Complete synchronization of Boolean networks

Asymptotical Feedback Set Stabilization of Probabilistic Boolean Control Networks

In this article, we investigate the asymptotical feedback set stabilization in distribution of probabilistic Boolean control networks (PBCNs). We prove that a PBCN is asymptotically feedback stabilizable to a given subset if and only if (iff) it constitutes asymptotically feedback stabilizable to the largest control-invariant subset (LCIS) contained in this subset. We proposed an algorithm to calculate the LCIS contained in any given subset with the necessary and sufficient condition for asymptotical set stabilizability in terms of obtaining the reachability matrix. In addition, we propose a method to design stabilizing feedback based on a state-space partition. Finally, the results were applied to solve asymptotical feedback output tracking and asymptotical feedback synchronization of PBCNs. Examples were detailed to demonstrate the feasibility of the proposed method and results.

Robust Event-Triggered Control Invariance of Probabilistic Boolean Control Networks

In this brief, the robust control invariance problem of probabilistic Boolean control networks (PBCNs) is investigated by a class of event-triggered control (ETC), which is an intermittent control scheme in essential. By resorting to the semi-tensor product (STP) technique, a PBCN with ETC can be equivalently described in a form of an algebraic linear system. Based on which, a matrix testing condition is derived to judge whether the given set can be a robust ETC invariant set (RETCIS). Subsequently, a necessary and sufficient condition is developed for the existence of event-triggered controllers. Meanwhile, all feasible event-triggered controllers are designed for guaranteeing the given set to be an RETCIS. Finally, a biological example is employed to demonstrate the availability of theoretical results.

Optimal State Estimation of Boolean Control Networks With Stochastic Disturbances

This paper presents an investigation of the optimal estimation of state for Boolean control networks subject to stochastic disturbances. The disturbances are modeled as independently and identically distributed processes that are assumed to be both mutually independent and independent of the current and the historical states. An iterative algorithm is proposed to calculate the conditional probability distribution of the state given the output measurements. This algorithm is applied to the problems of minimum mismatching estimation and maximum posterior estimation of the state. An example is provided to illustrate the proposed results.

Local Synchronization of Interconnected Boolean Networks With Stochastic Disturbances

This paper is concerned with the local synchronization problem for the interconnected Boolean networks (BNs) without and with stochastic disturbances. For the case without stochastic disturbances, first, the limit set and the transient period of the interconnected BNs are discussed by resorting to the properties of the reachable set for the global initial states set. Second, in terms of logical submatrices of a certain Boolean vector, a compact algebraic expression is presented for the limit set of the given initial states set. Based on it, several necessary and sufficient conditions are derived assuring the local synchronization of the interconnected BNs. Subsequently, an efficient algorithm is developed to calculate the largest domain of attraction. As for the interconnected BNs with stochastic disturbances, first, mutually independent two-valued random logical variables are introduced to describe the stochastic disturbances. Then, the corresponding local synchronization criteria are also established, and the algorithm to calculate the largest domain of attraction is designed. Finally, numerical examples are employed to illustrate the effectiveness of the obtained results/ algorithms.

Cluster Synchronization for Neutral Stochastic Delay Networks via Intermittent Adaptive Control

This paper studies the problem of cluster synchronization at exponential rates in both the mean square and almost sure senses for neutral stochastic coupled neural networks with time-varying delay via a periodically intermittent pinning adaptive control strategy. The network topology can be symmetric or asymmetric, with each network node being described by neutral stochastic delayed neural networks. When considering the exponential stabilization in the mean square sense for neutral stochastic delay system, the delay integral inequality approach is used to circumvent the obstacle arising from the coexistence of random disturbance, neutral item, and time-varying delay. The almost surely exponential stabilization is also analyzed with the nonnegative semimartingale convergence theorem. Sufficient criteria on cluster synchronization at exponential rates in both the mean square and almost sure senses of the underlying networks under the designed control scheme are derived. The effectiveness of the obtained theoretical results is illustrated by two examples.

