Sage Journals: Discover world-class research

Abstract

This paper studies the delay partial synchronization for mutual delay-coupled Boolean networks. First, the mutual delay-coupled Boolean network model is presented. Second, some necessary and sufficient conditions are derived to ensure the delay partial synchronization of the mutual delay-coupled Boolean networks. The upper bound of synchronization time is obtained. Finally, an example is provided to illustrate the efficiency of the theoretical analysis.

Keywords

Mutual delay-coupled Boolean network delay partial synchronization semi-tensor product control

Introduction

The Boolean network (BN) model is proposed by Kauffman, which mainly describes the gene regulatory network and the biological system.¹ In recent years, this model is widely used in the biology system, the physical system,^2,3 and so on. BN model is the discrete-time system, which is based on digraph.^4,5 In BN, the state of each gene is 0 or 1 at a certain moment. The state of each node at the next moment is determined by the states of the other nodes. The relationship between two nodes is the logical operation, which include AND, OR, NOT, and XOR. It is difficult to calculate and analyze in logical domain.

Semi-tensor product (STP) is a new matrix multiplication by which logical operations are converted to algebraic operations and BN is converted to the linear system.⁶ The STP has become a basic tool to study the BN.⁷ Classical control theories and methods can be used to describe and study BNs. In recent years, BN has become a research topic in the field of the automation and control. Ding et al. (2018) and Cheng et al. (2018) used the randomly generated BN to study the distribution of gene regulation models,^8,9 study the controllability of the BN,¹⁰ study the complete synchronization of BN,¹¹ design a variable structure controller to stabilize the BN,¹² study the sufficient and necessary conditions for synchronization of the switched BN with pulse,¹³ study the optimal control problem of Boolean control networks,¹⁴ discuss the relationship between the sensitivity of updating rules of BNs and the distribution of smoothness.

Synchronization is one of the natural phenomena which shows that the states of the two systems can eventually reach consensus. Synchronization is a basic characteristic of the complex systems, which exists in natural and artificial systems.¹⁵ In past years, there were other research results, for example, synchronization of neural networks via pinning control,^16–18 function projective synchronization of the complex networks,^19–21 outer synchronization between two hybrid-coupled delayed networks,²² synchronization of the delay partial differential systems with coupled neutral-type,²³ synchronization of the systems with different orders,²⁴ and robust synchronization control for chaotic systems with fractional-order.²⁵ With the rapid development of the system biology, BN is one of the important models for studying biological system and gene regulation network.²⁶ Synchronous control for BN is one of the research topics.^27,28 Synchronization of the BN is extensively applied to many fields, such as gene regulatory network, physics, and biology.^29,30 In recent years, synchronization of the BN has achieved many research results, such as the delay synchronization of the temporal BNs,³¹ the anti-synchronization of the two-coupled BNs,³² the synchronization of the mutual-coupled temporal BNs,²⁸ the synchronization of the BNs via state feedback controller,³³ and the synchronization of the switched BNs.¹²

There is a coupling relationship between systems, and delay is an unavoidable phenomenon.^8,34 Most of the BN models did not consider mutual delay-coupled. In recent works, one of the new BNs called mutual-coupled temporal BNs has been proposed, which can be used to describe biological systems (Wei and Xie, 2018).^28,35 The synchronization of mutual-coupled BNs has been investigated (Wei and Xie, 2018).^10,28 So far, most of the research on synchronization of BNs focuses on complete synchronization. In some cases, not all variables of the systems can be completely synchronized, which show partial synchronization.³⁶ Partial synchronization has a wide range of applications, such as communication, seismology, and neural networks.³⁷ The interconnected (mutual-coupled) systems show the phenomenon of delay which is called the delay coupling. There are few reports on delay partial synchronization of mutual delay-coupled BNs. Therefore, the delay partial synchronization of the mutual-coupled BNs is an important research content. In this paper, the mutual-coupled BNs are presented, and the delay partial synchronization for mutual delay-coupled BNs is investigated.

The rest of this paper is arranged follows. In section “Preliminaries,” some basic concepts of STP are introduced. In section “The model description,” two mutual delay-coupled BN models are presented. In section “Main results,” the main results are deduced. In section “Illustrative example,” an example is given to verify the validity of theoretical analysis. Finally, section “Conclusion” gives the conclusion of this paper.

