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

Asymptotic Stability of Delayed Boolean Networks With Random Data Dropouts

In real networks, communication constraints often prevent the full exchange of information between nodes, which is inevitable. This brief investigates the problem of time delay and randomly missing data in Boolean networks (BNs). A Bernoulli random variable is assigned to each node to characterize the probability of data packet dropout. Time delay and missing data are modeled by independent random variables. A novel data-sending rule that incorporates both communication constraints is proposed. An augmented system, comprising current states, delayed information, and successfully transmitted data, is established for theoretical analysis. Using the semitensor product (STP), the necessary and sufficient condition for asymptotic stability of delayed BNs with random data dropouts is derived. The convergence rate is also obtained.

A Nonaugmented Method for the Minimal Observability of Boolean Networks

This article proposes a nonaugmented method for investigating the minimal observability problem of Boolean networks (BNs). This method can be applied to more general BNs and reduce the computational and space complexity of existing results. First, unobservable states concerning an unobservable BN are classified into three categories using the vertex-colored state transition graph, each accompanied by a necessary and sufficient condition for determining additional measurements to make them distinguishable. Then, an algorithm is designed to identify the additional measurements that would render an unobservable BN observable using the conditions. Next, to determine the minimum added measurements, a necessary and sufficient condition and an algorithm based on a constructed matrix are presented. Finally, the results obtained are compared with existing literature and illustrated with examples.

Long-Run Behavior Estimation of Temporal Boolean Networks With Multiple Data Losses

This brief devotes to investigating the long-run behavior estimation of temporal Boolean networks (TBNs) with multiple data losses, especially the asymptotical stability. The information transmission is modeled by Bernoulli variables, based on which an augmented system is constructed to facilitate the analysis. A theorem guarantees that the asymptotical stability of the original system can be converted to that of the augmented system. Subsequently, one necessary and sufficient condition is obtained for asymptotical stability. Furthermore, an auxiliary system is derived to study the synchronization issue of the ideal TBNs with normal data transmission and TBNs with multiple data losses, as well as an effective criterion for verifying synchronization. Finally, numerical examples are given to illustrate the validity of the theoretical results.

Optimal preview pinning control of Boolean networks

This paper investigates the problem of optimal preview pinning control for Boolean networks. The primary objective is to design efficient control strategies that leverage future reference information for improved control performance. We propose a policy iteration algorithm specifically for Boolean networks based on an augmented error system, constructed using the state augmentation technique in combination with the Exclusive-Or operator approach. The algorithm effectively optimizes control policies using future information. Finally, an example is presented to illustrate the effectiveness of the proposed algorithm.

Multi-Sensor Fusion Boolean Bayesian Filtering for Stochastic Boolean Networks

Stochastic Boolean networks (SBNs) take process noise into account, so it is better to fit the actual situation and has a wider application background than Boolean networks (BNs). However, the presence of noise influences us to estimate the real state of the system. To minimize the inaccuracies caused by the presence of noise, an optimal state estimation problem is studied in this article. The multi-sensor fusion Boolean Bayesian filtering is proposed and a recursive algorithm is provided to calculate the prior and posterior belief of system state by fusing multi-sensor measurements based on the algebraic form of the SBN and Bayesian law. Then, the optimal state estimator is obtained, which minimizes the mean-square estimation error. Finally, a simulation example is carried out to demonstrate the performance of the proposed methodology. It has been shown through the simulation experiment that it increases the confidence level of the state estimation and improves the estimation performance using multi-sensor fusion compared with using single sensor.

