Research On Synchronization Of Coupled Boolean Networks

As the Human Genome Project has been raised,a new view of biology,called the systems biology,is emerging.That is,the researchers do not study individual cells,genes or proteins one at a time.Rather,they investigate the behavior and relationship of these organic matter as a whole.The group of cells,genes or proteins is called a cellular network.Undoubtedly,the Boolean network is a kind of the most effective model for the cellular networks,since it is convenient to use Boolean networks to describe,analyze,and simulate the cellular networks.The Boolean network is introduced by Kauffman for the first time and attracts a quick attention from many fields.It has been successfully applied to many other fields,for example social system,system control,circuit design,game theory,and so on.However,as we know,it is difficult to handle logical systems due to the shortage of efficient tools.In the early 2000' s,a powerful mathematical tool:semi-tensor product of matrices(STP)appeared,which was proposed and developed by Cheng et al.Later,STP was introduced by them into Boolean networks and significant and substantial progress has been made.Many excellent results about Boolean networks have been derived via STP and published on many top journals,but the theory of Boolean networks needs to be enriched and extended,and even there are still many problems unresolved,for example the synchronization problem of coupled Boolean networks.For the master-slave Boolean networks with exogenous input,the general synchronizability condition has not been derived or the general design method for a synchronizing controller has not been developed.On the other hand,although STP is an effective tool in dealing with logical systems,the greatest limitation of STP is the increase of computational complexity because of the exponential expan-sion of matrix dimension.So an interesting and significant topic is how to reduce the computational complexity involved in the STP technique when we study logical systems.This dissertation will discuss the topic and meanwhile solve the synchro-nization problem of master-slave networks.The main contributions are summarized as follows:(1)For an array of BNs coupled in the leader-follower configuration,a new method for analyzing the synchronization problem is proposed.Firstly,under the algebraic framework of Boolean networks,an error system is constructed by using the leader-follower Boolean networks.This error system is a switched system with the state of the leader Boolean network as the switching signal.Then the synchronization of leader-follower Boolean networks is equivalently transformed into a stabilization problem.Through the analysis of the error system,a new synchronization criterion is derived for leader-follower Boolean networks.Compared with the related results in the literature,our criterion reduces the computational complexity markedly.In addition,a constructive design method is provided for a synchronizing follower Boolean network.(2)A new concept:core input-state cycle is proposed to solve the state synchro-nization problem of master-slave Boolean networks.The master-slave Boolean network is an extended drive-response Boolean network with exogenous input.For the state synchronization of master-slave Boolean networks,a necessary and sufficient existence condition for a synchronizing state feedback controller is derived by using core input-state cycles.Furthermore,a constructive de-sign method for designing a feasible state feedback controller is developed.Compared with the existing related results,the above results is of obvious advantages.Finally,the above results are further extended into the output synchronization of master-slave Boolean networks.(3)A synchronization problem about a more complex kind of drive-response Boolean networks has been considered,where the drive system is a periodical time variable Boolean network.Since the periodical time variable Boolean network is of many properties different from those of time invariable Boolean networks and is more complex than time invariable Boolean networks,few results are reported about this kind of coupled Boolean network.This disser-tation gives a necessary and sufficient synchronization criterion for the above coupled Boolean network and provides a constructive design method for a synchronizing response Boolean network.(4)Under the algebraic framework of logical systems,a reverse transfer method is proposed to investigate the stabilization of k-valued logical control networks(LCNs)which are a more general kind of model than Boolean control net-works.Due to the specificity and super-exponential complexity of the existing results about open-loop stabilization of Boolean networks or multi-valued LC-Ns,the open-loop stabilization of k-valued LCNs is still a challenging problem.This dissertation derives a new necessary and sufficient criterion and provides a corresponding algorithm for the open-loop stabilization of k-valued LCNs.Through theoretical analysis,it can be shown that the criterion above reduces the computational complexity of the existing related results markedly.Later,the reverse transfer method is applied to the feedback stabilization of k-valued LCNs.Then a convenient and effective method for computing the largest sta-bility domain of a fixed point is given,and a constructive design approach to a stabilizing state feedback controller is developed.Finally,a concluding remark is provided and the development direction for Boolean networks is proposed.Furthermore,the prospects of the further investiga-tion are given.