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Synchronization of Boolean networks with time delays
Applied Mathematics and Computation 219 (2012) 917–927
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Synchronization of Boolean networks with time delays Rui Li, Meng Yang, Tianguang Chu ⇑ State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Keywords: Boolean network Synchronization Time delay Semi-tensor product
a b s t r a c t This paper studies analytically synchronization of two deterministic Boolean networks with time delays, which are unidirectionally coupled in the drive-response configuration. Necessary and sufficient conditions of synchronization are established based on the algebraic representations of logical dynamics in terms of the semi-tensor product of matrices. The synchronization conditions show that, for drive-response Boolean networks, different relations between the inherent state delay and the unidirectional coupling delay may result in distinct synchronization phenomena, and both kinds of time delays that ensure synchronization, if existent, are not unique. Examples are worked out to illustrate the present results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Boolean networks (BNs) have been successfully used in modeling complex systems as varied as gene regulatory networks, neural networks, biological evolution models, and physical spin models [1–5]. In a BN, at each discrete time point the node can take on one of two binary values, 1 or 0. A value of 1 represents that the node is ON, and a value of 0 means that it is OFF. The time evolution of each node is determined by a logical relationship with other nodes in the network, given in the form of a Boolean function. Although BNs illustrate relatively simple sketches of real systems, they may provide good approximations to nonlinear functions appearing in many complex systems [6,7]. Therefore, it is reasonable to investigate BNs for understanding and predicting real networks. A curious ability of some real systems is that they may evolve in perfect synchrony. In the past decades, synchronization phenomena have inspired a considerable amount of research [8–12]. Recently, interest has extend to synchronization of BNs, mostly because of their potential applications in biology, physics, as well as engineering. For example, the study of synchronized BNs could provide useful information on the coevolution of several biological species whose genetic dynamics influence each other [13]. The results obtained are also beneficial to the research on synchronization between two coupled lasers [14,15]. Besides, a detailed understanding of the synchronization dynamics of BNs may lead to new schemes for secure communications [16]. The existing studies on synchronization of BNs mainly focus on the delay-free case. In particular, synchronization between a pair of random BNs without time delays was considered in [17]. Later, based on a mean-field model, the study was further extended to mutual synchronization in a random network of delay-free random BNs [18]. However, owing to the different nature of coupling schemes, these results cannot be applied to deterministic BNs, which have often arisen in modeling complex systems such as the yeast cell-cycle network [19] and the lac operon in Escherichia coli [20]. In a very recent paper [21], the issue concerning complete synchronization of delay-free deterministic BNs was addressed, and some synchronization conditions were derived.
⇑ Corresponding author. E-mail addresses: [email protected] (R. Li), [email protected] (M. Yang), [email protected] (T. Chu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.071
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It is well known that the information flow in complex systems is not instantaneous in general, and the finite speed of information transmission thus gives rise to finite time delays. For example, in gene regulatory networks time delays are ubiquitous due primarily to the slow processes of transcription, translation, translocation, and the finite switching speed of amplifiers. To date, there have been several studies concerning delayed BNs. In [22], temporal BNs were proposed to model genetic networks, and the problem of model inference from noisy data was discussed. Recently, the problems of controllability and observability of deterministic BNs with state delays were also addressed [23,24]. However, to our knowledge, there have been few results on synchronization of delayed BNs in literature. In this paper, we intend to investigate synchronization between two deterministic BNs with time delays coupled in the drive-response configuration. Our discussion is based on the recently developed technique of the semi-tensor product (STP) of matrices that allows for an algebraic representation of a BN and thus facilitates rigorous analysis of the system dynamics [25–28]. Specifically, we will derive several necessary and sufficient conditions for synchronization of delayed BNs, and we will see from the conditions that different relations between the inherent state delay and the coupling delay may result in distinct phenomena of drive-response synchronization. The remainder of this paper is organized as follows. Section 2 gives some preliminary results about the STP technique and the algebraic representation of Boolean functions, as well as the problem formulation. Section 3 presents the main results on synchronization of drive-response BNs with time delays. Examples are treated in Section 4, and conclusions are drawn in the final section. 2. Preliminaries 2.1. STP of matrices Throughout this paper we will use the following notation. We define 1n to be the n-dimensional row vector whose entries are equal to 1, and we denote the n n identity matrix by In . Let din be the ith column of In and let Dn be the set of all n columns of In . The set of columns of a matrix A is denoted by ColðAÞ. An n m matrix A is called a logical matrix if ColðAÞ # Dn . The set of all n m logical matrices is denoted by Lnm . For an n n logical matrix A, define eðAÞ ¼ minfi : i P 1; Ai ¼ Aj for some j > ig, called the exponent of A [29]. Notice that the existence of eðAÞ follows from the observation that Ak 2 Lnn for all k P 1 and Lnn is a finite set. We simply write an n m logical matrix din1 din2 . . . dinm as dn ði1 ; i2 ; . . . ; im Þ, and sometimes use dn ðiÞ i in lieu of dn . Given two matrices A and B, we denote their Kronecker product by A B. Then with the Kronecker product a new matrix product, called the semi-tensor product (STP), can be defined as follows. Definition 1 [28]. For an n m matrix A and a p q matrix B, let ‘ be the least common multiple of m and p. Then the STP of A and B is
A n B ¼ ðA I‘=m ÞðB I‘=p Þ: Observe that the STP is associative [28], and it reduces to the standard matrix product when m ¼ p. We now present some basic facts about STPs and logical matrices that are useful in our later discussion. Lemma 1 [28]. (a) If r1 2 Dm and r2 2 Dn , then r1 n r2 ¼ r1 r2 . (b) If r 2 Dn , then r n A ¼ ðIn AÞ n r for every matrix A. (c) If r 2 D2n , then r n r ¼ Un r, where
Un ¼ d22n ð1; 2n þ 2; 2 2n þ 3; . . . ; ð2n 2Þ 2n þ 2n 1; 22n Þ: Lemma 2. If A 2 Lnn , then ColðAk Þ ¼ ColðAeðAÞ Þ for all k P eðAÞ. Proof. Let s ¼ minfi P 0 : AeðAÞþiþ1 ¼ AeðAÞ g. Then for every k P eðAÞ there is an integer ‘ such that eðAÞ 6 ‘ 6 eðAÞ þ s and A‘ ¼ Ak . Since
ColðAeðAÞ Þ ¼ ColðAeðAÞþsþ1 Þ # ColðA‘ Þ # ColðAeðAÞ Þ; we have ColðAk Þ ¼ ColðA‘ Þ ¼ ColðAeðAÞ Þ. h 2.2. Algebraic representation of Boolean functions By means of the STP, one can obtain an algebraic representation of logical dynamics. In so doing, we always identify the Boolean variables 1 and 0 with the vectors d12 and d22 , respectively. Namely, we regard a Boolean variable r 2 f1; 0g as a vector
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r 2 D2 , and a Boolean function of n variables f : f1; 0gn ! f1; 0g as a map f : ðD2 Þn ! D2 . The next result asserts that by utilizing STPs we can express a Boolean function in an algebraic form. Lemma 3 [27]. Let f : ðD2 Þn ! D2 be a Boolean function. Then there exists a unique matrix F 2 L22n such that
fðr1 ; r2 ; . . . ; rn Þ ¼ F n r1 n r2 n n rn for every ðr1 ; r2 ; . . . ; rn Þ 2 ðD2 Þn . F is called the structure matrix of f. Specific algorithms for calculating the structure matrix F from a Boolean function f, and vice versa, can be found in [28]. 2.3. Problem formulation In this paper, we consider two delayed BNs coupled in the drive-response configuration, with each network consisting of N nodes. The dynamics can be described as
xi ðt þ 1Þ ¼ fi ðx1 ðt sÞ; . . . ; xN ðt sÞÞ; yi ðt þ 1Þ ¼ g i ðx1 ðt s0 Þ; . . . ; xN ðt s0 Þ; y1 ðt sÞ; . . . ; yN ðt sÞÞ;
ð1Þ i ¼ 1; . . . ; N;
ð2Þ N