Leader-Follower Consensus of Multiagent Systems With Time Delays Over Finite Fields

This paper studies the leader-follower consensus of multiagent systems with time delays and switching topology over finite fields. First, an equivalent algebraic form is established for leader-follower multiagent systems with time delays over finite fields. Second, based on the algebraic form, a necessary and sufficient condition is presented for the finite-field leader-follower consensus with time delays. Third, the switching topology case is considered, and a new criterion is presented for the finite-field leader-follower consensus with time delays and switching topology. Finally, an example is worked out to illustrate the obtained results.

On the number of driver nodes for controlling a Boolean network when the targets are restricted to attractors

It is known that many driver nodes are required to control complex biological networks. Previous studies imply that O(N) driver nodes are required in both linear complex network and Boolean network models with N nodes if an arbitrary state is specified as the target. In order to cope with this intrinsic difficulty, we consider a special case of the control problem in which the targets are restricted to attractors. For this special case, we mathematically prove under the uniform distribution of states in basins that the expected number of driver nodes is only O(log₂N+log₂M) for controlling Boolean networks, where M is the number of attractors. Since it is expected that M is not very large in many practical networks, the new model requires a much smaller number of driver nodes. This result is based on discovery of novel relationships between control problems on Boolean networks and the coupon collector's problem, a well-known concept in combinatorics. We also provide lower bounds of the number of driver nodes as well as simulation results using artificial and realistic network data, which support our theoretical findings.

Observability of Boolean Control Networks Using Parallel Extension and Set Reachability

This brief reviews various definitions of observability for Boolean control networks (BCNs) and proposes a new one: output-feedback observability. This new definition applies to all BCNs whose initial states can be identified from the history of output measurements. A technique called parallel extension is then proposed to facilitate observability analysis. Furthermore, a technique called state transition graph reconstruction is proposed for analyzing the set reachability of BCNs, based on which new criteria for observability, single-input sequence observability, and arbitrary-input observability, are obtained. Using the proposed techniques, this brief proves that the problem of output-feedback observability can be recast as that of stabilizing a logic dynamical system with output feedback. Then, a necessary and sufficient condition for static output feedback observability is proposed. The relationships between the different definitions of observability are discussed, and the main results are illustrated with examples.

Synchronization for the Realization-Dependent Probabilistic Boolean Networks

This paper investigates the synchronization problem for the realization-dependent probabilistic Boolean networks (PBNs) coupled unidirectionally in the drive-response configuration. The realization of the response PBN is assumed to be uniquely determined by the realization signal generated by the drive PBN at each discrete time instant. First, the drive-response PBNs are expressed in their algebraic forms based on the semitensor product method, and then, a necessary and sufficient condition is presented for the synchronization of the PBNs. Second, by resorting to a newly defined matrix operator, the reachable set from any initial state is expressed by a column vector. Consequently, an easily computable algebraic criterion is derived assuring the synchronization of the drive-response PBNs. Finally, three illustrative examples are employed to demonstrate the applicability and usefulness of the developed theoretical results.

Controllability and Synchronization Analysis of Identical-Hierarchy Mixed-Valued Logical Control Networks

This paper investigates the controllability and synchronization problems for identical-hierarchy mixed-valued logical control networks. The logical network considered is hierarchical, and Boolean network is a special case of logical network. Here, identical-hierarchy means that there are identical number of nodes in each layer of logical network and corresponding nodes have the same dimension for any two layers of logical networks. Meanwhile, in each layer of logical networks, the dimensions of nodes are distinct, and it is called a mixed-valued logical network. First, the controllability problem is investigated and two notions of controllability are presented, i.e., group-controllability and simultaneously-controllability. By resorting to Perron-Frobenius theorem, some necessary and sufficient criteria are obtained to guarantee group-controllability and simultaneously-controllability, respectively. Second, based on the algebraic representation of the studied model, synchronization problems are analytically discussed for two types of controls, i.e., free control sequences and state-output feedback control. Finally, two numerical examples are presented to show the validness of our main results.