Preliminaries

In this section, the necessary concept and basic properties of STP are introduced. The contents of STP are introduced in detail in the literature.^35,38

Definition 1

The operation of the STP of two matrices $E$ and $F$ is defined as follows³⁵

E ⋉ F = (E \otimes I_{α / α n n}) (F \otimes I_{α / α p p})

where $α = lcm (n, p)$ is the least common multiple of $n and p$ , $E \in M_{m \times n}$ , $F \in M_{p \times q}$ , and ⊗ is the Kronecker product.

Remark 1

When $n = p$ , $E ⋉ F = (E \otimes I_{1}) (F \otimes I_{1}) = EF$ .³³ Therefore, the STP becomes the ordinary matrix multiplication. The symbol “⋉” is omitted in the following sections. The following Lemma 1 can be used in this paper.

Lemma 1

The STP has the following properties:³³

Let $C \in M_{m \times n} and D \in M_{np \times q}$ be given, then $C ⋉ D = (C \otimes I_{P}) D$ .

If $E \in Δ_{2^{k}}, EE = Φ_{k} E, where Φ_{k} = δ_{2^{2 k}} [1, 2^{k} + 2, \dots, (2^{k} - 2) 2^{k} + 2^{k} - 1, 2^{2 k}]$ .

Some STP operations are summarized as follows:³³

The swap matrix $W_{[m, n]}$ is defined as follows: $W_{[m, n]}$ is an $mn \times mn$ matrix in which rows and columns are all labeled by the double index $(i, j)$ , and columns and rows are arranged by the multi-index Id $(i, j : m, n)$ and Id $(j, i : n, m)$ . The element in the position $[(I, J), (i, j)]$ is

w_{(I, J), (i, j)} = δ_{i, j}^{I, J} = {\begin{matrix} 1, & I = i and J = j \\ 0, & otherwise \end{matrix}

$G \in R^{m} and H \in R^{n}$ , then

G ⋉ H = W_{[m, n]} H ⋉ G

(1)

2. $E_{d} \overset{Δ}{=} δ_{2} [1, 2, 1, 2]$ . $E_{d} cd = d$ for all $c, d \in Δ_{2}$ . $E_{d}$ is defined as the dummy matrix.

Lemma 2

A logical function is $f (x_{1}, x_{2}, \dots, x_{n}) : D^{n} \to D$ .³³ Then, there is a matrix $M_{f} \in L_{2 \times 2^{n}}, f (x_{1}, x_{2}, \dots, x_{n}) = M_{f} ⋉_{i = 1}^{n} x_{i}, x_{i} \in Δ_{2}$ , where $M_{f}$ is the structure matrix.

The model description

Two mutual delay-coupled BN models with n nodes are described as follows

x_{i} (t + 1) = r_{i} (X (t), Y (t - τ))

(2a)

\begin{matrix} y_{i} (t + 1) = h_{i} (Y (t), X (t - τ)), \\ i = 1, 2, \dots n, t = 0, 1, 2 \dots \end{matrix}

(2b)

where $X (t) = (x_{1} (t), x_{2} (t), \dots, x_{n} (t)) \in D^{n} and Y (t) = (y_{1} (t), y_{2} (t), \dots, y_{n} (t)) \in D^{n}$ , $r_{i} : D^{n} \to D$ and $h_{i} : D^{n} \to D$ are the Boolean functions. $τ (τ = 0, 1, 2, \dots n)$ is the time delay. The initial states of models (2a) and (2b) are $X (t, X_{0}, Y_{0})$ and $Y (t, Y_{0}, X_{0})$ , respectively. The BNs with n nodes achieve d nodes synchronization $(d < n)$ , which is called the partial synchronization. Let $l_{i} (1 \leq i \leq n)$ be the label of nodes of the delay partial synchronization for BNs (2a) and (2b). $1 \leq l_{1} \leq l_{2} \leq \dots \leq l_{b} \leq n, b = d$ . When $d = n$ , the partial synchronization becomes complete synchronization.

Definition 2

Two mutual delay-coupled BNs (2a) and (2b) are said to achieve delay partial synchronization if there is a positive integer $α$ such that

X_{i} (t, X_{0}, Y_{0}) = Y_{i} (t - μ, Y_{0}, X_{0}), \forall t \geq α, t \geq μ and t \geq τ

holds for all $X_{0}, Y_{0} \in D^{n}$ , any $i \leq d$ , where $X_{i} (t, X_{0}, Y_{0}) and Y_{i} (t, Y_{0}, X_{0})$ are the ith state of $X (t, X_{0}, Y_{0}) and Y (t, Y_{0}, X_{0})$ , and $μ$ is the synchronization time delay.