where xi and yi are the nodes of the drive BN (1) and the response BN (2), respectively, fi : f1; 0g ! f1; 0g and g i : f1; 0g2N ! f1; 0g are Boolean functions, s and s0 are nonnegative integers that denote, respectively, the inherent state delay of each BN and the unidirectional coupling delay between these two BNs and satisfy s 6 s0 , and t ¼ 0; 1; 2; . . .. For notational convenience, we denote by XðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xN ðtÞÞ and YðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; . . . ; yN ðtÞÞ the states of the drive BN (1) and the response BN (2) at time instant t, respectively. Notice that the state evolution of the delayed drive-response BNs depends on the initial state sequences Xðs0 Þ; Xð1 s0 Þ; . . . ; Xðs s0 Þ and YðsÞ; Yð1 sÞ; . . . ; Yð0Þ. Thus, we can define drive-response synchronization between BNs with time delays as follows. Definition 2. The drive BN (1) and the response BN (2) are said to be synchronized if for any Xðs0 Þ; Xð1 s0 Þ; . . . ; Xðs s0 Þ 2 f1; 0gN , there is a positive integer k such that t P k implies YðtÞ ¼ XðtÞ for all YðsÞ; Yð1 sÞ; . . . ; Yð0Þ 2 f1; 0gN . Notice that the integer k in the above definition can be chosen independent of initial state sequences of the system because of the finiteness of the state space. 3. Main results In this section we will establish some necessary and sufficient conditions of synchronization between delayed BNs. To present rigorous analysis, we shall use algebraic representations of BNs in terms of the STP theory introduced before. We first convert the drive-response BNs (1) and (2) to discrete time delay systems. Let xðtÞ ¼ x1 ðtÞ n n xN ðtÞ and yðtÞ ¼ y1 ðtÞ n n yN ðtÞ. Since the map from f1; 0gN to D2N sending ðr1 ; . . . ; rN Þ to r1 n n rN is a bijection [27], it follows that at every time instant t the state XðtÞ [resp., YðtÞ] of the drive BN (1) [resp., response BN (2)] can be well described by xðtÞ [resp., yðtÞ]. Let F i be the structure matrix of f i . Using first (1) and then Lemma 3, we obtain
xðt þ 1Þ ¼ ½F 1 xðt sÞ n n ½F N xðt sÞ: Since the STP is associative, it follows from Lemma 1(b) and (c) that
½F 1 xðt sÞ n n ½F N xðt sÞ ¼ ½F 1 n ðI2N F 2 Þ n UN n ðI2N F 3 Þ n UN n n ðI2N F N Þ n UN xðt sÞ: Writing F ¼ F 1 n f n Ni¼2 ½ðI2N F i ÞUN g, this gives us
xðt þ 1Þ ¼ Fxðt sÞ:
ð3Þ
Similarly, if Gi is the structure matrix of gi , let G ¼ G1 n f n Ni¼2 ½ðI22N Gi ÞU2N g. Then we have
yðt þ 1Þ ¼ G n xðt s0 Þ n yðt sÞ:
ð4Þ
Eqs. (3) and (4) are the algebraic representations of the BNs (1) and (2), respectively. Based on them we can derive some necessary and sufficient conditions of drive-response synchronization. Before stating the results, we need a few more facts about the solutions of the discrete time delay systems (3) and (4). Lemma 4. Suppose xðtÞ and yðtÞ are the solutions of (3) and (4), respectively. Let v ðtÞ ¼ xðt þ s s0 Þ n yðtÞ, let wðtÞ ¼ v ðtÞ n xðtÞ, and let a P 0 and 0 6 b 6 s be the unique integers such that s0 ¼ aðs þ 1Þ þ b. Then xðtÞ ¼ F p1 þa xðq1 s0 Þ for every t P b, where p1 P 0 and 0 6 q1 6 s are the unique integers such that t þ b ¼ p1 ðs þ 1Þ þ q1 ; (b) v ðtÞ ¼ Hp2 þ1 ½xðq2 s0 1Þ n yðq2 s 1Þ for every t P s, where p2 P 1 and 1 6 q2 6 s þ 1 are the unique integers such that t ¼ p2 ðs þ 1Þ þ q2 , and (a)
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H ¼ ðF GÞðUN I2N Þ;
ð5Þ
(c) yðtÞ ¼ GHp3 ½xðq3 s0 1Þ n yðq3 s 1Þ for every t P 1, where p3 P 0 and 1 6 q3 6 s þ 1 are the unique integers such that t ¼ p3 ðs þ 1Þ þ q3 , and H is as in (5); (d) wðtÞ ¼ ðHp3 þ1 F p3 þa Þ½xðq3 s0 1Þ n yðq3 s 1Þ n xðq3 þ b s0 Þ for every t P 1, where p3 and q3 are as in (c), and H is as in (5). Proof. Part (a): We see from (3) that xðs þ ðp1 þ aÞðs þ 1ÞÞ ¼ F p1 þa xðsÞ for every s P s0 . The special case s ¼ q1 s0 gives
xðtÞ ¼ xðp1 ðs þ 1Þ þ q1 þ aðs þ 1Þ s0 Þ ¼ F p1 þa xðq1 s0 Þ: Part (b): It follows from (3) and (4) that
vðs þ s þ 1Þ ¼ F n xðs þ s s0 Þ n G n xðs þ s s0 Þ n yðsÞ for every s P s. Then, by Lemma 1(b), (c), and Definition 1, we have
vðs þ s þ 1Þ ¼ F n ðI2
N
GÞ n UN n xðs þ s s0 Þ n yðsÞ ¼ Hv ðsÞ;
so that v ðs þ ðp2 þ 1Þðs þ 1ÞÞ ¼ Hp2 þ1 v ðsÞ. This implies the result since t ¼ q2 s 1 þ ðp2 þ 1Þðs þ 1Þ. Part (c): Since yðtÞ ¼ Gv ðt s 1Þ by (4), the result follows from (b) above. Part (d): By (3), xðs þ ðp3 þ aÞðs þ 1ÞÞ ¼ F p3 þa xðsÞ for every s P s0 . Taking s ¼ q3 þ b s0 , we obtain
xðtÞ ¼ F p3 þa xðq3 þ b s0 Þ:
ð6Þ
The result now follows from (6), part (b), and Lemma 1(a). h To establish the synchronization conditions, we will also need the following somewhat technical lemma. Lemma 5. Let F 2 L2N 2N , let G 2 L2N 22N , and let H be as in (5). Then
½F n ðGHn1 ÞðUN I2N Þ ¼ Hn for any n 2 N. Proof. We first prove that
ðF HÞðUN I2N Þ ¼ ðUN I2N ÞH: N
ð7Þ 2N
Let a1 ; . . . ; a2N 2 f1; . . . ; 2 g and h1 ; . . . ; h22N 2 f1; . . . ; 2 g be such that
F ¼ d2N ða1 ; . . . ; a2N Þ;
H ¼ d22N ðh1 ; . . . ; h22N Þ;
respectively. Then
ðF HÞðUN I2N Þ ¼ d23N ðða1 1Þ22N þ h1 ; . . . ; ða1 1Þ22N þ h2N ; ða2 1Þ22N þ h2N þ1 ; . . . ; ða2 1Þ22N þ h2Nþ1 ; . . . ; ða2N 1Þ22N þ hð2N 1Þ2N þ1 ; . . . ; ða2N 1Þ22N þ h22N Þ:
ð8Þ
Let b1 ; . . . ; b22N 2 f1; . . . ; 2N g be such that G ¼ d2N ðb1 ; . . . ; b22N Þ. It then follows from (5) that
H ¼ d22N ðða1 1Þ2N þ b1 ; . . . ; ða1 1Þ2N þ b2N ; ða2 1Þ2N þ b2N þ1 ; . . . ; ða2 1Þ2N þ b2Nþ1 ; . . . ; ða2N 1Þ2N þ bð2N 1Þ2N þ1 ; . . . ; ða2N 1Þ2N þ b22N Þ: 0
0
0
Hence, hi ¼ ðai0 1Þ2N þ bi for every 1 6 i 6 22N , where i is the unique integer such that 1 6 i 6 2N ; 1 6 i ði 1Þ2N 6 2N . This implies
ðUN I2N ÞH ¼ d23N ðða1 1Þð2N þ 1Þ2N þ b1 ; . . . ; ða1 1Þð2N þ 1Þ2N þ b2N ; ða2 1Þð2N þ 1Þ2N þ b2N þ1 ; . . . ; ða2 1Þ ð2N þ 1Þ2N þ b2Nþ1 ; . . . ; ða2N 1Þð2N þ 1Þ2N þ bð2N 1Þ2N þ1 ; . . . ; ða2N 1Þð2N þ 1Þ2N þ b22N Þ:
ð9Þ
Comparison of (8) and (9) gives (7). We shall use induction on n to complete the proof. The case n ¼ 1 is trivial. Suppose that the statement is true for n 2 N. Then we have
½F nþ1 ðGHn ÞðUN I2N Þ ¼ ½F n ðGHn1 ÞðF HÞðUN I2N Þ ¼ ½F n ðGHn1 ÞðUN I2N ÞH ¼ Hnþ1 ; via (7) and the induction hypothesis. h Now we are ready to present the synchronization conditions. We discuss them in two cases: s0 Xsðmod s þ 1Þ. Let us first consider the case s0 sðmod s þ 1Þ.
s0 sðmod s þ 1Þ and
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Theorem 1. Consider the drive and response BNs (1) and (2), and suppose s0 ¼ aðs þ 1Þ þ s for some nonnegative integer a. Let (3) and (4) be the algebraic representations of the BNs (1) and (2), respectively, and let H be as in (5). Then the following statements are equivalent: (a) The drive BN (1) and the response BN (2) are synchronized. (b) There is a positive integer n such that
ColððF a I2N ÞHn Þ # fd22N ðði 1Þ2N þ iÞ : i ¼ 1; . . . ; 2N g:
ð10Þ
(c) There is a positive integer m such that
F mþa 12N ¼ GHm1 :
ð11Þ
Proof. We will prove the equivalence of the above statements by showing the circular implications ðaÞ ) ðbÞ ) ðcÞ ) ðaÞ. ðaÞ ) ðbÞ: Suppose first that (a) holds. Since f1; 0gN is a finite set, there is k P 1 such that xðkÞ ¼ yðkÞ for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N . Let k1 P 0 and 1 6 k2 6 s þ 1 be the unique integers such that k ¼ k1 ðs þ 1Þ þ k2 , and put n ¼ k1 þ 1. Since xðkÞ ¼ F a xðk þ s s0 Þ, we obtain, by Lemma 4(b),
xðkÞ n yðkÞ ¼ F a n ½xðk þ s s0 Þ n yðkÞ ¼ ðF a I2N ÞHn ½xðk2 s0 1Þ n yðk2 s 1Þ: Fix xðk2 s0 1Þ; yðk2 s 1Þ 2 D2N . Let 1 6 r 6 2N be such that xðkÞ ¼ yðkÞ ¼ dr2N . Then
ðF a I2N ÞHn ½xðk2 s0 1Þ n yðk2 s 1Þ ¼ dr2N n dr2N ¼ d22N ððr 1Þ2N þ rÞ: This implies (10) since xðk2 s0 1Þ and yðk2 s 1Þ are chosen arbitrarily in D2N . Therefore ðaÞ ) ðbÞ. ðbÞ ) ðcÞ: Now suppose that (b) holds. By Lemma 5,
ðF a I2N ÞHn ¼ ½F nþa ðGHn1 ÞðUN I2N Þ:
ð12Þ
N
Let n1 ; . . . ; n2N ; g1 ; . . . ; g22N 2 f1; . . . ; 2 g be such that
F nþa ¼ d2N ðn1 ; . . . ; n2N Þ;
GHn1 ¼ d2N ðg1 ; . . . ; g22N Þ;
respectively. The right side of (12) is then
d22N ððn1 1Þ2N þ g1 ; . . . ; ðn1 1Þ2N þ g2N ; ðn2 1Þ2N þ g2N þ1 ; . . . ; ðn2 1Þ2N þ g2Nþ1 ; . . . ; ðn2N 1Þ2N þ gð2N 1Þ2N þ1 ; . . . ; ðn2N 1Þ2N þ g22N Þ: Hence ni ¼ gði1Þ2N þj for all 1 6 i; j 6 2N by hypothesis. It now follows that
GHn1 ¼ d2N ðg1 ; . . . ; g2N ; g2N þ1 ; . . . ; g2Nþ1 ; . . . ; gð2N 1Þ2N þ1 ; . . . ; g22N Þ ¼ d2N ðn1 ; . . . ; n1 ; n2 ; . . . ; n2 ; . . . ; n2N ; . . . ; n2N Þ ¼ F nþa 12N : Therefore ðbÞ ) ðcÞ. ðcÞ ) ðaÞ: Finally, suppose that (c) holds. Let k ¼ ðm 1Þðs þ 1Þ þ 1. It suffices to show that t P k implies yðtÞ ¼ xðtÞ for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N . Fix t P k. Let p P 0 and 0 6 q 6 s be the unique integers such that t þ s ¼ pðs þ 1Þ þ q. Since pðs þ 1Þ þ q P k þ s ¼ mðs þ 1Þ, it follows that p P m. If p ¼ m, then
yðtÞ ¼ GHm1 ½xðq s0 Þ n yðq sÞ ¼ ðF mþa 12N Þ½xðq s0 Þ yðq sÞ ¼ xðtÞ: The first of these equalities follows from Lemma 4(c). The second follows from (11) and Lemma 1(a). The last is Lemma 4(a). If p P m þ 1, then
Hpm ½xðq s0 Þ n yðq sÞ ¼ ½F pm ðGHpm1 Þ n UN n xðq s0 Þ n yðq sÞ ¼ ½F pm ðGHpm1 Þ½xðq s0 Þ n xðq s0 Þ n yðq sÞ ¼ ½F pm xðq s0 Þ ½ðGHpm1 Þ n xðq s0 Þ n yðq sÞ:
ð13Þ
The first equality holds by Lemma 5 and Definition 1, the second by Lemma 1(c), and the third by Lemma 1(a). Hence,
yðtÞ ¼ GHp1 ½xðq s0 Þ n yðq sÞ ¼ ðF mþa 12N Þf½F pm xðq s0 Þ ½ðGHpm1 Þ n xðq s0 Þ n yðq sÞg; pm1
by (11), (13), and Lemma 4(c). Since ðGH Þ n xðq s Þ n yðq sÞ 2 D2N , (14) implies yðtÞ ¼ F yðtÞ ¼ xðtÞ by Lemma 4(a). Therefore ðcÞ ) ðaÞ. h 0
pþa
ð14Þ
xðq s Þ, so that 0
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Remark 1. It is worth noting that Theorem 1 is also applicable to delay-free BNs, i.e., s ¼ s0 ¼ 0. Substituting a ¼ 0 into (11), one can obtain the result presented in [21]. Therefore, the synchronization criteria given in Theorem 1 improve and extend the previous result when reducing to the delay-free case. We now move onto the case s0 Xsðmod s þ 1Þ. Theorem 2. Consider the drive and response BNs (1) and (2), and suppose s0 Xsðmod s þ 1Þ. Let (3) and (4) be the algebraic representations of the BNs (1) and (2), respectively, and let H be as in (5). Then the following statements are equivalent: (a) The drive BN (1) and the response BN (2) are synchronized. (b) There is a positive integer m such that
ColðHmþ1 F mþa Þ # fd23N ðði 1Þ22N þ ðj 1Þ2N þ jÞ : i; j ¼ 1; . . . ; 2N g;
ð15Þ
where a is the unique positive integer such that 0 6 s aðs þ 1Þ 6 s 1. (c) There are positive integers m1 and m2 such that 0
ColðF m1 Þ ¼ ColðGHm2 Þ ¼ fdn2N g
ð16Þ
for some 1 6 n 6 2N . (d) There exist a state X 2 f1; 0gN and a positive integer k such that t P k implies YðtÞ ¼ XðtÞ ¼ X for any Xðs0 Þ; . . . ; Xðs s0 Þ; YðsÞ; . . . ; Yð0Þ 2 f1; 0gN .
