IEEE TRANSACTIONS ON FUZZY SYSTEMS 1

Short Papers

Exponential Adaptive Lag Synchronization of MemristiveNeural Networks via Fuzzy Method and Applications in

Pseudorandom Number Generators

Shiping Wen, Zhigang Zeng, Tingwen Huang,and Yide Zhang

Abstract—This paper investigates the problem of exponential lag syn-chronization control of memristive neural networks (MNNs) via the fuzzymethod and applications in pseudorandom number generators. Based onthe knowledge of memristor and recurrent neural networks, the model ofMNNs is established. Then, considering the state-dependent properties ofmemristor, a fuzzy model of MNNs is employed to provide a new way of an-alyzing the complicated MNNs with only two subsystems, and update lawsfor the connection weights of slave systems and controller gain are designedto make the slave systems exponentially lag synchronized with the mastersystems. Two examples about synchronization problems are presented toshow the effectiveness of the obtained results, and an application of theobtained theory is also given in the pseudorandom number generator.

Index Terms—Adaptive lag synchronization, fuzzy model, memristor,neural networks, pseudorandom number generator (PRNG).

I. INTRODUCTION

Although current digital computers can now possess computingspeed and complexity to emulate the brain functionality of animalslike a spiders, mice, and cats [1], [2], the associated energy dissipa-tion in the system grows exponentially along the hierarchy of animalintelligence, as the sequential processing of fetch, decode, and execu-tion of instructions through the classical von Neumann bottleneck ofconventional digital computers has resulted in less-efficient machinesas their ecosystems have grown to be increasingly complex [3]. Forexample, to perform certain cortical simulations at the cat scale evenat 83 times slower firing rate, the IBM team in [1] has to employ BlueGene/P (BG/P): a super computer equipped with 147456 CPUs and144 TBs of main memory. On the other hand, the human brain con-tains more than 100 billion neurons, and each neuron has more than20 000 synapses. Efficient circuit implementation of synapses, there-fore, is especially important to build a brain-like machine. However,

Manuscript received August 13, 2013; revised October 15, 2013; acceptedNovember 16, 2013. Date of publication December 11, 2013; date of currentversion. This work was supported by the Natural Science Foundation of Chinaunder Grant 61125303, National Basic Research Program of China (973 Pro-gram) under Grant 2011CB710606, Research Fund for the Doctoral Program ofHigher Education of China under Grant 20100142110021, the Excellent YouthFoundation of Hubei Province of China under Grant 2010CDA081, and the Na-tional Priority Research Project NPRP 4-451-2-168, funded by Qatar NationalResearch Fund.

S. P. Wen, Z. G. Zeng, and Y. D. Zhang are with the School of Au-tomation, Huazhong University of Science and Technology, and Key Lab-oratory of Image Processing and Intelligent Control of Education Ministryof China, Wuhan, Hubei, 430074, China (e-mail: wenshiping226@126.com;zgzeng527@126.com; edwardchang@hust.edu.cn).

T. W. Huang is with Texas A & M University at Qatar, Doha 23874, Qatar(e-mail: tingwen.huang@qatar.tamu.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2013.2294855

since shrinking the current transistor size is very difficult, introducinga more efficient approach is essential for further development of neuralnetwork implementations.

In 2008, the Williams group announced a successful fabrication of avery compact and nonvolatile nano scale memory called the memristor[4]. It was postulated by Chua [5] as the fourth basic circuit elementin electrical circuits. It is based on the nonlinear characteristics ofcharge and flux. By supplying a voltage or current to the memristor,its resistance can be altered [6]. This way, the memristor remembersinformation. Several examples of successful multichip networks ofspiking neurons have been recently proposed [7]–[9]; however, thereare still a number of practical problems that hinder the developmentof truly large-scale, distributed, massively parallel networks of verylarge scale integration (VLSI) neurons, such as how to set the weight ofindividual synapses in the network. It is well-known that changes in thesynaptic connections between neurons are widely believed to contributeto memory storage, and the activity-dependent development of neuralnetworks. These changes are thought to occur through correlated-based,or Hebbian plasticity.

In addition, we notice neural networks have been widely studiedin recent years, for their immense application prospective [10]–[23].Many applications have been developed in different areas such as com-binatorial optimization, knowledge acquisition, and pattern recogni-tion. Recently, the problem of lag synchronization of coupled neuralnetworks, which is one of the hottest research fields of complex net-works, has been a challenging issue because of its potential applicationsuch as information science, biological systems, and so on [24]–[36].

On the other hand, synchronization problem of neural networkshas attracted great attention because of its potential applicationsin many fields such as secure communications, biological sys-tems, information science, image encryption, and pseudorandomnumber generator (PRNG) [37], [38]. Currently, a wide variety ofsynchronization phenomena have been investigated, such as completesynchronization [39]–[41], generalized synchronization [42], phasesynchronization [43], and lag synchronization [44]. In the case of realapplications, it is very hard to directly get the identical parameters ofthe master and slave systems. Therefore, adaptive synchronization maybe a good choice for such cases. It is worth mentioning that in con-nected electronic networks, the occurrence of time delay is unavoidablebecause of finite signal transmission times, switching speeds, and someother reasons. Thus, the complete synchronization of neural networksis hard to implement effectively and it is more reasonable to considerthe lag synchronization problem.

However, to the best of the authors’ knowledge, the research onglobal exponential lag adaptive synchronization of memristive neuralnetworks is still an open problem that deserves further investigation.To shorten sup gap, we investigate the problem of global exponentiallag adaptive synchronization for a class of memristive neural networkswith time-varying delays. The main contributions of this paper can besummarized as follows: 1) A model of MNNs is established in accor-dance with the memristor-based electronic circuits; 2) a fuzzy model ofmemristive neural networks is employed to give a new way to analyzethe complicated MNNs with only two subsystems; 3) update laws aredesigned for the connection weights of slave systems and controllergain to make the slave systems exponentially lag synchronized withthe master systems; and 4) a simulation example is presented to showthe applications of the obtained results in the PRNG.

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2 IEEE TRANSACTIONS ON FUZZY SYSTEMS

Fig. 1. Circuit of memristive network, where xi (.) is the state of the ith subsystem; fj (.) is the amplifier; Rf ij is the connection resistor between the amplifierfj (.) and state xi (.); Mi and Ci are the memristor and capacitor; Ii is the external input; and ai , bi are the outputs i, j = 1, 2, . . . , n.

A. Circuit of Memristive Neural Networks

The memristive neural network can be implemented by VLSI circuitsas shown in Fig. 1. fj is the activation function, τj (t) is the time-varyingdelay, for the i-th subsystem, xi (t) is the voltage of the capacitorCi , fj (xj (t)), fj (xj (t − τj (t))) are the functions of xi (t) with orwithout time-varying delays respectively, Rf ij is the resistor betweenthe feedback function fj (xj (t)) and xi (t), Rg ij is the resistor betweenthe feedback function fj (xj (t − τj (t))) and xi (t), Mi is the memristorparallel to the capacitor Ci , and Ii is an external input or bias, wherei, j = 1, 2, . . . , n.