Setting $x (t) = ⋉_{i}^{n} x_{i} (t) and y (t) = ⋉_{i = 1}^{n} y_{i} (t)$ , BN models (2a) and (2b) are converted into the following algebraic forms

x (t + 1) = F x (t) y (t)

(3a)

y (t + 1) = G x (t) y (t)

(3b)

where $F \in L_{2^{n} \times 2^{2 n}} and G \in L_{2^{n} \times 2^{2 n}}$ are the transition matrices of BNs (2a) and (2b), respectively. In the following section, some sufficient and necessary conditions for delay partial synchronization of BNs (2a) and (2b) are calculated.

Main results

Before presenting some sufficient and necessary conditions for delay partial synchronization, some auxiliary results are introduced.

Lemma 3

Let $x^{1} = ⋉_{i = 1}^{b} x_{l_{i}} and x^{2} = ⋉_{j = 1}^{b_{1} - 1} x_{j} ⋉_{j = b_{1} + 1}^{b_{2} - 1} x_{j} \dots ⋉_{j = b_{a} + 1}^{h} x_{j}$ . Then, $x = T (x^{1} x^{2})$ , where $x \in Δ, T = T^{'} \otimes I_{2^{h - b_{r}}} and T^{'} = ⋉_{i = 1}^{a} T_{[2, 2^{b_{i} - 1 + a - i}]} W_{[2, 2^{b_{i} - 1 + a - i}]}, b = 1, 2, 3, \dots$ .

Proof

According to formula (1)

\begin{matrix} x = ⋉_{j = 1}^{b_{a} - 1} x_{j} ⋉ x_{b_{a}} ⋉_{j = b_{a} + 1}^{h} x_{j} = T_{[2, 2^{b_{a} - 1}]} ⋉ x_{b_{a}} ⋉_{j = 1}^{b_{a} - 1} x_{j} ⋉_{j = b_{a} + 1}^{h} x_{j} \\ = T_{[2, 2^{b_{a} - 1}]} ⋉ x_{b_{a}} ⋉_{j = 1}^{b_{a - 1} - 1} x_{j} ⋉ x_{b_{a}} ⋉_{j = b_{a - 1} + 1}^{b_{a} - 1} x_{j} ⋉_{b_{a} + 1}^{h} x_{j} \\ = \dots \\ = T^{'} ⋉ (x^{1} x^{2}) \end{matrix}

Since $T'$ is a $2^{b_{a} \times b_{a}}$ logical matrix, by Lemma 1, we can have

T' (x^{1} x^{2}) = (T' \otimes I_{2^{h - b_{a}}}) (x^{1} x^{2})

which completes the proof of Lemma 3.

Lemma 4

Let $x^{1} (t, x_{0}, y_{0}) and y^{1} (t, y_{0}, x_{0})$ be the initial states of the trajectories of $x^{1} (t) and y^{1} (t)$ . Then

x^{1} (t, x_{0}, y_{0}) = SV M^{t - 1} x_{0} y_{0}

(4)

y^{1} (t, y_{0}, x_{0}) = SR M^{t - 1} x_{0} y_{0}

(5)

where $x_{1} (t) \in Δ and y_{1} (t) \in Δ$ , $S = (E_{d})^{n - r} W_{[2^{r}, 2^{n - r}]} T' \in L_{2^{r} \times 2^{n}}$ and $M = V (I_{2^{2 n}} \otimes R) Φ_{2 n} \in L_{2^{2 n} \times 2^{2 n}}$ , where $T'$ is illustrated in Lemma 3, and $Φ_{2 n}$ is explained in Lemma 1.

Proof

According to the dummy matrix, the following equation holds

x^{1} (t) = {(E_{d})}^{n - r} W_{[2^{r}, 2^{n - r}]} x^{1} (t) x^{2} (t)

(6)

$T$ is invertible, according to formula (5)

x^{1} (t) = {(E_{d})}^{n - r} W_{[2^{r}, 2^{n - r}]} T^{- 1} x (t) = Sx (t)

(7)

According to formula (3a), we can yield

x^{1} (t) = SVx (t - 1) y (t - 1)

(8)

According to formulas (3a) and (3b)

x (t - 1) y (t - 1) = M x (t - 2) y (t - 2)

(9)

By iterating formula (9)

x (t - 1) y (t - 1) = M^{t - 1} x_{0} y_{0}

(10)

According to formulas (10) and (8), formula (4) holds. The proof process of formula (5) is similar to that of formula (4). The proof of formula (5) is omitted.