Proof. We will prove the equivalence of the above statements by showing the circular implications ðaÞ ) ðbÞ ) ðcÞ ) ðdÞ ) ðaÞ. ðaÞ ) ðbÞ: Suppose that (a) holds. Let a P 1 and 0 6 b 6 s 1 be the unique integers such that s0 ¼ aðs þ 1Þ þ b. By Definition 2, there is k P 1 such that t P k implies yðtÞ ¼ xðtÞ for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N , since the set f1; 0gN is finite. Let k1 P 0 and 1 6 k2 6 s þ 1 be the unique integers such that k ¼ k1 ðs þ 1Þ þ k2 . Put m ¼ k1 þ 1 and t0 ¼ mðs þ 1Þ þ 1. Then by Lemma 4(d), we have
xðt 0 þ s s0 Þ n yðt0 Þ n xðt 0 Þ ¼ ðHmþ1 F mþa Þ½xðs0 Þ n yðsÞ n xðb þ 1 s0 Þ: Since t 0 P k and 1 s0 6 b þ 1 s0 6 s s0 , it follows that yðt 0 Þ ¼ xðt 0 Þ for any xðs0 Þ; xðb þ 1 s0 Þ; yðsÞ 2 D2N . Fix 0 xðs0 Þ; xðb þ 1 s0 Þ; yðsÞ 2 D2N . Let 1 6 r; r 0 6 2N be such that xðt0 þ s s0 Þ ¼ dr2N and xðt0 Þ ¼ yðt0 Þ ¼ dr2N , respectively. Then
ðHmþ1 F mþa Þ½xðs0 Þ n yðsÞ n xðb þ 1 s0 Þ ¼ d23N ððr 1Þ22N þ ðr 0 1Þ2N þ r 0 Þ: This implies (15) since xðs0 Þ, xðb þ 1 s0 Þ, and yðsÞ are chosen arbitrarily in D2N . Therefore ðaÞ ) ðbÞ. ðpÞ ðpÞ ðpÞ ðpÞ ðbÞ ) ðcÞ: Now suppose that (b) holds. For every p P 1, let a1 ; . . . ; a2N 2 f1; . . . ; 2N g and h1 ; . . . ; h 2N 2 f1; . . . ; 22N g be 2 such that ðpÞ
ðpÞ
Hp ¼ d22N ðh1ðpÞ ; . . . ; hðpÞ Þ; 22N
F p ¼ d2N ða1 ; . . . ; a2N Þ;
respectively, and let b1 ; . . . ; b22N 2 f1; . . . ; 2N g be such that GHm ¼ d2N ðb1 ; . . . ; b22N Þ. Then by Lemma 5, ðmþ1Þ ðmþ1Þ Hmþ1 ¼ ½F mþ1 ðGHm ÞðUN I2N Þ ¼ d22N ððaðmþ1Þ 1Þ2N þ b1 ; . . . ; ða1 1Þ2N þ b2N ; ða2 1Þ2N 1 ðmþ1Þ
þ b2N þ1 ; . . . ; ða2
1Þ2N þ b2Nþ1 ; . . . ; ða2N
ðmþ1Þ
1Þ2N þ bð2N 1Þ2N þ1 ; . . . ; ða2N
ðmþ1Þ
1Þ2N þ b22N Þ:
This implies ðmþ1Þ
hi
ðmþ1Þ
¼ ðai0
1Þ2N þ bi 0
0
0
for all 1 6 i 6 22N , where i is the unique integer such that 1 6 i 6 2N ; 1 6 i ði 1Þ2N 6 2N . Hence ðmþ1Þ
ðhi
1Þ2N þ aj
ðmþaÞ
2N
for every 1 6 i 6 2 mþ1
H
F
mþa
ðmþ1Þ
¼ ðai0
1Þ22N þ ðbi 1Þ2N þ aj
ðmþaÞ
and 1 6 j 6 2 . Since ðmþ1Þ
¼ d23N ððh1
ðmþaÞ
þ a2N
1Þ2N þ a1
ðmþaÞ
ðmþ1Þ
; . . . ; ðh22N
it follows from (15) and (17) that bi ¼ a
ColðF
m
Þ ¼ ColðGH Þ ¼
Therefore ðbÞ ) ðcÞ.
fdb21N g:
ðmþ1Þ
; . . . ; ðh1
1Þ2N þ a1
ðmþaÞ j
mþa
ð17Þ
N
ðmþaÞ
1Þ2N þ a2N
ðmþaÞ
ðmþ1Þ
; . . . ; ðh22N 2N
for all 1 6 i 6 2
ðmþ1Þ
; ðh2
1Þ2N þ a2N
1Þ2N þ a1
ðmþaÞ N
ðmþaÞ
Þ;
and all 1 6 j 6 2 , so that
ðmþ1Þ
; . . . ; ðh2
1Þ2N
R. Li et al. / Applied Mathematics and Computation 219 (2012) 917–927
923
ðcÞ ) ðdÞ: Suppose that (c) holds. Let a P 1 and 0 6 b 6 s 1 be the unique integers such that s0 ¼ aðs þ 1Þ þ b. Put M ¼ maxfm1 a; m2 g and k ¼ Mðs þ 1Þ þ 1. It suffices to show that t P k implies xðtÞ ¼ yðtÞ ¼ dn2N for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N . Fix t P k. Let p1 P 0 and 0 6 q1 6 s be the unique integers such that t þ b ¼ p1 ðs þ 1Þ þ q1 ; let p2 P 0 and 1 6 q2 6 s þ 1 be the unique integers such that t ¼ p2 ðs þ 1Þ þ q2 . Then p1 P m1 a; p2 P m2 , since p1 ðs þ 1Þ þ q1 P Mðs þ 1Þ þ b þ 1; p2 ðs þ 1Þ þ q2 P Mðs þ 1Þ þ 1. By Lemma 4(a), xðtÞ ¼ F m1 ½F p1 þam1 xðq1 s0 Þ. Thus xðtÞ ¼ dn2N by (16), since F p1 þam1 xðq1 s0 Þ 2 D2N . Similarly,
yðtÞ ¼ GHm2 ½Hp2 m2 n xðq2 s0 1Þ n yðq2 s 1Þ ¼ dn2N ; by Lemma 4(c) and (16), since Hp2 m2 n xðq2 s0 1Þ n yðq2 s 1Þ 2 D2N . Therefore ðcÞ ) ðdÞ. ðdÞ ) ðaÞ: This follows directly from Definition 2. h Theorems 1 and 2 are our major results. They furnish several necessary and sufficient criteria for synchronization between two delayed BNs, which are unidirectionally coupled in the drive-response configuration. Our next result implies that one only needs to calculate a finite number of matrices when using these criteria. Corollary 1 (a) If n and m are the smallest positive integers satisfying (10) and (11), respectively, then n ¼ m 6 eðHÞ. (b) If m; m1 , and m2 are the smallest positive integers satisfying (15) and (16), respectively, then m 6 maxfeðFÞ a; eðHÞg þ 1; m1 6 eðFÞ, and m2 6 eðHÞ.
Proof. Part (a): From the proof that ðbÞ ) ðcÞ in Theorem 1, we see m 6 n. The proofs that ðcÞ ) ðaÞ and ðaÞ ) ðbÞ in Theorem 1 show that (10) holds with m in place of n, so that n 6 m. Suppose, to get a contradiction, that n > eðHÞ. Then, by Lemma 2, ColðHeðHÞ Þ ¼ ColðHn Þ, so that
ColððF a I2N ÞHeðHÞ Þ ¼ ColððF a I2N ÞHn Þ: Hence (10) holds with eðHÞ in place of n. This contradicts the minimality of n. Part (b): If m1 > eðFÞ, then by Lemma 2 we would have ColðF eðFÞ Þ ¼ ColðF m1 Þ ¼ fdn2N g, contradicting the minimality of m1 . The proof that m2 6 eðHÞ is analogous. Let M ¼ maxfm1 a; m2 g. Then from the proofs that ðcÞ ) ðdÞ and ðaÞ ) ðbÞ in Theorem 2 we see that (15) holds with M þ 1 in place of m. Hence m 6 M þ 1 6 maxfeðFÞ a; eðHÞg þ 1. h Remark 2. From Theorems 1 and 2, we see clearly that different relations between the inherent state delay s and the unidirectional coupling delay s0 may lead to quite distinct synchronization conditions and phenomena. In particular, Theorem 2 implies that in the case s0 Xsðmod s þ 1Þ drive-response synchronization can only occur between two BNs that globally converge to a common fixed point, and the fixed point is precisely the synchronized state of these two BNs. On the other hand, in the case s0 sðmod s þ 1Þ synchronous behavior between response and drive BNs can be relatively complex; see Example 1 in the next section. As an immediate consequence of Theorems 1 and 2, the following result asserts that in both cases time delays that result in drive-response synchronization, if existent, are not unique. Corollary 2. Suppose s0 ; s00 are nonnegative integers, s00 P s0 , and the drive BN (1) and the response BN (2) are synchronized with
s ¼ s0 ; s0 ¼ s00 .
(a) If s00 s0 ðmod s0 þ 1Þ, then the drive BN (1) and the response BN (2) are synchronized for all s P 0 and s0 ¼ aðs þ 1Þ þ s, where a is the unique integer such that s00 ¼ aðs0 þ 1Þ þ s0 . (b) If s00 Xs0 ðmod s0 þ 1Þ, then the drive BN (1) and the response BN (2) are synchronized for all s0 P s P 0.
Proof. Part (a): The result follows from Theorem 1. Part (b): By Theorem 2, there exist m1 and m2 such that (16) holds for some 1 6 n 6 2N . If s0 Xsðmod s þ 1Þ, then the BNs (1) and (2) are synchronized by Theorem 2. If s0 sðmod s þ 1Þ, let a P 0 be the unique integer such that s0 ¼ aðs þ 1Þ þ s, and put m ¼ maxfm1 a; m2 þ 1g. Then ColðF mþa Þ # ColðF m1 Þ ¼ fdn2N g and ColðGHm1 Þ # ColðGHm2 Þ ¼ fdn2N g, so that F mþa 12N ¼ GHm1 . Hence the BNs (1) and (2) are synchronized by Theorem 1. h 4. Illustrative examples Now we present some illustrative examples to demonstrate the applications of the main results. Example 1. We take the following delayed BN as the drive BN
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x1
1
y
1
0 0
5
10
15
20
25
30
35
40
45
x
1
2
y2
0 0
5
10
15
20
25
30
35
40
45
x3
1
y3
0 0
5
10
15
20
t
25
30
35
40
45
Fig. 1. Time evolution of the BNs (18) and (19) with initial state sequences Xð17Þ ¼ ð1; 1; 1Þ; Xð16Þ ¼ ð1; 0; 0Þ; Xð15Þ ¼ ð1; 0; 1Þ, and Yð2Þ ¼ ð1; 1; 0Þ; Yð1Þ ¼ ð0; 0; 0Þ; Yð0Þ ¼ ð0; 1; 0Þ.