The memductance of the memristors can be depicted as in Fig. 2 [45],which are bounded. Thus, by Kirchoff’s current law, the equation ofthe i-th subsystem is written as follows:

Ci xi (t) = −[

n∑j=1

(1

Rf ij

+1

Rg ij

)+ Wi (xi (t))

]xi (t)

+n∑

j=1

signij fj (xj (t))Rf ij

+n∑

j=1

signij fj (xj (t − τj (t)))Rg ij

+ Ii (1)

where

signij ={

1, i �= j−1, i = j

and Wi are the memductances of the memristors Mi , and

Wi (xi (t)) ={

W ′i , xi (t) ≤ 0

W ′′i , xi (t) > 0.

Therefore

xi (t) = −di (xi (t))xi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si (2)

where

aij =signij

CiRf ij

, bij =signij

CiRg ij

, si =Ii

Ci

di (xi (t)) =1Ci

[n∑

j=1

(1

Rf ij

+1

Rg ij

)+ Wi (xi (t))

]

={

d1 i , xi (t) ≤ 0d2 i , xi (t) > 0.

IEEE TRANSACTIONS ON FUZZY SYSTEMS 3

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5x 10−4

V

I

Fig. 2. Typical I-V characteristic of memristor [45]. The pinched hysteresisloop occurs because of the nonlinear relationship between the memristancecurrent and voltage. The memristor exhibits the feature of pinched hysteresis,which means that a lag occurs between the application and the removal of afield and its subsequent effect, just like the neurons in the human brain.

Then, we can get

x(t) = −D(x(t))x(t) + Af (x(t)) + Bf (x(t − τ (t))) + s (3)

where

D(x(t)) = diag{d1 (x1 (t)), d2 (x2 (t)), . . . , dn (xn (t))}

A = [aij ]n×n , B = [bij ]n×n , s = (s1 , s2 , . . . , sn )T

f (x(t)) = (f1 (x1 (t)), . . . , fn (xn (t)))T

f (x(t − τ (t))) = (f1 (x1 (t − τ1 (t))), . . . , fn (t − τn (t)))T .

B. Fuzzy Model of Memristive Neural Networks

To solve the problem about nonlinear control, fuzzy logic has at-tracted much attention as a powerful tool. Among various kinds of fuzzymethods, the Takagi–Sugeno fuzzy systems are widely accepted as auseful tool for design and analysis of fuzzy control system [46]–[53].Currently, some control methods for memristive systems have beenproposed [54], in which the number of the linear subsystems is decidedby how many minimum nonlinear terms should be linearized in orig-inal system. Then, the memristive neural network (2) can be exactlyrepresented by the fuzzy model as follows:

Rule 1: IF xi (t) is N1 i , THEN

xi (t) = −d1 i xi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si

Rule 2: IF xi (t) is N2 i , THEN

xi (t) = −d2 i xi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si

where N1 i is xi (t) ≤ 0, and N2 i is xi (t) > 0. With a center-averagedefuzzier, the over fuzzy system is represented as

xi (t) = −2∑

l=1

ϑli (t)dlixi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si (4)

where

ϑ1 i (t) ={

1, xi (t) ≤ 0,0, xi (t) > 0,

ϑ2 i (t) ={

0, xi (t) ≤ 01, xi (t) > 0.

When the system becomes complicated with n memristors, there are2n subsystems (according to 2n fuzzy rules) and 2n equations in theT–S fuzzy system. If n is large, the number of linear subsystems in theT–S fuzzy system is huge. For this problem, Li and Ge proposed a fuzzymodeling method and applied in the lag synchronization problem of twototally different chaotic systems [55]. Based on this work, a new fuzzymodel is proposed to simplify memristive systems, in which only twosubsystems are included. Furthermore, through this model, the idea ofPDC can be applied to achieve between subsystems. Therefore, system(4) can be represented by

x(t) = −2∑

l=1

Πl (t)Dlx(t) + Af (x(t))

+ Bf (x(t − τ (t))) + s (5)

where Πl (t) = diag{ϑl1 (t), . . . , ϑln (t)},∑2

l=1 ϑli (t) = 1, i =1, . . . , n, l = 1, 2, and

Dl = diag{dl1 , dl2 , . . . , dln }.The initial conditions of system (5) is in the form of x(t) = φ(t) ∈C([−τ, 0],Rn ), τ = max1≤i≤n {τi (t)}.

II. PRELIMINARIES

Denote u = (u1 , . . . , un )T , |u| as the absolute-value vector; i.e.,|u| = (|u1 |, |u2 |, . . . , |un |)T , ‖x‖p as the p-norm of the vector x withp, 1 ≤ p < ∞. ‖x‖∞ = maxi∈{1 ,2 , . . . ,n }|xi | is the vector infinity norm.Denote ‖D‖p as the p-norm of the matrix D with p. Denote C asthe set of continuous functions. In addition, we assume the followingthroughout the paper:

A1. For i ∈ {1, 2, . . . , n}, the activation function fi is Lipschitzcontinuous, and ∀r1 , r2 ∈ R, there exists real number ιi such that

0 ≤ fi (r1 ) − fi (r2 )r1 − r2

≤ ιi

where fi (0) = 0, r1 , r2 ∈ R, and r1 �= r2 .A2. For i ∈ {1, 2, . . . , n}, the time-varying delay τi (t) satisfies the

following inequalities:

0 ≤ τi (t) ≤ τ

τi (t) ≤ μ. (6)

In this paper, we consider system (5) as the master system, andthrough electronic inductors, the values of memristor will be presentedin the corresponding slave system; then, the slave system is given as

zi (t) = −2∑

l=1

ϑli (t)dli zi (t) +n∑

j=1

aij (t)fj (zj (t))

+n∑

j=1

bij (t)fj (zj (t − τj (t))) + si + ui (t) (7)

4 IEEE TRANSACTIONS ON FUZZY SYSTEMS

or in compact form

z(t) = −2∑

l=1

Πl (t)Dlz(t) + A(t)f (z(t))

+ B(t)f (z(t − τ (t))) + s + u(t), t ≥ 0 (8)

where A(t) = (a(t))n×n , B(t) = (b(t))n×n are unknown connectionweights, z(t) = (z1 (t), z2 (t), . . . , zn (t))T , Πl (t) is related to the mas-ter system, and u(t) = (u1 (t), u2 (t), . . . , un (t))T is the control inputwith the following form:

ui (t) = �i (t)(zi (t) − xi (t − ν)) (9)

where �i (t) is the adaptive control gain which needs to be designed,and the initial condition of system (8) is in the form of z(t) = ψ(t) ∈C([−τ, 0],Rn ).

Definition 1: The master system (4) and slave system (7) are saidto be globally exponentially synchronized with lag ν , if there existpositive constants λ and μ, such that

‖z(t) − x(t − ν)‖ ≤ ωe−λt , t ≥ 0. (10)

If ν = 0, the synchronization is complete synchronization.Notation: The notation used here is fairly standard. The superscript

“T ” stands for matrix transposition, Rn denotes the n-dimensionalEuclidean space, and Rm ×n is the set of all real matrices of di-mension m × n, I and 0 represent the identity matrix and zeromatrix, respectively. Matrices, if their dimensions are not explic-itly stated, are assumed to be compatible for algebraic operations.Denote u = (u1 , . . . , un )T , |u| as the absolute-value vector, i.e.,|u| = (|u1 |, |u2 |, . . . , |un |)T , ‖x‖p as the p-norm of the vector x withp, 1 ≤ p < ∞. ‖x‖∞ = maxi∈{1 ,2 , . . . ,n }|xi | is the vector infinity norm.Denote ‖D‖p as the p-norm of the matrix D with p. Denote C as theset of continuous functions.

