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[14] Z. Uykan, “Spectral based solutions for (Near) optimum channel/frequency allocation,” in Proc. 18th Int. Conf. Syst. Signals Image Process., Sarajevo, Bosnia and Herzegovina, Jun. 2011, pp. 1–4. [15] M. E. J. Newman, Networks–An Introduction, London, U.K.: Oxford Univ. Press, 2011, pp. 358–380. [16] I. Stojmenovic, “Handbook of Wireless Networks and Mobile Computing,” New York, USA: Wiley, 2002. [17] O. Lazaro, and D. Girma, “Real-time operational aspects of Hopfield neural network based dynamic channel allocation scheme,” Electron. Lett., vol. 40, no. 18, pp. 1141–1143, 2004. [18] C. W. Ahn, and R. S. Ramakrishna, “QoS provisioning dynamic connection-admission control for multimedia wireless networks using Hopfield neural networks,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 106–117, Jan. 2004. [19] D. Calabuig, J. F. Monserrat, D. G. Barquero, and N. Cardona, “Hopfield neural network algorithm for dynamic resource allocation in WCDMA systems,” in Proc. IEEE 3rd Int. Symp. Wireless Commun. Syst., Valencia, Spain, Sep. 2006, pp. 40–44. [20] D. Calabuig, J. F. Monserrat, D. G. Barquero, and O. Lazaro, “User bandwidth usage-driven HNN neuron excitation method for maximum resource utilization within packet-switched communication networks,” IEEE Commun. Lett., vol. 10, no. 11, pp. 766–768, Nov. 2006. [21] G. Joya, M. A. Atencia, and F. Sandoval, “Hopfield neural network applied to optimization problems: Some theoretical and simulation results,” in Proc. Biol. Artif. Comput., Neurosci. Technol, Lect. Notes Comput. Sci., vol. 1240, pp. 556–565, 1997. [22] T. S. Rappaport, Wireless Communications: Principles & Practice, 2nd ed., Englewood Cliffs, NJ: Prentice Hall, 2002. [23] Z. Zeng and W. X. Zheng, “Multistability of neural networks with timevarying delays and concave-convex characteristics,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 293–305, Feb. 2012.
Synchronization Design of Boolean Networks via the Semi-Tensor Product Method Rui Li, Meng Yang, and Tianguang Chu
Abstract— We provide a general approach for the design of a response Boolean network (BN) to achieve complete synchronization with a given drive BN. The approach is based on the algebraic representation of BNs in terms of the semitensor product of matrices. Instead of designing the logical dynamic equations of a response BN directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. The results are applied to a three-neuron network in order to illustrate the effectiveness of the proposed approach. Index Terms— Algebraic representation, Boolean network, complete synchronization, semi-tensor product.
0 means that it is OFF. These nodes interact with each other according to some logical rules, specified through a set of Boolean functions that determine the states of the nodes at the next time-step, and thereby give the dynamics of the network. Even though BNs seem to be oversimplified models of real networks, they retain in most cases meaningful information that can be used to make inferences regarding the systems they model, and hence have been investigated widely. Recently, an increasing interest has been focused on synchronization of coupled BNs, motivated by many potential applications in biology, physics, and engineering [10], [11]. In [12], synchronization between a pair of stochastically coupled random BNs was considered. Later, the study was further extended to mutual synchronization in a random network of random BNs [13]. The spectral characterization of two synchronized deterministic BNs coupled in a drive–response configuration was discussed in [14]. In our recent paper [15], the concept of complete synchronization was introduced for the case of deterministic BNs in a drive–response configuration, and a necessary and sufficient algebraic criterion for complete synchronization was presented. In this brief, we further consider the problem of synchronization design for BNs. This brief constitutes a continuation of [15], providing a general approach for constructing a response BN to achieve complete synchronization with a given drive BN. Similar to [15], our discussion is based on the recently developed technique of the semi-tensor product of matrices that allows an algebraic representation of logical dynamics of BNs [16]–[19]. To demonstrate the application of the design approach, we apply the results to a three-neuron network studied in [20] and [21]. The remainder of this brief is organized as follows. Section II gives the problem formulation and some preliminary results. Section III presents our approach for constructing synchronized response BNs. An illustrative example is treated in Section IV, and a brief conclusion is drawn in the final section. Notation: Rn×m denotes the set of all n × m real matrices. In is the n × n identity matrix. δn (i ) stands for the i th column of In . n stands for the set of all n columns of In . Col(A) denotes the set of columns of a matrix A. 1n represents the n-dimensional row vector whose entries are equal to 1. ⊗ is the Kronecker product of matrices.
I. I NTRODUCTION
II. P ROBLEM F ORMULATION AND P RELIMINARIES
Boolean networks (BNs) have been successfully used in modeling complex systems such as neural networks and gene regulatory networks [1]–[9]. In a BN, the nodes can take on one of two binary values, 1 or 0, at each discrete time point. A value of 1 represents that the node is ON and a value of
Let us consider two BNs with N nodes respectively coupled in a drive–response configuration and described as follows:
Manuscript received May 14, 2012; revised November 19, 2012; accepted February 11, 2013. Date of publication March 7, 2013; date of current version April 5, 2013. This work was supported in part by the National Basic Research Program of China under Grant 2012CB821200 and the National Natural Science Foundation of China under Grant 61273111 and Grant 60974064. The authors are with the State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TNNLS.2013.2248092
x i (t + 1) = f i (x 1 (t), . . . , x N (t)) yi (t + 1) = gi (x 1 (t), . . . , x N (t), y1 (t), . . . , y N (t)),
(1) (2)
i = 1, . . . , N where x i and yi are the nodes of the drive BN (1) and the response BN (2), respectively, f i and gi are Boolean functions, and t = 0, 1, 2, . . .. We succinctly denote by X = (x 1 , . . . , x N ) and Y = (y1 , . . . , y N ) the states of the drive BN (1) and the response BN (2), respectively. Let X(t, X0 ) be the trajectory of (1) with X(0, X0 ) = X0 , and let
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Y(t, Y0 , X0 ) be the trajectory of (2) with Y(0, Y0 , X0 ) = Y0 corresponding to the driving trajectory X(t, X0 ). Before formulating the problem to be investigated, we first recall the following definition. Definition 1 ( [15]): The drive BN (1) and the response BN (2) are said to be completely synchronized if for every X0 ∈ {1, 0} N there exists a positive integer k such that t ≥ k implies Y(t, Y0 , X0 ) = X(t, X0 ) for all Y0 ∈ {1, 0} N . The objective of this brief is to design a response BN of the form (2) to achieve complete synchronization with the given drive BN (1). We will approach this problem by using algebraic representations of logical dynamics based on the technique of the semi-tensor product of matrices. Definition 2 ( [19]): Let A ∈ Rn×m and B ∈ R p×q . The semi-tensor product of A and B is
and Hi is the structure matrix of h i for i = 1, 2. It is worth mentioning that every column of the above matrices F, G, Fi , and G i has a unique nonzero entry and all nonzero entries are equal to 1. We call such a matrix a logical matrix, and simply write an n × m logical matrix [δn (i 1 ) δn (i 2 ) . . . δn (i m )] as δn (i 1 , i 2 , . . . , i m ).
A B = (A ⊗ Il/m )(B ⊗ Il/ p )
III. M AIN R ESULT
where l is the least common multiple of m and p. The semi-tensor product allows representing a BN in an algebraic form. In so doing, we regard a Boolean variable σ ∈ {1, 0} as a vector σ ∈ 2 by identifying 1 and 0 with δ2 (1) and δ2 (2), respectively. Since the map from {1, 0} N to 2 N sending (σ1 , . . . , σ N ) to σ1 · · · σ N is a bijection [18], it follows that at every time instant t the states of the drive BN (1) and the response BN (2) can be completely described by x(t) = x1 (t) · · · x N (t) and y(t) = y1 (t) · · · y N (t), respectively. Then by utilizing the semi-tensor product we can express the drive BN (1) and the response BN (2) as [19] xi (t + 1) = Fi x(t)
(3)
yi (t + 1) = G i x(t) y(t) i = 1, . . . , N
(4)
N
2N
respectively, where Fi ∈ R2×2 and G i ∈ R2×2 for each i . We call Fi (resp., G i ) the structure matrix of f i (resp., gi ). Moreover, (3) and (4) can further be converted into the following discrete-time systems [19]: x(t + 1) = Fx(t)
(5)
y(t + 1) = G x(t) y(t)
(6)
2 N ×2 N
2 N ×22N
respectively, where F ∈ R and G ∈ R . Equations (5) and (6) are called the algebraic representations of the BNs (1) and (2), respectively. It was proved that (5) and (6) are equivalent to the original BNs (1) and (2), respectively [19]. Hence, instead of designing the logical dynamics of the response BN (2) directly, we will first construct its algebraic representation (6) to realize complete synchronization and then convert the algebraic representation back to the logical form. The following result provides a procedure for calculating the logical form (2) from the algebraic representation (6). Proposition 1 ([22]): 1) Let (6) be the algebraic representation of the BN (2), and let G i be the structure matrix of gi for i = 1, . . . , N. Then G i = Si G, i = 1, . . . , N where Si = 12i−1 ⊗ I2 ⊗ 12 N−i for each i .