8 > < x1 ðt þ 1Þ ¼ :x2 ðt 2Þ; x2 ðt þ 1Þ ¼ x1 ðt 2Þ ^ x3 ðt 2Þ; > : x3 ðt þ 1Þ ¼ x1 ðt 2Þ
ð18Þ
and the response BN is given by
8 > < y1 ðt þ 1Þ ¼ :x2 ðt 17Þ _ :y2 ðt 2Þ _ :y3 ðt 2Þ; y2 ðt þ 1Þ ¼ ½x1 ðt 17Þ _ y1 ðt 2Þ ^ y3 ðt 2Þ; > : y3 ðt þ 1Þ ¼ ½:x2 ðt 17Þ _ x3 ðt 17Þ $ y1 ðt 2Þ:
ð19Þ
Denoting xðtÞ ¼ x1 ðtÞ n x2 ðtÞ n x3 ðtÞ and yðtÞ ¼ y1 ðtÞ n y2 ðtÞ n y3 ðtÞ, then we can express the drive BN (18) and the response BN (19) in their algebraic forms as
xðt þ 1Þ ¼ Fxðt 2Þ; yðt þ 1Þ ¼ G n xðt 17Þ n yðt 2Þ; respectively, where
F ¼ d8 ð5; 7; 1; 3; 8; 8; 4; 4Þ; G ¼ d8 ð5; 3; 1; 3; 6; 4; 2; 4; 6; 4; 2; 4; 5; 3; 1; 3; 1; 3; 1; 3; 2; 4; 2; 4; 1; 3; 1; 3; 2; 4; 2; 4; 5; 3; 1; 3; 8; 4; 4; 4; 6; 4; 2; 4; 7; 3; 3; 3; 1; 3; 1; 3; 4; 4; 4; 4; 1; 3; 1; 3; 4; 4; 4; 4Þ: In order to examine whether these two BNs can be synchronized or not, we compute ðF 5 I8 Þ½ðF GÞðU3 I8 Þn according to Theorem 1, since 17 ¼ 5 ð2 þ 1Þ þ 2. Simple calculations lead to
ColððF 5 I8 Þ½ðF GÞðU3 I8 Þ6 Þ ¼ fdi64 : i ¼ 1; 19; 28; 37; 64g # fdi64 : i ¼ 1; 10; 19; 28; 37; 46; 55; 64g: Therefore, the response BN (19) can synchronize with the drive BN (18) by Theorem 1. Fig. 1 shows the time evolution of the drive-response BNs (18) and (19), where Xð17Þ ¼ ð1; 1; 1Þ; Xð16Þ ¼ ð1; 0; 0Þ; Xð15Þ ¼ ð1; 0; 1Þ, and Yð2Þ ¼ ð1; 1; 0Þ; Yð1Þ ¼ ð0; 0; 0Þ; Yð0Þ ¼ ð0; 1; 0Þ. The Hamming distance between the states of the BNs (18) and (19) P HðtÞ ¼ 3i¼1 jxi ðtÞ yi ðtÞj versus the time t is plotted in Fig. 2. As seen in Fig. 1, neither of these two BNs converges globally to any fixed points. Example 2. Let us consider the following drive and response BNs with each network consisting of four nodes:
8 x1 ðt þ 1Þ ¼ x2 ðt sÞ _ x3 ðt sÞ; > > > < x ðt þ 1Þ ¼ :x ðt sÞ ^ x ðt sÞ ^ x ðt sÞ; 2 1 2 4 > x ðt þ 1Þ ¼ :x ðt s Þ; 3 4 > > : x4 ðt þ 1Þ ¼ x1 ðt sÞ ! :x2 ðt sÞ;
ð20Þ
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R. Li et al. / Applied Mathematics and Computation 219 (2012) 917–927
4
3
H
2
1
0
−1
0
5
10
15
20
t
25
30
35
40
45
Fig. 2. Hamming distance between the states of the BNs (18) and (19) versus time, with Xð17Þ ¼ ð1; 1; 1Þ; Xð16Þ ¼ ð1; 0; 0Þ; Xð15Þ ¼ ð1; 0; 1Þ, and Yð2Þ ¼ ð1; 1; 0Þ; Yð1Þ ¼ ð0; 0; 0Þ; Yð0Þ ¼ ð0; 1; 0Þ.
4
3
H
2
1
0
−1
0
10
20
30
t
40
50
60
Fig. 3. Hamming distance between the states of the BNs (20) and (21) versus time, with s ¼ 3; s0 ¼ 5; Xð5Þ ¼ ð1; 1; 1; 1Þ; Xð4Þ ¼ ð1; 1; 0; 1Þ; Xð3Þ ¼ ð0; 1; 1; 1Þ; Xð2Þ ¼ ð0; 1; 0; 1Þ, and Yð3Þ ¼ ð0; 1; 0; 1Þ; Yð2Þ ¼ ð1; 0; 1; 1Þ; Yð1Þ ¼ ð1; 1; 1; 1Þ; Yð0Þ ¼ ð1; 0; 0; 0Þ.
8 y1 ðt þ 1Þ ¼ ½x1 ðt s0 Þ ^ x2 ðt s0 Þ _ y2 ðt sÞ; > > > < y2 ðt þ 1Þ ¼ x3 ðt s0 Þ $ y4 ðt sÞ; 0 > > y3 ðt þ 1Þ ¼ x2 ðt s Þ _ y1 ðt sÞ; > : y4 ðt þ 1Þ ¼ x4 ðt s0 Þ _ y3 ðt sÞ _ :y4 ðt sÞ;
ð21Þ
where s0 P s P 0. Let xðtÞ ¼ x1 ðtÞ n x2 ðtÞ n x3 ðtÞ n x4 ðtÞ and yðtÞ ¼ y1 ðtÞ n y2 ðtÞ n y3 ðtÞ n y4 ðtÞ. Then the algebraic representations of the drive BN (20) and the response BN (21) are
xðt þ 1Þ ¼ Fxðt sÞ; yðt þ 1Þ ¼ G n xðt s0 Þ n yðt sÞ; respectively, where
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R. Li et al. / Applied Mathematics and Computation 219 (2012) 917–927
F ¼ d16 ð8; 6; 8; 6; 7; 5; 15; 13; 3; 5; 3; 5; 7; 5; 15; 13Þ; G ¼ d16 ð1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 2; 5; 1; 5; 2; 5; 1; 5; 2; 5; 1; 5; 2; 5; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 6; 1; 5; 1; 6; 1; 5; 1; 6; 1; 5; 1; 6; 1; 1; 5; 1; 5; 9; 13; 9; 13; 3; 7; 3; 7; 11; 15; 11; 15; 1; 5; 2; 5; 9; 13; 10; 13; 3; 7; 4; 7; 11; 15; 12; 15; 5; 1; 5; 1; 13; 9; 13; 9; 7; 3; 7; 3; 15; 11; 15; 11; 5; 1; 6; 1; 13; 9; 14; 9; 7; 3; 8; 3; 15; 11; 16; 11; 1; 5; 1; 5; 9; 13; 9; 13; 1; 5; 1; 5; 9; 13; 9; 13; 1; 5; 2; 5; 9; 13; 10; 13; 1; 5; 2; 5; 9; 13; 10; 13; 5; 1; 5; 1; 13; 9; 13; 9; 5; 1; 5; 1; 13; 9; 13; 9; 5; 1; 6; 1; 13; 9; 14; 9; 5; 1; 6; 1; 13; 9; 14; 9; 1; 5; 1; 5; 9; 13; 9; 13; 3; 7; 3; 7; 11; 15; 11; 15; 1; 5; 2; 5; 9; 13; 10; 13; 3; 7; 4; 7; 11; 15; 12; 15; 5; 1; 5; 1; 13; 9; 13; 9; 7; 3; 7; 3; 15; 11; 15; 11; 5; 1; 6; 1; 13; 9; 14; 9; 7; 3; 8; 3; 15; 11; 16; 11Þ: It is easy to check that
ColðF 5 Þ ¼ ColðG½ðF GÞðU4 I16 Þ6 Þ ¼ fd15 16 g: So applying Theorem 2 and Corollary 2(b), we conclude that the drive BN (20) and the response BN (21) are synchronized for all s0 P s P 0. In fact, for any s and s0 , both of these BNs will converge to the state XP ¼ ð0; 0; 0; 1Þ, since d22 n d22 n d22 n d12 ¼ d15 16 . In Fig. 3, the Hamming distance between the states of the BNs (20) and (21) HðtÞ ¼ 4i¼1 jxi ðtÞ yi ðtÞj versus the time t is plotted, where s ¼ 3; s0 ¼ 5; Xð5Þ ¼ ð1; 1; 1; 1Þ; Xð4Þ ¼ ð1; 1; 0; 1Þ; Xð3Þ ¼ ð0; 1; 1; 1Þ; Xð2Þ ¼ ð0; 1; 0; 1Þ, and Yð3Þ ¼ ð0; 1; 0; 1Þ; Yð2Þ ¼ ð1; 0; 1; 1Þ; Yð1Þ ¼ ð1; 1; 1; 1Þ; Yð0Þ ¼ ð1; 0; 0; 0Þ. 5. Conclusions In this paper, we have discussed synchronization of two deterministic BNs with time delays coupled unidirectionally in the drive-response configuration. Our emphasis has focused on rigorous analysis in terms of algebraic representations of the logical dynamics of BNs. The main results furnish some necessary and sufficient algebraic criteria for synchronization. The criteria also reveal that, for drive-response BNs, different relations between the inherent state delay and the unidirectional coupling delay may lead to quite distinct synchronization phenomena. Furthermore, as an application of the main results, we have found that both the inherent state delay and the coupling delay with which drive-response synchronization occurs, if existent, are not unique. Acknowledgments The authors thank the editor and the reviewers for their valuable comments and suggestions. This work was supported by the National Basic Research Program of China under Grant 2012CB821200 and the National Natural Science Foundation of China under Grant 60974064. References [1] R. Albert, H. Othmer, The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster, J. Theor. Biol. 223 (2003) 1–18. [2] K.E. Kürten, Correspondence between neural threshold networks and Kauffman Boolean cellular automata, J. Phys. A Math. Gen. 21 (1988) L615–L619. [3] S. Bornholdt, T. Rohlf, Topological evolution of dynamical networks: global criticality from local dynamics, Phys. Rev. Lett. 84 (2000) 6114–6117. [4] M.D. Stern, Emergence of homeostasis and noise imprinting in an evolution model, Proc. Natl. Acad. Sci. USA 96 (1999) 10746–10751. [5] B. Derrida, H. Flyvbjerg, Multivalley structure in Kauffman’s model: analogy with spin glasses, J. Phys. A Math. Gen. 19 (1986) L1003–L1008. [6] L.A.N. Amaral, A. Dı´az-Guilera, A.A. Moreira, A.L. Goldberger, L.A. Lipsitz, Emergence of complex dynamics in a simple model of signaling networks, Proc. Natl. Acad. Sci. USA 101 (2004) 15551–15555. [7] J. Heidel, J. Maloney, C. Farrow, J.A. Rogers, Finding cycles in synchronous Boolean networks with applications to biochemical systems, Int. J. Bifurcat. Chaos 13 (2003) 535–552. [8] J.H. Park, S.M. Lee, H.Y. Jung, LMI optimization approach to synchronization of stochastic delayed discrete-time complex networks, J. Optim. Theory Appl. 143 (2009) 357–367. [9] D.H. Ji, J.H. Park, W.J. Yoo, S.C. Won, S.M. Lee, Synchronization criterion for Lur’e type complex dynamical networks with time-varying delay, Phys. Lett. A 374 (2010) 1218–1227. [10] D.H. Ji, D.W. Lee, J.H. Koo, S.C. Won, S.M. Lee, J.H. Park, Synchronization of neutral complex dynamical networks with coupling time-varying delays, Nonlinear Dynam. 65 (2011) 349–358. [11] D.H. Ji, S.C. Jeong, J.H. Park, S.M. Lee, S.C. Won, Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling, Appl. Math. Comput. 218 (2012) 4872–4880. [12] T.H. Lee, J.H. Park, D.H. Ji, O.M. Kwon, S.M. Lee, Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control, Appl. Math. Comput. 218 (2012) 6469–6481. [13] L.G. Morelli, D.H. Zanette, Synchronization of Kauffman networks, Phys. Rev. E 63 (2001) 036204. [14] J.L. Guisado, F. Jiménez-Morales, J.M. Guerra, Cellular automaton model for the simulation of laser dynamics, Phys. Rev. E 67 (2003) 066708. [15] M.-C. Ho, Y.-C. Hung, I.-M. Jiang, Stochastic coupling of two random Boolean networks, Phys. Lett. A 344 (2005) 36–42. [16] J. Parriaux, P. Guillot, G. Millérioux, Towards a spectral approach for the design of self-synchronizing stream ciphers, Cryptogr. Commun. 3 (2011) 259– 274. [17] Y.-C. Hung, M.-C. Ho, J.-S. Lih, I.-M. Jiang, Chaos synchronization of two stochastically coupled random Boolean networks, Phys. Lett. A 356 (2006) 35– 43. [18] Y.-C. Hung, Microscopic interactions lead to mutual synchronization in a network of networks, Phys. Lett. A 375 (2011) 2809–2814. [19] D.J. Irons, Logical analysis of the budding yeast cell cycle, J. Theor. Biol. 257 (2009) 543–559. [20] A. Veliz-Cuba, B. Stigler, Boolean models can explain bistability in the lac operon, J. Comput. Biol. 18 (2011) 783–794.