The algebraic criterion of the delay partial synchronization for BNs (2a) and (2b) is presented in the following.

Theorem 1

Logic functions (2a) and (2b) can be transformed into algebraic representations of the network models (3a) and (3b). Then, the BNs (2a) and (2b) realize delay partial synchronization if and only if there is a positive integer z such that

SV M^{z - 1} = SR M^{z - 1}

(11)

Proof

Sufficiency: when formula (11) holds for z, then $x^{1} (z, x_{0}, y_{0}) = y^{1} (z, y_{0}, x_{0})$ for every $x_{0} \in Δ_{2^{n}} and y_{0} \in Δ_{2^{n}}$ . According to Lemma 4, for any positive integer $τ$ and any $x_{0} and y_{0}$ , the following formulas can be derived

x^{1} (k + τ) = SV M^{k - 1} M^{τ} x_{0} y_{0}

According to formula (11)

SV M^{k - 1} M^{τ} x_{0} y_{0} = SR M^{k - 1} M^{τ} x_{0} y_{0}

x^{1} (k + τ) = y^{1} (k + τ)

(12)

Therefore, formulas (2a) and (2b) can achieve the partial synchronization.

2. Necessity: according to Definition 2, if formulas (2a) and (2b) achieve delay partial synchronization, there exists a positive integer k such that $x^{1} (k, x_{0}, y_{0}) = y^{1} (k, y_{0}, x_{0})$ for any $x_{0} and y_{0}$ . According to Lemma 4, the following formula holds

S V M x_{0} y_{0} = S R M x_{0} y_{0}

(13)

which shows the validity of formula (11). The proof is completed. In the following section, the upper bound k is derived by another synchronization method.

Let a new variable $Q (t)$ be $Q (t) = x (t) y (t)$ . Then, according to formula (9)

Q (t + 1) = M Q (t)

(14)

In the following, the preliminary criterion is presented.

Lemma 5

If $x_{i} (t) = y_{i} (t)$ is the delay partial synchronization node, then $Q (t) \in Clo (B)$ , where $B = T (I_{2^{n}} \otimes T) W_{[2^{r}, 2^{n}]} Φ_{r} \in L_{2^{2 n} \times 2^{2 n - r}}$ . And each column of the matrix $B$ is different and so $| C o l (B) | \overset{Δ}{=} p = 2^{2 n - r}$ .

Proof

The map from $(Δ_{2})^{r}$ to $Δ_{2^{r}}$ is bijective, so $x^{1} (t) = y^{1} (t) .$ According to Lemma 1 and Lemma 3, then

\begin{matrix} Q (t) = T x^{1} (t) x^{2} (t) T y^{1} (t) y^{2} (t) \\ = T (I_{2^{n}} \otimes T) x^{1} (t) x^{2} (t) y^{1} (t) y^{2} (t) \\ = T (I_{2^{n}} \otimes T) W_{[2^{r}, 2^{n}]} y^{1} (t) x^{1} (t) x^{2} (t) y^{2} (t) \\ = B x^{1} (t) x^{2} (t) y^{2} (t) \end{matrix}

(15)

which shows $Q (T) \in C o l (B)$ .

By setting $\bar{B} = W (I_{2^{n}} \otimes W) W_{[2^{r}, 2^{n}]} \in L_{2^{2 n} \times 2^{2 n}}$ , $C o l (\bar{B}) = Δ_{2^{2 n}}$ due to $W$ is invertible. Each column of the matrix $B = \bar{B} Φ_{r}$ is different by the definition of $Φ_{r}$ , so $| C o l (B) | \overset{Δ}{=} q = 2^{2 n - r}$ .

Remark 2

On the basis of Lemma 5, two mutual delay-coupled BNs can achieve delay partial synchronization from the kth time step, then $Q (t) \in C o l (B)$ for all $t > k, t \geq μ, and t \geq τ$ . Hence, the set $C o l (B)$ is an essential factor for analyzing delay partial synchronization of BNs (2a) and (2b). Lemma 5 gives a method to deduce the delay partial synchronization state set $C o l (B)$ .