2) Suppose h(σ1 , σ2 , . . . , σn ) is a Boolean function, and n H ∈ R2×2 is the structure matrix of h. Partition H as n−1 H1, H2 ∈ R2×2 . H = H1 H 2 , Then h(σ1 , σ2 , . . . , σn ) = (σ1 ∧ h 1 (σ2 , . . . , σn )) ∨(¬σ1 ∧ h 2 (σ2 , . . . , σn ))
We first give some lemmas that will be used in the proof of our main result. Lemma 1 ( [15]): Suppose (5) and (6) are the algebraic representations of the BNs (1) and (2), respectively. Define N = δ22N (1, 2 N +2, 2·2 N +3, . . . , (2 N −2)2 N +2 N −1, 22N ) and = (F ⊗ G)( N ⊗ I2 N ).
(7)
Then the drive BN (1) and the response BN (2) are completely synchronized if and only if there is a positive integer m such that Gm−1 = F m ⊗ 12 N . Let α1 , α2 , . . . , α2 N , β1 , β2 , . . . , β22N ∈ {1, 2, . . . , 2 N } be such that F = δ2 N (α1 , α2 , . . . , α2 N ) G = δ2 N (β1 , β2 , . . . , β22N ). For every positive integer n, put (n)
αi
= αα (n−1) i
for each i , where by convention αi(0) = i . Then (n)
(n)
(n)
F n = δ2 N (α1 , α2 , . . . , α2 N ) by induction. We have the following lemma. Lemma 2: Let be defined as in (7), and let two integers 1 ≤ i, r ≤ 2 N be given. Then for every nonnegative integer n there is an integer 1 ≤ rn ≤ 2 N such that n δ22N ((αi − 1)2 N + β(i−1)2 N +r ) (n+1)
− 1)2 N + β(α (n)−1)2 N +r ). = δ22N ((αi n i Proof: We shall use induction on n. The case n = 0 is trivial. Suppose that the statement is true for n ≥ 0. It follows from (7) that = δ22N ((α1 − 1)2 N + β1 , . . . , (α1 − 1)2 N + β2 N , (α2 − 1)2 N + β2 N +1 , . . . , (α2 − 1)2 N + β2 N+1 , . . . , (α2 N − 1)2 N + β(2 N −1)2 N +1 , . . . , (α2 N − 1)2 N + β22N ).
(8)
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for j = 0, 1, 2, . . ., λ = 1, 2, . . . , p. This is true for k = 1 by (9). If k ≥ 2, then by (8) and (10)
( j +1) − 1)2 N + β(α ( j )−1)2 N +a ( j ) δ22N (αiλ iλ λ,k
( j +1) ( j +1) N = δ22N (αiλ − 1)2 + aλ,k−1
( j +2) = δ22N (αiλ − 1)2 N + β(α ( j +1)−1)2 N +a ( j +1) .
Let rn+1 = β(α (n) −1)2 N +r . n
i
Then we have
n+1 δ22N (αi − 1)2 N + β(i−1)2 N +r
= δ22N (αi(n+1) − 1)2 N + β(α (n)−1)2 N +r n i
(n+2) N = δ22N (αi − 1)2 + β(α (n+1)−1)2 N +r n+1
i
via the induction hypothesis and (8). (l) A sequence {δ2 N (i ), δ2 N (αi ), . . . , δ2 N (αi )} is called a period l cycle of the BN (1) if i, αi , . . . , αi(l−1) are distinct and αi(l) = i [18]. A period 1 cycle is also called a fixed point. Suppose that the drive BN (1) has p cycles (l ) {δ2 N (i λ ), δ2 N (αiλ ), . . . , δ2 N (αiλλ )}, λ = 1, 2, . . . , p. For j = ( j) ( j) ( j) 0, 1, 2, . . ., λ = 1, 2, . . . , p, define aλ,1 , aλ,2 , . . . , aλ,2 N −1 by ( j)
aλ,k =
k, if k < k + 1, if k ≥
( j) αiλ ( j) αiλ
k = 1, 2, . . . , 2 N − 1.
We are now ready to present the main result of this brief. Theorem 3: The drive BN (1) and the response BN (2) are completely synchronized if β(α ( j )−1)2 N +α ( j ) = β(α ( j )−1)2 N +a ( j ) = iλ
iλ
λ,1
iλ
( j +1)
β(α ( j )−1)2 N +a ( j ) = aλ,k−1 , iλ
λ,k
( j +1) αiλ
(9)
k = 2, 3, . . . , 2 N − 1
(10)
j = 0, 1, . . . , lλ − 1, λ = 1, 2, . . . , p. Proof: Let be defined as in (7). We first show that, for every n ≥ 0 ( j +1)
n δ22N ((αiλ
(n+ j +1)
= δ22N ((αiλ
− 1)2 N + β(α ( j )−1)2 N +α ( j ) ) iλ
(n+ j +1)
− 1)2 N + αiλ
iλ
)
(11)
Therefore, by induction
( j +1) k−1 N δ22N (αiλ − 1)2 + β(α ( j )−1)2 N +a ( j ) iλ λ,k
( j +2) = k−2 δ22N (αiλ − 1)2 N + β(α ( j +1)−1)2 N +a ( j +1) iλ λ,k−1
(k+ j ) = δ22N (αiλ − 1)2 N + β(α (k+ j −1) −1)2 N +α (k+ j −1) . iλ
iλ
Let = {δ22N ((i − 1)2 N + i ): i = 1, 2, . . . , 2 N }.
(13)
It then follows from (11) and (12) that
N ( j +1) 2 −2 δ22N (αiλ − 1)2 N + β(α ( j )−1)2 N +a ( j ) iλ λ,k
(2 N −1+ j ) (2 N −1+ j ) N ∈ = δ22N (αiλ − 1)2 + αiλ for j = 0, 1, . . . , lλ −1, λ = 1, 2, . . . , p, k = 1, 2, . . . , 2 N −1. This, combined with (11), shows that
N ( j +1) − 1)2 N + β(α ( j ) −1)2 N +r ∈ (14) 2 −2 δ22N (αiλ iλ
for j = 0, 1, . . . , lλ − 1, λ = 1, 2, . . . , p, r = 1, 2, . . . , 2 N . Since there are only a finite number of states for the drive BN (1), every state must eventually end up on one of the cycles {δ2 N (i λ ), δ2 N (αiλ ), . . . , δ2 N (αi(lλλ ) )}, λ = 1, 2, . . . , p. Let t0 be the minimal time for all states of the BN (1) to enter into the cycles, and let ( j)
for j = 0, 1, 2, . . ., λ = 1, 2, . . . , p. We shall use induction on n. The case n = 0 is trivial, so we take n ≥ 1 and assume that the statement holds for n − 1. Then
( j +1) n N δ22N (αiλ − 1)2 + β(α ( j )−1)2 N +α ( j ) iλ i
λ (n+ j ) (n+ j ) = δ22N (αiλ − 1)2 N + αiλ
(n+ j +1) N = δ22N (αiλ − 1)2 + β(α (n+ j )−1)2 N +α (n+ j ) iλ iλ
(n+ j +1) (n+ j +1) = δ22N (αiλ − 1)2 N + αiλ by (8), (9), and the induction hypothesis. Next, we use induction to show that, for every 1 ≤ k ≤ 2N − 1
( j +1) k−1 δ22N (αiλ − 1)2 N + β(α ( j )−1)2 N +a ( j ) iλ λ,k
(k+ j ) N = δ22N (αiλ − 1)2 + β(α (k+ j −1) −1)2 N +α (k+ j −1) (12) iλ
λ,k−1
iλ
iλ
S = {αiλ : j = 0, 1, . . . , lλ − 1, λ = 1, 2, . . . , p}
(15)
and put m = t0 + 2 N − 1. Then F t0 δ2 N (i ) ∈
p λ=1
( j)
{δ2 N (αiλ ): j = 0, 1, . . . , lλ − 1} (t )
for i = 1, 2, . . . , 2 N , so that αi 0 ∈ S for each i . Now let 1 ≤ i, r ≤ 2 N be given. By Lemma 2, there exists 1 ≤ rt0 ≤ 2 N such that
t0 δ22N (αi − 1)2 N + β(i−1)2 N +r
. (16) = δ22N (αi(t0 +1) − 1)2 N + β (t0 ) N (αi
−1)2 +rt0
Therefore
m−1 δ22N (αi − 1)2 N + β(i−1)2 N +r N = 2 −2 δ22N (αi(t0 +1) −1)2 N +β (t0) (αi
−1)2 N +rt0
∈ (17)
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σ
by (14) and (16). Since i and r are arbitrary, (8) and (17) imply that Col(m ) ⊆ .