III. MAIN RESULTS

In practice, lag exists, when the synchronization happens betweenthe master and slave systems, which can be characterized as z(t) =x(t − υ) for some constant lag time υ > 0. The lag synchronizationerror between the master and slave systems can be presented as

e(t) = z(t) − x(t − ν). (11)

Then, we can get the adaptive lag synchronization algorithm for mem-ristive neural networks with unknown connection weights of the slavesystems.

Theorem 1: System (7) will be globally exponentially synchronizedwith system (4) with lag ν , if the connection weights of system (7)aij (t), bij (t) and controller gain �i (t), i, j = 1, 2, . . . , n, are adaptedin accordance with the following update law:

˙aij (t) = −�ij fj (zj (t))sign(ei (t))eεt

˙bij (t) = −αij fj (zj (t − τj (t)))sign(ei (t))eεt

�i (t) = −βi |ei (t)|eεt (12)

where �ij , αij and βi , i, j = 1, 2, . . . , n are arbitrary positive con-stants.

Proof: Let Vi (t) = eεt |ei (t)|, and d−i = minl=1 ,2{dli}. Calculating

the derivative of Vi (t) along systems (4) and (7), we can get

Vi (t) ≤ eεt

{(ε + �i (t) − d−

i )|ei (t)|

+

∥∥∥∥∥n∑

j=1

aij (fj (zj (t)) − fj (xj (t − ν)))

∥∥∥∥∥+

∥∥∥∥∥n∑

j=1

bij (fj (zj (t − τj (t))) − fj (xj (t − τj (t) − ν)))

∥∥∥∥∥+

n∑j=1

(aij (t) − aij )fj (zj (t))sign(ei (t))

+n∑

j=1

(bij (t) − bij )fj (zj (t − τj (t)))sign(ei (t))

}

≤ eεt

{(ε + �i (t) − d−

i )|ei (t)| +n∑

j=1

|aij |ιj |ej (t)|

+n∑

j=1

|bij |ιj |ej (t − τj (t))|

+n∑

j=1

(aij (t) − aij )fj (zj (t))sign(ei (t))

+n∑

j=1

(bij (t) − bij )fj (zj (t − τj (t)))sign(ei (t))

}. (13)

Define a Lyapunov functional as

V (t) =n∑

i=1

{Vi (t) +

n∑j=1

|bij |ιj eετ j (t)

∫ t

t−τ j (t)|ej (s)|eεs ds

+n∑

j=1

12�ij

(aij (t) − aij )2 +n∑

j=1

12αij

(bij (t) − bij )2

+n∑

j=1

12βi

(�i (t) + ωi )2

}(14)

where ωi is a constant, and aij (t), bij (t), and �i (t) are adapted by theupdate law (12). Then

V (t) =n∑

i=1

{Vi (t) + eεt

n∑j=1

|bij |ιj (eετ j (t) |ej (t)|−|ej (t − τj (t))|)

+n∑

j=1

1�ij

(aij (t) − aij ) ˙aij (t)+n∑

j=1

1αij

(bij (t)−bij )˙bij (t)

+1βi

(�i (t) + ωi )�i (t)

}

≤n∑

i=1

{Vi (t) + eεt

n∑j=1

|bij |ιj (eετ |ej (t)| − |ej (t − τj (t))|)

+n∑

j=1

1�ij

(aij (t) − aij ) ˙aij (t)+n∑

j=1

1αij

(bij (t)−bij )˙bij (t)

+1βi

(�i (t) + ωi )�i (t)

}

IEEE TRANSACTIONS ON FUZZY SYSTEMS 5

=n∑

i=1

{Vi (t) + eεt

n∑j=1

|bij |ιj (eετ |ej (t)| − |ej (t − τj (t))|)

−n∑

j=1

(aij (t) − aij )fj (zj (t))sign(ei (t))eεt

−n∑

j=1

(bij (t) − bij )fj (zj (t − τj (t)))sign(ei (t))eεt

− (�i (t) + ωi )|ei (t)|eεt

}. (15)

Combining the derivatives (13) and (15)

V (t) ≤ eεt

n∑i=1

{(ε − ωi − d−

i )|ei (t)|

+n∑

j=1

(|aij | + |bij |eετ )ιj |ej (t)|}

= eεt

n∑i=1

(ε − d−

i +n∑

j=1

(|aij | + |bij |eετ )ιj − ωi

)|ei (t)|.

(16)

Set

ωi > −d−i +

n∑j=1

(|aij | + |bij |eετ )ιj . (17)

If we let ε be small enough, we can get

eεt

n∑i=1

(ε − d−

i +n∑

j=1

(|aij | + |bij |eετ )ιj − ωi ≤ 0. (18)

HenceV (t) ≤ 0 (19)

and with (13) and (15), we can get

eεt

n∑i=1

|ei (t)| ≤ V (t) ≤ V (0) (20)

where

V (0) ≤n∑

i=1

{ (1 +

n∑j=1

|bi j|ιj τ eετ

)max

s∈[−τ + ν ,ν ]|ϕ(s) − φ(s)|

+ �i (0)

}

in which

�i (0) =n∑

j=1

12�ij

(aij (0) − aij )2 +n∑

j=1

12αij

(bij (0) − bij )2

+n∑

j=1

12βi

(�i (0) + ωi )2

}

≡ Φ.

Consequently, the following inequality holds:

n∑i=1

|ei (t)| ≤ Φe−εt , t ≥ 0. (21)

This completes the proof.When the connection weights aij (t) and bij (t) are known, then we

can get update law of the adaptive controller gain as follows.Corollary 1: System (7) will be globally exponentially synchronized

with system (4) with lag ν , if controller gains �i (t), i = 1, 2, . . . , n areadaptive iterating in accordance with the following update law:

�i (t) = −βi |ei (t)|eεt (22)

where βi , i = 1, 2, . . . , n are arbitrary positive constants.The proof is the same as in Theorem 1, and therefore, it is omitted.

IV. NUMERICAL EXAMPLES

In this section, several numerical examples are utilized to demon-strate the effectiveness and applications of the obtained results.

Example 1: Consider memristive system (5) with

A =[

2 −0.1−5 4.5

], B =

[−1.5 −0.1−0.2 −4

]

fi (xi ) = tanh(xi ), τi (t) = 1, si = 0, i = 1, 2.

Let

d1 (x1 (t)) ={

0.9, x1 (t) ≤ 01.1, x1 (t) > 0,

d2 (x2 (t)) ={

1.1, x2 (t) ≤ 00.9, x2 (t) > 0.

The initial values of master system (5) are set to be [0.4 0.6]. And thedynamical behaviors of this system are shown as in Fig. 3, which arechaotic and can be used in secure communications.

Without loss of generality, let

A(t) = A, B(t) =[

b11 (t) −0.1−0.2 b22 (t)

]in which

b11 (0) = b22 (0) = 1.

Set ε = 0.02, �i (0) = 1, α11 = α22 = 0.9, β1 = β2 = 0.8, the lagtime ν = 1.5, and the initial values of the slave system is set to be[−0.5 − 0.5]. Then, we can get the simulation results as shown inFig. 4.

Example 2: Consider memristive system (5) with

A =[

1 + π/4 200.1 1 + π/4

], B =

[−1.3

√2π/4 0.1

0.1 −1.3√

2π/4

]

fi (xi ) = 0.5(|xi + 1| − |xi − 1|), τi (t) = 1, si = 0, i = 1, 2.