State Estimation for Probabilistic Boolean Networks via Outputs Observation

This article studies the state estimation for probabilistic Boolean networks via observing output sequences. Detectability describes the ability of an observer to uniquely estimate system states. By defining the probability of an observed output sequence, a new concept called detectability measure is proposed. The detectability measure is defined as the limit of the sum of probabilities of all detectable output sequences when the length of output sequences goes to infinity, and it can be regarded as a quantitative assessment of state estimation. A stochastic state estimator is designed by defining a corresponding nondeterministic stochastic finite automaton, which combines the information of state estimation and probability of output sequences. The proposed concept of detectability measure further performs the quantitative analysis on detectability. Furthermore, by defining a Markov chain, the calculation of detectability measure is converted to the calculation of the sum of probabilities of certain specific states in Markov chain. Finally, numerical examples are given to illustrate the obtained theoretical results.

Event-Triggered Sampled Feedback Synchronization in an Array of Output-Coupled Boolean Control Networks

In this article, synchronization problem in an array of output-coupled Boolean control networks (BCNs) is studied by using event-triggered sampled feedback control. Algebraic forms of an array of output-coupled BCNs are presented via the semitensor product (STP) of matrices. Based on the algebraic forms, a necessary and sufficient condition is obtained for the synchronization of an array of output-coupled BCNs. Furthermore, an algorithm is proposed to design event-triggered sampled feedback controllers. Finally, the obtained results are well illustrated by numerical examples.

On Optimal Time-Varying Feedback Controllability for Probabilistic Boolean Control Networks

This brief studies controllability for probabilistic Boolean control network (PBCN) with time-varying feedback control laws. The concept of feedback controllability with an arbitrary probability for PBCNs is formulated first, and a control problem to maximize the probability of time-varying feedback controllability is investigated afterward. By introducing semitensor product (STP) technique, an equivalent multistage decision problem is deduced, and then a novel optimization algorithm is proposed to obtain the maximum probability of controllability and the corresponding optimal feedback law simultaneously. The advantages of the time-varying optimal controller obtained by the proposed algorithm, compared to the time-invariant one, are illustrated by numerical simulations.

Tractable Learning and Inference for Large-Scale Probabilistic Boolean Networks

Probabilistic Boolean networks (PBNs) have previously been proposed so as to gain insights into complex dynamical systems. However, identification of large networks and their underlying discrete Markov chain which describes their temporal evolution still remains a challenge. In this paper, we introduce an equivalent representation for PBNs, the stochastic conjunctive normal form network (SCNFN), which enables a scalable learning algorithm and helps predict long-run dynamic behavior of large-scale systems. State-of-the-art methods turn out to be 400 times slower for middle-sized networks (i.e., containing 100 nodes) and incapable of terminating for large networks (i.e., containing 1000 nodes) compared to the SCNFN-based learning, when attempting to achieve comparable accuracy. In addition, in contrast to the currently used methods which introduce strict restrictions on the structure of the learned PBNs, the hypothesis space of our training paradigm is the set of all possible PBNs. Moreover, SCNFNs enable efficient sampling so as to statistically infer multistep transition probabilities which can provide information on the activity levels of individual nodes in the long run. Extensive experimental results showcase the scalability of the proposed approach both in terms of sample and runtime complexity. In addition, we provide examples to study large and complex cell signaling networks to show the potential of our model. Finally, we suggest several directions for future research on model variations, theoretical analysis, and potential applications of SCNFNs.

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.

Function Perturbation Impact on Feedback Stabilization of Boolean Control Networks

Function perturbation analysis of Boolean networks is an important topic in the study of gene regulation due to gene mutation or immeasurable variables. This brief studies the function perturbation impact on feedback stabilization of Boolean control networks (BCNs) by using the algebraic state space representation approach. First, the state feedback stabilization control design of BCNs is recalled and the function perturbation problem is formulated. Second, given a state feedback stabilizer, it is robust to the considered function perturbation if one of the following three cases happens: 1) the block where the function perturbation occurs is different from the block which is affected by the state feedback stabilizer (Case 1); 2) when Case 1 does not happen, the perturbed column converges to the equilibrium faster than the original column (Case 2); 3) when Cases 1 and 2 do not happen, the perturbed column does not belong to the reachable set of the original column (Case 3). Third, when the perturbed column belongs to the reachable set of the original column, a constructive procedure is proposed to modify the given state feedback stabilizer to be robust to the function perturbation. Finally, the obtained new results are applied to the function perturbation analysis of lactose operon in Escherichia coli. The main novelty of this brief is to develop a new theoretical framework for the robustness of feedback controllers of BCNs with respect to function perturbation, which is not solved in the existing literature.