Remark 3

If BNs (2a) and (2b) achieve delay complete synchronization, then $B = Φ_{n}$ in Lemma 5, and $C o l (B) = C o l (Φ_{n}) = {δ_{2^{2 n}}^{1}, δ_{2^{2 n}}^{2^{n} + 2}, \dots, δ_{2^{2 n}}^{2^{2 n}}}$ . So, in general

| C o l (B) | = q \geq 2^{n} = | C o l (Φ_{n}) |

Therefore

| C o l (B) | \supseteq | C o l (Φ_{n}) |

So, in the following, $C o l (B) = {δ_{2^{2 n}}^{j_{1}}, δ_{2^{2 n}}^{j_{2}}, \dots, δ_{2^{2 n}}^{j_{q}}}$ with $j_{1} < j_{2} < \dots < j_{q} and j_{1} = 1, j_{q} = 2^{2 n}$ .

Theorem 2

The delay partial synchronization occurs between BNs (2a) and (2b) if and only if there exists a positive integer k such as

C o l (M^{k}) \subseteq {δ_{2^{2 n}}^{j_{1}}, δ_{2^{2 n}}^{j_{2}}, \dots, δ_{2^{2 n}}^{j_{q}}}

(16)

If k₀ is the smallest positive integer and satisfies formula (16), then

k_{0} \leq k = min {d : d \geq 1, M^{d} = M^{r} for some r > d}

(17)

Proof

The proof is composed of sufficiency and necessity:

Sufficiency: if formula (16) holds for some positive integer k, then $Q (k) = M^{k} Q (0) \in C o l (B)$ . And for any positive integer $v$ , the following formula (18) holds

Q (k + v) = M^{k} M^{v} Q (0)

(18)

Since $M^{k}, M^{v} \in L_{2^{2 n} \times 2^{2 n}}$ , $C o l (M^{k} M^{v}) \in C o l (B)$ . Hence, for all $Q (0) \in Δ_{2^{2 n}}$ , the following formula holds

Q (k + v) \in C o l (M^{k} M^{v}) \subseteq C o l (B)

So, BNs (2a) and (2b) achieve delay partial synchronization.

2. Necessity: it holds based on the proof of sufficiency obviously.

In the following section, the validity of formula (16) is illustrated.

The matrix $M$ has only $N \overset{Δ}{=} 2^{2 n} \times 2^{2 n}$ possible independent value. So, if a sequence of N + 1 matrices is constructed as ${M^{0}, M^{1}, \dots, M^{N}}$ , there exist two equal matrices, then $k$ is well-defined. If $k_{0} > k$ , let

d = min {i \geq 0 : M^{k} = M^{k + i + 1}}

Then, there is an integer $f$ such that $k \leq f \leq k + l and M^{f} = M^{k_{0}}$ . Notice

C o l (M^{k + l + 1}) = C o l (M^{k}) \subseteq C o l (M^{f}) \subseteq C o l (M^{k})

which can show

C o l (M^{k}) = C o l (M^{f}) = C o l (M^{k_{0}}) \subseteq C o l (B)

This contradicts the minimality of $k_{0}$ . The proof is completed.

Illustrative example

The time delay may take different values as required. In this example, model delay and synchronization delay are 1. However, the model delay and synchronization delay may take other values as required. We consider the following BNs (19a) and (19b) as mutual delay-coupled BNs

{\begin{matrix} x_{1} (t + 1) = - x_{2} (t) \\ x_{2} (t + 1) = x_{1} (t) \land x_{3} (t) \\ x_{3} (t + 1) = x_{3} (t) \lor y_{1} (t - τ) \end{matrix}

(19a)

{\begin{matrix} y_{1} (t + 1) = - y_{3} (t) \\ y_{2} (t + 1) = - y_{1} (t) \land y_{2} (t) \\ y_{3} (t + 1) = - y_{3} (t) \lor x_{1} (t - τ) \end{matrix}

(19b)

where $τ$ is the delay-coupled time. If there are BNs (19a) and (19b) with three nodes, among the three nodes, two nodes achieve synchronization, $τ = μ = 1$ , then $k_{0} \leq 4$ . BNs (19a) and (19b) can be expressed as follows

x (t + 1) = Vx (t) y (t)

(20a)

y (t + 1) = Rx (t) y (t)

