Feedback Controller Design for the Synchronization of Boolean Control Networks

Edge Removal and Q-Learning for Stabilizability of Boolean Networks

This article develops a new edge removal mechanism for the global stabilizability of Boolean networks (BNs). In order to achieve the edge removal control, several control variables are properly placed into the dynamics of BNs based on the fundamental logical operators. On the basis of the new edge removal mechanism, several necessary and sufficient conditions are obtained for the global stabilizability and set stabilizability of BNs. Furthermore, a kind of stable edge removal control is proposed and achieved via the -learning algorithm to optimize the edge removal mechanism. As an application, the edge removal control is used to verify whether or not the mammalian cortical area development model can be made stabilizable to the expected stable states.

Feedback Stabilization of Boolean Control Networks With Missing Data

Data loss is often random and unavoidable in realistic networks due to transmission failure or node faults. When it comes to Boolean control networks (BCNs), the model actually becomes a delayed system with unbounded time delays. It is difficult to find a suitable way to model it and transform it into a familiar form, so there have been no available results so far. In this article, the stabilization of BCNs is studied with Bernoulli-distributed missing data. First, an augmented probabilistic BCN (PBCN) is constructed to estimate the appearance of data loss items in the model form. Based on this model, some necessary and sufficient conditions are proposed based on the construction of reachable matrices and one-step state transition probability matrices. Moreover, algorithms are proposed to complete the state feedback stabilizability analysis. In addition, a constructive method is developed to design all feasible state feedback controllers. Finally, illustrative examples are given to show the effectiveness of the proposed results.

Weak Stabilization of Boolean Networks Under State-Flipped Control

In this brief, stabilization of Boolean networks (BNs) by flipping a subset of nodes is considered, here we call such action state-flipped control. The state-flipped control implies that the logical variables of certain nodes are flipped from 1 to 0 or 0 to 1 as time flows. Under state-flipped control on certain nodes, a state-flipped-transition matrix is defined to describe the impact on the state transition space. Weak stabilization is first defined and then some criteria are presented to judge the same. An algorithm is proposed to find a stabilizing kernel such that BNs can achieve weak stabilization to the desired state with in-degree more than 0. By defining a reachable set, another approach is proposed to verify weak stabilization, and an algorithm is given to obtain a flip sequence steering an initial state to a given target state. Subsequently, the issue of finding flip sequences to steer BNs from weak stabilization to global stabilization is addressed. In addition, a model-free reinforcement algorithm, namely the Q -learning ( [Formula: see text]) algorithm, is developed to find flip sequences to achieve global stabilization. Finally, several numerical examples are given to illustrate the obtained theoretical results.

On the Compressive Power of Boolean Threshold Autoencoders

An autoencoder is a layered neural network whose structure can be viewed as consisting of an encoder, which compresses an input vector to a lower dimensional vector, and a decoder, which transforms the low-dimensional vector back to the original input vector (or one that is very similar). In this article, we explore the compressive power of autoencoders that are Boolean threshold networks by studying the numbers of nodes and layers that are required to ensure that each vector in a given set of distinct input binary vectors is transformed back to its original. We show that for any set of n distinct vectors there exists a seven-layer autoencoder with the optimal compression ratio, (i.e., the size of the middle layer is logarithmic in n ), but that there is a set of n vectors for which there is no three-layer autoencoder with a middle layer of logarithmic size. In addition, we present a kind of tradeoff: if the compression ratio is allowed to be considerably larger than the optimal, then there is a five-layer autoencoder. We also study the numbers of nodes and layers required only for encoding, and the results suggest that the decoding part is the bottleneck of autoencoding. For example, there always is a three-layer Boolean threshold encoder that compresses n vectors into a dimension that is twice the logarithm of n .

On the Distribution of Successor States in Boolean Threshold Networks

We study the distribution of successor states in Boolean networks (BNs). The state vector y is called a successor of x if y = F(x) holds, where x, y ∈ {0,1}ⁿ are state vectors and F is an ordered set of Boolean functions describing the state transitions. This problem is motivated by analyzing how information propagates via hidden layers in Boolean threshold networks (discrete model of neural networks) and is kept or lost during time evolution in BNs. In this article, we measure the distribution via entropy and study how entropy changes via the transition from x to y , assuming that x is given uniformly at random. We focus on BNs consisting of exclusive OR (XOR) functions, canalyzing functions, and threshold functions. As a main result, we show that there exists a BN consisting of d -ary XOR functions, which preserves the entropy if d is odd and , whereas there does not exist such a BN if d is even. We also show that there exists a specific BN consisting of d -ary threshold functions, which preserves the entropy if [Formula: see text]. Furthermore, we theoretically analyze the upper and lower bounds of the entropy for BNs consisting of canalyzing functions and perform computational experiments using BN models of real biological networks.

Feedback Pinning Control of Successive Lag Synchronization on a Dynamical Network

In nature and human society, successive lag synchronization (SLS) is an important synchronization phenomenon. Compared with other synchronization patterns, the control theory of SLS is very lacking. To this end, we first introduce a complex dynamical network model with distributed delayed couplings, and design both the linear feedback pinning control and adaptive feedback pinning control to push SLS to the desired trajectories. Second, we obtain a series of sufficient conditions to achieve SLS to a desired trajectory with global stability. What is more, the control flow of SLS is given to show how to pick the pinned nodes accurately and set the feedback gains as well. Finally, since time-varying delay is common, we extend the constant time delay in SLS to be time varying. We find that the proposed pinning control schemes are still feasible if the coupling terms are appropriately adjusted. The theoretical results are verified on a neural network and the coupled Chua's circuits.