Distributed Impulsive Quasi-Synchronization of Lur'e Networks With Proportional Delay

This paper investigates the exponential synchronization of nonidentically coupled Lur'e dynamical networks with proportional delay. Since the heterogeneities existed in different Lur'e systems, quasi-synchronization rather than complete synchronization is thus discussed. Different from general time delay, the proportional delay is a type of unbounded time-varying delay, which tremendously increases the requirements on network synchronization. Based on distributed impulsive pinning control protocol and different roles that impulsive effects play, the criteria for quasi-synchronization of nonidentically coupled Lur'e dynamical networks are derived by jointly applying the delayed impulsive comparison principle, the extended formula for the variation of parameters, and the definition of an average impulsive interval. Moreover, synchronization errors for different impulsive effects with different functions are evaluated and simultaneously, the corresponding exponential convergence rates are obtained. In addition, three numerical examples are presented to illustrate the validity of the control scheme and the theoretical analysis.

Set Stabilization of Probabilistic Boolean Networks Using Pinning Control

Probabilistic Boolean network (PBN) is a kind of stochastic logical system in which update functions are randomly selected from a set of candidate Boolean functions according to a prescribed probability distribution at each time step. In this brief, a pinning controller design algorithm is proposed to set stabilize any PBN with probability one. First, an algorithm is given to change the columns of its transition matrix. Then, according to the newly obtained transition matrix, a fraction of nodes can be selected as pinning nodes to inject control inputs to achieve set stabilization. The problem is challenging since the Boolean functions in a PBN are not deterministic but are randomly chosen among several Boolean functions. Furthermore, the structure matrices of the pinning controllers are given by solving some logical matrices equations based on which a pinning controller design algorithm is provided to set stabilize the PBN with probability one. Finally, the theoretical results are validated using several examples.

A Boolean network control algorithm guided by forward dynamic programming

Control problem in a biological system is the problem of finding an interventional policy for changing the state of the biological system from an undesirable state, e.g. disease, into a desirable healthy state. Boolean networks are utilized as a mathematical model for gene regulatory networks. This paper provides an algorithm to solve the control problem in Boolean networks. The proposed algorithm is implemented and applied on two biological systems: T-cell receptor network and Drosophila melanogaster network. Results show that the proposed algorithm works faster in solving the control problem over these networks, while having similar accuracy, in comparison to previous exact methods. Source code and a simple web service of the proposed algorithm is available at http://goliaei.ir/net-control/www/.

Sampled-Data Control for the Synchronization of Boolean Control Networks

In this paper, we investigate the sampled-data state feedback control (SDSFC) for the synchronization of Boolean control networks (BCNs) under the configuration of drive-response coupling. Necessary and sufficient conditions for the complete synchronization of BCNs are obtained by the algebraic representations of logical dynamics. Based on the analysis of the sampling periods, we establish an algorithm to guarantee the synchronization of drive-response coupled BCNs by SDSFC. An example is given to illustrate the significance of the obtained results.

Event-Triggered Control for the Disturbance Decoupling Problem of Boolean Control Networks

This paper investigates the disturbance decoupling problem (DDP) of Boolean control networks (BCNs) by event-triggered control. Using the semi-tensor product of matrices, algebraic forms of BCNs can be achieved, based on which, event-triggered controllers are designed to solve the DDP of BCNs. In addition, the DDP of Boolean partial control networks is also derived by event-triggered control. Finally, two illustrative examples demonstrate the effectiveness of proposed methods.