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Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Synchronization of Boolean networks with time delays Rui Li, Meng Yang, Tianguang Chu ⇑ State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Keywords: Boolean network Synchronization Time delay Semi-tensor product
a b s t r a c t This paper studies analytically synchronization of two deterministic Boolean networks with time delays, which are unidirectionally coupled in the drive-response configuration. Necessary and sufficient conditions of synchronization are established based on the algebraic representations of logical dynamics in terms of the semi-tensor product of matrices. The synchronization conditions show that, for drive-response Boolean networks, different relations between the inherent state delay and the unidirectional coupling delay may result in distinct synchronization phenomena, and both kinds of time delays that ensure synchronization, if existent, are not unique. Examples are worked out to illustrate the present results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Boolean networks (BNs) have been successfully used in modeling complex systems as varied as gene regulatory networks, neural networks, biological evolution models, and physical spin models [1–5]. In a BN, at each discrete time point the node can take on one of two binary values, 1 or 0. A value of 1 represents that the node is ON, and a value of 0 means that it is OFF. The time evolution of each node is determined by a logical relationship with other nodes in the network, given in the form of a Boolean function. Although BNs illustrate relatively simple sketches of real systems, they may provide good approximations to nonlinear functions appearing in many complex systems [6,7]. Therefore, it is reasonable to investigate BNs for understanding and predicting real networks. A curious ability of some real systems is that they may evolve in perfect synchrony. In the past decades, synchronization phenomena have inspired a considerable amount of research [8–12]. Recently, interest has extend to synchronization of BNs, mostly because of their potential applications in biology, physics, as well as engineering. For example, the study of synchronized BNs could provide useful information on the coevolution of several biological species whose genetic dynamics influence each other [13]. The results obtained are also beneficial to the research on synchronization between two coupled lasers [14,15]. Besides, a detailed understanding of the synchronization dynamics of BNs may lead to new schemes for secure communications [16]. The existing studies on synchronization of BNs mainly focus on the delay-free case. In particular, synchronization between a pair of random BNs without time delays was considered in [17]. Later, based on a mean-field model, the study was further extended to mutual synchronization in a random network of delay-free random BNs [18]. However, owing to the different nature of coupling schemes, these results cannot be applied to deterministic BNs, which have often arisen in modeling complex systems such as the yeast cell-cycle network [19] and the lac operon in Escherichia coli [20]. In a very recent paper [21], the issue concerning complete synchronization of delay-free deterministic BNs was addressed, and some synchronization conditions were derived.
⇑ Corresponding author. E-mail addresses: [email protected] (R. Li), [email protected] (M. Yang), [email protected] (T. Chu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.071
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It is well known that the information flow in complex systems is not instantaneous in general, and the finite speed of information transmission thus gives rise to finite time delays. For example, in gene regulatory networks time delays are ubiquitous due primarily to the slow processes of transcription, translation, translocation, and the finite switching speed of amplifiers. To date, there have been several studies concerning delayed BNs. In [22], temporal BNs were proposed to model genetic networks, and the problem of model inference from noisy data was discussed. Recently, the problems of controllability and observability of deterministic BNs with state delays were also addressed [23,24]. However, to our knowledge, there have been few results on synchronization of delayed BNs in literature. In this paper, we intend to investigate synchronization between two deterministic BNs with time delays coupled in the drive-response configuration. Our discussion is based on the recently developed technique of the semi-tensor product (STP) of matrices that allows for an algebraic representation of a BN and thus facilitates rigorous analysis of the system dynamics [25–28]. Specifically, we will derive several necessary and sufficient conditions for synchronization of delayed BNs, and we will see from the conditions that different relations between the inherent state delay and the coupling delay may result in distinct phenomena of drive-response synchronization. The remainder of this paper is organized as follows. Section 2 gives some preliminary results about the STP technique and the algebraic representation of Boolean functions, as well as the problem formulation. Section 3 presents the main results on synchronization of drive-response BNs with time delays. Examples are treated in Section 4, and conclusions are drawn in the final section. 2. Preliminaries 2.1. STP of matrices Throughout this paper we will use the following notation. We define 1n to be the n-dimensional row vector whose entries are equal to 1, and we denote the n n identity matrix by In . Let din be the ith column of In and let Dn be the set of all n columns of In . The set of columns of a matrix A is denoted by ColðAÞ. An n m matrix A is called a logical matrix if ColðAÞ # Dn . The set of all n m logical matrices is denoted by Lnm . For an n n logical matrix A, define eðAÞ ¼ minfi : i P 1; Ai ¼ Aj for some j > ig, called the exponent of A [29]. Notice that the existence of eðAÞ follows from the observation that Ak 2 Lnn for all k P 1 and Lnn is a finite set. We simply write an n m logical matrix din1 din2 . . . dinm as dn ði1 ; i2 ; . . . ; im Þ, and sometimes use dn ðiÞ i in lieu of dn . Given two matrices A and B, we denote their Kronecker product by A B. Then with the Kronecker product a new matrix product, called the semi-tensor product (STP), can be defined as follows. Definition 1 [28]. For an n m matrix A and a p q matrix B, let ‘ be the least common multiple of m and p. Then the STP of A and B is
A n B ¼ ðA I‘=m ÞðB I‘=p Þ: Observe that the STP is associative [28], and it reduces to the standard matrix product when m ¼ p. We now present some basic facts about STPs and logical matrices that are useful in our later discussion. Lemma 1 [28]. (a) If r1 2 Dm and r2 2 Dn , then r1 n r2 ¼ r1 r2 . (b) If r 2 Dn , then r n A ¼ ðIn AÞ n r for every matrix A. (c) If r 2 D2n , then r n r ¼ Un r, where
Un ¼ d22n ð1; 2n þ 2; 2 2n þ 3; . . . ; ð2n 2Þ 2n þ 2n 1; 22n Þ: Lemma 2. If A 2 Lnn , then ColðAk Þ ¼ ColðAeðAÞ Þ for all k P eðAÞ. Proof. Let s ¼ minfi P 0 : AeðAÞþiþ1 ¼ AeðAÞ g. Then for every k P eðAÞ there is an integer ‘ such that eðAÞ 6 ‘ 6 eðAÞ þ s and A‘ ¼ Ak . Since
ColðAeðAÞ Þ ¼ ColðAeðAÞþsþ1 Þ # ColðA‘ Þ # ColðAeðAÞ Þ; we have ColðAk Þ ¼ ColðA‘ Þ ¼ ColðAeðAÞ Þ. h 2.2. Algebraic representation of Boolean functions By means of the STP, one can obtain an algebraic representation of logical dynamics. In so doing, we always identify the Boolean variables 1 and 0 with the vectors d12 and d22 , respectively. Namely, we regard a Boolean variable r 2 f1; 0g as a vector
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r 2 D2 , and a Boolean function of n variables f : f1; 0gn ! f1; 0g as a map f : ðD2 Þn ! D2 . The next result asserts that by utilizing STPs we can express a Boolean function in an algebraic form. Lemma 3 [27]. Let f : ðD2 Þn ! D2 be a Boolean function. Then there exists a unique matrix F 2 L22n such that
fðr1 ; r2 ; . . . ; rn Þ ¼ F n r1 n r2 n n rn for every ðr1 ; r2 ; . . . ; rn Þ 2 ðD2 Þn . F is called the structure matrix of f. Specific algorithms for calculating the structure matrix F from a Boolean function f, and vice versa, can be found in [28]. 2.3. Problem formulation In this paper, we consider two delayed BNs coupled in the drive-response configuration, with each network consisting of N nodes. The dynamics can be described as
xi ðt þ 1Þ ¼ fi ðx1 ðt sÞ; . . . ; xN ðt sÞÞ; yi ðt þ 1Þ ¼ g i ðx1 ðt s0 Þ; . . . ; xN ðt s0 Þ; y1 ðt sÞ; . . . ; yN ðt sÞÞ;
ð1Þ i ¼ 1; . . . ; N;
ð2Þ N
