global exponential stability of discrete time multidirectional associative memory neural network with variable delays

Volume 2012, Article ID 831715, 10 pagesdoi:10.5402/2012/831715 Research Article Global Exponential Stability of Discrete-Time Multidirectional Associative Memory Neural Network with Var

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Volume 2012, Article ID 831715, 10 pages

doi:10.5402/2012/831715

Research Article

Global Exponential Stability of

Discrete-Time Multidirectional Associative

Memory Neural Network with Variable Delays

Min Wang, Tiejun Zhou, and Xiaolan Zhang

College of Science, Hunan Agricultural University, Hunan, Changsha 410128, China

Correspondence should be addressed to Tiejun Zhou,hntjzhou@126.com

Received 3 July 2012; Accepted 20 September 2012

Academic Editors: C.-K Lin and W F Smyth

Copyrightq 2012 Min Wang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A discrete-time multidirectional associative memory neural networks model with varying time delays is formulated by employing the semidiscretization method A suﬃcient condition for the existence of an equilibrium point is given By calculating diﬀerence and using inequality technique,

a suﬃcient condition for the global exponential stability of the equilibrium point is obtained The results are helpful to design global exponentially stable multidirectional associative memory neural networks An example is given to illustrate the eﬀectiveness of the results

1 Introduction

The multidirectional associative memory MAM neural networks were first proposed by the Japanese scholar M Hagiwara in 19901 The MAM neural networks have found wide applications in areas of speech recognition, image denoising, pattern recognition, and other more complex intelligent information processing So they have attracted the attention of many researchers2 7 In 5, we proposed a mathematical model of multidirectional

asso-ciative memory neural network with varying time delays as follows, which consists of the m fields, and there are n k neurons in the field k k 1, 2, , m:

dx ki

dt I ki − a ki x ki t m

p 1,p / k

n p

j1

w ki pj f pj

x pj

t − τ ki

pj t, 1.1

where k 1, 2, , m, i 1, 2, , n k , x ki t denote the membrane voltage of the ith neuron

in the field k at time t, a ki > 0 denote the decay rate of the ith neuron in the field k, f pj· is

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a neuronal activation function of the ith neuron in the field k, w ki pj is the connection weight

from the jth neuron in the field p to the ith neuron in the field k, I kiis the external input of

the ith neuron in the field k, and τ pj ki t is the time delay of the synapse from the j neuron in the field p to the ith neuron in the field k at time t We studied the existence of an equilibrium

point by using Brouwer fixed-point theorem and obtained a suﬃcient condition for the global exponential stability of an equilibrium point by constructing a suitable Lyapunov function However, discrete-time neural networks are more important than their continuous-time counterparts in applications of neural networks One can refer to8 10 in order to find out the research significance of discrete-time neural networks To the best of our knowledge, few studies have considered the stability of discrete-time MAM neural networks

In this paper, we first formulate a discrete-time analogues of the continuous-time net-work1.1, and in a next study the existence and the global exponential stability of an equi-librium point for the discrete-time MAM neural network

2 Discrete-Time MAM Neural Network Model and Some Notations

In this section we formulate a discrete-time MAM neural network model with time-varying delays by employing the semidiscretization technique8

Let N m

k1n k,Z denote the integers set, Z  {1, 2, 3, }, Z

0  {0, 1, 2, }, and Za, b {a, a 1, , b} where a, b ∈ Z.