σ
(18)
Let ξ1 , . . . , ξ22N ∈ {1, . . . , 2 N } be such that Gm−1 = δ2 N (ξ1 , . . . , ξ22N ). Since by [15]
σ ∧ ¬σ (a)
m = (F m ⊗ (Gm−1 ))( N ⊗ I2 N ) direct computation leads to (m)
m = δ22N ((α1
(m)
− 1)2 N + ξ1 , . . . , (α1
− 1)2 N
+ ξ2 N , (α2(m) − 1)2 N + ξ2 N +1 , . . . , (α2(m) − 1)2 N N + ξ2 N+1 , . . . , (α2(m) N − 1)2
+
(m) ξ(2 N −1)2 N +1 , . . . , (α2 N
(b) N
− 1)2 + ξ22N ).
Hence αi(m) = ξ(i−1)2 N +r ,
Fig. 1. (a) Model of a neuron in the human brain. From the Boolean input variables σ1 and σ2 , the neuron generates the Boolean function σ1 ∧ ¬σ2 . (b) Network of three neurons.
i, r = 1, 2, . . . , 2 N
by (18). It now follows that: Gm−1 = δ2 N (ξ1 , . . . , ξ2 N , . . . , ξ(2 N −1)2 N +1 , . . . , ξ22N ) (m) = δ2 N (α1(m) , . . . , α1(m) , . . . , α2(m) N , . . . , α2 N )
= F m ⊗ 12 N .
Therefore, the BNs (1) and (2) are completely synchronized by Lemma 1. This completes the proof. Remark 1: Theorem 3 gives a constructive approach for designing the response BN to achieve complete synchronization. According to the approach, if the drive BN has transient states, then, for every 1 ≤ r ≤ 2 N and 1 ≤ s ≤ 2 N for which s ∈ S [defined in (15)], β(s−1)2 N +r can be chosen arbitrarily in {1, 2, . . . , 2 N }, and hence the obtained response BN is not unique. It should be pointed out that, for this case, by properly choosing β(s−1)2 N +r with 1 ≤ r ≤ 2 N , 1 ≤ s ≤ 2 N , s ∈ S, one can adjust the speed of synchronization. To establish this claim, let us consider the synchronization time of the BNs (1) and (2), which is defined to be the smallest positive integer T such that t ≥ T implies Y(t, Y0 , X0 ) = X(t, X0 ) for all X0 ∈ {1, 0} N , Y0 ∈ {1, 0} N . It is easily verified that T is the smallest positive integer such that Col(T ) ⊆ , where is defined as in (13). Since, under conditions (9) and (10)
( j +1) k−2 δ22N (αiλ − 1)2 N + β(α ( j )−1)2 N +a ( j ) iλ λ,k
(k+ j −1) (k+ j −1) N = δ22N (αiλ − 1)2 + aλ,1 ∈ for j = 0, 1, . . . , lλ −1, λ = 1, 2, . . . , p, k = 2, 3, . . . , 2 N −1, it follows from the proof of Theorem 3 that 2 N − 1 ≤ T ≤ 2 N + t0 − 1, where t0 is the minimal time for all states of the BN (1) to enter into the cycles. Observe that, to every 1 ≤ s ≤ 2 N with s ∈ S and αs ∈ S, there correspond unique (j ) integers 1 ≤ λs ≤ p and 0 ≤ js ≤ lλs −1 such that αs = αiλ s . s Then if we take (j )
β(s−1)2 N +r = αiλ s s
for 1 ≤ r ≤ 2 N and 1 ≤ s ≤ 2 N for which s ∈ S and αs ∈ S, we have
N 2 −2 δ22N (αi − 1)2 N + β(i−1)2 N +r
N N = δ22N (αi(2 −1) − 1)2 N + αi(2 −1) ∈ for all 1 ≤ i ≤ 2 N and 1 ≤ r ≤ 2 N , so that T = 2 N − 1. Similarly, for k = 2 N , 2 N + 1, . . . , 2 N + t0 − 1, if (j )
β(s−1)2 N +r = aλss,k−t0 (which makes sense since t0 ≤ 2 N − 1) for 1 ≤ r ≤ 2 N and 1 ≤ s ≤ 2 N for which s ∈ S and αs ∈ S, then we have T = k, establishing the desired claim. Remark 2: The approach presented in Theorem 3 is not the only way to design response BNs. For example, if we take β(α ( j ) −1)2 N +α ( j ) = β(α ( j )−1)2 N +a ( j ) iλ
iλ
iλ
( j +1)
λ,2 N −1
( j +1)
= αiλ
β(α ( j )−1)2 N +a ( j ) = aλ,k+1 iλ
λ,k
for k = 1, 2, . . . , 2 N −2, j = 0, 1, . . . , lλ −1, λ = 1, 2, . . . , p, then analogous to the proof of Theorem 3, it can be shown that the designed BN also completely synchronizes the drive BN (1). We leave the details of the proof to the reader. IV. I LLUSTRATIVE E XAMPLE In this section, we present an example to demonstrate the application of the main result. Let us consider a network consisting of three identical neurons [20], [21], shown schematically in Fig. 1. The logical equations for this network are x 1 (t + 1) = x 2 (t) ∧ ¬x 3 (t) x 2 (t + 1) = x 1 (t) ∧ ¬x 3 (t) x 3 (t + 1) = ¬x 1 (t) ∧ ¬x 2 (t)
(19)
and the state space graph is shown in Fig. 2. We now construct a response BN to completely synchronize this BN.
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3
H(t)
2
1
Fig. 2. State space graph of the BN (19). The entire state space consists of two fixed points, a period 2 cycle, and four transient states.
Let x(t) = x1 (t) x2 (t) x3 (t). Then we can express the BN (19) in its algebraic form as x(t + 1) = Fx(t), where F = δ8 (8, 2, 8, 6, 8, 4, 7, 7). Since this BN contains three cycles {δ8 (2), δ8 (2)}, {δ8 (7), δ8 (7)}, {δ8 (4), δ8 (6), δ8 (4)}, we have i 1 = 2, i 2 = 7,
αi1 = 2, αi2 = 7,
i 3 = 4,
αi3 = 6,
l1 = 1, l2 = 1, αi(2) = 4, 3
l3 = 2.
(0) (1) (0) (1) (0) (1) = a1,1 = 1, a1,2 = a1,2 = 3, a1,3 = a1,3 =4 a1,1 (0) (1) (0) (1) (0) (1) a1,4 = a1,4 = 5, a1,5 = a1,5 = 6, a1,6 = a1,6 =7
= = = = =
(1) a1,7 (1) a2,3 (1) a2,6 (2) a3,2 (2) a3,5
= 8, = 3, = 6, = 2, = 6,
(0) a2,1 (0) a2,4 (0) a2,7 (0) a3,3 (0) a3,6
(1)
= 1, a3,2 = 2, (1)
= 5, a3,6 = 7,
(1) = a2,1 (1) = a2,4 (1) = a2,7 (2) = a3,3 (2) = a3,6 (1) a3,3 = (1) a3,7 =
= = = = = 3,
(0) 1, a2,2 (0) 4, a2,5 (0) 8, a3,1 (0) 3, a3,4 (0) 7, a3,7 (1) a3,4 =
= = = = =
(1) a2,2 (1) a2,5 (2) a3,1 (2) a3,4 (2) a3,7
2
4
6
8
10 t
12
14
16
18
20
Fig. 3. Hamming distance between the states of the BNs (19) and (20) versus time t with initial values (x1 (0), x2 (0), x3 (0)) = (1, 1, 1) and (y1 (0), y2 (0), y3 (0)) = (0, 0, 1).
G 1 = δ2 (1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2) G 2 = δ2 (1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2,
=2 =5
G 3 = δ2 (2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1,
=1
2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2)
=5 =8
4
8.