Let

d1 (x1 (t)) ={

1.0, x1 (t) ≤ 01.2, x1 (t) > 0,

d2 (x2 (t)) ={

1.2, x2 (t) ≤ 01.0, x2 (t) > 0.

The initial values of master system (5) are set to be [0.2 − 0.2], andthe dynamical behaviors of this system are shown in Fig. 5.

LetA(t) = A, B(t) = B

and ε = 0.01, �i (0) = 0.6, β1 = β2 = 0.8, the lag time ν = 1.5, andthe initial values of the slave system are set to be [−0.5 0.5]. Then, wecan get the simulation results as shown in Fig. 6.

V. MEMRISTIVE NEURAL NETWORKS IN THE PSUEDORANDOM

NUMBER GENERATOR

Based on the above discussion, this section will discuss the appli-cations of the exponential lag synchronization between MNNs in thefield of the PRNG.

6 IEEE TRANSACTIONS ON FUZZY SYSTEMS

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 1(t)

0 20 40 60 80 100−10

−5

0

5

10

t

x 2(t)

−1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8

x1(t)

x 2(t)

(a) (b)

Fig. 3. Transient behavior of memristive system (4).

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 1(t),z

1(t)

x1(t)

z1(t)

0 20 40 60 80 100−10

−5

0

5

10

t

x 2(t),z

2(t)

x2(t)

z2(t)

(a)

−1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8

x1(t),z1(t)

x 2(t),z

2(t)

(b)

Fig. 4. State trajectories of master system (4) and slave system (7) when lag time ν = 1.5.

0 20 40 60 80 100−20

−10

0

10

20

t

x 1(t)

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 2(t)

(a)

−15 −10 −5 0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1(t)

x 2(t)

(b)

Fig. 5. Transient behavior of memristive system (4).

IEEE TRANSACTIONS ON FUZZY SYSTEMS 7

0 20 40 60 80 100−20

−10

0

10

20

t

x 1(t),z

1(t)

x1(t)

z1(t)

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 2(t),z

2(t)

x2(t)

z2(t)

(a)

−15 −10 −5 0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1(t),z1(t)

x 2(t),z

2(t)

(b)

Fig. 6. State trajectories of master system (4) and slave system (7) when lag time ν = 1.5.

PRNG

Pseudu-random outputs

Unpredictable inputs

Fig. 7. Black-box view of a PRNG.

As random number generation plays an important role in cryptog-raphy and software testing, PRNG are intended to be general-purposevehicles for the creation of random data used in these areas as inFig. 7 [65]. In addition, many different methods exist to generate pseu-dorandom numbers like Blum-Blum-Shub, Mersenne Twister algo-rithms, etc. It is well known that pseudorandomness is the basis forcryptography and is essential for the achievement of any cryptographicfunction such as encryption, authentication, and identification. Neuralnetworks can be used to generate random numbers as they are highlynonlinear mathematical systems. Based on the dynamics of neural net-works, pseudorandom numbers are generated via neural plasticity.

Meanwhile, it is important to produce a perfect random number gen-erator that gets a series of independent identically distributed continu-ous random variables in [0 1] [66]. One can produce a perfect randomnumber generator only using nondeterministic physical phenomena. Itis a practical way to employ a computer to produce a random-lookingsequence of numbers in the way of a recursive rule. However, there existunavoidable problems such as numerical algorithms are deterministic,the sequence of numbers cannot be “really random.” To solve suchlimitations, chaotic systems provide a clue to produce random numbergenerators as the deterministic systems may have a time evolution thatappears rather “irregular” with the typical features of genuine randomprocesses.

In this paper, utilizing the complex dynamics of chaotic MNNs andthe algorithms of synchronization control, memristive neural networksare used to generate pseudorandom numbers to achieve encryption anddecryption functions.

If we define a pseudorandom number sequence k(t) = h(y1

(t), y2 (t)), t ∈ [tstart, tend], [tstart, tend] is the operating interval, and

h(y1 (t), y2 (t)) ={

1, y1 (t) ≤ y2 (t)0, y1 (t) > y2 (t)

(23)

where

y1 (t)=x1 (t)

maxt∈[tstart ,tend]{x1 (t)}, y2 (t)=

x2 (t)maxt∈[tstart ,tend ]{x2 (t)}

.

Then, we can get the PRNG by the chaotic MNNs in Examples 1and 2 as in Fig. 8(a) and (b), respectively.

Let s(t) be the transmitted signal, which is operated with the signalsgenerated by PRNG, we can get the encrypted signals as follows:

p(t) = s(t) ⊗ k(t). (24)

The original signal and correspondingly encrypted signals by sys-tems in Examples 1 and 2 are shown in Figs. 9 and 10, respectively.

Remark 1: From these simulation results, it is obvious that the en-crypted signals produced by the PRNG is quite different from theoriginal signals because of the chaotic properties of the MNNs, andthey can be easily retrieved through the synchronization of the chaoticMNNs in the receipt termination.

Remark 2: The decryption process is the same as the encryptionprocess and as the existence of lag time, the decryption PRNG shouldbe adopted after ν/h signals, where h is the length of the iterative step.

Remark 3: As the kernel of a new generation of cipher dreams,the hardware implementation of PRNGs based on memristive neuralnetworks will come true in the future for their great applications inthe field of the signal communication. Meanwhile, it is meaningful toinvestigate the design of the algorithm of digital image encryption anddecryption via PRNGs based on memristive neural networks.

8 IEEE TRANSACTIONS ON FUZZY SYSTEMS

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

k(t)

(a)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

k(t)

(b)

Fig. 8. (a) PRNG produced by chaotic memristive neural networks in Example 1. (b) PRNG produced by chaotic memristive neural networks in Example 2.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

s(t)

(a)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

p(t)

(b)

Fig. 9. (a) Original signal. (b) Encrypted signals by the chaotic memristive neural network in Example 1.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

s(t)

(a)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

p(t)

(b)

Fig. 10. (a) Original signal. (b) Encrypted signals by the chaotic memristive neural network in Example 2.

IEEE TRANSACTIONS ON FUZZY SYSTEMS 9

VI. CONCLUSION

In this paper, the problem of exponential lag adaptive synchroniza-tion control of MNNs was investigated via fuzzy method and appliedin a pseudorandom number generator. A model of fuzzy MNNs wasestablished with only two subsystems, and the update laws for the con-nection weights of slave systems and controller gain are designed tomake the slave systems exponentially lag synchronized with the mastersystems. Illustrative examples were given to demonstrate the effective-ness of the obtained results, which can be extended into the field ofPRNG as an encryption method.

In the future, there are some issues that deserve further investigation,such as 1) how to design the optimal update laws of the connectionweights of slave systems to achieve desired results; 2) how to extendthe applications of memristive neural networks into the fields of optimalcomputation, biological systems, and secure communications; and 3)how to deal with the problem of synchronization of memristive neuralnetworks with discrete and distributed time-varying delays.

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10 IEEE TRANSACTIONS ON FUZZY SYSTEMS

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IEEE TRANSACTIONS ON FUZZY SYSTEMS 1

Short Papers

Exponential Adaptive Lag Synchronization of MemristiveNeural Networks via Fuzzy Method and Applications in

Pseudorandom Number Generators

Shiping Wen, Zhigang Zeng, Tingwen Huang,and Yide Zhang

Abstract—This paper investigates the problem of exponential lag syn-chronization control of memristive neural networks (MNNs) via the fuzzymethod and applications in pseudorandom number generators. Based onthe knowledge of memristor and recurrent neural networks, the model ofMNNs is established. Then, considering the state-dependent properties ofmemristor, a fuzzy model of MNNs is employed to provide a new way of an-alyzing the complicated MNNs with only two subsystems, and update lawsfor the connection weights of slave systems and controller gain are designedto make the slave systems exponentially lag synchronized with the mastersystems. Two examples about synchronization problems are presented toshow the effectiveness of the obtained results, and an application of theobtained theory is also given in the pseudorandom number generator.