Let h be a fixed positive real number denoting a uniform discretionary step size

and u denote the integer part of the real number u If t ∈ nh, n 1h n ∈ Z

0, then

t/h n By replacing the time t of the network 1.1 with nh, we can formulate the following

approximation of the network1.1:

dx ki

dt I ki − a ki x ki t m

p 1,p / k

n p

j1

w ki pj × f pj

⎛

⎝x pj

⎛

⎝nh −

⎡

⎣τ pj ki nh

h

⎤

⎦h

⎞

⎠

⎞

⎠ 2.1

for t ∈ nh, n 1h n ∈ Z

0 Denote τ ki

pj nh/h κ ki

pj n, we rewrite 2.1 as follows:

dx ki

dt a ki x ki t I ki m

p 1,p / k

n p

j1

w ki

pj f pj

x pj

nh − κ ki

pj nh. 2.2

Multiplying both sides of2.2 by e a ki t, we obtain

d

x ki e a ki t

dt e a ki t

⎛

⎝I ki m

p 1,p / k

n p

j1

w pj ki × f pj

x pj

nh − κ ki

pj nh

⎞

⎠. 2.3

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Integrating the2.3 over nh, tt ≤ n 1h, we have

x ki te a ki t − x ki nhe a ki nh e a ki t − e a ki nh

a ki

×

⎛

⎝I ki m

p 1,p / k

n p

j1

w ki pj f pj

x pj

nh − κ ki

pj nh

⎞

⎠.

2.4

Letting t → n 1h in 2.4, we obtain

x ki n 1h x ki nhe −a ki h 1− e −a ki h

a ki

×

⎛

⎝I ki m

p 1,p / k

n p

j1

w ki pj f pj

x pj

nh − κ ki

pj nh

⎞

⎠.

2.5

If we adopt the notations

x ki n x ki nh, α ki n e −a ki h ,

w pj ki w ki

pj

1− e −a ki h

a ki , I ki I ki1− e −a ki h

a ki ,

2.6

then we obtain the discrete-time analogue of the continuous-time network1.1 as follows:

x ki n 1 I ki α ki x ki n m

p 1,p / k

n p

j1

w pj ki f pj

x pj

n − κ ki

pj n 2.7

for n∈ Z

0, k ∈ Z1, m, i ∈ Z1, n k Obviously, 0 < α ki < 1.

Throughout this paper, for any k ∈ Z1, m, p ∈ Z1, m p / k, i ∈ Z1, n k , j ∈ Z1, n p , we assume that the neuronal activation functions f kiand the time delays sequences

κ ki

pj n satisfy the following conditions, respectively:

H1 There exist L ki > 0 such that |f ki x − f ki y| ≤ L ki |x − y| for each x, y ∈ R,

H2 0 < κ ki

pj supn∈Z

0κ ki pj n < ∞.

The initial conditions associated with2.7 are of the form

x ki n ϕ ki n, 2.8

where k ∈ Z1, m, i ∈ Z1, n k , n ∈ Z−κ ki , 0 , κ ki max1≤p≤m,p / kmax1≤j≤np κ ki pj

For convenience sake, set colbki b11, , b 1n1, b21, , b 2n2, , b m1 , , b mn mT , x colxki , fx colf ki x ki For any matrixes U u ij and V v ij, we use the notation

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U ≥ V to mean that u ij ≥ v ij for all i, j, and |U| |u ij | Let matrix A diag1 − α11, , 1−

α 1n1, 1 − α21, , 1 − α m1 , , 1 − α mn m , L diagL11, , L 1n1, L21, , L m1 , , L mn m,

W

⎛

⎜

O11 W12 · · · W 1m

W21 O22 · · · W 2m

· · · ·

W m1 W m2 · · · O mm

⎞

⎟

where

W kp

⎛

⎜

⎝

w p1 k1 w k1 p2 · · · w k1

pn p

w k2 p1 w k2 p2 · · · w k2

pn p

· · · ·

w kn k

p1 w kn k

p2 · · · w kn k

pn p

⎞

⎟

⎠

O kkare zero matrixesk, p ∈ Z1, m

3 The Existence and Global Exponential Stability of

an Equilibrium Point

In this section, we will give two theorems about the existence and the global exponential stability of an equilibrium point of the discrete-time MAM neural network2.7

Lemma 3.1 If 0 < b < 1, 0 ≤ x ≤ 1 − b, then be x ≤ x b.