where xi and yi are the nodes of the drive BN (1) and the response BN (2), respectively, fi : f1; 0g ! f1; 0g and g i : f1; 0g2N ! f1; 0g are Boolean functions, s and s0 are nonnegative integers that denote, respectively, the inherent state delay of each BN and the unidirectional coupling delay between these two BNs and satisfy s 6 s0 , and t ¼ 0; 1; 2; . . .. For notational convenience, we denote by XðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xN ðtÞÞ and YðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; . . . ; yN ðtÞÞ the states of the drive BN (1) and the response BN (2) at time instant t, respectively. Notice that the state evolution of the delayed drive-response BNs depends on the initial state sequences Xðs0 Þ; Xð1 s0 Þ; . . . ; Xðs s0 Þ and YðsÞ; Yð1 sÞ; . . . ; Yð0Þ. Thus, we can define drive-response synchronization between BNs with time delays as follows. Definition 2. The drive BN (1) and the response BN (2) are said to be synchronized if for any Xðs0 Þ; Xð1 s0 Þ; . . . ; Xðs s0 Þ 2 f1; 0gN , there is a positive integer k such that t P k implies YðtÞ ¼ XðtÞ for all YðsÞ; Yð1 sÞ; . . . ; Yð0Þ 2 f1; 0gN . Notice that the integer k in the above definition can be chosen independent of initial state sequences of the system because of the finiteness of the state space. 3. Main results In this section we will establish some necessary and sufficient conditions of synchronization between delayed BNs. To present rigorous analysis, we shall use algebraic representations of BNs in terms of the STP theory introduced before. We first convert the drive-response BNs (1) and (2) to discrete time delay systems. Let xðtÞ ¼ x1 ðtÞ n n xN ðtÞ and yðtÞ ¼ y1 ðtÞ n n yN ðtÞ. Since the map from f1; 0gN to D2N sending ðr1 ; . . . ; rN Þ to r1 n n rN is a bijection [27], it follows that at every time instant t the state XðtÞ [resp., YðtÞ] of the drive BN (1) [resp., response BN (2)] can be well described by xðtÞ [resp., yðtÞ]. Let F i be the structure matrix of f i . Using first (1) and then Lemma 3, we obtain
xðt þ 1Þ ¼ ½F 1 xðt sÞ n n ½F N xðt sÞ: Since the STP is associative, it follows from Lemma 1(b) and (c) that
½F 1 xðt sÞ n n ½F N xðt sÞ ¼ ½F 1 n ðI2N F 2 Þ n UN n ðI2N F 3 Þ n UN n n ðI2N F N Þ n UN xðt sÞ: Writing F ¼ F 1 n f n Ni¼2 ½ðI2N F i ÞUN g, this gives us
xðt þ 1Þ ¼ Fxðt sÞ:
ð3Þ
Similarly, if Gi is the structure matrix of gi , let G ¼ G1 n f n Ni¼2 ½ðI22N Gi ÞU2N g. Then we have
yðt þ 1Þ ¼ G n xðt s0 Þ n yðt sÞ:
ð4Þ
Eqs. (3) and (4) are the algebraic representations of the BNs (1) and (2), respectively. Based on them we can derive some necessary and sufficient conditions of drive-response synchronization. Before stating the results, we need a few more facts about the solutions of the discrete time delay systems (3) and (4). Lemma 4. Suppose xðtÞ and yðtÞ are the solutions of (3) and (4), respectively. Let v ðtÞ ¼ xðt þ s s0 Þ n yðtÞ, let wðtÞ ¼ v ðtÞ n xðtÞ, and let a P 0 and 0 6 b 6 s be the unique integers such that s0 ¼ aðs þ 1Þ þ b. Then xðtÞ ¼ F p1 þa xðq1 s0 Þ for every t P b, where p1 P 0 and 0 6 q1 6 s are the unique integers such that t þ b ¼ p1 ðs þ 1Þ þ q1 ; (b) v ðtÞ ¼ Hp2 þ1 ½xðq2 s0 1Þ n yðq2 s 1Þ for every t P s, where p2 P 1 and 1 6 q2 6 s þ 1 are the unique integers such that t ¼ p2 ðs þ 1Þ þ q2 , and (a)
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H ¼ ðF GÞðUN I2N Þ;
ð5Þ
(c) yðtÞ ¼ GHp3 ½xðq3 s0 1Þ n yðq3 s 1Þ for every t P 1, where p3 P 0 and 1 6 q3 6 s þ 1 are the unique integers such that t ¼ p3 ðs þ 1Þ þ q3 , and H is as in (5); (d) wðtÞ ¼ ðHp3 þ1 F p3 þa Þ½xðq3 s0 1Þ n yðq3 s 1Þ n xðq3 þ b s0 Þ for every t P 1, where p3 and q3 are as in (c), and H is as in (5). Proof. Part (a): We see from (3) that xðs þ ðp1 þ aÞðs þ 1ÞÞ ¼ F p1 þa xðsÞ for every s P s0 . The special case s ¼ q1 s0 gives
xðtÞ ¼ xðp1 ðs þ 1Þ þ q1 þ aðs þ 1Þ s0 Þ ¼ F p1 þa xðq1 s0 Þ: Part (b): It follows from (3) and (4) that
vðs þ s þ 1Þ ¼ F n xðs þ s s0 Þ n G n xðs þ s s0 Þ n yðsÞ for every s P s. Then, by Lemma 1(b), (c), and Definition 1, we have
vðs þ s þ 1Þ ¼ F n ðI2
N
GÞ n UN n xðs þ s s0 Þ n yðsÞ ¼ Hv ðsÞ;
so that v ðs þ ðp2 þ 1Þðs þ 1ÞÞ ¼ Hp2 þ1 v ðsÞ. This implies the result since t ¼ q2 s 1 þ ðp2 þ 1Þðs þ 1Þ. Part (c): Since yðtÞ ¼ Gv ðt s 1Þ by (4), the result follows from (b) above. Part (d): By (3), xðs þ ðp3 þ aÞðs þ 1ÞÞ ¼ F p3 þa xðsÞ for every s P s0 . Taking s ¼ q3 þ b s0 , we obtain
xðtÞ ¼ F p3 þa xðq3 þ b s0 Þ:
ð6Þ
The result now follows from (6), part (b), and Lemma 1(a). h To establish the synchronization conditions, we will also need the following somewhat technical lemma. Lemma 5. Let F 2 L2N 2N , let G 2 L2N 22N , and let H be as in (5). Then
½F n ðGHn1 ÞðUN I2N Þ ¼ Hn for any n 2 N. Proof. We first prove that
ðF HÞðUN I2N Þ ¼ ðUN I2N ÞH: N
ð7Þ 2N
Let a1 ; . . . ; a2N 2 f1; . . . ; 2 g and h1 ; . . . ; h22N 2 f1; . . . ; 2 g be such that
F ¼ d2N ða1 ; . . . ; a2N Þ;
H ¼ d22N ðh1 ; . . . ; h22N Þ;
respectively. Then
ðF HÞðUN I2N Þ ¼ d23N ðða1 1Þ22N þ h1 ; . . . ; ða1 1Þ22N þ h2N ; ða2 1Þ22N þ h2N þ1 ; . . . ; ða2 1Þ22N þ h2Nþ1 ; . . . ; ða2N 1Þ22N þ hð2N 1Þ2N þ1 ; . . . ; ða2N 1Þ22N þ h22N Þ:
ð8Þ
Let b1 ; . . . ; b22N 2 f1; . . . ; 2N g be such that G ¼ d2N ðb1 ; . . . ; b22N Þ. It then follows from (5) that
H ¼ d22N ðða1 1Þ2N þ b1 ; . . . ; ða1 1Þ2N þ b2N ; ða2 1Þ2N þ b2N þ1 ; . . . ; ða2 1Þ2N þ b2Nþ1 ; . . . ; ða2N 1Þ2N þ bð2N 1Þ2N þ1 ; . . . ; ða2N 1Þ2N þ b22N Þ: 0
0
0
Hence, hi ¼ ðai0 1Þ2N þ bi for every 1 6 i 6 22N , where i is the unique integer such that 1 6 i 6 2N ; 1 6 i ði 1Þ2N 6 2N . This implies
ðUN I2N ÞH ¼ d23N ðða1 1Þð2N þ 1Þ2N þ b1 ; . . . ; ða1 1Þð2N þ 1Þ2N þ b2N ; ða2 1Þð2N þ 1Þ2N þ b2N þ1 ; . . . ; ða2 1Þ ð2N þ 1Þ2N þ b2Nþ1 ; . . . ; ða2N 1Þð2N þ 1Þ2N þ bð2N 1Þ2N þ1 ; . . . ; ða2N 1Þð2N þ 1Þ2N þ b22N Þ:
ð9Þ
Comparison of (8) and (9) gives (7). We shall use induction on n to complete the proof. The case n ¼ 1 is trivial. Suppose that the statement is true for n 2 N. Then we have
½F nþ1 ðGHn ÞðUN I2N Þ ¼ ½F n ðGHn1 ÞðF HÞðUN I2N Þ ¼ ½F n ðGHn1 ÞðUN I2N ÞH ¼ Hnþ1 ; via (7) and the induction hypothesis. h Now we are ready to present the synchronization conditions. We discuss them in two cases: s0 Xsðmod s þ 1Þ. Let us first consider the case s0 sðmod s þ 1Þ.
s0 sðmod s þ 1Þ and
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Theorem 1. Consider the drive and response BNs (1) and (2), and suppose s0 ¼ aðs þ 1Þ þ s for some nonnegative integer a. Let (3) and (4) be the algebraic representations of the BNs (1) and (2), respectively, and let H be as in (5). Then the following statements are equivalent: (a) The drive BN (1) and the response BN (2) are synchronized. (b) There is a positive integer n such that
ColððF a I2N ÞHn Þ # fd22N ðði 1Þ2N þ iÞ : i ¼ 1; . . . ; 2N g:
ð10Þ
(c) There is a positive integer m such that
F mþa 12N ¼ GHm1 :
ð11Þ
Proof. We will prove the equivalence of the above statements by showing the circular implications ðaÞ ) ðbÞ ) ðcÞ ) ðaÞ. ðaÞ ) ðbÞ: Suppose first that (a) holds. Since f1; 0gN is a finite set, there is k P 1 such that xðkÞ ¼ yðkÞ for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N . Let k1 P 0 and 1 6 k2 6 s þ 1 be the unique integers such that k ¼ k1 ðs þ 1Þ þ k2 , and put n ¼ k1 þ 1. Since xðkÞ ¼ F a xðk þ s s0 Þ, we obtain, by Lemma 4(b),
xðkÞ n yðkÞ ¼ F a n ½xðk þ s s0 Þ n yðkÞ ¼ ðF a I2N ÞHn ½xðk2 s0 1Þ n yðk2 s 1Þ: Fix xðk2 s0 1Þ; yðk2 s 1Þ 2 D2N . Let 1 6 r 6 2N be such that xðkÞ ¼ yðkÞ ¼ dr2N . Then
ðF a I2N ÞHn ½xðk2 s0 1Þ n yðk2 s 1Þ ¼ dr2N n dr2N ¼ d22N ððr 1Þ2N þ rÞ: This implies (10) since xðk2 s0 1Þ and yðk2 s 1Þ are chosen arbitrarily in D2N . Therefore ðaÞ ) ðbÞ. ðbÞ ) ðcÞ: Now suppose that (b) holds. By Lemma 5,
ðF a I2N ÞHn ¼ ½F nþa ðGHn1 ÞðUN I2N Þ:
ð12Þ
N
Let n1 ; . . . ; n2N ; g1 ; . . . ; g22N 2 f1; . . . ; 2 g be such that
F nþa ¼ d2N ðn1 ; . . . ; n2N Þ;
GHn1 ¼ d2N ðg1 ; . . . ; g22N Þ;
respectively. The right side of (12) is then
d22N ððn1 1Þ2N þ g1 ; . . . ; ðn1 1Þ2N þ g2N ; ðn2 1Þ2N þ g2N þ1 ; . . . ; ðn2 1Þ2N þ g2Nþ1 ; . . . ; ðn2N 1Þ2N þ gð2N 1Þ2N þ1 ; . . . ; ðn2N 1Þ2N þ g22N Þ: Hence ni ¼ gði1Þ2N þj for all 1 6 i; j 6 2N by hypothesis. It now follows that
GHn1 ¼ d2N ðg1 ; . . . ; g2N ; g2N þ1 ; . . . ; g2Nþ1 ; . . . ; gð2N 1Þ2N þ1 ; . . . ; g22N Þ ¼ d2N ðn1 ; . . . ; n1 ; n2 ; . . . ; n2 ; . . . ; n2N ; . . . ; n2N Þ ¼ F nþa 12N : Therefore ðbÞ ) ðcÞ. ðcÞ ) ðaÞ: Finally, suppose that (c) holds. Let k ¼ ðm 1Þðs þ 1Þ þ 1. It suffices to show that t P k implies yðtÞ ¼ xðtÞ for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N . Fix t P k. Let p P 0 and 0 6 q 6 s be the unique integers such that t þ s ¼ pðs þ 1Þ þ q. Since pðs þ 1Þ þ q P k þ s ¼ mðs þ 1Þ, it follows that p P m. If p ¼ m, then
yðtÞ ¼ GHm1 ½xðq s0 Þ n yðq sÞ ¼ ðF mþa 12N Þ½xðq s0 Þ yðq sÞ ¼ xðtÞ: The first of these equalities follows from Lemma 4(c). The second follows from (11) and Lemma 1(a). The last is Lemma 4(a). If p P m þ 1, then
Hpm ½xðq s0 Þ n yðq sÞ ¼ ½F pm ðGHpm1 Þ n UN n xðq s0 Þ n yðq sÞ ¼ ½F pm ðGHpm1 Þ½xðq s0 Þ n xðq s0 Þ n yðq sÞ ¼ ½F pm xðq s0 Þ ½ðGHpm1 Þ n xðq s0 Þ n yðq sÞ:
ð13Þ
The first equality holds by Lemma 5 and Definition 1, the second by Lemma 1(c), and the third by Lemma 1(a). Hence,
yðtÞ ¼ GHp1 ½xðq s0 Þ n yðq sÞ ¼ ðF mþa 12N Þf½F pm xðq s0 Þ ½ðGHpm1 Þ n xðq s0 Þ n yðq sÞg; pm1
by (11), (13), and Lemma 4(c). Since ðGH Þ n xðq s Þ n yðq sÞ 2 D2N , (14) implies yðtÞ ¼ F yðtÞ ¼ xðtÞ by Lemma 4(a). Therefore ðcÞ ) ðaÞ. h 0
pþa
ð14Þ
xðq s Þ, so that 0
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Remark 1. It is worth noting that Theorem 1 is also applicable to delay-free BNs, i.e., s ¼ s0 ¼ 0. Substituting a ¼ 0 into (11), one can obtain the result presented in [21]. Therefore, the synchronization criteria given in Theorem 1 improve and extend the previous result when reducing to the delay-free case. We now move onto the case s0 Xsðmod s þ 1Þ. Theorem 2. Consider the drive and response BNs (1) and (2), and suppose s0 Xsðmod s þ 1Þ. Let (3) and (4) be the algebraic representations of the BNs (1) and (2), respectively, and let H be as in (5). Then the following statements are equivalent: (a) The drive BN (1) and the response BN (2) are synchronized. (b) There is a positive integer m such that
ColðHmþ1 F mþa Þ # fd23N ðði 1Þ22N þ ðj 1Þ2N þ jÞ : i; j ¼ 1; . . . ; 2N g;
ð15Þ
where a is the unique positive integer such that 0 6 s aðs þ 1Þ 6 s 1. (c) There are positive integers m1 and m2 such that 0
ColðF m1 Þ ¼ ColðGHm2 Þ ¼ fdn2N g
ð16Þ
for some 1 6 n 6 2N . (d) There exist a state X 2 f1; 0gN and a positive integer k such that t P k implies YðtÞ ¼ XðtÞ ¼ X for any Xðs0 Þ; . . . ; Xðs s0 Þ; YðsÞ; . . . ; Yð0Þ 2 f1; 0gN .