Index Terms—Adaptive lag synchronization, fuzzy model, memristor,neural networks, pseudorandom number generator (PRNG).

I. INTRODUCTION

Although current digital computers can now possess computingspeed and complexity to emulate the brain functionality of animalslike a spiders, mice, and cats [1], [2], the associated energy dissipa-tion in the system grows exponentially along the hierarchy of animalintelligence, as the sequential processing of fetch, decode, and execu-tion of instructions through the classical von Neumann bottleneck ofconventional digital computers has resulted in less-efficient machinesas their ecosystems have grown to be increasingly complex [3]. Forexample, to perform certain cortical simulations at the cat scale evenat 83 times slower firing rate, the IBM team in [1] has to employ BlueGene/P (BG/P): a super computer equipped with 147456 CPUs and144 TBs of main memory. On the other hand, the human brain con-tains more than 100 billion neurons, and each neuron has more than20 000 synapses. Efficient circuit implementation of synapses, there-fore, is especially important to build a brain-like machine. However,

Manuscript received August 13, 2013; revised October 15, 2013; acceptedNovember 16, 2013. Date of publication December 11, 2013; date of currentversion. This work was supported by the Natural Science Foundation of Chinaunder Grant 61125303, National Basic Research Program of China (973 Pro-gram) under Grant 2011CB710606, Research Fund for the Doctoral Program ofHigher Education of China under Grant 20100142110021, the Excellent YouthFoundation of Hubei Province of China under Grant 2010CDA081, and the Na-tional Priority Research Project NPRP 4-451-2-168, funded by Qatar NationalResearch Fund.

S. P. Wen, Z. G. Zeng, and Y. D. Zhang are with the School of Au-tomation, Huazhong University of Science and Technology, and Key Lab-oratory of Image Processing and Intelligent Control of Education Ministryof China, Wuhan, Hubei, 430074, China (e-mail: wenshiping226@126.com;zgzeng527@126.com; edwardchang@hust.edu.cn).

T. W. Huang is with Texas A & M University at Qatar, Doha 23874, Qatar(e-mail: tingwen.huang@qatar.tamu.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2013.2294855

since shrinking the current transistor size is very difficult, introducinga more efficient approach is essential for further development of neuralnetwork implementations.

In 2008, the Williams group announced a successful fabrication of avery compact and nonvolatile nano scale memory called the memristor[4]. It was postulated by Chua [5] as the fourth basic circuit elementin electrical circuits. It is based on the nonlinear characteristics ofcharge and flux. By supplying a voltage or current to the memristor,its resistance can be altered [6]. This way, the memristor remembersinformation. Several examples of successful multichip networks ofspiking neurons have been recently proposed [7]–[9]; however, thereare still a number of practical problems that hinder the developmentof truly large-scale, distributed, massively parallel networks of verylarge scale integration (VLSI) neurons, such as how to set the weight ofindividual synapses in the network. It is well-known that changes in thesynaptic connections between neurons are widely believed to contributeto memory storage, and the activity-dependent development of neuralnetworks. These changes are thought to occur through correlated-based,or Hebbian plasticity.

In addition, we notice neural networks have been widely studiedin recent years, for their immense application prospective [10]–[23].Many applications have been developed in different areas such as com-binatorial optimization, knowledge acquisition, and pattern recogni-tion. Recently, the problem of lag synchronization of coupled neuralnetworks, which is one of the hottest research fields of complex net-works, has been a challenging issue because of its potential applicationsuch as information science, biological systems, and so on [24]–[36].

On the other hand, synchronization problem of neural networkshas attracted great attention because of its potential applicationsin many fields such as secure communications, biological sys-tems, information science, image encryption, and pseudorandomnumber generator (PRNG) [37], [38]. Currently, a wide variety ofsynchronization phenomena have been investigated, such as completesynchronization [39]–[41], generalized synchronization [42], phasesynchronization [43], and lag synchronization [44]. In the case of realapplications, it is very hard to directly get the identical parameters ofthe master and slave systems. Therefore, adaptive synchronization maybe a good choice for such cases. It is worth mentioning that in con-nected electronic networks, the occurrence of time delay is unavoidablebecause of finite signal transmission times, switching speeds, and someother reasons. Thus, the complete synchronization of neural networksis hard to implement effectively and it is more reasonable to considerthe lag synchronization problem.

However, to the best of the authors’ knowledge, the research onglobal exponential lag adaptive synchronization of memristive neuralnetworks is still an open problem that deserves further investigation.To shorten sup gap, we investigate the problem of global exponentiallag adaptive synchronization for a class of memristive neural networkswith time-varying delays. The main contributions of this paper can besummarized as follows: 1) A model of MNNs is established in accor-dance with the memristor-based electronic circuits; 2) a fuzzy model ofmemristive neural networks is employed to give a new way to analyzethe complicated MNNs with only two subsystems; 3) update laws aredesigned for the connection weights of slave systems and controllergain to make the slave systems exponentially lag synchronized withthe master systems; and 4) a simulation example is presented to showthe applications of the obtained results in the PRNG.

1063-6706 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

2 IEEE TRANSACTIONS ON FUZZY SYSTEMS

Fig. 1. Circuit of memristive network, where xi (.) is the state of the ith subsystem; fj (.) is the amplifier; Rf ij is the connection resistor between the amplifierfj (.) and state xi (.); Mi and Ci are the memristor and capacitor; Ii is the external input; and ai , bi are the outputs i, j = 1, 2, . . . , n.

A. Circuit of Memristive Neural Networks

The memristive neural network can be implemented by VLSI circuitsas shown in Fig. 1. fj is the activation function, τj (t) is the time-varyingdelay, for the i-th subsystem, xi (t) is the voltage of the capacitorCi , fj (xj (t)), fj (xj (t − τj (t))) are the functions of xi (t) with orwithout time-varying delays respectively, Rf ij is the resistor betweenthe feedback function fj (xj (t)) and xi (t), Rg ij is the resistor betweenthe feedback function fj (xj (t − τj (t))) and xi (t), Mi is the memristorparallel to the capacitor Ci , and Ii is an external input or bias, wherei, j = 1, 2, . . . , n.

The memductance of the memristors can be depicted as in Fig. 2 [45],which are bounded. Thus, by Kirchoff’s current law, the equation ofthe i-th subsystem is written as follows:

Ci xi (t) = −[

n∑j=1

(1

Rf ij

+1

Rg ij

)+ Wi (xi (t))

]xi (t)

+n∑

j=1

signij fj (xj (t))Rf ij

+n∑

j=1

signij fj (xj (t − τj (t)))Rg ij

+ Ii (1)

where

signij ={

1, i �= j−1, i = j

and Wi are the memductances of the memristors Mi , and

Wi (xi (t)) ={

W ′i , xi (t) ≤ 0

W ′′i , xi (t) > 0.