Proof Define a function g t 1 − t ln t 0 < t ≤ 1 From gt 1 − t/t > 0 for 0 < t < 1,

we know that the function gt is increasing on the interval 0, 1 Therefore, gb < g1 0 for 0 < b < 1 So we have 1 − b < − ln b.

Define a function hx be x − x − b for x ≥ 0 again Obviously hx be x − 1,

hx be x From hx > 0, the hx is increasing Because there exists an unique critical number x0  − ln b, we know that hx < h− ln b 0 for 0 ≤ x ≤ − ln b It shows that hx is

deceasing on0, − ln b So we have hx be x − x − b ≤ h0 0 In view of 1 − b < − ln b, we obtain be x ≤ x b for 0 ≤ x ≤ 1 − b.

With a similar method of5, we can prove the followingTheorem 3.2

Theorem 3.2 Suppose that all the neuronal activation functions f ki · k ∈ Z1, m, i ∈ Z1, n k are continuous and the condition (H1) holds If B A − |W|L is a nonsingular M matrix,

then there exists an equilibrium point of the discrete-time MAM neural network2.7.

Proof Let β ki m

p 1, / kn p

j1|w ki

pj f pj 0|/1 − α ki |I ki |/1 − α ki Obviously, β ki ≥ 0 Denote

β colβ ki , r colr ki A − |W|L−1Aβ Because Aβ ≥ 0 and B A−|W|L is a nonsingular

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M matrix, by Lemma A3 in11, we have r A − |W|L−1Aβ ≥ 0 That is r ki ≥ 0 for any

k ∈ Z1, m, i ∈ Z1, n k From the definition of r, we have E − A−1|W|Lr β Therefore,

1

1− α ki

⎡

⎣ m

p 1,p / k

n p

j1

w ki

pjL

pj r pj

⎤

⎦ β ki r ki 3.1

LetΩ {x colx ki | x ki ∈ −r ki , r ki } with a norm x max1≤k≤mmax1≤i≤nk {|x ki|} Obviously,Ω is a bounded closed compact subset Define a function F : Ω → R N as Fx 

colFki x, where

F ki x  1

1− α ki

⎡

⎣ m

p 1,p / k

n p

j1

w ki

pj f pj

x pj

I ki

⎤

From the conditionH1 and 3.1, we have

|F ki x| ≤ 1

1− α ki

⎡

⎣ m

p 1,p / k

n p

j1

w ki

pj

L pjx pj f pj0 |Iki|

⎤

⎦

≤ 1

1− α ki

⎡

⎣ m

p 1,p / k

n p

j1

w ki

pj

L pj r pj f pj0 |Iki|

⎤

⎦

1

1− α ki

⎡

⎣ m

p 1,p / k

n p

j1

w ki

pjL

pj r pj

⎤

⎦ β ki r ki

3.3

Thus Fx is a self-map from Ω to Ω By Brouwer fixed-point theorem, there exists at least a

x∗∈ Ω, such that Fx∗ x∗ That is

x∗ki I ki α ki x ki∗ m

p 1,p / k

n p

j1

w ki pj f pj

x∗pj

Therefore x∗is an equilibrium point of the MAM neural network2.7

Next we prove the global exponential stability of the equilibrium point of the discrete-time MAM neural network2.7

Theorem 3.3 Suppose that all the neuronal activation functions f ki ·k ∈ Z1, m, i ∈ Z1, n k

are continuous and the conditions (H1) and (H2) hold If B A − |W|L is a nonsingular M matrix,

then the equilibrium point of the MAM neural network2.7 is global exponential stable.