β9 = 2, β13 = 4,
β10 = 2, β14 = 5,
β11 = 1, β15 = 6,
β12 = 3, β16 = 7,
β49 = 7, β53 = 4,
β50 = 1, β54 = 5,
β51 = 2, β55 = 7,
β52 = 3, β56 = 6,
β25 = 6, β29 = 3,
β26 = 1, β30 = 4,
β27 = 2, β31 = 5,
β28 = 6, β32 = 7,
β41 = 4,
β42 = 1,
β43 = 2,
β44 = 3,
β45 = 5,
β46 = 4,
β47 = 6,
β48 = 7,
and β1 , . . . , β8 , β17 , . . . , β24 , β33 , . . . , β40 , β57 , . . . , β64 can be chosen arbitrarily in {1, 2, . . . , 8}. For example, we can take βi+16 = βi+24 , βi+56 = βi+48 ,
0
2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1)
According to Theorem 3, we take
βi = βi+8 , βi+32 = βi+40 ,
−1
Then the structure matrices of the Boolean functions for the designed response BN are
Hence
(0) a1,7 (0) a2,3 (0) a2,6 (0) a3,2 (0) a3,5 (1) a3,1 (1) a3,5
0
i = 1, 2, . . . , 8.
and the logical equations for the response BN are y1 (t + 1) = (x 1 (t) ∧ x 2 (t) ∧ (y1 (t) ∨ (y2 (t) ∧ y3 (t)))) ¯ 2 (t))) ∨(x 1 (t) ∧ ¬x 2 (t) ∧ ((y1 (t) ∧ y3 (t))∨y ∨(¬x 1 (t) ∧ x 2 (t) ∧ (y1 (t) ∨ (y2 (t) ∧ ¬y3 (t)))) ¯ 2 (t) ∧ y3 (t)))) ∨(¬x 1 (t) ∧ ¬x 2 (t) ∧ (y1 (t)∨(y y2 (t + 1) ¯ 3 (t)))) = (x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∧ y2 (t)) ∨ (y2 (t)∨y ∨(x 1 (t) ∧ ¬x 2 (t) ∧ (y1 (t) ∨ (¬y2 (t) ∧ y3 (t)))) ¯ 3 (t))) ∨(¬x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∧ y2 (t))∨y ¯ 3 (t))) ∨(¬x 1 (t) ∧ ¬x 2 (t) ∧ ((y1 (t) ∧ ¬y2 (t))∨¬y y3 (t + 1) = (x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∧ ¬y2 (t)) ∨ (¬y1 (t) ∧ ¬y3 (t)))) ¯ 2 (t) ∧ ¬y3 (t)))) ∨(x 1 (t) ∧ ¬x 2 (t) ∧ (¬y1 (t)∨(y ¯ 3 (t))) ∨(¬x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∨ ¬y2 (t))∨y ¯ 2 (t) ∨(¬x 1 (t) ∧ ¬x 2 (t) ∧ ((y1 (t) ∧ (¬y2 (t) ∨ y3 (t)))∨y ¯ 3 (t))) ∨y
(20)
by Proposition 1. In Fig. 3, the Hamming distance between the states of the BNs (19) and (20) H (t) = 3i=1 |x i (t) − yi (t)|
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versus the time t is plotted, where (x 1 (0), x 2 (0), x 3 (0)) = (1, 1, 1) and (y1 (0), y2 (0), y3 (0)) = (0, 0, 1). We see that the designed response BN (20) completely synchronizes the drive BN (19) from the seventh step. V. C ONCLUSION In this brief, we addressed the problem of synchronization design for BNs. Our discussion was based on the technique of the semi-tensor product of matrices that allows an algebraic representation of BNs and thus facilitates rigorous analysis of the system dynamics. The main result provided a constructive method for designing a response BN to completely synchronize a given drive BN. An illustrative example was included to depict the design method. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. R EFERENCES [1] B. Derrida, E. Gardner, and A. Zippelius, “An exactly solvable asymmetric neural network model,” Eur. Lett., vol. 4, no. 2, pp. 167–173, Jul. 1987. [2] K. E. Kurten, “Correspondence between neural threshold networks and Kauffman Boolean cellular automata,” J. Phys. A, Math. General, vol. 21, no. 11, pp. L615–L619, Jun. 1988. [3] L. Wang, E. E. Pichler, and J. Ross, “Oscillations and chaos in neural networks: An exactly solvable model,” Proc. Nat. Acad. Sci. United States Amer., vol. 87, no. 23, pp. 9467–9471, Dec. 1990. [4] M. H. Hassoun, Fundamentals of Artificial Neural Networks. Cambridge, MA, USA: MIT Press, 1995. [5] S. Bornholdt and T. Rohlf, “Topological evolution of dynamical networks: Global criticality from local dynamics,” Phys. Rev. Lett., vol. 84, no. 26, pp. 6114–6117, Jun. 2000. [6] S. Huang and D. E. Ingber, “Shape-dependent control of cell growth, differentiation, and apoptosis: Switching between attractors in cell regulatory networks,” Exp. Cell Res., vol. 261, no. 1, pp. 91–103, Nov. 2000. [7] R. Albert and H. G. Othmer, “The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster,” J. Theoretical Biol., vol. 223, no. 1, pp. 1–18, Jul. 2003. [8] D. J. Irons, “Logical analysis of the budding yeast cell cycle,” J. Theoretical Biol., vol. 257, no. 4, pp. 543–559, Apr. 2009. [9] A. Veliz-Cuba and B. Stigler, “Boolean models can explain bistability in the lac operon,” J. Comput. Biol., vol. 18, no. 6, pp. 783–794, Jun. 2011. [10] L. G. Morelli and D. H. Zanette, “Synchronization of Kauffman networks,” Phys. Rev. E, vol. 63, no. 3, pp. 036204-1–036204-10, Mar. 2001. [11] M.-C. Ho, Y.-C. Hung, and I.-M. Jiang, “Stochastic coupling of two random Boolean networks,” Phys. Lett. A, vol. 344, no. 1, pp. 36–42, Aug. 2005. [12] Y.-C. Hung, M.-C. Ho, J.-S. Lih, and I.-M. Jiang, “Chaos synchronization of two stochastically coupled random Boolean networks,” Phys. Lett. A, vol. 356, no. 1, pp. 35–43, Jul. 2006. [13] Y.-C. Hung, “Microscopic interactions lead to mutual synchronization in a network of networks,” Phys. Lett. A, vol. 375, nos. 30–31, pp. 2809–2814, Jul. 2011. [14] J. Parriaux, P. Guillot, and G. Millérioux, “Synchronization of Boolean dynamical systems: A spectral characterization,” in Proc. 6th Int. Conf. Sequences Their Appl., 2010, pp. 373–386. [15] R. Li and T. Chu, “Complete synchronization of Boolean networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 5, pp. 840–846, May 2012. [16] D. Cheng, “Input-state approach to Boolean networks,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 512–521, Mar. 2009. [17] D. Cheng and H. Qi, “State-space analysis of Boolean networks,” IEEE Trans. Neural Netw., vol. 21, no. 4, pp. 584–594, Apr. 2010.
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[18] D. Cheng and H. Qi, “A linear representation of dynamics of Boolean networks,” IEEE Trans. Autom. Control, vol. 55, no. 10, pp. 2251–2258, Oct. 2010. [19] D. Cheng, H. Qi, and Z. Li, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach. New York, USA: Springer-Verlag, 2011. [20] F. Robert, Discrete Iterations: A Metric Study. New York, USA: Springer-Verlag, 1986. [21] J. Heidel, J. Maloney, C. Farrow, and J. A. Rogers, “Finding cycles in synchronous Boolean networks with applications to biochemical systems,” Int. J. Bifurcat. Chaos, vol. 13, no. 3, pp. 535–552, Mar. 2003. [22] D. Cheng, H. Qi, and Z. Li, “Model construction of Boolean network via observed data,” IEEE Trans. Neural Netw., vol. 22, no. 4, pp. 525–536, Apr. 2011.
Bogdanov–Takens Singularity in Tri-Neuron Network With Time Delay Xing He, Chuandong Li, Senior Member, IEEE, Tingwen Huang, and Chaojie Li
Abstract— This brief reports a retarded functional differential equation modeling tri-neuron network with time delay. The Bogdanov–Takens (B-T) bifurcation is investigated by using the center manifold reduction and the normal form method. We get the versal unfolding of the norm forms at the B-T singularity and show that the model can exhibit pitchfork, Hopf, homoclinic, and double-limit cycles bifurcations. Some numerical simulations are given to support the analytic results and explore chaotic dynamics. Finally, an algorithm is given to show that chaotic tri-neuron networks can be used for encrypting a color image. Index Terms— Bogdanov–Takens bifurcation, tri-neuron network.