Therefore

xi (t) = −di (xi (t))xi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si (2)

where

aij =signij

CiRf ij

, bij =signij

CiRg ij

, si =Ii

Ci

di (xi (t)) =1Ci

[n∑

j=1

(1

Rf ij

+1

Rg ij

)+ Wi (xi (t))

]

={

d1 i , xi (t) ≤ 0d2 i , xi (t) > 0.

IEEE TRANSACTIONS ON FUZZY SYSTEMS 3

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5x 10−4

V

I

Fig. 2. Typical I-V characteristic of memristor [45]. The pinched hysteresisloop occurs because of the nonlinear relationship between the memristancecurrent and voltage. The memristor exhibits the feature of pinched hysteresis,which means that a lag occurs between the application and the removal of afield and its subsequent effect, just like the neurons in the human brain.

Then, we can get

x(t) = −D(x(t))x(t) + Af (x(t)) + Bf(x(t − τ (t))) + s (3)

where

D(x(t)) = diag{d1 (x1 (t)), d2 (x2 (t)), . . . , dn (xn (t))}

A = [aij ]n×n , B = [bij ]n×n , s = (s1 , s2 , . . . , sn )T

f (x(t)) = (f1 (x1 (t)), . . . , fn (xn (t)))T

f (x(t − τ (t))) = (f1 (x1 (t − τ1 (t))), . . . , fn (t − τn (t)))T .

B. Fuzzy Model of Memristive Neural Networks

To solve the problem about nonlinear control, fuzzy logic has at-tracted much attention as a powerful tool. Among various kinds of fuzzymethods, the Takagi–Sugeno fuzzy systems are widely accepted as auseful tool for design and analysis of fuzzy control system [46]–[53].Currently, some control methods for memristive systems have beenproposed [54], in which the number of the linear subsystems is decidedby how many minimum nonlinear terms should be linearized in orig-inal system. Then, the memristive neural network (2) can be exactlyrepresented by the fuzzy model as follows:

Rule 1: IF xi (t) is N1 i , THEN

xi (t) = −d1 i xi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si

Rule 2: IF xi (t) is N2 i , THEN

xi (t) = −d2 i xi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si

where N1 i is xi (t) ≤ 0, and N2 i is xi (t) > 0. With a center-averagedefuzzier, the over fuzzy system is represented as

xi (t) = −2∑

l=1

ϑli (t)dlixi (t) +n∑

j=1

aij fj (xj (t))

+n∑

j=1

bij fj (xj (t − τj (t))) + si (4)

where

ϑ1 i (t) ={

1, xi (t) ≤ 0,0, xi (t) > 0,

ϑ2 i (t) ={

0, xi (t) ≤ 01, xi (t) > 0.

When the system becomes complicated with n memristors, there are2n subsystems (according to 2n fuzzy rules) and 2n equations in theT–S fuzzy system. If n is large, the number of linear subsystems in theT–S fuzzy system is huge. For this problem, Li and Ge proposed a fuzzymodeling method and applied in the lag synchronization problem of twototally different chaotic systems [55]. Based on this work, a new fuzzymodel is proposed to simplify memristive systems, in which only twosubsystems are included. Furthermore, through this model, the idea ofPDC can be applied to achieve between subsystems. Therefore, system(4) can be represented by

x(t) = −2∑

l=1

Πl (t)Dlx(t) + Af (x(t))

+ Bf(x(t − τ (t))) + s (5)

where Πl (t) = diag{ϑl1 (t), . . . , ϑln (t)},∑2

l=1 ϑli (t) = 1, i =1, . . . , n, l = 1, 2, and

Dl = diag{dl1 , dl2 , . . . , dln }.The initial conditions of system (5) is in the form of x(t) = φ(t) ∈C([−τ, 0],Rn ), τ = max1≤i≤n {τi (t)}.

II. PRELIMINARIES

Denote u = (u1 , . . . , un )T , |u| as the absolute-value vector; i.e.,|u| = (|u1 |, |u2 |, . . . , |un |)T , ‖x‖p as the p-norm of the vector x withp, 1 ≤ p < ∞. ‖x‖∞ = maxi∈{1 ,2 , . . . ,n }|xi | is the vector infinity norm.Denote ‖D‖p as the p-norm of the matrix D with p. Denote C asthe set of continuous functions. In addition, we assume the followingthroughout the paper:

A1. For i ∈ {1, 2, . . . , n}, the activation function fi is Lipschitzcontinuous, and ∀r1 , r2 ∈ R, there exists real number ιi such that

0 ≤ fi (r1 ) − fi (r2 )r1 − r2

≤ ιi

where fi (0) = 0, r1 , r2 ∈ R, and r1 �= r2 .A2. For i ∈ {1, 2, . . . , n}, the time-varying delay τi (t) satisfies the

following inequalities:

0 ≤ τi (t) ≤ τ

τi (t) ≤ μ. (6)

In this paper, we consider system (5) as the master system, andthrough electronic inductors, the values of memristor will be presentedin the corresponding slave system; then, the slave system is given as

zi (t) = −2∑

l=1

ϑli (t)dli zi (t) +n∑

j=1

aij (t)fj (zj (t))

+n∑

j=1

bij (t)fj (zj (t − τj (t))) + si + ui (t) (7)

4 IEEE TRANSACTIONS ON FUZZY SYSTEMS

or in compact form

z(t) = −2∑

l=1

Πl (t)Dlz(t) + A(t)f (z(t))

+ B(t)f (z(t − τ (t))) + s + u(t), t ≥ 0 (8)

where A(t) = (a(t))n×n , B(t) = (b(t))n×n are unknown connectionweights, z(t) = (z1 (t), z2 (t), . . . , zn (t))T , Πl (t) is related to the mas-ter system, and u(t) = (u1 (t), u2 (t), . . . , un (t))T is the control inputwith the following form:

ui (t) = �i (t)(zi (t) − xi (t − ν)) (9)

where �i (t) is the adaptive control gain which needs to be designed,and the initial condition of system (8) is in the form of z(t) = ψ(t) ∈C([−τ, 0],Rn ).

Definition 1: The master system (4) and slave system (7) are saidto be globally exponentially synchronized with lag ν , if there existpositive constants λ and μ, such that

‖z(t) − x(t − ν)‖ ≤ ωe−λt , t ≥ 0. (10)

If ν = 0, the synchronization is complete synchronization.Notation: The notation used here is fairly standard. The superscript

“T ” stands for matrix transposition, Rn denotes the n-dimensionalEuclidean space, and Rm ×n is the set of all real matrices of di-mension m × n, I and 0 represent the identity matrix and zeromatrix, respectively. Matrices, if their dimensions are not explic-itly stated, are assumed to be compatible for algebraic operations.Denote u = (u1 , . . . , un )T , |u| as the absolute-value vector, i.e.,|u| = (|u1 |, |u2 |, . . . , |un |)T , ‖x‖p as the p-norm of the vector x withp, 1 ≤ p < ∞. ‖x‖∞ = maxi∈{1 ,2 , . . . ,n }|xi | is the vector infinity norm.Denote ‖D‖p as the p-norm of the matrix D with p. Denote C as theset of continuous functions.

III. MAIN RESULTS

In practice, lag exists, when the synchronization happens betweenthe master and slave systems, which can be characterized as z(t) =x(t − υ) for some constant lag time υ > 0. The lag synchronizationerror between the master and slave systems can be presented as

e(t) = z(t) − x(t − ν). (11)

Then, we can get the adaptive lag synchronization algorithm for mem-ristive neural networks with unknown connection weights of the slavesystems.

