Synchronization of Coupled Boolean Networks With Different Update Scheme

In this paper, the synchronization of coupled Boolean networks with asynchronous update scheme is investigated. First, the asynchronous Boolean dynamics is transformed into a linear expression with the semi-tensor product technique. Then, the synchronization cycle is proposed to establish the synchronization conditions of asynchronous Boolean networks. Furthermore, an algorithm is given, which can be used to quickly check whether there exists the synchronization cycle in the coefficient matrix of the Boolean function. Finally, an example is provided to verify the efficiency of the theoretical analysis.


I. INTRODUCTION
Since Boolean network (BN) can effectively capture the dynamics behaviors in complex systems, it has been extensively used for analyzing the gene regulatory network [1]- [14]. In 1969, the Boolean network was first used to describe the gene regulatory network by Kauffman [15]. In the model, a biological gene is abstractly represented as a node with logic values 0 and 1, and the interaction of genes is expressed by a Boolean function [16].
During the past few years, considerable attention has been paid to the synchronization phenomenon and many important results have been obtained [17]- [24]. After the logical expression of the BN was transformed into an algebraic form using the semi-tensor product (STP) technique by Cheng and Qi [25], Cheng et al. [26], the synchronization of Boolean networks has been important and meaningful topic [27]- [39]. Li and Chu finished the complete synchronization of BNs in [40]. The core input-state cycle was proposed to study the coupled synchronization of BNs [41]. Chen et al. [42] investigated the synchronization problem for switched BNs under arbitrary switching signals. Using the algebraic state space representation method, the synchronization of BNs The associate editor coordinating the review of this manuscript and approving it for publication was Jianquan Lu . with probabilistic time delays was achieved in [43]. The synchronization of Boolean control networks with impulsive disturbances was studied in [44]. The state feedback controller was designed to achieve the synchronization of coupled BNs with time delays [45]. With the event-triggered control design, Li et al. [46] investigated the synchronization of switched k-valued logical control networks. The globally exponential synchronization of BNs was considered in [47]. Zhong et al. [48] designed an output feedback stabilizer to achieve the global stabilization of Boolean networks, even without using the state transition matrix.
In the above Boolean network models, the update schemes are all synchronous. However, many biological phenomena in the real world are asynchronous. For example, asynchronous dynamics is generated by ''Type A'' spiking neurons. In addition, it has been indicated that asynchronous modeling is a realistic approach to biological information processing in [49]. With the STP technique, Luo and Wang [16], Luo et al. [50] investigated the dynamics of asynchronous multiple-valued networks and the controllability of asynchronous Boolean network (ABN). Zhang et al. [51], [52] achieved the complete synchronization between an ABN and a response BN, with the assumption that the linear representations of ABN was determined by a Boolean function versus time. To date, there have been no reports on the VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ synchronization of coupled Boolean Network models, with the condition that asynchronous update sequence is designated or known. This issue is investigated in our work. The concept of the synchronization cycle is proposed to establish the conditions of synchronization for BNs with asynchronous update sequence, and an algorithm is designed, which can be used to quickly check whether there exists the synchronization cycle in the coefficient matrix of the Boolean function. Finally, an example is provided to verify the efficiency of the theoretical analysis.

II. PRELIMINARIES
Some main definitions and theorems are first introduced as follows. Definition 1 ( [25]): If matrices X ∈ R r×s and Y ∈ R p×q , then STP of X andY is defined as Eq. (1).
where a is the least common multiple of s and p, ⊗ represents the Kronecker product, and I n is an n-dimensional identity matrix.

III. PROBLEM FORMULATION
According to the asynchronous update scheme [16], an ABN with n nodes can be expressed as Eq. (4).
where Y i (t) is the state variable of node i, Y i ∈ D 2 , and F i is a logic function of the i-th node, which is selected at random for an update. Therefore, for the ABN (4) with n nodes, there are 2 n different selection strategies in every update process. At time step t + 1, the values of nodes that are selected to be updated, are determined by the logic function F i and the input values of Y i (t). While the remainder of the nodes keep the values at time step t. In addition, the response BN coupled with ABN (4) is defined as Eq. (5).
In the following, the logical forms of ABN (4) and BN (5) are transformed into the linear expressions, respectively.
For the ABN (4), suppose that are the values of nodes at time step t, and C i is the structure matrix of the logical function F i (Y 1 (t), . . . , Y n (t)). If the i-th node is updated, then from Lemma1, we obtain the following: , and M r = δ 4 [1,4]. If 2 n nodes are all updated, from Lemma1, we obtain the following: where C 1 (I 2 n ⊗ C 2 ) n (I 2 n ⊗ C 3 ) n . . . (I 2 n ⊗ C n ) n is called the transition matrix. Then, there are 2 n different transition matrices for the ABN (4) with n nodes. Let transition matrices L i F (i = 1, 2, · · · , 2 n ), and set that all transition matrices L i F constitute of a block matrix L F = [L 1 F L 2 F · · · L 2 n F ]. Therefore, the linear expression of the ABN (4) is described as Eq. (8).
where s(t) is called the transition selection matrix, and set Accordingly, the linear expression of the BN (5) can be obtained as follows.
where L G is the coefficient matrix and L G ∈ L 2 2n ×2 2n . This means that the complete synchronization of the ABN (4) and BN (5) can be converted into the one between ABN (8) and BN (9). The synchronization definition for the ABN (8) and BN (9) is given as follows.
Definition 3: The response BN (9) can achieve the complete synchronization with the drive ABN (8) if for any states Y i (t) and Z i (t), i ∈ [1, n], there exist a positive integer k and an asynchronous update sequence π , such that Y i (t) = Z i (t), when t ≥ k. 79320 VOLUME 8, 2020