bifurcation,
homoclinic
I. I NTRODUCTION Neural networks (NNs) have great potential in practical applications, such as associative memory, optimization, and pattern recognition. In application of NNs, multistable NNs can be viewed as models for associative memory [1], and attractors can be interpreted as stored memories. For example, an equilibrium point can be considered as a single storage or memory pattern, or an optimum object, while a limit cycle can also restore various complex patterns. In order to realize a memory system, [4] examined the dynamical phenomenon Manuscript received January 17, 2012; revised October 22, 2012; accepted December 19, 2012. Date of publication March 12, 2013; date of current version April 5, 2013. This work was supported in part by the National Natural Science Foundation of China, under Grant 60974020, the Fundamental Research Funds for the Central Universities of China, under Project CDJZR10 18 55 01, and the National Priority Research Project, under Grant NPRP 4-1162-1-181 funded by Qatar National Research Fund, Qatar. X. He and C. Li are with the School of Electronics and Information Engineering, Southwest University, Chongqing 400715, China (e-mail: [email protected]; [email protected]). T. Huang is with Texas A&M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). C. J. Li is with the School of Science, Information Technology and Engineering, University of Ballarat, Mount St. Helen, VIC 3350, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2238681
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[14] Z. Uykan, “Spectral based solutions for (Near) optimum channel/frequency allocation,” in Proc. 18th Int. Conf. Syst. Signals Image Process., Sarajevo, Bosnia and Herzegovina, Jun. 2011, pp. 1–4. [15] M. E. J. Newman, Networks–An Introduction, London, U.K.: Oxford Univ. Press, 2011, pp. 358–380. [16] I. Stojmenovic, “Handbook of Wireless Networks and Mobile Computing,” New York, USA: Wiley, 2002. [17] O. Lazaro, and D. Girma, “Real-time operational aspects of Hopfield neural network based dynamic channel allocation scheme,” Electron. Lett., vol. 40, no. 18, pp. 1141–1143, 2004. [18] C. W. Ahn, and R. S. Ramakrishna, “QoS provisioning dynamic connection-admission control for multimedia wireless networks using Hopfield neural networks,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 106–117, Jan. 2004. [19] D. Calabuig, J. F. Monserrat, D. G. Barquero, and N. Cardona, “Hopfield neural network algorithm for dynamic resource allocation in WCDMA systems,” in Proc. IEEE 3rd Int. Symp. Wireless Commun. Syst., Valencia, Spain, Sep. 2006, pp. 40–44. [20] D. Calabuig, J. F. Monserrat, D. G. Barquero, and O. Lazaro, “User bandwidth usage-driven HNN neuron excitation method for maximum resource utilization within packet-switched communication networks,” IEEE Commun. Lett., vol. 10, no. 11, pp. 766–768, Nov. 2006. [21] G. Joya, M. A. Atencia, and F. Sandoval, “Hopfield neural network applied to optimization problems: Some theoretical and simulation results,” in Proc. Biol. Artif. Comput., Neurosci. Technol, Lect. Notes Comput. Sci., vol. 1240, pp. 556–565, 1997. [22] T. S. Rappaport, Wireless Communications: Principles & Practice, 2nd ed., Englewood Cliffs, NJ: Prentice Hall, 2002. [23] Z. Zeng and W. X. Zheng, “Multistability of neural networks with timevarying delays and concave-convex characteristics,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 293–305, Feb. 2012.
Synchronization Design of Boolean Networks via the Semi-Tensor Product Method Rui Li, Meng Yang, and Tianguang Chu
Abstract— We provide a general approach for the design of a response Boolean network (BN) to achieve complete synchronization with a given drive BN. The approach is based on the algebraic representation of BNs in terms of the semitensor product of matrices. Instead of designing the logical dynamic equations of a response BN directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. The results are applied to a three-neuron network in order to illustrate the effectiveness of the proposed approach. Index Terms— Algebraic representation, Boolean network, complete synchronization, semi-tensor product.
0 means that it is OFF. These nodes interact with each other according to some logical rules, specified through a set of Boolean functions that determine the states of the nodes at the next time-step, and thereby give the dynamics of the network. Even though BNs seem to be oversimplified models of real networks, they retain in most cases meaningful information that can be used to make inferences regarding the systems they model, and hence have been investigated widely. Recently, an increasing interest has been focused on synchronization of coupled BNs, motivated by many potential applications in biology, physics, and engineering [10], [11]. In [12], synchronization between a pair of stochastically coupled random BNs was considered. Later, the study was further extended to mutual synchronization in a random network of random BNs [13]. The spectral characterization of two synchronized deterministic BNs coupled in a drive–response configuration was discussed in [14]. In our recent paper [15], the concept of complete synchronization was introduced for the case of deterministic BNs in a drive–response configuration, and a necessary and sufficient algebraic criterion for complete synchronization was presented. In this brief, we further consider the problem of synchronization design for BNs. This brief constitutes a continuation of [15], providing a general approach for constructing a response BN to achieve complete synchronization with a given drive BN. Similar to [15], our discussion is based on the recently developed technique of the semi-tensor product of matrices that allows an algebraic representation of logical dynamics of BNs [16]–[19]. To demonstrate the application of the design approach, we apply the results to a three-neuron network studied in [20] and [21]. The remainder of this brief is organized as follows. Section II gives the problem formulation and some preliminary results. Section III presents our approach for constructing synchronized response BNs. An illustrative example is treated in Section IV, and a brief conclusion is drawn in the final section. Notation: Rn×m denotes the set of all n × m real matrices. In is the n × n identity matrix. δn (i ) stands for the i th column of In . n stands for the set of all n columns of In . Col(A) denotes the set of columns of a matrix A. 1n represents the n-dimensional row vector whose entries are equal to 1. ⊗ is the Kronecker product of matrices.
I. I NTRODUCTION
II. P ROBLEM F ORMULATION AND P RELIMINARIES
Boolean networks (BNs) have been successfully used in modeling complex systems such as neural networks and gene regulatory networks [1]–[9]. In a BN, the nodes can take on one of two binary values, 1 or 0, at each discrete time point. A value of 1 represents that the node is ON and a value of
Let us consider two BNs with N nodes respectively coupled in a drive–response configuration and described as follows:
Manuscript received May 14, 2012; revised November 19, 2012; accepted February 11, 2013. Date of publication March 7, 2013; date of current version April 5, 2013. This work was supported in part by the National Basic Research Program of China under Grant 2012CB821200 and the National Natural Science Foundation of China under Grant 61273111 and Grant 60974064. The authors are with the State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TNNLS.2013.2248092
x i (t + 1) = f i (x 1 (t), . . . , x N (t)) yi (t + 1) = gi (x 1 (t), . . . , x N (t), y1 (t), . . . , y N (t)),
(1) (2)
i = 1, . . . , N where x i and yi are the nodes of the drive BN (1) and the response BN (2), respectively, f i and gi are Boolean functions, and t = 0, 1, 2, . . .. We succinctly denote by X = (x 1 , . . . , x N ) and Y = (y1 , . . . , y N ) the states of the drive BN (1) and the response BN (2), respectively. Let X(t, X0 ) be the trajectory of (1) with X(0, X0 ) = X0 , and let
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997
Y(t, Y0 , X0 ) be the trajectory of (2) with Y(0, Y0 , X0 ) = Y0 corresponding to the driving trajectory X(t, X0 ). Before formulating the problem to be investigated, we first recall the following definition. Definition 1 ( [15]): The drive BN (1) and the response BN (2) are said to be completely synchronized if for every X0 ∈ {1, 0} N there exists a positive integer k such that t ≥ k implies Y(t, Y0 , X0 ) = X(t, X0 ) for all Y0 ∈ {1, 0} N . The objective of this brief is to design a response BN of the form (2) to achieve complete synchronization with the given drive BN (1). We will approach this problem by using algebraic representations of logical dynamics based on the technique of the semi-tensor product of matrices. Definition 2 ( [19]): Let A ∈ Rn×m and B ∈ R p×q . The semi-tensor product of A and B is
and Hi is the structure matrix of h i for i = 1, 2. It is worth mentioning that every column of the above matrices F, G, Fi , and G i has a unique nonzero entry and all nonzero entries are equal to 1. We call such a matrix a logical matrix, and simply write an n × m logical matrix [δn (i 1 ) δn (i 2 ) . . . δn (i m )] as δn (i 1 , i 2 , . . . , i m ).
A B = (A ⊗ Il/m )(B ⊗ Il/ p )
III. M AIN R ESULT
where l is the least common multiple of m and p. The semi-tensor product allows representing a BN in an algebraic form. In so doing, we regard a Boolean variable σ ∈ {1, 0} as a vector σ ∈ 2 by identifying 1 and 0 with δ2 (1) and δ2 (2), respectively. Since the map from {1, 0} N to 2 N sending (σ1 , . . . , σ N ) to σ1 · · · σ N is a bijection [18], it follows that at every time instant t the states of the drive BN (1) and the response BN (2) can be completely described by x(t) = x1 (t) · · · x N (t) and y(t) = y1 (t) · · · y N (t), respectively. Then by utilizing the semi-tensor product we can express the drive BN (1) and the response BN (2) as [19] xi (t + 1) = Fi x(t)
(3)
yi (t + 1) = G i x(t) y(t) i = 1, . . . , N
(4)
N
2N
respectively, where Fi ∈ R2×2 and G i ∈ R2×2 for each i . We call Fi (resp., G i ) the structure matrix of f i (resp., gi ). Moreover, (3) and (4) can further be converted into the following discrete-time systems [19]: x(t + 1) = Fx(t)
(5)
y(t + 1) = G x(t) y(t)
(6)
2 N ×2 N
2 N ×22N
respectively, where F ∈ R and G ∈ R . Equations (5) and (6) are called the algebraic representations of the BNs (1) and (2), respectively. It was proved that (5) and (6) are equivalent to the original BNs (1) and (2), respectively [19]. Hence, instead of designing the logical dynamics of the response BN (2) directly, we will first construct its algebraic representation (6) to realize complete synchronization and then convert the algebraic representation back to the logical form. The following result provides a procedure for calculating the logical form (2) from the algebraic representation (6). Proposition 1 ([22]): 1) Let (6) be the algebraic representation of the BN (2), and let G i be the structure matrix of gi for i = 1, . . . , N. Then G i = Si G, i = 1, . . . , N where Si = 12i−1 ⊗ I2 ⊗ 12 N−i for each i .
