Neural networks letterComplete synchronization of temporal Boolean networks
Introduction
With the development of systems biology, genetic regulatory networks received much attention. Genetic regulatory networks can be modeled in quite different ways, such as differential equations (Shen, Wang, and Liang, 2011, Wang et al., 2008), Bayesian networks (Martínez-Rodríguez, May, & Vargas, 2008), and Boolean networks (BNs), (Kauffman, 1969).
Boolean network (BN) was first introduced by Kauffman (1969). In a Boolean network, 0 (or 1) corresponds to the inactive (or active) state of the gene. Every Boolean variable updates its state according to a Boolean function, which is a logical function. Since Boolean networks can provide a general description of the behavior of many living organisms at system level, they have attracted much attention in recent years. The topological structure of Boolean networks including attractors and transient time has been studied by Drossel, Mihaljev, and Greil (2005) and Samuelsson and Troein (2003), etc. Recently, the study has been extended to Boolean control networks, such as stability, controllability, observability, optimal control, etc. (Cheng and Qi, 2009, Cheng, Qi, Li, and Liu, 2011, Li and Sun, 2011, Li and Sun, 2012, Li et al., 2011, Li and Wang, 2012, Zhao et al., 2011, Zhao et al., 2010).
The synchronization problem consists of making two systems oscillate in a synchronized manner, which can explain many natural phenomena. In the past decades, synchronization phenomena have been investigated experimentally, numerically and theoretically by many researchers; see e.g. Huang and Feng (2009), Shen, Wang, and Liu, 2011, Shen et al., 2012, Su, Wang, and Lin (2009) and Zhu and Cui (2010). Recently, interest has extended to synchronization of Boolean networks (BNs) (Li and Chu, 2012, Li et al., 2012), mostly due to their potential applications in biology, physics, etc. For example, the study of synchronized BNs could provide useful information on the coevolution of several biological species whose genetic dynamics influence each other (Morelli & Zanette, 2001).
The complete synchronization of two delay-free deterministic BNs has been studied in Li and Chu (2012). Later, the study was further extended in Li et al. (2012), yielding complete synchronization between two deterministic BNs with time delays coupled in the drive-response configuration. However, we can note that the time delays of every Boolean variables in Li et al. (2012) are the same. Compared with (Li et al., 2012), temporal Boolean network has more complex structure because temporal Boolean networks allow the time delays in each Boolean variable to be different. Hence, it is meaningful and challenging to investigate complete synchronization of two temporal Boolean networks. However, there has been no result studying the complete synchronization of two temporal Boolean networks, to the best of our knowledge.
Motivated by the above, in this letter, we focus on theoretical framework and strict analysis of complete synchronization between two temporal BNs coupled in the drive-response configuration. The main tool in this letter is the semi-tensor product of matrices, which can be used in many fields (Cheng, Qi, and Li, 2011, Wang et al., 2012). Based on the semi-tensor product of matrices, the algebraic representation of a temporal BN can be obtained. Then necessary and sufficient conditions for complete synchronization of two temporal BNs are obtained. Moreover, the upper bound to check the criterion is given.
The rest of the letter is organized as follows. Section 2 gives a brief review for the semi-tensor product of matrices. In Section 3, main results on synchronization of drive-response temporal BNs are presented. Necessary and sufficient conditions are obtained. Examples are given to show the non-emptiness of the obtained results in Section 4. Finally, Section 5 presents the conclusions.
Section snippets
Preliminaries
In this section, we give a brief introduction of the semi-tensor product (STP) and algebraic representation of Boolean functions which are proposed first by Daizhan Cheng and his colleagues (Cheng and Qi, 2010, Cheng, Qi, and Li, 2011).
Main results
In this section, we will establish some necessary and sufficient conditions for complete synchronization between two temporal BNs.
Example
Assume that the drive BN is given as
By letting , where , we have where are the structure matrices of logical function “” and “” respectively. Then where . We can verify that
Hence, the algebraic
Conclusions
Complete synchronization of two temporal Boolean networks coupled in the drive-response configuration has been investigated in this letter. We have given the necessary and sufficient conditions based on the algebraic representation of Boolean networks. Moreover, the upper bound to check the criterion has been obtained. Examples have been worked out to illustrate the proposed results.
Temporal Boolean networks are very complex; many control related problems, such as optimal control, etc., remain
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant 61174039, the China Postdoctoral Science Foundation funded project 2012M520849 and the Research Fund for the Doctoral Program of China20120074110021.
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