Proof Let x∗  colx∗

ki be an equilibrium point for the MAM neural network 2.7, xn

colxki n, be an arbitrary solution of 2.7 Set un colu ki n, where

u ki n x ki n − x∗

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for k ∈ Z1, m, i ∈ Z1, n k Define functions

F pj z f pj

z x∗

pj

− f pj

x∗pj

where p ∈ Z1, m, j ∈ Z1, n p Obviously, F pj0 0 and from the condition H1, we have

F pj z ≤ L pj |z| 3.7

for any z∈ R By 3.5, the MAM neural network 2.7 is reduced to the form

u ki n 1 α ki u ki n m

p 1,p / k

n p

j1

w pj ki F pj

u pj

n − κ ki

pj n, 3.8

where k ∈ Z1, m, i ∈ Z1, n k Obviously, there exists an equilibrium point u∗ 0 of the system3.8 From 2.8 and 3.5, the initial conditions associated with 3.8 are of the form

u ki n ψ ki n ϕ ki n − x∗

where k ∈ Z1, m, i ∈ Z1, n k , n ∈ Z−κ ki , 0 Let ψ colψ ki n, ψ 

maxk ∈Z1,mmaxi ∈Z1,n ksup−κ ki ≤n≤0 |ψ ki n|.

Because B A − |W|L is a nonsingular M matrix, then there exist constants ξ ki > 0

i ∈ Z1, n k , k ∈ Z1, m such that

1 − α ki ξ ki− m

p 1,p / k

n p

j1

w ki

pjL

Define the functions

H ki λ 1 − α ki − λξ ki e −λ− m

p 1,p / k

n p

j1

w ki

pjL

pj ξ pj e λκ ki , 3.11

where λ ∈ R , k ∈ Z1, m, i ∈ Z1, n k Apparently H ki λ is strictly monotone decreasing

and continuous function In view of3.10, it is clear that H ki 0 > 0, H ki 1−α ki < 0 Therefore there exist λ ki ∈ 0, 1 − α ki such that H ki λ ki 0 k ∈ Z1, m, i ∈ Z1, n k Taking α

mink ∈Z1,mmini ∈Z1,n k{λ ki } < min k ∈Z1,mmini ∈Z1,n k{1 − α ki}, we have

H ki α 1 − α ki − αξ ki e −α− m

p 1,p / k

n p

j1

w ki

pjL

pj ξ pj e ακ ki≥ 0 3.12

for i ∈ Z1, n k , k ∈ Z1, m.

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of system3.8, we have

Δy ki n e α n 1 |u ki n 1| − e αn |u ki n|

≤ α ki − e −α

α ki

e α n 1 |u ki n 1| e αn

α ki

× m

p 1,p / k

n p

j1

w ki

pjF

pj

u pj

n − κ ki

pj n.

3.13

By using the inequality3.7, we have

Δy ki n ≤ α ki − e −α

α ki e

α n 1 |u ki n 1| e αn

α ki

× m

p 1,p / k

n p

j1

w ki

pjL

pju

pj

n − κ ki

pj n

α ki − e −α

α ki y ki n 1 1 α

ki

m

p 1,p / k

n p

j1

w ki

pj

× L pj e ακ ki ny

pj

n − κ ki

pj n

≤ α ki − e −α

α ki y ki n 1 1 α

ki

m

p 1,p / k

n p

j1

w ki

pj

× L pj e ακ kiy

pj

n − κ ki

pj n

≤ α ki e α− 1

α ki e α y ki n 1 1

α ki

m

p 1,p / k

n p

j1

w ki

pj

× L pj e ακ ki sup

n −κ ki ≤s≤n

y pj s.

3.14

ByLemma 3.1, we have

Δy ki n ≤ α α ki− 1

α ki e α y ki n 1 1 α

ki

m

p 1,p / k

n p

j1

w ki

pj

× L pj e ακ ki sup

n −κ ki ≤s≤n

y pj s. 3.15

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Let ξ mink ∈Z1,mmini ∈Z1,n k{ξ ki }, ξ max k ∈Z1,mmaxi ∈Z1,n k{ξ ki } and l0  1 δψ/ξ, where δ is a positive constant Therefore, when s ∈ Z−κ ki , 0, we have