2) Suppose h(σ1 , σ2 , . . . , σn ) is a Boolean function, and n H ∈ R2×2 is the structure matrix of h. Partition H as n−1 H1, H2 ∈ R2×2 . H = H1 H 2 , Then h(σ1 , σ2 , . . . , σn ) = (σ1 ∧ h 1 (σ2 , . . . , σn )) ∨(¬σ1 ∧ h 2 (σ2 , . . . , σn ))
We first give some lemmas that will be used in the proof of our main result. Lemma 1 ( [15]): Suppose (5) and (6) are the algebraic representations of the BNs (1) and (2), respectively. Define N = δ22N (1, 2 N +2, 2·2 N +3, . . . , (2 N −2)2 N +2 N −1, 22N ) and = (F ⊗ G)( N ⊗ I2 N ).
(7)
Then the drive BN (1) and the response BN (2) are completely synchronized if and only if there is a positive integer m such that Gm−1 = F m ⊗ 12 N . Let α1 , α2 , . . . , α2 N , β1 , β2 , . . . , β22N ∈ {1, 2, . . . , 2 N } be such that F = δ2 N (α1 , α2 , . . . , α2 N ) G = δ2 N (β1 , β2 , . . . , β22N ). For every positive integer n, put (n)
αi
= αα (n−1) i
for each i , where by convention αi(0) = i . Then (n)
(n)
(n)
F n = δ2 N (α1 , α2 , . . . , α2 N ) by induction. We have the following lemma. Lemma 2: Let be defined as in (7), and let two integers 1 ≤ i, r ≤ 2 N be given. Then for every nonnegative integer n there is an integer 1 ≤ rn ≤ 2 N such that n δ22N ((αi − 1)2 N + β(i−1)2 N +r ) (n+1)
− 1)2 N + β(α (n)−1)2 N +r ). = δ22N ((αi n i Proof: We shall use induction on n. The case n = 0 is trivial. Suppose that the statement is true for n ≥ 0. It follows from (7) that = δ22N ((α1 − 1)2 N + β1 , . . . , (α1 − 1)2 N + β2 N , (α2 − 1)2 N + β2 N +1 , . . . , (α2 − 1)2 N + β2 N+1 , . . . , (α2 N − 1)2 N + β(2 N −1)2 N +1 , . . . , (α2 N − 1)2 N + β22N ).
(8)
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for j = 0, 1, 2, . . ., λ = 1, 2, . . . , p. This is true for k = 1 by (9). If k ≥ 2, then by (8) and (10)
( j +1) − 1)2 N + β(α ( j )−1)2 N +a ( j ) δ22N (αiλ iλ λ,k
( j +1) ( j +1) N = δ22N (αiλ − 1)2 + aλ,k−1
( j +2) = δ22N (αiλ − 1)2 N + β(α ( j +1)−1)2 N +a ( j +1) .
Let rn+1 = β(α (n) −1)2 N +r . n
i
Then we have
n+1 δ22N (αi − 1)2 N + β(i−1)2 N +r
= δ22N (αi(n+1) − 1)2 N + β(α (n)−1)2 N +r n i
(n+2) N = δ22N (αi − 1)2 + β(α (n+1)−1)2 N +r n+1
i
via the induction hypothesis and (8). (l) A sequence {δ2 N (i ), δ2 N (αi ), . . . , δ2 N (αi )} is called a period l cycle of the BN (1) if i, αi , . . . , αi(l−1) are distinct and αi(l) = i [18]. A period 1 cycle is also called a fixed point. Suppose that the drive BN (1) has p cycles (l ) {δ2 N (i λ ), δ2 N (αiλ ), . . . , δ2 N (αiλλ )}, λ = 1, 2, . . . , p. For j = ( j) ( j) ( j) 0, 1, 2, . . ., λ = 1, 2, . . . , p, define aλ,1 , aλ,2 , . . . , aλ,2 N −1 by ( j)
aλ,k =
k, if k < k + 1, if k ≥
( j) αiλ ( j) αiλ
k = 1, 2, . . . , 2 N − 1.
We are now ready to present the main result of this brief. Theorem 3: The drive BN (1) and the response BN (2) are completely synchronized if β(α ( j )−1)2 N +α ( j ) = β(α ( j )−1)2 N +a ( j ) = iλ
iλ
λ,1
iλ
( j +1)
β(α ( j )−1)2 N +a ( j ) = aλ,k−1 , iλ
λ,k
( j +1) αiλ
(9)
k = 2, 3, . . . , 2 N − 1
(10)
j = 0, 1, . . . , lλ − 1, λ = 1, 2, . . . , p. Proof: Let be defined as in (7). We first show that, for every n ≥ 0 ( j +1)
n δ22N ((αiλ
(n+ j +1)
= δ22N ((αiλ
− 1)2 N + β(α ( j )−1)2 N +α ( j ) ) iλ
(n+ j +1)
− 1)2 N + αiλ
iλ
)
(11)
Therefore, by induction
( j +1) k−1 N δ22N (αiλ − 1)2 + β(α ( j )−1)2 N +a ( j ) iλ λ,k
( j +2) = k−2 δ22N (αiλ − 1)2 N + β(α ( j +1)−1)2 N +a ( j +1) iλ λ,k−1
(k+ j ) = δ22N (αiλ − 1)2 N + β(α (k+ j −1) −1)2 N +α (k+ j −1) . iλ
iλ
Let = {δ22N ((i − 1)2 N + i ): i = 1, 2, . . . , 2 N }.
(13)
It then follows from (11) and (12) that
N ( j +1) 2 −2 δ22N (αiλ − 1)2 N + β(α ( j )−1)2 N +a ( j ) iλ λ,k
(2 N −1+ j ) (2 N −1+ j ) N ∈ = δ22N (αiλ − 1)2 + αiλ for j = 0, 1, . . . , lλ −1, λ = 1, 2, . . . , p, k = 1, 2, . . . , 2 N −1. This, combined with (11), shows that
N ( j +1) − 1)2 N + β(α ( j ) −1)2 N +r ∈ (14) 2 −2 δ22N (αiλ iλ
for j = 0, 1, . . . , lλ − 1, λ = 1, 2, . . . , p, r = 1, 2, . . . , 2 N . Since there are only a finite number of states for the drive BN (1), every state must eventually end up on one of the cycles {δ2 N (i λ ), δ2 N (αiλ ), . . . , δ2 N (αi(lλλ ) )}, λ = 1, 2, . . . , p. Let t0 be the minimal time for all states of the BN (1) to enter into the cycles, and let ( j)
for j = 0, 1, 2, . . ., λ = 1, 2, . . . , p. We shall use induction on n. The case n = 0 is trivial, so we take n ≥ 1 and assume that the statement holds for n − 1. Then
( j +1) n N δ22N (αiλ − 1)2 + β(α ( j )−1)2 N +α ( j ) iλ i
λ (n+ j ) (n+ j ) = δ22N (αiλ − 1)2 N + αiλ
(n+ j +1) N = δ22N (αiλ − 1)2 + β(α (n+ j )−1)2 N +α (n+ j ) iλ iλ
(n+ j +1) (n+ j +1) = δ22N (αiλ − 1)2 N + αiλ by (8), (9), and the induction hypothesis. Next, we use induction to show that, for every 1 ≤ k ≤ 2N − 1
( j +1) k−1 δ22N (αiλ − 1)2 N + β(α ( j )−1)2 N +a ( j ) iλ λ,k
(k+ j ) N = δ22N (αiλ − 1)2 + β(α (k+ j −1) −1)2 N +α (k+ j −1) (12) iλ
λ,k−1
iλ
iλ
S = {αiλ : j = 0, 1, . . . , lλ − 1, λ = 1, 2, . . . , p}
(15)
and put m = t0 + 2 N − 1. Then F t0 δ2 N (i ) ∈
p λ=1
( j)
{δ2 N (αiλ ): j = 0, 1, . . . , lλ − 1} (t )
for i = 1, 2, . . . , 2 N , so that αi 0 ∈ S for each i . Now let 1 ≤ i, r ≤ 2 N be given. By Lemma 2, there exists 1 ≤ rt0 ≤ 2 N such that
t0 δ22N (αi − 1)2 N + β(i−1)2 N +r
. (16) = δ22N (αi(t0 +1) − 1)2 N + β (t0 ) N (αi
−1)2 +rt0
Therefore
m−1 δ22N (αi − 1)2 N + β(i−1)2 N +r N = 2 −2 δ22N (αi(t0 +1) −1)2 N +β (t0) (αi
−1)2 N +rt0
∈ (17)
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999
σ
by (14) and (16). Since i and r are arbitrary, (8) and (17) imply that Col(m ) ⊆ .