Synchronization of Arbitrarily Switched Boolean Networks

This paper investigates the complete synchronization problem for the drive-response switched Boolean networks (SBNs) under arbitrary switching signals, where the switching signals of the response SBN follow those generated by the drive SBN at each time instant. First, the definition of complete synchronization is introduced for the drive-response SBNs under arbitrary switching signals. Second, the concept of switching reachable set starting from a given initial state set is put forward. Based on it, a necessary and sufficient condition is derived for the complete synchronization of the drive-response SBNs. Last, we give a simple algebraic expression for the switching reachable set in a given number of time steps, and two computable algebraic criteria are obtained for the complete synchronization of the SBNs. A biological example is given to demonstrate the effectiveness of the obtained main results.

Partial Synchronization of Interconnected Boolean Networks

This paper addresses the partial synchronization problem for the interconnected Boolean networks (BNs) via the semi-tensor product (STP) of matrices. First, based on an algebraic state space representation of BNs, a necessary and sufficient criterion is presented to ensure the partial synchronization of the interconnected BNs. Second, by defining an induced digraph of the partial synchronized states set, an equivalent graphical description for the partial synchronization of the interconnected BNs is established. Consequently, the second partial synchronization criterion is derived in terms of adjacency matrix of the induced digraph. Finally, two examples (including an epigenetic model) are provided to illustrate the efficiency of the obtained results.

Synchronization of Asynchronous Switched Boolean Network

In this paper, the complete synchronizations for asynchronous switched Boolean network with free Boolean sequence controllers and close-loop controllers are studied. First, the basic asynchronous switched Boolean network model is provided. With the method of semi-tensor product, the Boolean dynamics is translated into linear representation. Second, necessary and sufficient conditions for ASBN synchronization with free Boolean sequence control and close-loop control are derived, respectively. Third, some illustrative examples are provided to show the efficiency of the proposed methods.

Synchronization Analysis of Master-Slave Probabilistic Boolean Networks

In this paper, we analyze the synchronization problem of master-slave probabilistic Boolean networks (PBNs). The master Boolean network (BN) is a deterministic BN, while the slave BN is determined by a series of possible logical functions with certain probability at each discrete time point. In this paper, we firstly define the synchronization of master-slave PBNs with probability one, and then we investigate synchronization with probability one. By resorting to new approach called semi-tensor product (STP), the master-slave PBNs are expressed in equivalent algebraic forms. Based on the algebraic form, some necessary and sufficient criteria are derived to guarantee synchronization with probability one. Further, we study the synchronization of master-slave PBNs in probability. Synchronization in probability implies that for any initial states, the master BN can be synchronized by the slave BN with certain probability, while synchronization with probability one implies that master BN can be synchronized by the slave BN with probability one. Based on the equivalent algebraic form, some efficient conditions are derived to guarantee synchronization in probability. Finally, several numerical examples are presented to show the effectiveness of the main results.

Forced synchronization of autonomous dynamical Boolean networks

We present the design of an autonomous time-delay Boolean network realized with readily available electronic components. Through simulations and experiments that account for the detailed nonlinear response of each circuit element, we demonstrate that a network with five Boolean nodes displays complex behavior. Furthermore, we show that the dynamics of two identical networks display near-instantaneous synchronization to a periodic state when forced by a common periodic Boolean signal. A theoretical analysis of the network reveals the conditions under which complex behavior is expected in an individual network and the occurrence of synchronization in the forced networks. This research will enable future experiments on autonomous time-delay networks using readily available electronic components with dynamics on a slow enough time-scale so that inexpensive data collection systems can faithfully record the dynamics.

