 − 共1 − ␦兲共xi1共t − ␶共t兲兲 − x01共t − ␶共t兲兲兲 兴 − 兺 ␣共xik1共t兲 − x01共t兲兲2 艋 ␩
共t兲Q␩共t兲, 2

2

i=1

k=1

where ␩共t兲 = 共兩x11共t兲 − x01共t兲兩 , . . . , 兩xN1共t兲 − x01共t兲兩 , 兩x11共t − ␶兲 − x01共t − ␶兲兩 , . . . , 兩xN1共t − ␶兲 − x01共t − ␶兲兩 , 兩x12共t兲 − x02共t兲兩 , . . . , 兩xN2共t兲 − x02共t兲兩 , 兩x13共t兲 − x03共t兲兩 , . . . , 兩xN3共t兲 − x03共t兲兩兲T,

Q=

冢

冉

Q1

1¯ B⬘ 2

1¯ B⬘ 2

− ␥共1 − ␦兲INN

冊

冉

共␹1 + ␬1兲M + 0

冊

1 ␬ 2␹ 2 + 1 INN INN 2 2 0

1 共␹1 + ␬1兲M + INN 2

0

− INN

0

␬ 2␹ 2 + 1 INN 2

0

0

− ␬2INN

冣

,

¯ ⬘ + 共2␹ M + ␥兲I − ⌳, ⌳ is a diagonal matrix whose i th elements are ␣ and the others are 0. Q1 = A 1 NN k 2 ¯ ¯ 共共 Provided with ␭m共A⬘ + 1 / 4␥共1 − ␦兲B⬘ 兲l ⬍ − 共␹1 + ␬1兲M + 21 兲2 + 1 / ␬2共␬2␹2 + 1 / 2兲2 + 2␹1M + ␥兲 in Theorem 2, one has Downloaded 12 Nov 2008 to 144.214.130.8. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp

043111-6

Q1 +

Chaos 18, 043111 共2008兲

Zhou et al.

1 B ⬘2 + 4␥共1 − ␦兲

冉冉

共␹1 + ␬1兲M +

1 2

冊 冉 2

+

1 ␬ 2␹ 2 + 1 2 ␬2

冊冊 2

INN ⬍ 0

when ␣ is large enough according to Lemma 2. It follows that

冢

Q1 −

冉冉

共␹1 + ␬1兲M +

1 2

冊 冉 2

+

1 ␬ 2␹ 2 + 1 2 ␬2

冊冊 2

1¯ B⬘ 2

1¯ B⬘ 2

INN

− ␥共1 − ␦兲INN

冣

⬍ 0,

namely,

Q1

1¯ B⬘ − ␥共1 − ␦兲INN 2

冣

A= − Q
2

冢

− INN 0

冣

0 1 Q2 ⬍ 0, − INN ␬2

where

Q2 =

冢

冉

共␹1 + ␬1兲M +

1 2

冊

INN 0

␬ 2␹ 2 + 1 INN 2

0

冣

.

As a result, one gets Q ⬍ 0 according to Lemma 1. Then we have limt→⬁ ␩共t兲 = 0, and further, limt→⬁ xir共t兲 − x0r共t兲 = 0共1 艋 i 艋 N , 1 艋 r 艋 n兲. Therefore, the synchronous solution of the controlled weighted delayed neural network 共5兲 is globally asymptotically stable with adaptive pinning controllers 共6兲. Thus the proof is completed. ¯ ⬘ + 1 / 4␥ Remark 1: It is revealed that if ␭m共A 1 2 2 2 ¯ 共1 − ␦兲B⬘ 兲l ⬍ −共共共␹1 + ␬1兲M + 2 兲 + 1 / ␬2共␬2␹2 + 1 / 2兲 + 2␹1M + ␥兲 for an appropriate ␥ ⬎ 0, globally asymptotic synchronization of this type of neural network can be reached by controlling only the membrane potential of the i1th, i2th, …, ilth neurons. Remark 2: Similarly, it should be noted that the condition in the theorem is just sufficient.