Proof. We will prove the equivalence of the above statements by showing the circular implications ðaÞ ) ðbÞ ) ðcÞ ) ðdÞ ) ðaÞ. ðaÞ ) ðbÞ: Suppose that (a) holds. Let a P 1 and 0 6 b 6 s 1 be the unique integers such that s0 ¼ aðs þ 1Þ þ b. By Definition 2, there is k P 1 such that t P k implies yðtÞ ¼ xðtÞ for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N , since the set f1; 0gN is finite. Let k1 P 0 and 1 6 k2 6 s þ 1 be the unique integers such that k ¼ k1 ðs þ 1Þ þ k2 . Put m ¼ k1 þ 1 and t0 ¼ mðs þ 1Þ þ 1. Then by Lemma 4(d), we have
xðt 0 þ s s0 Þ n yðt0 Þ n xðt 0 Þ ¼ ðHmþ1 F mþa Þ½xðs0 Þ n yðsÞ n xðb þ 1 s0 Þ: Since t 0 P k and 1 s0 6 b þ 1 s0 6 s s0 , it follows that yðt 0 Þ ¼ xðt 0 Þ for any xðs0 Þ; xðb þ 1 s0 Þ; yðsÞ 2 D2N . Fix 0 xðs0 Þ; xðb þ 1 s0 Þ; yðsÞ 2 D2N . Let 1 6 r; r 0 6 2N be such that xðt0 þ s s0 Þ ¼ dr2N and xðt0 Þ ¼ yðt0 Þ ¼ dr2N , respectively. Then
ðHmþ1 F mþa Þ½xðs0 Þ n yðsÞ n xðb þ 1 s0 Þ ¼ d23N ððr 1Þ22N þ ðr 0 1Þ2N þ r 0 Þ: This implies (15) since xðs0 Þ, xðb þ 1 s0 Þ, and yðsÞ are chosen arbitrarily in D2N . Therefore ðaÞ ) ðbÞ. ðpÞ ðpÞ ðpÞ ðpÞ ðbÞ ) ðcÞ: Now suppose that (b) holds. For every p P 1, let a1 ; . . . ; a2N 2 f1; . . . ; 2N g and h1 ; . . . ; h 2N 2 f1; . . . ; 22N g be 2 such that ðpÞ
ðpÞ
Hp ¼ d22N ðh1ðpÞ ; . . . ; hðpÞ Þ; 22N
F p ¼ d2N ða1 ; . . . ; a2N Þ;
respectively, and let b1 ; . . . ; b22N 2 f1; . . . ; 2N g be such that GHm ¼ d2N ðb1 ; . . . ; b22N Þ. Then by Lemma 5, ðmþ1Þ ðmþ1Þ Hmþ1 ¼ ½F mþ1 ðGHm ÞðUN I2N Þ ¼ d22N ððaðmþ1Þ 1Þ2N þ b1 ; . . . ; ða1 1Þ2N þ b2N ; ða2 1Þ2N 1 ðmþ1Þ
þ b2N þ1 ; . . . ; ða2
1Þ2N þ b2Nþ1 ; . . . ; ða2N
ðmþ1Þ
1Þ2N þ bð2N 1Þ2N þ1 ; . . . ; ða2N
ðmþ1Þ
1Þ2N þ b22N Þ:
This implies ðmþ1Þ
hi
ðmþ1Þ
¼ ðai0
1Þ2N þ bi 0
0
0
for all 1 6 i 6 22N , where i is the unique integer such that 1 6 i 6 2N ; 1 6 i ði 1Þ2N 6 2N . Hence ðmþ1Þ
ðhi
1Þ2N þ aj
ðmþaÞ
2N
for every 1 6 i 6 2 mþ1
H
F
mþa
ðmþ1Þ
¼ ðai0
1Þ22N þ ðbi 1Þ2N þ aj
ðmþaÞ
and 1 6 j 6 2 . Since ðmþ1Þ
¼ d23N ððh1
ðmþaÞ
þ a2N
1Þ2N þ a1
ðmþaÞ
ðmþ1Þ
; . . . ; ðh22N
it follows from (15) and (17) that bi ¼ a
ColðF
m
Þ ¼ ColðGH Þ ¼
Therefore ðbÞ ) ðcÞ.
fdb21N g:
ðmþ1Þ
; . . . ; ðh1
1Þ2N þ a1
ðmþaÞ j
mþa
ð17Þ
N
ðmþaÞ
1Þ2N þ a2N
ðmþaÞ
ðmþ1Þ
; . . . ; ðh22N 2N
for all 1 6 i 6 2
ðmþ1Þ
; ðh2
1Þ2N þ a2N
1Þ2N þ a1
ðmþaÞ N
ðmþaÞ
Þ;
and all 1 6 j 6 2 , so that
ðmþ1Þ
; . . . ; ðh2
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ðcÞ ) ðdÞ: Suppose that (c) holds. Let a P 1 and 0 6 b 6 s 1 be the unique integers such that s0 ¼ aðs þ 1Þ þ b. Put M ¼ maxfm1 a; m2 g and k ¼ Mðs þ 1Þ þ 1. It suffices to show that t P k implies xðtÞ ¼ yðtÞ ¼ dn2N for any xðs0 Þ; . . . ; xðs s0 Þ; yðsÞ; . . . ; yð0Þ 2 D2N . Fix t P k. Let p1 P 0 and 0 6 q1 6 s be the unique integers such that t þ b ¼ p1 ðs þ 1Þ þ q1 ; let p2 P 0 and 1 6 q2 6 s þ 1 be the unique integers such that t ¼ p2 ðs þ 1Þ þ q2 . Then p1 P m1 a; p2 P m2 , since p1 ðs þ 1Þ þ q1 P Mðs þ 1Þ þ b þ 1; p2 ðs þ 1Þ þ q2 P Mðs þ 1Þ þ 1. By Lemma 4(a), xðtÞ ¼ F m1 ½F p1 þam1 xðq1 s0 Þ. Thus xðtÞ ¼ dn2N by (16), since F p1 þam1 xðq1 s0 Þ 2 D2N . Similarly,
yðtÞ ¼ GHm2 ½Hp2 m2 n xðq2 s0 1Þ n yðq2 s 1Þ ¼ dn2N ; by Lemma 4(c) and (16), since Hp2 m2 n xðq2 s0 1Þ n yðq2 s 1Þ 2 D2N . Therefore ðcÞ ) ðdÞ. ðdÞ ) ðaÞ: This follows directly from Definition 2. h Theorems 1 and 2 are our major results. They furnish several necessary and sufficient criteria for synchronization between two delayed BNs, which are unidirectionally coupled in the drive-response configuration. Our next result implies that one only needs to calculate a finite number of matrices when using these criteria. Corollary 1 (a) If n and m are the smallest positive integers satisfying (10) and (11), respectively, then n ¼ m 6 eðHÞ. (b) If m; m1 , and m2 are the smallest positive integers satisfying (15) and (16), respectively, then m 6 maxfeðFÞ a; eðHÞg þ 1; m1 6 eðFÞ, and m2 6 eðHÞ.
Proof. Part (a): From the proof that ðbÞ ) ðcÞ in Theorem 1, we see m 6 n. The proofs that ðcÞ ) ðaÞ and ðaÞ ) ðbÞ in Theorem 1 show that (10) holds with m in place of n, so that n 6 m. Suppose, to get a contradiction, that n > eðHÞ. Then, by Lemma 2, ColðHeðHÞ Þ ¼ ColðHn Þ, so that
ColððF a I2N ÞHeðHÞ Þ ¼ ColððF a I2N ÞHn Þ: Hence (10) holds with eðHÞ in place of n. This contradicts the minimality of n. Part (b): If m1 > eðFÞ, then by Lemma 2 we would have ColðF eðFÞ Þ ¼ ColðF m1 Þ ¼ fdn2N g, contradicting the minimality of m1 . The proof that m2 6 eðHÞ is analogous. Let M ¼ maxfm1 a; m2 g. Then from the proofs that ðcÞ ) ðdÞ and ðaÞ ) ðbÞ in Theorem 2 we see that (15) holds with M þ 1 in place of m. Hence m 6 M þ 1 6 maxfeðFÞ a; eðHÞg þ 1. h Remark 2. From Theorems 1 and 2, we see clearly that different relations between the inherent state delay s and the unidirectional coupling delay s0 may lead to quite distinct synchronization conditions and phenomena. In particular, Theorem 2 implies that in the case s0 Xsðmod s þ 1Þ drive-response synchronization can only occur between two BNs that globally converge to a common fixed point, and the fixed point is precisely the synchronized state of these two BNs. On the other hand, in the case s0 sðmod s þ 1Þ synchronous behavior between response and drive BNs can be relatively complex; see Example 1 in the next section. As an immediate consequence of Theorems 1 and 2, the following result asserts that in both cases time delays that result in drive-response synchronization, if existent, are not unique. Corollary 2. Suppose s0 ; s00 are nonnegative integers, s00 P s0 , and the drive BN (1) and the response BN (2) are synchronized with
s ¼ s0 ; s0 ¼ s00 .
(a) If s00 s0 ðmod s0 þ 1Þ, then the drive BN (1) and the response BN (2) are synchronized for all s P 0 and s0 ¼ aðs þ 1Þ þ s, where a is the unique integer such that s00 ¼ aðs0 þ 1Þ þ s0 . (b) If s00 Xs0 ðmod s0 þ 1Þ, then the drive BN (1) and the response BN (2) are synchronized for all s0 P s P 0.
Proof. Part (a): The result follows from Theorem 1. Part (b): By Theorem 2, there exist m1 and m2 such that (16) holds for some 1 6 n 6 2N . If s0 Xsðmod s þ 1Þ, then the BNs (1) and (2) are synchronized by Theorem 2. If s0 sðmod s þ 1Þ, let a P 0 be the unique integer such that s0 ¼ aðs þ 1Þ þ s, and put m ¼ maxfm1 a; m2 þ 1g. Then ColðF mþa Þ # ColðF m1 Þ ¼ fdn2N g and ColðGHm1 Þ # ColðGHm2 Þ ¼ fdn2N g, so that F mþa 12N ¼ GHm1 . Hence the BNs (1) and (2) are synchronized by Theorem 1. h 4. Illustrative examples Now we present some illustrative examples to demonstrate the applications of the main results. Example 1. We take the following delayed BN as the drive BN
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x1
1
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1
0 0
5
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x
1
2
y2
0 0
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x3
1
y3
0 0
5
10
15
20
t
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Fig. 1. Time evolution of the BNs (18) and (19) with initial state sequences Xð17Þ ¼ ð1; 1; 1Þ; Xð16Þ ¼ ð1; 0; 0Þ; Xð15Þ ¼ ð1; 0; 1Þ, and Yð2Þ ¼ ð1; 1; 0Þ; Yð1Þ ¼ ð0; 0; 0Þ; Yð0Þ ¼ ð0; 1; 0Þ.