Controllability and observability of Boolean networks arising from biology

Boolean networks are currently receiving considerable attention as a computational scheme for system level analysis and modeling of biological systems. Studying control-related problems in Boolean networks may reveal new insights into the intrinsic control in complex biological systems and enable us to develop strategies for manipulating biological systems using exogenous inputs. This paper considers controllability and observability of Boolean biological networks. We propose a new approach, which draws from the rich theory of symbolic computation, to solve the problems. Consequently, simple necessary and sufficient conditions for reachability, controllability, and observability are obtained, and algorithmic tests for controllability and observability which are based on the Gröbner basis method are presented. As practical applications, we apply the proposed approach to several different biological systems, namely, the mammalian cell-cycle network, the T-cell activation network, the large granular lymphocyte survival signaling network, and the Drosophila segment polarity network, gaining novel insights into the control and/or monitoring of the specific biological systems.

Synchronization in an array of output-coupled Boolean networks with time delay

This brief presents an analytical study of synchronization in an array of coupled deterministic Boolean networks (BNs) with time delay. Two kinds of models are considered. In one model, the outputs contain time delay, while in another one, the outputs do not. One restriction in this brief is that the state delay and output delay are restricted to be equal. By referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices, some necessary and sufficient conditions are derived for the synchronization of delay-coupled BNs. Examples including a practical epigenetic example are given for illustration.

Synchronization in output-coupled temporal Boolean networks

This paper presents an analytical study of synchronization in an array of output-coupled temporal Boolean networks. A temporal Boolean network (TBN) is a logical dynamic system developed to model Boolean networks with regulatory delays. Both state delay and output delay are considered, and these two delays are assumed to be different. By referring to the algebraic representations of logical dynamics and using the semi-tensor product of matrices, the output-coupled TBNs are firstly converted into a discrete-time algebraic evolution system, and then the relationship between the states of coupled TBNs and the initial state sequence is obtained. Then, some necessary and sufficient conditions are derived for the synchronization of an array of TBNs with an arbitrary given initial state sequence. Two numerical examples including one epigenetic model are finally given to illustrate the obtained results.

Synchronization of Boolean Networks with Different Update Schemes

In this paper, the synchronizations of Boolean networks with different update schemes (synchronized Boolean networks and asynchronous Boolean networks) are investigated. All nodes in Boolean network are represented in terms of semi-tensor product. First, we give the concept of inner synchronization and observe that all nodes in a Boolean network are synchronized with each other. Second, we investigate the outer synchronization between a driving Boolean network and a corresponding response Boolean network. We provide not only the concept of traditional complete synchronization, but also the anti-synchronization and get the anti-synchronization in simulation. Third, we extend the outer synchronization to asynchronous Boolean network and get the complete synchronization between an asynchronous Boolean network and a response Boolean network. Consequently, theorems for synchronization of Boolean networks and asynchronous Boolean networks are derived. Examples are provided to show the correctness of our theorems.

Synchronization of coupled large-scale Boolean networks

This paper investigates the complete synchronization and partial synchronization of two large-scale Boolean networks. First, the aggregation algorithm towards large-scale Boolean network is reviewed. Second, the aggregation algorithm is applied to study the complete synchronization and partial synchronization of large-scale Boolean networks. Finally, an illustrative example is presented to show the efficiency of the proposed results.

Synchronization design of Boolean networks via the semi-tensor product method

We provide a general approach for the design of a response Boolean network (BN) to achieve complete synchronization with a given drive BN. The approach is based on the algebraic representation of BNs in terms of the semi-tensor product of matrices. Instead of designing the logical dynamic equations of a response BN directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. The results are applied to a three-neuron network in order to illustrate the effectiveness of the proposed approach.

Complete synchronization of temporal Boolean networks

This letter studies complete synchronization of two temporal Boolean networks coupled in the drive-response configuration. Necessary and sufficient conditions are provided based on the algebraic representation of Boolean networks. Moreover, the upper bound to check the criterion is given. Finally, an illustrative example shows the efficiency of the proposed results.