Stability and Guaranteed Cost Analysis of Time-Triggered Boolean Networks

This paper investigates stability and guaranteed cost of time-triggered Boolean networks (BNs) based on the semitensor product of matrices. The time triggering is generated by mode-dependent average dwell-time switching signals in the BNs. With the help of the copositive Lyapunov function, a sufficient condition is derived to ensure that the considered network is globally stable under a designed average dwell-time switching signal. Subsequently, an infinite time cost function is further discussed and its bound is presented according to the obtained stability result. Numerical examples are finally given to show the feasibility of the theoretical results.

Normalization and Solvability of Dynamic-Algebraic Boolean Networks

In this brief, we first study the normalization of dynamic-algebraic Boolean networks (DABNs). A new expression for the normalized DABNs is obtained. As applications of this result, the solvability and uniqueness of the solution to DABNs are then investigated. Necessary and sufficient conditions for the solvability and the uniqueness are obtained. In addition, pinning control to ensure the solvability and uniqueness of the solution to DABNs is also studied. Numerical examples are given to illustrate the efficiency of the proposed results.

Single-Input Pinning Controller Design for Reachability of Boolean Networks

This brief is concerned with the problem of a single-input pinning control design for reachability of Boolean networks (BNs). Specifically, the transition matrix of a BN is designed to steer the BN from an initial state to a desirable one. In addition, some nodes are selected as the pinning nodes by solving some logical matrix equations. Furthermore, a single-input pinning control algorithm is given. Eventually, a genetic regulatory network is provided to demonstrate the effectiveness and feasibility of the developed method.

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.

Robust Consensus of Networked Evolutionary Games with Attackers and Forbidden Profiles †

Using the algebraic state space representation, this paper studies the robust consensus of networked evolutionary games (NEGs) with attackers and forbidden profiles. Firstly, an algebraic form is established for NEGs with attackers and forbidden profiles. Secondly, based on the algebraic form, a necessary and sufficient condition is presented for the robust constrained reachability of NEGs. Thirdly, a series of robust reachable sets is constructed by using the robust constrained reachability, based on which a constructive procedure is proposed to design state feedback controls for the robust consensus of NEGs with attackers and forbidden profiles. Finally, an illustrative example is given to show that the main results are effective.

Adaptive Exponential Synchronization of Multislave Time-Delayed Recurrent Neural Networks With Lévy Noise and Regime Switching

This paper discusses the problem of adaptive exponential synchronization in mean square for a new neural network model with the following features: 1) the noise is characterized by the Lévy process and the parameters of the model change in line with the Markovian process; 2) the master system is also disturbed by the same Lévy noise; and 3) there are multiple slave systems, and the state matrix of each slave system is an affine function of the state matrices of all slave systems. Based on the Lyapunov functional theory, the generalized Itô's formula, -matrix method, and the adaptive control technique, some criteria are established to ensure the adaptive exponential synchronization in the mean square of the master system and each slave system. Moreover, the update law of the control gain and the dynamic variation of the parameters of the slave systems are provided. Finally, the effectiveness of the synchronization criteria proposed in this paper is verified by a practical example.

Variable structure controller design for Boolean networks

The paper investigates the variable structure control for stabilization of Boolean networks (BNs). The design of variable structure control consists of two steps: determine a switching condition and determine a control law. We first provide a method to choose states from the reaching mode. Using this method, we can guarantee that the number of nodes which should be controlled is minimized. According to the selected states, we determine the switching condition to guarantee that the time of global stabilization in the BN is the shortest. A control law is then determined to ensure that all selected states can enter into the sliding mode, such that any initial state can arrive in the steady-state mode. Some examples are provided to illustrate the 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.

Robust Reachability of Boolean Control Networks

Boolean networks serve a powerful tool in analysis of genetic regulatory networks since it emphasizes the fundamental principles and establishes a nature framework for capturing the dynamics of regulation of cellular states. In this paper, the robust reachability of Boolean control networks is investigated by means of semi-tensor product. Necessary and sufficient conditions for the robust reachability of Boolean control networks are provided, in which control inputs relying on disturbances or not are considered, respectively. Besides, the corresponding control algorithms are developed for these two cases. A reduced model of the lac operon in the Escherichia coli is presented to show the effectiveness of the presented results.