(20b)

where

\begin{matrix} V = δ_{8} [\begin{matrix} 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 5, 5, \\ 3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 \end{matrix}] \\ R = δ_{8} [\begin{matrix} 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 5, 5, \\ 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1 \end{matrix}] \end{matrix}

M = δ_{64} [\begin{matrix} 1, 1, 1, 1, 34, 53, 4, 40, 40, 39, 36, 33, 9, 4, 56, 58, 52, 53, 32, 63, 63, 25, 32, 42, \\ 42, 8, 8, 25, 25, 8, 9, 61, 61, 62, 62, 63, 63, 58, 58, 56, 56, 54, 54, 52, 52, 53, 53, 58, 55, \\ 8, 57, 57, 5, 15, 1, 40, 40, 12, 12, 13, 13 \end{matrix}]

Therefore, the nodes $x_{i} and y_{i} (i = 1, 3)$ achieve the delay partial synchronization. Figure 1 illustrates the state trajectory of BNs (19a) and (19b) when $X_{0} = (0, 1, 0) and Y_{0} = (1, 0, 1)$ , which shows delay partial synchronization from the fourth step.

Figure 1.

The state trajectory of the BNs (19a) and (19b) when $X_{0} = (0, 1, 0) and Y_{0} = (1, 0, 1)$ .

Conclusion

In this paper, the mutual delay-coupled BNs have been proposed, and the delay partial synchronization of mutual delay-coupled BNs has been investigated based on the STP technique. The sufficient and necessary conditions for delay partial synchronization of the mutual delay-coupled BNs have been established. The mutual delay-coupled BNs realize the delay partial synchronization when STP of the nodes of the two mutual delay-coupled BNs is equal. The upper bound of time for delay partial synchronization has been deduced by recursive relation and proof by contradiction. An example is provided to verify the efficiency of the theoretical analysis.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Q.W. got support from the National Natural Science Foundation of China (no. 61640223), the 13th five-year science and technology project of Education Department of Jilin Province (no. JJKH20190646KJ), and the Doctor Start-up capital of the Beihua University (no. 199500096); C.-J.X. got support from the key scientific research projects in the Science and Technology Development Plan of Jilin Province (no. 20160204033GX); and W.S. got support from the 13th five-year science and technology project of Education Department of Jilin Province (no. JJKH20190647KJ).

ORCID iD

Qiang Wei

References

Kauffman

. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 1969; 22: 437–467.

Dnyane

Puntambekar

Gadgil

. Method for identification of sensitive nodes in Boolean models of biological networks. IET Syst Biol 2018; 12: 1–6.

Bloomingdale

Nguyen

Niu

, et al. Boolean network modeling in systems pharmacology. J Pharmacokinet Phar 2018; 45: 159–180.

Luo

Wang

. Algebraic representation of asynchronous multiple-valued networks and its dynamics. IEEE ACM Trans Comput Biol 2013; 10(4): 927–938.

Zheng

Alsaadi

. Algebraic formulation and topological structure of Boolean networks with state-dependent delay. J Comput Appl Math 2019; 350: 87–97.

. Pinning control design for the synchronization of two coupled Boolean networks. IEEE Trans Circuits Syst II Express Briefs 2016; 63: 309–313.

Liu

, et al. Pinning control for the disturbance decoupling problem of Boolean networks. IEEE Trans Automat Contr 2017; 62(12): 6595–6601.

Ding

Wang

. Set stability and synchronization of logical networks with probabilistic time delays. J Frankl Inst 2018; 355(15): 7735–7748.

Cheng

Zhang

, et al. Controllability of Boolean networks via mixed controls. IEEE Contr Syst Letters 2018; 2: 254–259.

10.

Chen

Liang

Huang

, et al. Synchronization of arbitrarily switched Boolean Networks. IEEE Trans Neural Netw Learn Syst 2017; 28: 612–619.

11.

Sun

Liu

, et al. Variable structure controller design for Boolean networks. Neural Networks 2018; 97: 107–115.

12.

Liu

, et al. Synchronization of switched Boolean networks with impulsive effects. Inter J Biomat 2018; 11: 1850080.

13.

Zhu

Liu

, et al. On the optimal control of Boolean control networks. SIAM J Contr Optim 2018; 56(2): 1321–1341.

14.

Taichi

. Evolution of activity-dependent adaptive Boolean networks towards criticality: an analytic approach. J Complex Netw 2018; 6: 914–926.