8 > < x1 ðt þ 1Þ ¼ :x2 ðt 2Þ; x2 ðt þ 1Þ ¼ x1 ðt 2Þ ^ x3 ðt 2Þ; > : x3 ðt þ 1Þ ¼ x1 ðt 2Þ
ð18Þ
and the response BN is given by
8 > < y1 ðt þ 1Þ ¼ :x2 ðt 17Þ _ :y2 ðt 2Þ _ :y3 ðt 2Þ; y2 ðt þ 1Þ ¼ ½x1 ðt 17Þ _ y1 ðt 2Þ ^ y3 ðt 2Þ; > : y3 ðt þ 1Þ ¼ ½:x2 ðt 17Þ _ x3 ðt 17Þ $ y1 ðt 2Þ:
ð19Þ
Denoting xðtÞ ¼ x1 ðtÞ n x2 ðtÞ n x3 ðtÞ and yðtÞ ¼ y1 ðtÞ n y2 ðtÞ n y3 ðtÞ, then we can express the drive BN (18) and the response BN (19) in their algebraic forms as
xðt þ 1Þ ¼ Fxðt 2Þ; yðt þ 1Þ ¼ G n xðt 17Þ n yðt 2Þ; respectively, where
F ¼ d8 ð5; 7; 1; 3; 8; 8; 4; 4Þ; G ¼ d8 ð5; 3; 1; 3; 6; 4; 2; 4; 6; 4; 2; 4; 5; 3; 1; 3; 1; 3; 1; 3; 2; 4; 2; 4; 1; 3; 1; 3; 2; 4; 2; 4; 5; 3; 1; 3; 8; 4; 4; 4; 6; 4; 2; 4; 7; 3; 3; 3; 1; 3; 1; 3; 4; 4; 4; 4; 1; 3; 1; 3; 4; 4; 4; 4Þ: In order to examine whether these two BNs can be synchronized or not, we compute ðF 5 I8 Þ½ðF GÞðU3 I8 Þn according to Theorem 1, since 17 ¼ 5 ð2 þ 1Þ þ 2. Simple calculations lead to
ColððF 5 I8 Þ½ðF GÞðU3 I8 Þ6 Þ ¼ fdi64 : i ¼ 1; 19; 28; 37; 64g # fdi64 : i ¼ 1; 10; 19; 28; 37; 46; 55; 64g: Therefore, the response BN (19) can synchronize with the drive BN (18) by Theorem 1. Fig. 1 shows the time evolution of the drive-response BNs (18) and (19), where Xð17Þ ¼ ð1; 1; 1Þ; Xð16Þ ¼ ð1; 0; 0Þ; Xð15Þ ¼ ð1; 0; 1Þ, and Yð2Þ ¼ ð1; 1; 0Þ; Yð1Þ ¼ ð0; 0; 0Þ; Yð0Þ ¼ ð0; 1; 0Þ. The Hamming distance between the states of the BNs (18) and (19) P HðtÞ ¼ 3i¼1 jxi ðtÞ yi ðtÞj versus the time t is plotted in Fig. 2. As seen in Fig. 1, neither of these two BNs converges globally to any fixed points. Example 2. Let us consider the following drive and response BNs with each network consisting of four nodes:
8 x1 ðt þ 1Þ ¼ x2 ðt sÞ _ x3 ðt sÞ; > > > < x ðt þ 1Þ ¼ :x ðt sÞ ^ x ðt sÞ ^ x ðt sÞ; 2 1 2 4 > x ðt þ 1Þ ¼ :x ðt s Þ; 3 4 > > : x4 ðt þ 1Þ ¼ x1 ðt sÞ ! :x2 ðt sÞ;
ð20Þ
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3
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−1
0
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Fig. 2. Hamming distance between the states of the BNs (18) and (19) versus time, with Xð17Þ ¼ ð1; 1; 1Þ; Xð16Þ ¼ ð1; 0; 0Þ; Xð15Þ ¼ ð1; 0; 1Þ, and Yð2Þ ¼ ð1; 1; 0Þ; Yð1Þ ¼ ð0; 0; 0Þ; Yð0Þ ¼ ð0; 1; 0Þ.
4
3
H
2
1
0
−1
0
10
20
30
t
40
50
60
Fig. 3. Hamming distance between the states of the BNs (20) and (21) versus time, with s ¼ 3; s0 ¼ 5; Xð5Þ ¼ ð1; 1; 1; 1Þ; Xð4Þ ¼ ð1; 1; 0; 1Þ; Xð3Þ ¼ ð0; 1; 1; 1Þ; Xð2Þ ¼ ð0; 1; 0; 1Þ, and Yð3Þ ¼ ð0; 1; 0; 1Þ; Yð2Þ ¼ ð1; 0; 1; 1Þ; Yð1Þ ¼ ð1; 1; 1; 1Þ; Yð0Þ ¼ ð1; 0; 0; 0Þ.
8 y1 ðt þ 1Þ ¼ ½x1 ðt s0 Þ ^ x2 ðt s0 Þ _ y2 ðt sÞ; > > > < y2 ðt þ 1Þ ¼ x3 ðt s0 Þ $ y4 ðt sÞ; 0 > > y3 ðt þ 1Þ ¼ x2 ðt s Þ _ y1 ðt sÞ; > : y4 ðt þ 1Þ ¼ x4 ðt s0 Þ _ y3 ðt sÞ _ :y4 ðt sÞ;
ð21Þ
where s0 P s P 0. Let xðtÞ ¼ x1 ðtÞ n x2 ðtÞ n x3 ðtÞ n x4 ðtÞ and yðtÞ ¼ y1 ðtÞ n y2 ðtÞ n y3 ðtÞ n y4 ðtÞ. Then the algebraic representations of the drive BN (20) and the response BN (21) are
xðt þ 1Þ ¼ Fxðt sÞ; yðt þ 1Þ ¼ G n xðt s0 Þ n yðt sÞ; respectively, where
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F ¼ d16 ð8; 6; 8; 6; 7; 5; 15; 13; 3; 5; 3; 5; 7; 5; 15; 13Þ; G ¼ d16 ð1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 2; 5; 1; 5; 2; 5; 1; 5; 2; 5; 1; 5; 2; 5; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 5; 1; 6; 1; 5; 1; 6; 1; 5; 1; 6; 1; 5; 1; 6; 1; 1; 5; 1; 5; 9; 13; 9; 13; 3; 7; 3; 7; 11; 15; 11; 15; 1; 5; 2; 5; 9; 13; 10; 13; 3; 7; 4; 7; 11; 15; 12; 15; 5; 1; 5; 1; 13; 9; 13; 9; 7; 3; 7; 3; 15; 11; 15; 11; 5; 1; 6; 1; 13; 9; 14; 9; 7; 3; 8; 3; 15; 11; 16; 11; 1; 5; 1; 5; 9; 13; 9; 13; 1; 5; 1; 5; 9; 13; 9; 13; 1; 5; 2; 5; 9; 13; 10; 13; 1; 5; 2; 5; 9; 13; 10; 13; 5; 1; 5; 1; 13; 9; 13; 9; 5; 1; 5; 1; 13; 9; 13; 9; 5; 1; 6; 1; 13; 9; 14; 9; 5; 1; 6; 1; 13; 9; 14; 9; 1; 5; 1; 5; 9; 13; 9; 13; 3; 7; 3; 7; 11; 15; 11; 15; 1; 5; 2; 5; 9; 13; 10; 13; 3; 7; 4; 7; 11; 15; 12; 15; 5; 1; 5; 1; 13; 9; 13; 9; 7; 3; 7; 3; 15; 11; 15; 11; 5; 1; 6; 1; 13; 9; 14; 9; 7; 3; 8; 3; 15; 11; 16; 11Þ: It is easy to check that
ColðF 5 Þ ¼ ColðG½ðF GÞðU4 I16 Þ6 Þ ¼ fd15 16 g: So applying Theorem 2 and Corollary 2(b), we conclude that the drive BN (20) and the response BN (21) are synchronized for all s0 P s P 0. In fact, for any s and s0 , both of these BNs will converge to the state XP ¼ ð0; 0; 0; 1Þ, since d22 n d22 n d22 n d12 ¼ d15 16 . In Fig. 3, the Hamming distance between the states of the BNs (20) and (21) HðtÞ ¼ 4i¼1 jxi ðtÞ yi ðtÞj versus the time t is plotted, where s ¼ 3; s0 ¼ 5; Xð5Þ ¼ ð1; 1; 1; 1Þ; Xð4Þ ¼ ð1; 1; 0; 1Þ; Xð3Þ ¼ ð0; 1; 1; 1Þ; Xð2Þ ¼ ð0; 1; 0; 1Þ, and Yð3Þ ¼ ð0; 1; 0; 1Þ; Yð2Þ ¼ ð1; 0; 1; 1Þ; Yð1Þ ¼ ð1; 1; 1; 1Þ; Yð0Þ ¼ ð1; 0; 0; 0Þ. 5. Conclusions In this paper, we have discussed synchronization of two deterministic BNs with time delays coupled unidirectionally in the drive-response configuration. Our emphasis has focused on rigorous analysis in terms of algebraic representations of the logical dynamics of BNs. The main results furnish some necessary and sufficient algebraic criteria for synchronization. The criteria also reveal that, for drive-response BNs, different relations between the inherent state delay and the unidirectional coupling delay may lead to quite distinct synchronization phenomena. Furthermore, as an application of the main results, we have found that both the inherent state delay and the coupling delay with which drive-response synchronization occurs, if existent, are not unique. Acknowledgments The authors thank the editor and the reviewers for their valuable comments and suggestions. This work was supported by the National Basic Research Program of China under Grant 2012CB821200 and the National Natural Science Foundation of China under Grant 60974064. References [1] R. Albert, H. Othmer, The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster, J. Theor. Biol. 223 (2003) 1–18. [2] K.E. Kürten, Correspondence between neural threshold networks and Kauffman Boolean cellular automata, J. Phys. A Math. Gen. 21 (1988) L615–L619. [3] S. Bornholdt, T. Rohlf, Topological evolution of dynamical networks: global criticality from local dynamics, Phys. Rev. Lett. 84 (2000) 6114–6117. [4] M.D. Stern, Emergence of homeostasis and noise imprinting in an evolution model, Proc. Natl. Acad. Sci. USA 96 (1999) 10746–10751. [5] B. Derrida, H. Flyvbjerg, Multivalley structure in Kauffman’s model: analogy with spin glasses, J. Phys. A Math. Gen. 19 (1986) L1003–L1008. [6] L.A.N. Amaral, A. Dı´az-Guilera, A.A. Moreira, A.L. Goldberger, L.A. Lipsitz, Emergence of complex dynamics in a simple model of signaling networks, Proc. Natl. Acad. Sci. USA 101 (2004) 15551–15555. [7] J. Heidel, J. Maloney, C. Farrow, J.A. Rogers, Finding cycles in synchronous Boolean networks with applications to biochemical systems, Int. J. Bifurcat. Chaos 13 (2003) 535–552. [8] J.H. Park, S.M. Lee, H.Y. Jung, LMI optimization approach to synchronization of stochastic delayed discrete-time complex networks, J. Optim. Theory Appl. 143 (2009) 357–367. [9] D.H. Ji, J.H. Park, W.J. Yoo, S.C. Won, S.M. Lee, Synchronization criterion for Lur’e type complex dynamical networks with time-varying delay, Phys. Lett. A 374 (2010) 1218–1227. [10] D.H. Ji, D.W. Lee, J.H. Koo, S.C. Won, S.M. Lee, J.H. Park, Synchronization of neutral complex dynamical networks with coupling time-varying delays, Nonlinear Dynam. 65 (2011) 349–358. [11] D.H. Ji, S.C. Jeong, J.H. Park, S.M. Lee, S.C. Won, Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling, Appl. Math. Comput. 218 (2012) 4872–4880. [12] T.H. Lee, J.H. Park, D.H. Ji, O.M. Kwon, S.M. Lee, Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control, Appl. Math. Comput. 218 (2012) 6469–6481. [13] L.G. Morelli, D.H. Zanette, Synchronization of Kauffman networks, Phys. Rev. E 63 (2001) 036204. [14] J.L. Guisado, F. Jiménez-Morales, J.M. Guerra, Cellular automaton model for the simulation of laser dynamics, Phys. Rev. E 67 (2003) 066708. [15] M.-C. Ho, Y.-C. Hung, I.-M. Jiang, Stochastic coupling of two random Boolean networks, Phys. Lett. A 344 (2005) 36–42. [16] J. Parriaux, P. Guillot, G. Millérioux, Towards a spectral approach for the design of self-synchronizing stream ciphers, Cryptogr. Commun. 3 (2011) 259– 274. [17] Y.-C. Hung, M.-C. Ho, J.-S. Lih, I.-M. Jiang, Chaos synchronization of two stochastically coupled random Boolean networks, Phys. Lett. A 356 (2006) 35– 43. [18] Y.-C. Hung, Microscopic interactions lead to mutual synchronization in a network of networks, Phys. Lett. A 375 (2011) 2809–2814. [19] D.J. Irons, Logical analysis of the budding yeast cell cycle, J. Theor. Biol. 257 (2009) 543–559. [20] A. Veliz-Cuba, B. Stigler, Boolean models can explain bistability in the lac operon, J. Comput. Biol. 18 (2011) 783–794.
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