IV. COMPLETE SYNCHRONIZATION OF ABNS
To describe the conditions of synchronization, some relevant definitions and descriptions are given as follows. First, we calculate the synchronized state set of ABN (8) and BN (9). Let X (t) = Y (t)Z (t), the following equations are established.
where L is called the coefficient matrix and s(t) is the transition selection matrix. From Definition 3, if the ABN (8) and BN(9) achieve complete synchronization from the k-th time step, then it can be hold that and t ≥ k, (12) and from Eq. (11), it can be derived that X (t, X (0)) = δ (i−1)2 n +i 2 2n , ∀t ≥ k. Therefore, the synchronized state set θ = {δ , then there exists a cycle in matrix L. Moreover, if all U l (l = j, . . . , (k − 1)) ⊆ θ , then the cycle is called the synchronization cycle (SC) and can be expressed as the following form.
Based on Definition 4, an algorithm is given, which can be used to quickly check whether there exists the synchronization cycle in a coefficient matrix, with an asynchronous update sequence π = {s(0), s(1), . . . , s(i), . . .}.
Step 1. Set i = 1 and check whether U 0 ⊆ θ or not. If yes, set = {U 0 } and go to step 2. Else set = {φ} and go to step 2.
Step 2. Calculate L b = Ls(t i ) L a , U i = Col(L b ) and L a = L b , then check whether U i ⊆ θ or not. If yes, go to step 3. Else set = {φ} and go to step 4.
Step 3. Check whether U i ∈ or not. If yes, go to step 5. Else, add U i to , and go to step 4.
Step 4. Check whether i < h or not. If yes, set i = i + 1 and go to step 2. Else, stop.
Step 5. Print , stop. There exists the synchronization cycle in matrix L.
For coupled BNs with random update scheme, it is hard to achieve synchronization without any precondition. Therefore, the assumption is given as follows.
Assumption 1: If there is an SC in the coefficient matrix L of the system (10), then, selection matrix s(t) will repeat the selection sequence s(j), s(j + 1), . . . , s(k − 1), s(k) of the SC from time step t = k.
Based on the concept of the synchronization cycle, the synchronization condition of the asynchronous Boolean networks is given as follows.
Theorem 1: When the ABN (8) satisfies Assumption 1, the ABN (8) and BN (9) will be synchronized if and only if for initial values X (0) ∈ 2 2n , there are a positive integer k and corresponding update sequence π , such that there is the synchronization cycle in matrix L.
Proof:(Necessity) From Theorem 1, if the ABN (8) and BN (9) are synchronized, then the system (11) can reach the state set θ after k time steps. Additionally, when time step t > k, the trajectory of the system (11) should stay in the set θ . Since the elements in the set θ are limited, there must be a repeat path in the trajectory of the system (11). From Definition 4, there must be a synchronization cycle in matrix L.
(Sufficiency) If there exists a synchronization cycle in matrix L, then, we can find a positive integer k and the corresponding update sequence π such that system (11) can reach the state set θ after k time steps. From Assumption 1, the trajectory of the system (11) will be stay forever in θ when t > k. Therefore, for t ≥ k, it can be verified that That is, the ABN (8) and BN (9) are completely synchronized for t ≥ k. The proof is complete.

V. NUMERICAL SIMULATIONS
An asynchronous Boolean network [51] is selected as an example, which is expressed as Eq. (15).
The response Boolean network is defined as Eq. (16).

VI. CONCLUSION
In this paper, the synchronization of asynchronous Boolean networks is investigated. The concept of the synchronization cycle was proposed to establish the synchronization conditions. Additionally, an algorithm was designed, which can be used to quickly check whether there exists a synchronization cycle in the coefficient matrix of the Boolean function. Finally, an example is provided and simulation results verify the efficiency of the theoretical analysis.
In further work, the computational complexity will be considered, and we will study the synchronization of asynchronous Boolean networks, without calculating the state transition matrix.