σ
(18)
Let ξ1 , . . . , ξ22N ∈ {1, . . . , 2 N } be such that Gm−1 = δ2 N (ξ1 , . . . , ξ22N ). Since by [15]
σ ∧ ¬σ (a)
m = (F m ⊗ (Gm−1 ))( N ⊗ I2 N ) direct computation leads to (m)
m = δ22N ((α1
(m)
− 1)2 N + ξ1 , . . . , (α1
− 1)2 N
+ ξ2 N , (α2(m) − 1)2 N + ξ2 N +1 , . . . , (α2(m) − 1)2 N N + ξ2 N+1 , . . . , (α2(m) N − 1)2
+
(m) ξ(2 N −1)2 N +1 , . . . , (α2 N
(b) N
− 1)2 + ξ22N ).
Hence αi(m) = ξ(i−1)2 N +r ,
Fig. 1. (a) Model of a neuron in the human brain. From the Boolean input variables σ1 and σ2 , the neuron generates the Boolean function σ1 ∧ ¬σ2 . (b) Network of three neurons.
i, r = 1, 2, . . . , 2 N
by (18). It now follows that: Gm−1 = δ2 N (ξ1 , . . . , ξ2 N , . . . , ξ(2 N −1)2 N +1 , . . . , ξ22N ) (m) = δ2 N (α1(m) , . . . , α1(m) , . . . , α2(m) N , . . . , α2 N )
= F m ⊗ 12 N .
Therefore, the BNs (1) and (2) are completely synchronized by Lemma 1. This completes the proof. Remark 1: Theorem 3 gives a constructive approach for designing the response BN to achieve complete synchronization. According to the approach, if the drive BN has transient states, then, for every 1 ≤ r ≤ 2 N and 1 ≤ s ≤ 2 N for which s ∈ S [defined in (15)], β(s−1)2 N +r can be chosen arbitrarily in {1, 2, . . . , 2 N }, and hence the obtained response BN is not unique. It should be pointed out that, for this case, by properly choosing β(s−1)2 N +r with 1 ≤ r ≤ 2 N , 1 ≤ s ≤ 2 N , s ∈ S, one can adjust the speed of synchronization. To establish this claim, let us consider the synchronization time of the BNs (1) and (2), which is defined to be the smallest positive integer T such that t ≥ T implies Y(t, Y0 , X0 ) = X(t, X0 ) for all X0 ∈ {1, 0} N , Y0 ∈ {1, 0} N . It is easily verified that T is the smallest positive integer such that Col(T ) ⊆ , where is defined as in (13). Since, under conditions (9) and (10)
( j +1) k−2 δ22N (αiλ − 1)2 N + β(α ( j )−1)2 N +a ( j ) iλ λ,k
(k+ j −1) (k+ j −1) N = δ22N (αiλ − 1)2 + aλ,1 ∈ for j = 0, 1, . . . , lλ −1, λ = 1, 2, . . . , p, k = 2, 3, . . . , 2 N −1, it follows from the proof of Theorem 3 that 2 N − 1 ≤ T ≤ 2 N + t0 − 1, where t0 is the minimal time for all states of the BN (1) to enter into the cycles. Observe that, to every 1 ≤ s ≤ 2 N with s ∈ S and αs ∈ S, there correspond unique (j ) integers 1 ≤ λs ≤ p and 0 ≤ js ≤ lλs −1 such that αs = αiλ s . s Then if we take (j )
β(s−1)2 N +r = αiλ s s
for 1 ≤ r ≤ 2 N and 1 ≤ s ≤ 2 N for which s ∈ S and αs ∈ S, we have
N 2 −2 δ22N (αi − 1)2 N + β(i−1)2 N +r
N N = δ22N (αi(2 −1) − 1)2 N + αi(2 −1) ∈ for all 1 ≤ i ≤ 2 N and 1 ≤ r ≤ 2 N , so that T = 2 N − 1. Similarly, for k = 2 N , 2 N + 1, . . . , 2 N + t0 − 1, if (j )
β(s−1)2 N +r = aλss,k−t0 (which makes sense since t0 ≤ 2 N − 1) for 1 ≤ r ≤ 2 N and 1 ≤ s ≤ 2 N for which s ∈ S and αs ∈ S, then we have T = k, establishing the desired claim. Remark 2: The approach presented in Theorem 3 is not the only way to design response BNs. For example, if we take β(α ( j ) −1)2 N +α ( j ) = β(α ( j )−1)2 N +a ( j ) iλ
iλ
iλ
( j +1)
λ,2 N −1
( j +1)
= αiλ
β(α ( j )−1)2 N +a ( j ) = aλ,k+1 iλ
λ,k
for k = 1, 2, . . . , 2 N −2, j = 0, 1, . . . , lλ −1, λ = 1, 2, . . . , p, then analogous to the proof of Theorem 3, it can be shown that the designed BN also completely synchronizes the drive BN (1). We leave the details of the proof to the reader. IV. I LLUSTRATIVE E XAMPLE In this section, we present an example to demonstrate the application of the main result. Let us consider a network consisting of three identical neurons [20], [21], shown schematically in Fig. 1. The logical equations for this network are x 1 (t + 1) = x 2 (t) ∧ ¬x 3 (t) x 2 (t + 1) = x 1 (t) ∧ ¬x 3 (t) x 3 (t + 1) = ¬x 1 (t) ∧ ¬x 2 (t)
(19)
and the state space graph is shown in Fig. 2. We now construct a response BN to completely synchronize this BN.
1000
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3
H(t)
2
1
Fig. 2. State space graph of the BN (19). The entire state space consists of two fixed points, a period 2 cycle, and four transient states.
Let x(t) = x1 (t) x2 (t) x3 (t). Then we can express the BN (19) in its algebraic form as x(t + 1) = Fx(t), where F = δ8 (8, 2, 8, 6, 8, 4, 7, 7). Since this BN contains three cycles {δ8 (2), δ8 (2)}, {δ8 (7), δ8 (7)}, {δ8 (4), δ8 (6), δ8 (4)}, we have i 1 = 2, i 2 = 7,
αi1 = 2, αi2 = 7,
i 3 = 4,
αi3 = 6,
l1 = 1, l2 = 1, αi(2) = 4, 3
l3 = 2.
(0) (1) (0) (1) (0) (1) = a1,1 = 1, a1,2 = a1,2 = 3, a1,3 = a1,3 =4 a1,1 (0) (1) (0) (1) (0) (1) a1,4 = a1,4 = 5, a1,5 = a1,5 = 6, a1,6 = a1,6 =7
= = = = =
(1) a1,7 (1) a2,3 (1) a2,6 (2) a3,2 (2) a3,5
= 8, = 3, = 6, = 2, = 6,
(0) a2,1 (0) a2,4 (0) a2,7 (0) a3,3 (0) a3,6
(1)
= 1, a3,2 = 2, (1)
= 5, a3,6 = 7,
(1) = a2,1 (1) = a2,4 (1) = a2,7 (2) = a3,3 (2) = a3,6 (1) a3,3 = (1) a3,7 =
= = = = = 3,
(0) 1, a2,2 (0) 4, a2,5 (0) 8, a3,1 (0) 3, a3,4 (0) 7, a3,7 (1) a3,4 =
= = = = =
(1) a2,2 (1) a2,5 (2) a3,1 (2) a3,4 (2) a3,7
2
4
6
8
10 t
12
14
16
18
20
Fig. 3. Hamming distance between the states of the BNs (19) and (20) versus time t with initial values (x1 (0), x2 (0), x3 (0)) = (1, 1, 1) and (y1 (0), y2 (0), y3 (0)) = (0, 0, 1).
G 1 = δ2 (1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2) G 2 = δ2 (1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2,
=2 =5
G 3 = δ2 (2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1,
=1
2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2)
=5 =8
4
8.