Theorem 1: System (7) will be globally exponentially synchronizedwith system (4) with lag ν , if the connection weights of system (7)aij (t), bij (t) and controller gain �i (t), i, j = 1, 2, . . . , n, are adaptedin accordance with the following update law:

˙aij (t) = −�ij fj (zj (t))sign(ei (t))eεt

˙bij (t) = −αij fj (zj (t − τj (t)))sign(ei (t))eεt

�i (t) = −βi |ei (t)|eεt (12)

where �ij , αij and βi , i, j = 1, 2, . . . , n are arbitrary positive con-stants.

Proof: Let Vi (t) = eεt |ei (t)|, and d−i = minl=1 ,2{dli}. Calculating

the derivative of Vi (t) along systems (4) and (7), we can get

Vi (t) ≤ eεt

{(ε + �i (t) − d−

i )|ei (t)|

+

∥∥∥∥∥n∑

j=1

aij (fj (zj (t)) − fj (xj (t − ν)))

∥∥∥∥∥+

∥∥∥∥∥n∑

j=1

bij (fj (zj (t − τj (t))) − fj (xj (t − τj (t) − ν)))

∥∥∥∥∥+

n∑j=1

(aij (t) − aij )fj (zj (t))sign(ei (t))

+n∑

j=1

(bij (t) − bij )fj (zj (t − τj (t)))sign(ei (t))

}

≤ eεt

{(ε + �i (t) − d−

i )|ei (t)| +n∑

j=1

|aij |ιj |ej (t)|

+n∑

j=1

|bij |ιj |ej (t − τj (t))|

+n∑

j=1

(aij (t) − aij )fj (zj (t))sign(ei (t))

+n∑

j=1

(bij (t) − bij )fj (zj (t − τj (t)))sign(ei (t))

}. (13)

Define a Lyapunov functional as

V (t) =n∑

i=1

{Vi (t) +

n∑j=1

|bij |ιj eετ j (t)

∫ t

t−τ j (t)|ej (s)|eεs ds

+n∑

j=1

12�ij

(aij (t) − aij )2 +n∑

j=1

12αij

(bij (t) − bij )2

+n∑

j=1

12βi

(�i (t) + ωi )2

}(14)

where ωi is a constant, and aij (t), bij (t), and �i (t) are adapted by theupdate law (12). Then

V (t) =n∑

i=1

{Vi (t) + eεt

n∑j=1

|bij |ιj (eετ j (t) |ej (t)|−|ej (t − τj (t))|)

+n∑

j=1

1�ij

(aij (t) − aij ) ˙aij (t)+n∑

j=1

1αij

(bij (t)−bij )˙bij (t)

+1βi

(�i (t) + ωi )�i (t)

}

≤n∑

i=1

{Vi (t) + eεt

n∑j=1

|bij |ιj (eετ |ej (t)| − |ej (t − τj (t))|)

+n∑

j=1

1�ij

(aij (t) − aij ) ˙aij (t)+n∑

j=1

1αij

(bij (t)−bij )˙bij (t)

+1βi

(�i (t) + ωi )�i (t)

}

IEEE TRANSACTIONS ON FUZZY SYSTEMS 5

=n∑

i=1

{Vi (t) + eεt

n∑j=1

|bij |ιj (eετ |ej (t)| − |ej (t − τj (t))|)

−n∑

j=1

(aij (t) − aij )fj (zj (t))sign(ei (t))eεt

−n∑

j=1

(bij (t) − bij )fj (zj (t − τj (t)))sign(ei (t))eεt

− (�i (t) + ωi )|ei (t)|eεt

}. (15)

Combining the derivatives (13) and (15)

V (t) ≤ eεt

n∑i=1

{(ε − ωi − d−

i )|ei (t)|

+n∑

j=1

(|aij | + |bij |eετ )ιj |ej (t)|}

= eεt

n∑i=1

(ε − d−

i +n∑

j=1

(|aij | + |bij |eετ )ιj − ωi

)|ei (t)|.

(16)

Set

ωi > −d−i +

n∑j=1

(|aij | + |bij |eετ )ιj . (17)

If we let ε be small enough, we can get

eεt

n∑i=1

(ε − d−

i +n∑

j=1

(|aij | + |bij |eετ )ιj − ωi ≤ 0. (18)

HenceV (t) ≤ 0 (19)

and with (13) and (15), we can get

eεt

n∑i=1

|ei (t)| ≤ V (t) ≤ V (0) (20)

where

V (0) ≤n∑

i=1

{ (1 +

n∑j=1

|bi j|ιj τ eετ

)max

s∈[−τ + ν ,ν ]|ϕ(s) − φ(s)|

+ �i (0)

}

in which

�i (0) =n∑

j=1

12�ij

(aij (0) − aij )2 +n∑

j=1

12αij

(bij (0) − bij )2

+n∑

j=1

12βi

(�i (0) + ωi )2

}

≡ Φ.

Consequently, the following inequality holds:

n∑i=1

|ei (t)| ≤ Φe−εt , t ≥ 0. (21)

This completes the proof.When the connection weights aij (t) and bij (t) are known, then we

can get update law of the adaptive controller gain as follows.Corollary 1: System (7) will be globally exponentially synchronized

with system (4) with lag ν , if controller gains �i (t), i = 1, 2, . . . , n areadaptive iterating in accordance with the following update law:

�i (t) = −βi |ei (t)|eεt (22)

where βi , i = 1, 2, . . . , n are arbitrary positive constants.The proof is the same as in Theorem 1, and therefore, it is omitted.

IV. NUMERICAL EXAMPLES

In this section, several numerical examples are utilized to demon-strate the effectiveness and applications of the obtained results.

Example 1: Consider memristive system (5) with

A =[

2 −0.1−5 4.5

], B =

[−1.5 −0.1−0.2 −4

]

fi (xi ) = tanh(xi ), τi (t) = 1, si = 0, i = 1, 2.

Let

d1 (x1 (t)) ={

0.9, x1 (t) ≤ 01.1, x1 (t) > 0,

d2 (x2 (t)) ={

1.1, x2 (t) ≤ 00.9, x2 (t) > 0.

The initial values of master system (5) are set to be [0.4 0.6]. And thedynamical behaviors of this system are shown as in Fig. 3, which arechaotic and can be used in secure communications.

Without loss of generality, let

A(t) = A, B(t) =[

b11 (t) −0.1−0.2 b22 (t)

]in which

b11 (0) = b22 (0) = 1.

Set ε = 0.02, �i (0) = 1, α11 = α22 = 0.9, β1 = β2 = 0.8, the lagtime ν = 1.5, and the initial values of the slave system is set to be[−0.5 − 0.5]. Then, we can get the simulation results as shown inFig. 4.

Example 2: Consider memristive system (5) with

A =[

1 + π/4 200.1 1 + π/4

], B =

[−1.3

√2π/4 0.1

0.1 −1.3√

2π/4

]

fi (xi ) = 0.5(|xi + 1| − |xi − 1|), τi (t) = 1, si = 0, i = 1, 2.

Let

d1 (x1 (t)) ={

1.0, x1 (t) ≤ 01.2, x1 (t) > 0,

d2 (x2 (t)) ={

1.2, x2 (t) ≤ 01.0, x2 (t) > 0.

The initial values of master system (5) are set to be [0.2 − 0.2], andthe dynamical behaviors of this system are shown in Fig. 5.