Observability of Boolean multiplex control networks

Boolean multiplex (multilevel) networks (BMNs) are currently receiving considerable attention as theoretical arguments for modeling of biological systems and system level analysis. Studying control-related problems in BMNs may not only provide new views into the intrinsic control in complex biological systems, but also enable us to develop a method for manipulating biological systems using exogenous inputs. In this article, the observability of the Boolean multiplex control networks (BMCNs) are studied. First, the dynamical model and structure of BMCNs with control inputs and outputs are constructed. By using of Semi-Tensor Product (STP) approach, the logical dynamics of BMCNs is converted into an equivalent algebraic representation. Then, the observability of the BMCNs with two different kinds of control inputs is investigated by giving necessary and sufficient conditions. Finally, examples are given to illustrate the efficiency of the obtained theoretical 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.

Generalized Higher Order Orthogonal Iteration for Tensor Learning and Decomposition

Low-rank tensor completion (LRTC) has successfully been applied to a wide range of real-world problems. Despite the broad, successful applications, existing LRTC methods may become very slow or even not applicable for large-scale problems. To address this issue, a novel core tensor trace-norm minimization (CTNM) method is proposed for simultaneous tensor learning and decomposition, and has a much lower computational complexity. In our solution, first, the equivalence relation of trace norm of a low-rank tensor and its core tensor is induced. Second, the trace norm of the core tensor is used to replace that of the whole tensor, which leads to two much smaller scale matrix TNM problems. Finally, an efficient alternating direction augmented Lagrangian method is developed to solve our problems. Our CTNM formulation needs only O((R^N+NRI)log(√{I^N})) observations to reliably recover an N th-order I×I×…×I tensor of n -rank (r,r,…,r) , compared with O(rI^N-1) observations required by those tensor TNM methods ( I >> R ≥ r ). Extensive experimental results show that CTNM is usually more accurate than them, and is orders of magnitude faster.

Disturbance Decoupling of Singular Boolean Control Networks

This paper investigates the controller designing for disturbance decoupling problem (DDP) of singular Boolean control networks (SBCNs). Using semi-tensor product (STP) of matrices and the Implicit Function Theorem, a SBCN is converted into the standard BCN. Based on the redundant variable separation technique, both state feedback and output feedback controllers are designed to solve the DDP of the SBCN. Sufficient conditions are also given to analyze the invariance of controllers concerning the DDP of the SBCN with function perturbation. Two illustrative examples are presented to support the effectiveness of these obtained results.

Feedback Controller Design for the Synchronization of Boolean Control Networks

This brief investigates the partial and complete synchronization of two Boolean control networks (BCNs). Necessary and sufficient conditions for partial and complete synchronization are established by the algebraic representations of logical dynamics. An algorithm is obtained to construct the feedback controller that guarantees the synchronization of master and slave BCNs. Two biological examples are provided to illustrate the effectiveness of the obtained results.

Sampled-Data State Feedback Stabilization of Boolean Control Networks

In this letter, we investigate the sampled-data state feedback control (SDSFC) problem of Boolean control networks (BCNs). Some necessary and sufficient conditions are obtained for the global stabilization of BCNs by SDSFC. Different from conventional state feedback controls, new phenomena observed the study of SDSFC. Based on the controllability matrix, we derive some necessary and sufficient conditions under which the trajectories of BCNs can be stabilized to a fixed point by piecewise constant control (PCC). It is proved that the global stabilization of BCNs under SDSFC is equivalent to that by PCC. Moreover, algorithms are given to construct the sampled-data state feedback controllers. Numerical examples are given to illustrate the efficiency of the obtained results.

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.

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.