0

−6

6

5

6

− 11

冣

and

B=

冢

−2

1

1

1

−1

0

1

0

−1

冣

,

0 −2 0 5

xi2 , x02

100

200

300

400

500

100

200

300

400

500

100

200

300

400

500

2 −1 0 2

xi3 , x03

In this subsection, some numerical examples are simulated to show the effectiveness of the adaptive pinning control presented in Sec. IV. For simplicity, three neurons are considered to form a neurobiological network 共4兲 with coupling delay ␶共t兲 = 1, and the dynamics of each neuron is a HR equation with parameters ␹1 = 0.2, ␹2 = 1.5, ␹3 = 0.1, ␬1 = 1.9, and ␬2 = 0.1. In this case, the bound M of the first variable in the HR equation is 0.8. Suppose that the instantaneous and delayed outercoupling matrices are

5

2

V. NUMERICAL SIMULATION A. A detailed example with network size 3

冢

0

respectively. Moreover, g1共x j1兲 = h1共x j1兲 = x j1共1 艋 j 艋 N兲 are assumed to be the instantaneous and delayed inner-coupling function. It is easy to check that H1, H3⬘, and H4⬘ naturally hold where d1⬘ = d2⬘ = e1⬘ = e2⬘ = 1 and ␦ = 0. Selecting ␥ = 1 / 2共1 − ␦兲, we get −共共共␹1 + ␬1兲M + 21 兲2 + 1 / ␬2共␬2b + 1 / 2兲2 + 2␹1M + 1 / 2共1 − ␦兲兲 = −9.5187 by simple ¯ + 1B ¯ 2兲 = −10⬍ −9.5187 when i computation. Since ␭m共A 1 2 2 = 1, i2 = 2, the first two neurons can be pinned to reach global synchronization with adaptive controllers 共6兲 according to Theorem 2. Choose initial values as xi共0兲 = 共0.1+ 0.1i , 0.2+ 0.1i , 0.3 + 0.1i兲T for 0 艋 i 艋 N and ␣k共0兲 = ␤k = 1 for 1 艋 k 艋 l with N = 3, l = 2 in the numerical simulation. The state time evolutions of the neurons xir共t兲共1 艋 i 艋 3 , 1 艋 r 艋 3兲 and of the desired synchronous state x0r共t兲 without control are depicted in Fig. 1 and those with adaptive pinning control are exhibited in Fig. 2. Synchronization errors xir − x0r without control and with adaptive pinning control are displayed in Fig. 3 and Fig. 4, respectively. The figures reveal that these neurons cannot synchronize with the desired state when no controllers are

xi1 , x01

冢

1¯ B⬘ 2

−5

0 −2 0

t

FIG. 1. 共Color online兲 The state time evolutions of the neurons xir共t兲共1 艋 i 艋 3 , 1 艋 r 艋 3兲 and of the desired synchronous state x0r共t兲 共red line兲 without control.

Downloaded 12 Nov 2008 to 144.214.130.8. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp

xi3 − x03

Chaos 18, 043111 共2008兲

Pinning synchronization

2

1.16

0

1.14

−2 0 2

100

200

300

400

1.12

500

1.1 αik

xi2 − x02

xi1 − x01

043111-7

0 −2 0 1

100

200

300

400

1.06

500

1.04 1.02

0 −1 0

100

200

300

t

400

1 0

500

FIG. 2. 共Color online兲 The state time evolutions of the neurons xir共t兲共1 艋 i 艋 3 , 1 艋 r 艋 3兲 and of the desired synchronous state x0r共t兲 共red line兲 for pinning the first two neurons.

0

t

300

400

500

20

100

200

300

400

500

2 −1 0 2

15 xi1 , x01

xi1 , x01

200

25

−2 0 5 xi2 , x02

100

FIG. 5. 共Color online兲 Adaptive feedback gains ␣ik共ik 苸 兵1 , 2其兲 for pinning the first two neurons.

2

xi3 , x03

1.08

100

200

300

400

5

500

0

0 −2 0

10

100

200

300

t

400

−5 0

500

FIG. 3. 共Color online兲 Synchronization errors xir − x0r共1 艋 i 艋 3 , 1 艋 r 艋 3兲 without control.

20

40

t

60

80

100

FIG. 6. 共Color online兲 The state time evolutions of the neurons xi1共1 艋 i 艋 200兲 and of the desired synchronous state x01共t兲 共red line兲 for pinning the first 12 neurons.

xi3 − x03

20

0 −0.5 0 1.2

100

200

300

400

500

xi2 , x02

xi2 − x02

xi1 − x01

25 0.5

0 −1.2 0 0.5

100

200

300

400

500

0 −0.5 0

100

200

t

300

400

500

FIG. 4. 共Color online兲 Synchronization errors xir − x0r共1 艋 i 艋 3 , 1 艋 r 艋 3兲 for pinning the first two neurons.

15 10 5 0 0

20

40

t

60

80

100

FIG. 7. 共Color online兲 The state time evolutions of the neurons xi2共1 艋 i 艋 200兲 and of the desired synchronous state x02共t兲 共red line兲 for pinning the first 12 neurons.