LetA(t) = A, B(t) = B

and ε = 0.01, �i (0) = 0.6, β1 = β2 = 0.8, the lag time ν = 1.5, andthe initial values of the slave system are set to be [−0.5 0.5]. Then, wecan get the simulation results as shown in Fig. 6.

V. MEMRISTIVE NEURAL NETWORKS IN THE PSUEDORANDOM

NUMBER GENERATOR

Based on the above discussion, this section will discuss the appli-cations of the exponential lag synchronization between MNNs in thefield of the PRNG.

6 IEEE TRANSACTIONS ON FUZZY SYSTEMS

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 1(t)

0 20 40 60 80 100−10

−5

0

5

10

t

x 2(t)

−1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8

x1(t)

x 2(t)

(a) (b)

Fig. 3. Transient behavior of memristive system (4).

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 1(t),z

1(t)

x1(t)

z1(t)

0 20 40 60 80 100−10

−5

0

5

10

t

x 2(t),z

2(t)

x2(t)

z2(t)

(a)

−1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8

x1(t),z1(t)

x 2(t),z

2(t)

(b)

Fig. 4. State trajectories of master system (4) and slave system (7) when lag time ν = 1.5.

0 20 40 60 80 100−20

−10

0

10

20

t

x 1(t)

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 2(t)

(a)

−15 −10 −5 0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1(t)

x 2(t)

(b)

Fig. 5. Transient behavior of memristive system (4).

IEEE TRANSACTIONS ON FUZZY SYSTEMS 7

0 20 40 60 80 100−20

−10

0

10

20

t

x 1(t),z

1(t)

x1(t)

z1(t)

0 20 40 60 80 100−1

−0.5

0

0.5

1

t

x 2(t),z

2(t)

x2(t)

z2(t)

(a)

−15 −10 −5 0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1(t),z1(t)

x 2(t),z

2(t)

(b)

Fig. 6. State trajectories of master system (4) and slave system (7) when lag time ν = 1.5.

PRNG

Pseudu-random outputs

Unpredictable inputs

Fig. 7. Black-box view of a PRNG.

As random number generation plays an important role in cryptog-raphy and software testing, PRNG are intended to be general-purposevehicles for the creation of random data used in these areas as inFig. 7 [65]. In addition, many different methods exist to generate pseu-dorandom numbers like Blum-Blum-Shub, Mersenne Twister algo-rithms, etc. It is well known that pseudorandomness is the basis forcryptography and is essential for the achievement of any cryptographicfunction such as encryption, authentication, and identification. Neuralnetworks can be used to generate random numbers as they are highlynonlinear mathematical systems. Based on the dynamics of neural net-works, pseudorandom numbers are generated via neural plasticity.

Meanwhile, it is important to produce a perfect random number gen-erator that gets a series of independent identically distributed continu-ous random variables in [0 1] [66]. One can produce a perfect randomnumber generator only using nondeterministic physical phenomena. Itis a practical way to employ a computer to produce a random-lookingsequence of numbers in the way of a recursive rule. However, there existunavoidable problems such as numerical algorithms are deterministic,the sequence of numbers cannot be “really random.” To solve suchlimitations, chaotic systems provide a clue to produce random numbergenerators as the deterministic systems may have a time evolution thatappears rather “irregular” with the typical features of genuine randomprocesses.

In this paper, utilizing the complex dynamics of chaotic MNNs andthe algorithms of synchronization control, memristive neural networksare used to generate pseudorandom numbers to achieve encryption anddecryption functions.

If we define a pseudorandom number sequence k(t) = h(y1

(t), y2 (t)), t ∈ [tstart, tend], [tstart, tend] is the operating interval, and

h(y1 (t), y2 (t)) ={

1, y1 (t) ≤ y2 (t)0, y1 (t) > y2 (t)

(23)

where

y1 (t)=x1 (t)

maxt∈[tstart ,tend]{x1 (t)}, y2 (t)=

x2 (t)maxt∈[tstart ,tend ]{x2 (t)}

.

Then, we can get the PRNG by the chaotic MNNs in Examples 1and 2 as in Fig. 8(a) and (b), respectively.

Let s(t) be the transmitted signal, which is operated with the signalsgenerated by PRNG, we can get the encrypted signals as follows:

p(t) = s(t) ⊗ k(t). (24)

The original signal and correspondingly encrypted signals by sys-tems in Examples 1 and 2 are shown in Figs. 9 and 10, respectively.

Remark 1: From these simulation results, it is obvious that the en-crypted signals produced by the PRNG is quite different from theoriginal signals because of the chaotic properties of the MNNs, andthey can be easily retrieved through the synchronization of the chaoticMNNs in the receipt termination.

Remark 2: The decryption process is the same as the encryptionprocess and as the existence of lag time, the decryption PRNG shouldbe adopted after ν/h signals, where h is the length of the iterative step.

Remark 3: As the kernel of a new generation of cipher dreams,the hardware implementation of PRNGs based on memristive neuralnetworks will come true in the future for their great applications inthe field of the signal communication. Meanwhile, it is meaningful toinvestigate the design of the algorithm of digital image encryption anddecryption via PRNGs based on memristive neural networks.

8 IEEE TRANSACTIONS ON FUZZY SYSTEMS

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

k(t)

(a)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

k(t)

(b)

Fig. 8. (a) PRNG produced by chaotic memristive neural networks in Example 1. (b) PRNG produced by chaotic memristive neural networks in Example 2.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

s(t)

(a)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

p(t)

(b)

Fig. 9. (a) Original signal. (b) Encrypted signals by the chaotic memristive neural network in Example 1.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

s(t)

(a)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

p(t)

(b)

Fig. 10. (a) Original signal. (b) Encrypted signals by the chaotic memristive neural network in Example 2.

IEEE TRANSACTIONS ON FUZZY SYSTEMS 9

VI. CONCLUSION

In this paper, the problem of exponential lag adaptive synchroniza-tion control of MNNs was investigated via fuzzy method and appliedin a pseudorandom number generator. A model of fuzzy MNNs wasestablished with only two subsystems, and the update laws for the con-nection weights of slave systems and controller gain are designed tomake the slave systems exponentially lag synchronized with the mastersystems. Illustrative examples were given to demonstrate the effective-ness of the obtained results, which can be extended into the field ofPRNG as an encryption method.

In the future, there are some issues that deserve further investigation,such as 1) how to design the optimal update laws of the connectionweights of slave systems to achieve desired results; 2) how to extendthe applications of memristive neural networks into the fields of optimalcomputation, biological systems, and secure communications; and 3)how to deal with the problem of synchronization of memristive neuralnetworks with discrete and distributed time-varying delays.

REFERENCES

[1] R. Ananthanarayanan, S. Eser, H. Simon, and D. Modha, presented atthe IEEE/ACM Conf. High Perform. Netw. Comput., Portland, OR, USA,Nov. 2009.

[2] L. Smith, Handbook of Nature-Inspired and Innovative Computing: Inte-grating Classical Models with Emerging Technologies. New York, NY,USA: Springer-Verlag, pp. 433–475.

[3] S. Jo, T. Chang, I. Ebong, B. Bhadviya, P. Mazumder, and W. Lu,“Nanoscale memristor device as synapse in neuromorphic systems,” Nan-otech. Lett., vol. 10, pp. 1297–1301, 2010.

[4] D. Strukov, G. Snider, D. Stewart, and R. Williams, “The missing mem-ristor found,” Nature, vol. 453, pp. 80–83, 2008.

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