15.

Zhang

Wang

Lin

. Synchronization of asynchronous switched Boolean network. IEEE ACM Trans Comput Biol 2015; 12(6): 1449–1456.

16.

Huang

Chen

Ren

, et al. Analysis and pinning control for generalized synchronization of delayed coupled neural networks with different dimensional nodes. J Frankl Inst 2018; 355: 5968–5997.

17.

Botmart

Yotha

Niamsup

, et al. Hybrid adaptive pinning control for function projective synchronization of delayed neural networks with mixed uncertain couplings. Complexity 2017: 4654020.

18.

Guo

, et al. Weighted average pinning synchronization for a class of coupled neural networks with time-varying delays. Neural Process Letters 2017; 45: 95–108.

19.

Niamsup

Botmart

Weera

. Modified function projective synchronization of complex dynamical networks with mixed time-varying and asymmetric coupling delays via new hybrid pinning adaptive control. Adv Diff Equ 2017; 2017: 124.

20.

Shi

Zhu

Zhong

, et al. Function projective synchronization of complex networks with asymmetric coupling via adaptive and pinning feedback control. ISA Trans 2016; 65: 81–87.

21.

. Function projective synchronization in complex dynamical networks with or without external disturbances via error feedback control. Neurocomputing 2016; 173: 1443–1449.

22.

Cai

Lei

Liu

. Outer synchronization between two hybrid-coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control. Complexity 2016; 21: 593–605.

23.

Zhao

Yao

. Synchronization of coupled neutral-type delay partial differential systems. Circ Syst Signal Process 2016; 35: 1–16.

24.

Yang

Liu

. Synchronization of fractional-order and integer-order chaotic (hyper-chaotic) systems with different dimensions. Adv Diff Equ 2017; 2017: 344.

25.

Chen

Shao

Shi

, et al. Disturbance-observer-based robust synchronization control for a class of fractional-order chaotic systems. IEEE Trans Circuits Syst II Express Briefs 2017; 64: 417–421.

26.

Gao

Chen

Başar

. Controllability of conjunctive Boolean networks with application to gene regulation. IEEE Trans Contr of Netw Syst 2017; 5: 770–781.

27.

Colley

Keller

Halpern

. Working memory and auditory imagery predict sensorimotor synchronisation with expressively timed music. Tran Inst Meas Contr 2018; 71: 1781–1796.

28.

Wei

Xie

. Synchronization in mutual-coupled temporal Boolean networks. Tran Inst Meas Contr 2018; 40: 2211–2216.

29.

Zhong

Huang

, et al. Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks. IEEE Trans Cybernetics 2017; 47: 3482–3493.

30.

Chen

Sun

Liu

. Partial stability and stabilization of Boolean networks. Int J Syst Sci 2016; 47: 2119–2127.

31.

Wei

Xie

Liang

. Delay synchronization of temporal Boolean networks. AIP Adv 2016; 6: 67–103.

32.

. Anti-synchronization of two coupled Boolean networks. J Frankl Inst 2016; 353: 5013–5024.

33.

Shen

. State feedback controller design for the synchronization of Boolean networks with time delays. Physica A Stat Mech Appl 2018; 490: 1267–1276.

34.

Ding

, et al. Leader-follower consensus of multiagent systems with time delays over finite fields. IEEE Trans Cybernetics 2019; 49(8): 3203–3208.

35.

Chen

Liang

, et al. Synchronization for the realization-dependent probabilistic Boolean networks. IEEE Trans Neural Netw Learn Syst 2018; 29: 819–831.

36.

Chen

Liang

. Partial synchronization of interconnected Boolean networks. IEEE Trans Cybernetics 2017; 47: 258–266.

37.

Zhang

Cerdeira

, et al. Partial synchronization and spontaneous spatial ordering in coupled chaotic systems. Physical Rev 2001; 63: 026211.

38.

Liu

Chen

, et al. Function perturbations on singular Boolean networks. Automatica 2017; 84: 36–42.

Delay Partial Synchronization of Mutual Delay Coupled Boolean Networks

Abstract

Keywords

Introduction

Preliminaries

Definition 1

Remark 1

Lemma 1

Lemma 2

The model description

Definition 2

Main results

Lemma 3

Proof

Lemma 4

Proof

Theorem 1

Proof

Lemma 5

Proof

Remark 2

Remark 3

Theorem 2

Proof

Illustrative example

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References