Downloaded 12 Nov 2008 to 144.214.130.8. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp

Chaos 18, 043111 共2008兲

Zhou et al.

25

30

20

25

15

20

10

αik

xi3 , x03

043111-8

15

5

10

0

5

−5 0

20

40

t

60

80

100

FIG. 8. 共Color online兲 The state time evolutions of the neurons xi3共1 艋 i 艋 200兲 and of the desired synchronous state x03共t兲 共red line兲 for pinning the first 12 neurons.

applied, while the network approaches synchronization with the desired state asymptotically with the pinning technique presented in Sec. IV. In addition, the adaptive feedback gains ␣ik共ik 苸 兵1 , 2其兲 are shown in Fig. 5, from which it is seen that ␣ik do not increase after a short period of time. B. An example with large-scale neural network

Over the past two decades, small-world20 and scale-free21 networks have been intensively investigated as large-scale complex networks. Barabási and Albert 共BA兲 model of preferential attachment has become the standard mechanism to explain the emergence of scale-free networks. Nodes are added to the network with a preferential bias toward attachment to nodes with already high degree. This naturally gives rise to hubs with a degree distribution following a power law. In this subsection, a BA neural network consisting of 200 HR neurons with m0 = 5 and m = 5 are considered, where m0 is the size of the initial network, m is the number of edges added in each step. Choose A = 10 A*, B = 0.5 A*, where A* is the adjacency matrix in which aii* = −兺 j苸Nia*ij, a*ij 苸 兵0 , 1其. Let the parameters and initial values in the network be the same as those in the previous subsection with N = 200, l = 12. After controlling the membrane potential of only 6% neurons with the largest degree, synchronization is achieved asymptotically. The state time evolutions of the neurons and of the desired synchronous state can be seen in Figs. 6–8. Besides, the adaptive parameters ␣ik共ik 苸 兵1 , 2 , . . . , 12其兲 are shown in Fig. 9. From that, it is obvious that ␣ik do not increase after a short interval of time.

0 0

20

40

t

60

80

100

FIG. 9. 共Color online兲 Adaptive feedback gains ␣ik共ik 苸 兵1 , 2 , . . . , 12其兲 for pinning the first 12 neurons.

VI. CONCLUSIONS

In this paper, we have studied the problem that which neurons in a delayed neurobiological network should be controlled to achieve adaptive pinning synchronization. In view of Schur complement and the Lyapunov function methods, we have proven that under a mild topology-based condition, synchronization of this general neural network can be reached using the criterion presented in Sec. III. Moreover, for a concrete neurobiological network consisting of identical HR neurons, we have proposed a specific pinning control technique to synchronize it. Finally, we have exhibited simple computational examples to illustrate the effectiveness of the proposed approach. In addition to the wide applications in neurobiological networks, this technique can also be applied to many other complex networks. ACKNOWLEDGMENTS

This work is supported with the funding of a Competitive Earmarked Research Grant No. PolyU 5269/06E from the University Grants Council of the Hong Kong government. Jin Zhou and Professor Jun-an Lu are supported by the National Natural Science Foundation of China under Grant Nos. 70771084, 60574045, and the National Basic Research Program of China under Grant No. 2007CB310805. APPENDIX: A PROPERTY OF HR MODEL

For any two vectors z1共t兲 = 共z11共t兲 , z12共t兲 , z13共t兲兲T and z2共t兲 = 共z21共t兲 , z22共t兲 , z23共t兲兲T in the HR equation, we have

2 共z21共t兲 − z11共t兲兲共f 1共z2共t兲兲 − f 1共z1共t兲兲兲 = 共z21共t兲 − z11共t兲兲关␹1共z21共t兲 + z11共t兲兲共z21共t兲 − z11共t兲兲 − 共z21共t兲 − z11共t兲兲共z21 共t兲 + z21共t兲z11共t兲 2 共t兲兲 − 共z22共t兲 − z12共t兲兲 − 共z23共t兲 − z13共t兲兲兴 + z11

艋 2␹1M共z21共t兲 − z11共t兲兲2 + 共z21共t兲 − z11共t兲兲共z22共t兲 − z12共t兲兲 + 共z21共t兲 − z11共t兲兲共z23共t兲 − z13共t兲兲 艋 2␹1M兩z21共t兲 − z11共t兲兩2 + 兩z21共t兲 − z11共t兲兩兩z22共t兲 − z12共t兲兩 + 兩z21共t兲 − z11共t兲兩兩z23共t兲 − z13共t兲兩, Downloaded 12 Nov 2008 to 144.214.130.8. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp

043111-9

Chaos 18, 043111 共2008兲

Pinning synchronization

共z22共t兲 − z12共t兲兲共f 2共z2共t兲兲 − f 2共z1共t兲兲兲 = 共z22共t兲 − z12共t兲兲关共␹1 + ␬1兲共z21共t兲 + z11共t兲兲共z21共t兲 − z11共t兲兲 − 共z22共t兲 − z12共t兲兲兴 艋 2共␹1 + ␬1兲M共z21共t兲 − z11共t兲兲共z22共t兲 − z12共t兲兲 − 共z22共t兲 − z12共t兲兲2 艋 2共␹1 + ␬1兲M兩z21共t兲 − z11共t兲兩z22共t兲 − z12共t兲兩− 兩z22共t兲 − z12共t兲兩2 and that 共z23共t兲 − z13共t兲兲共f 3共z2共t兲兲 − f 3共z1共t兲兲兲 = ␬2共z23共t兲 − z13共t兲兲关␹2共z21共t兲 − z11共t兲兲 − 共z23共t兲 − z13共t兲兲兴 艋 ␬2␹2兩共z21共t兲 − z11共t兲兲兩z23共t兲 − z13共t兲兩− ␬2兩z23共t兲 − z13共t兲兩2 , where M is a positive constant representing the boundary of the first variable in the HR equation. Thus the following inequality holds:

N

共xi − x0兲 兺 i=1

N




共f共xi兲 − f共x0兲兲 艋 兺 i=1

冢 冣 兩xi1 − x01兩

兩xi2 − x02兩

兩xi3 − x03兩




冢

共␹1 + ␬1兲M +

␬ 2␹ 2 + 1 2

H. Nozawa, Chaos 2, 377 共1992兲. I. Belykh, E. de Lange, and M. Hasler, Phys. Rev. Lett. 94, 188101 共2005兲. 3 Y. Y. Hou, T. L. Liao, C. H. Lien, and J. J. Yan, Chaos 17, 033120 共2007兲. 4 X. Q. Wu, Physica A 387, 997 共2008兲. 5 W. W. Yu and J. D. Cao, Chaos 16, 023119 共2006兲. 6 W. W. Yu, J. D. Cao, and J. H. Lü, SIAM J. Appl. Dyn. Syst. 7, 108 共2008兲. 7 W. L. Lu and T. P. Chen, IEEE Trans. Circuits Syst., I: Regul. Pap. 51, 2491 共2004兲. 8 F. Sorrentino and E. Ott, Phys. Rev. Lett. 100, 114101 共2008兲. 9 J. Zhou, J. A. Lu, and J. H. Lü, IEEE Trans. Autom. Control 51, 652 共2006兲. 10 J. D. Cao and J. Q. Lu, Chaos 16, 013133 共2006兲. 11 A. Raffone and C. V. Leeuwen, Chaos 13, 1090 共2003兲. 12 M. di Bernardo, F. Garofalo, and F. Sorrentino, Int. J. Bifurcation Chaos 1 2

共␹1 + ␬1兲M +

2␹2 M 1 2

1 ␬ 2␹ 2 + 1 2 2

−1

0

0

− ␬2

冣

冢 冣 兩xi1 − x01兩

兩xi2 − x02兩 . 兩xi3 − x03兩

Appl. Sci. Eng. 17, 3499 共2007兲. X. Wang and G. Chen, Physica A 310, 521 共2002兲. 14 X. Li, X. F. Wang, and G. R. Chen, IEEE Trans. Circuits Syst., I: Regul. Pap. 51, 2074 共2004兲. 15 F. Sorrentino, M. di Bernardo, F. Garofalo, and G. R. Chen, Phys. Rev. E 75, 046103 共2007兲. 16 P. A. Regalia and M. K. Sanjit, SIAM Rev. 31, 586 共1989兲. 17 S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory 共SIAM, Philadelphia, 1994兲. 18 W. W. Yu, J. D. Cao, and J. Wang, Neural Networks 20, 810 共2007兲. 19 K. K. Hassan, Nonlinear Systems, 3rd ed. 共Prentice-Hall, New York, 2002兲. 20 D. J. Watts and S. H. Strogatz, Nature 共London兲 393, 440 共1998兲. 21 A-L. Barabási and R. Albert, Science 286, 509 共1999兲. 22 W. W. Yu, G. R. Chen, and J. H. Lü, “On pinning synchronization of complex dynamical networks,” Automatica 共in press兲. 13

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