β9 = 2, β13 = 4,
β10 = 2, β14 = 5,
β11 = 1, β15 = 6,
β12 = 3, β16 = 7,
β49 = 7, β53 = 4,
β50 = 1, β54 = 5,
β51 = 2, β55 = 7,
β52 = 3, β56 = 6,
β25 = 6, β29 = 3,
β26 = 1, β30 = 4,
β27 = 2, β31 = 5,
β28 = 6, β32 = 7,
β41 = 4,
β42 = 1,
β43 = 2,
β44 = 3,
β45 = 5,
β46 = 4,
β47 = 6,
β48 = 7,
and β1 , . . . , β8 , β17 , . . . , β24 , β33 , . . . , β40 , β57 , . . . , β64 can be chosen arbitrarily in {1, 2, . . . , 8}. For example, we can take βi+16 = βi+24 , βi+56 = βi+48 ,
0
2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1)
According to Theorem 3, we take
βi = βi+8 , βi+32 = βi+40 ,
−1
Then the structure matrices of the Boolean functions for the designed response BN are
Hence
(0) a1,7 (0) a2,3 (0) a2,6 (0) a3,2 (0) a3,5 (1) a3,1 (1) a3,5
0
i = 1, 2, . . . , 8.
and the logical equations for the response BN are y1 (t + 1) = (x 1 (t) ∧ x 2 (t) ∧ (y1 (t) ∨ (y2 (t) ∧ y3 (t)))) ¯ 2 (t))) ∨(x 1 (t) ∧ ¬x 2 (t) ∧ ((y1 (t) ∧ y3 (t))∨y ∨(¬x 1 (t) ∧ x 2 (t) ∧ (y1 (t) ∨ (y2 (t) ∧ ¬y3 (t)))) ¯ 2 (t) ∧ y3 (t)))) ∨(¬x 1 (t) ∧ ¬x 2 (t) ∧ (y1 (t)∨(y y2 (t + 1) ¯ 3 (t)))) = (x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∧ y2 (t)) ∨ (y2 (t)∨y ∨(x 1 (t) ∧ ¬x 2 (t) ∧ (y1 (t) ∨ (¬y2 (t) ∧ y3 (t)))) ¯ 3 (t))) ∨(¬x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∧ y2 (t))∨y ¯ 3 (t))) ∨(¬x 1 (t) ∧ ¬x 2 (t) ∧ ((y1 (t) ∧ ¬y2 (t))∨¬y y3 (t + 1) = (x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∧ ¬y2 (t)) ∨ (¬y1 (t) ∧ ¬y3 (t)))) ¯ 2 (t) ∧ ¬y3 (t)))) ∨(x 1 (t) ∧ ¬x 2 (t) ∧ (¬y1 (t)∨(y ¯ 3 (t))) ∨(¬x 1 (t) ∧ x 2 (t) ∧ ((y1 (t) ∨ ¬y2 (t))∨y ¯ 2 (t) ∨(¬x 1 (t) ∧ ¬x 2 (t) ∧ ((y1 (t) ∧ (¬y2 (t) ∨ y3 (t)))∨y ¯ 3 (t))) ∨y
(20)
by Proposition 1. In Fig. 3, the Hamming distance between the states of the BNs (19) and (20) H (t) = 3i=1 |x i (t) − yi (t)|
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 6, JUNE 2013
versus the time t is plotted, where (x 1 (0), x 2 (0), x 3 (0)) = (1, 1, 1) and (y1 (0), y2 (0), y3 (0)) = (0, 0, 1). We see that the designed response BN (20) completely synchronizes the drive BN (19) from the seventh step. V. C ONCLUSION In this brief, we addressed the problem of synchronization design for BNs. Our discussion was based on the technique of the semi-tensor product of matrices that allows an algebraic representation of BNs and thus facilitates rigorous analysis of the system dynamics. The main result provided a constructive method for designing a response BN to completely synchronize a given drive BN. An illustrative example was included to depict the design method. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. R EFERENCES [1] B. Derrida, E. Gardner, and A. Zippelius, “An exactly solvable asymmetric neural network model,” Eur. Lett., vol. 4, no. 2, pp. 167–173, Jul. 1987. [2] K. E. Kurten, “Correspondence between neural threshold networks and Kauffman Boolean cellular automata,” J. Phys. A, Math. General, vol. 21, no. 11, pp. L615–L619, Jun. 1988. [3] L. Wang, E. E. Pichler, and J. Ross, “Oscillations and chaos in neural networks: An exactly solvable model,” Proc. Nat. Acad. Sci. United States Amer., vol. 87, no. 23, pp. 9467–9471, Dec. 1990. [4] M. H. Hassoun, Fundamentals of Artificial Neural Networks. Cambridge, MA, USA: MIT Press, 1995. [5] S. Bornholdt and T. Rohlf, “Topological evolution of dynamical networks: Global criticality from local dynamics,” Phys. Rev. Lett., vol. 84, no. 26, pp. 6114–6117, Jun. 2000. [6] S. Huang and D. E. Ingber, “Shape-dependent control of cell growth, differentiation, and apoptosis: Switching between attractors in cell regulatory networks,” Exp. Cell Res., vol. 261, no. 1, pp. 91–103, Nov. 2000. [7] R. Albert and H. G. Othmer, “The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster,” J. Theoretical Biol., vol. 223, no. 1, pp. 1–18, Jul. 2003. [8] D. J. Irons, “Logical analysis of the budding yeast cell cycle,” J. Theoretical Biol., vol. 257, no. 4, pp. 543–559, Apr. 2009. [9] A. Veliz-Cuba and B. Stigler, “Boolean models can explain bistability in the lac operon,” J. Comput. Biol., vol. 18, no. 6, pp. 783–794, Jun. 2011. [10] L. G. Morelli and D. H. Zanette, “Synchronization of Kauffman networks,” Phys. Rev. E, vol. 63, no. 3, pp. 036204-1–036204-10, Mar. 2001. [11] M.-C. Ho, Y.-C. Hung, and I.-M. Jiang, “Stochastic coupling of two random Boolean networks,” Phys. Lett. A, vol. 344, no. 1, pp. 36–42, Aug. 2005. [12] Y.-C. Hung, M.-C. Ho, J.-S. Lih, and I.-M. Jiang, “Chaos synchronization of two stochastically coupled random Boolean networks,” Phys. Lett. A, vol. 356, no. 1, pp. 35–43, Jul. 2006. [13] Y.-C. Hung, “Microscopic interactions lead to mutual synchronization in a network of networks,” Phys. Lett. A, vol. 375, nos. 30–31, pp. 2809–2814, Jul. 2011. [14] J. Parriaux, P. Guillot, and G. Millérioux, “Synchronization of Boolean dynamical systems: A spectral characterization,” in Proc. 6th Int. Conf. Sequences Their Appl., 2010, pp. 373–386. [15] R. Li and T. Chu, “Complete synchronization of Boolean networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 5, pp. 840–846, May 2012. [16] D. Cheng, “Input-state approach to Boolean networks,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 512–521, Mar. 2009. [17] D. Cheng and H. Qi, “State-space analysis of Boolean networks,” IEEE Trans. Neural Netw., vol. 21, no. 4, pp. 584–594, Apr. 2010.
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Bogdanov–Takens Singularity in Tri-Neuron Network With Time Delay Xing He, Chuandong Li, Senior Member, IEEE, Tingwen Huang, and Chaojie Li
Abstract— This brief reports a retarded functional differential equation modeling tri-neuron network with time delay. The Bogdanov–Takens (B-T) bifurcation is investigated by using the center manifold reduction and the normal form method. We get the versal unfolding of the norm forms at the B-T singularity and show that the model can exhibit pitchfork, Hopf, homoclinic, and double-limit cycles bifurcations. Some numerical simulations are given to support the analytic results and explore chaotic dynamics. Finally, an algorithm is given to show that chaotic tri-neuron networks can be used for encrypting a color image. Index Terms— Bogdanov–Takens bifurcation, tri-neuron network.
bifurcation,
homoclinic
I. I NTRODUCTION Neural networks (NNs) have great potential in practical applications, such as associative memory, optimization, and pattern recognition. In application of NNs, multistable NNs can be viewed as models for associative memory [1], and attractors can be interpreted as stored memories. For example, an equilibrium point can be considered as a single storage or memory pattern, or an optimum object, while a limit cycle can also restore various complex patterns. In order to realize a memory system, [4] examined the dynamical phenomenon Manuscript received January 17, 2012; revised October 22, 2012; accepted December 19, 2012. Date of publication March 12, 2013; date of current version April 5, 2013. This work was supported in part by the National Natural Science Foundation of China, under Grant 60974020, the Fundamental Research Funds for the Central Universities of China, under Project CDJZR10 18 55 01, and the National Priority Research Project, under Grant NPRP 4-1162-1-181 funded by Qatar National Research Fund, Qatar. X. He and C. Li are with the School of Electronics and Information Engineering, Southwest University, Chongqing 400715, China (e-mail: [email protected]; [email protected]). T. Huang is with Texas A&M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). C. J. Li is with the School of Science, Information Technology and Engineering, University of Ballarat, Mount St. Helen, VIC 3350, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2238681
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