y ki s e αsψ ki s ≤ ψ < ξ ki l0 3.16

for i ∈ Z1, n k , k ∈ Z1, m We assert that

y ki n< ξ ki l0 3.17

n ∈ Z−κ ki , t0 From 3.12 and 3.15, and noticed that α α ki − 1 < 0, we obtain

Δy ki t0 ≤ 1

α ki

⎡

⎣α α ki − 1ξ ki e −α m

p 1,p / k

n p

j1

w ki

pjL

pj ξ pj e ακ ki

⎤

⎦l0< 0. 3.18

It conflicts withΔ|y ki t0| ≥ 0 Therefore,

y ki n< ξ ki l0 3.19

for n∈ Z Then we have

|u ki n| < ξ ki l0e −αn≤ 1 δξ

ξ ψe −αn Mψe −αn , 3.20

where M 1 δξ/ξ > 1 So the zero solution of the system 3.8 is global exponential stable; thus the equilibrium point of the discrete-time MAM neural network2.7 is global exponential stable

4 An Example

Consider the following discrete-time MAM neural network with three fields:

x11n 1 I11 α11x11n w11

21f21

x21

n − κ11

21n

w11

31f31

x31

t − κ11 31

,

x12n 1 I12 α12x12n w12

21f21

x21

n − κ12

21n

w12

31f31

x31

t − κ12 31

,

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− 5 0 10 20 30 40 0

1

2

3

The state trajectories of the 1th neuron on the first field 1 with three initial values

0

1

2

3

0

1

2

3 The state trajectories of the 1th neuron on the first field 2 with three initial values

0

1

2

3

Figure 1: The globally exponential stability of the equilibrium of the MAM network 4.1 with 10 cases random initial values

x21n 1 I21 α21x21n w21

11f11

x11

n − κ21

11n

w21

12f12

x12

n − κ21

12n w21

31f31

x31

n − κ21

31n,

x31n 1 I31 α31x31n w31

11f11

x11

n − κ31

11n

w31

12f12

x12

n − κ31

12n w31

21f21

x21

n − κ31

21n,

4.1

where the neuronal signal decay rates α11  0.1, α12  0.2, α21  0.2, α31  0.1, the external inputs I11  1, I21 1, I21  1, I31  1, and connection weights w11

21  0.2, w11

31  0.3, w12

21  0.2,

w2131  0.4, w21

11  0.25, w21

12  −0.2, w21

31  0.3, w31

11  −0.2, w31

12  0.2, w31

21  0.3 The neuronal activation functions f ki x x, the time delays κ ki

pj n 5 sinπ/2n.

Obviously, signal transfer functions f ki x are continuous, and they satisfy the

con-ditionH1, and the constant L ki 1 The time delays κ ki

pj n satisfy the condition H2, and

κ ki 6

By calculating, we have

B A − |W|L 

⎛

⎜

⎝

0.9 0 −0.2 −0.3

0 0.8 −0.2 −0.4

−0.25 −0.2 0.8 −0.3

−0.2 −0.2 −0.3 0.9

⎞

⎟

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It is easy to verify that the matrix B is a nonsingular M matrix Then by Theorems3.2and3.3, there exists an equilibrium point which is globally exponentially stable for the MAM neural network4.1

The numerical simulation is given inFigure 1in which ten cases of initial values are taken at random FromFigure 1, we can know that the MAM neural network4.1 converges globally exponentially to the equilibrium point x∗

11, x∗12, x∗21, x31∗ T ≈ 2.1735, 2.68, 1.9635, 1.8782 Tno matter what it starts from the initial states

5 Conclusions

In this paper, we have formulated a discrete-time analogue of the continuous-time multi-directional associative memory neural network with time-varying delays by using semidis-cretization method Some suﬃcient conditions for the existence and the global exponential stability of an equilibrium point have been obtained Our results have shown that the dis-crete-time analogue inherits the existence and global exponential stability of equilibrium point for the continuous-time MAM neural network

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