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.

Identification of periodic attractors in Boolean networks using a priori information

Boolean networks (BNs) have been developed to describe various biological processes, which requires analysis of attractors, the long-term stable states. While many methods have been proposed to detection and enumeration of attractors, there are no methods which have been demonstrated to be theoretically better than the naive method and be practically used for large biological BNs. Here, we present a novel method to calculate attractors based on a priori information, which works much and verifiably faster than the naive method. We apply the method to two BNs which differ in size, modeling formalism, and biological scope. Despite these differences, the method presented here provides a powerful tool for the analysis of both networks. First, our analysis of a BN studying the effect of the microenvironment during angiogenesis shows that the previously defined microenvironments inducing the specialized phalanx behavior in endothelial cells (ECs) additionally induce stalk behavior. We obtain this result from an extended network version which was previously not analyzed. Second, we were able to heuristically detect attractors in a cell cycle control network formalized as a bipartite Boolean model (bBM) with 3158 nodes. These attractors are directly interpretable in terms of genotype-to-phenotype relationships, allowing network validation equivalent to an in silico mutagenesis screen. Our approach contributes to the development of scalable analysis methods required for whole-cell modeling efforts.

Systems biology requires not only scalable formalization methods, but also means to analyze complex networks. Although Boolean networks (BNs) are a convenient way to formalize biological processes, their analysis suffers from the combinatorial complexity with increasing number of nodes n. Hence, the long standing O(2ⁿ) barrier for detection of periodic attractors in BNs has obstructed the development of large, biological BNs. We break this barrier by introducing a novel algorithm using a priori information. We show that the proposed algorithm enables systematic analysis of BNs formulated as bipartite models in the form of in silico mutagenesis screens.

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.

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.

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.

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.

Anti-Synchronization of a Class of Chaotic Systems with Application to Lorenz System: A Unified Analysis of the Integer Order and Fractional Order

The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply our results to the integer-order and fractional-order Lorenz system, respectively. Finally, numerical simulations are presented to show the feasibility of the proposed control scheme. At the same time, through the numerical simulation results, it is show that for the Lorenz chaotic system, when the order is greater, the more quickly is anti-synchronization achieved.

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.

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.

Synchronization of Nonlinearly and Stochastically Coupled Markovian Switching Networks via Event-Triggered Sampling

This paper studies the exponential synchronization problem for a new array of nonlinearly and stochastically coupled networks via events-triggered sampling (ETS) by self-adaptive learning. The networks include the following features: 1) a Bernoulli stochastic variable is introduced to describe the random structural coupling; 2) a stochastic variable with positive mean is used to model the coupling strength; and 3) a continuous time homogeneous Markov chain is employed to characterize the dynamical switching of the coupling structure and pinned node sets. The proposed network model is capable to capture various stochastic effect of an external environment during the network operations. In order to reduce networks' workload, different ETS strategies for network self-adaptive learning are proposed under continuous and discrete monitoring, respectively. Based on these ETS approaches, several sufficient conditions for synchronization are derived by employing stochastic Lyapunov-Krasovskii functions, the properties of stochastic processes, and some linear matrix inequalities. Numerical simulations are provided to demonstrate the effectiveness of the theoretical results and the superiority of the proposed ETS approach.

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.

Policy Iteration Algorithm for Optimal Control of Stochastic Logical Dynamical Systems

This brief investigates the infinite horizon optimal control problem for stochastic multivalued logical dynamical systems with discounted cost. Applying the equivalent descriptions of stochastic logical dynamics in term of Markov decision process, the discounted infinite horizon optimal control problem is presented in an algebraic form. Then, employing the method of semitensor product of matrices and the increasing-dimension technique, a succinct algebraic form of the policy iteration algorithm is derived to solve the optimal control problem. To show the effectiveness of the proposed policy iteration algorithm, an optimization problem of p53-Mdm2 gene network is investigated.

Attractor controllability of Boolean networks by flipping a subset of their nodes

The controllability analysis of Boolean networks (BNs), as models of biomolecular regulatory networks, has drawn the attention of researchers in recent years. In this paper, we aim at governing the steady-state behavior of BNs using an intervention method which can easily be applied to most real system, which can be modeled as BNs, particularly to biomolecular regulatory networks. To this end, we introduce the concept of attractor controllability of a BN by flipping a subset of its nodes, as the possibility of making a BN converge from any of its attractors to any other one, by one-time flipping members of a subset of BN nodes. Our approach is based on the algebraic state-space representation of BNs using semi-tensor product of matrices. After introducing some new matrix tools, we use them to derive necessary and sufficient conditions for the attractor controllability of BNs. A forward search algorithm is then suggested to identify the minimal perturbation set for attractor controllability of a BN. Next, a lower bound is derived for the cardinality of this set. Two new indices are also proposed for quantifying the attractor controllability of a BN and the influence of each network variable on the attractor controllability of the network and the relationship between them is revealed. Finally, we confirm the efficiency of the proposed approach by applying it to the BN models of some real biomolecular networks.

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.

An Application of Invertibility of Boolean Control Networks to the Control of the Mammalian Cell Cycle

In Fauré et al. (2006), the dynamics of the core network regulating the mammalian cell cycle is formulated as a Boolean control network (BCN) model consisting of nine proteins as state nodes and a tenth protein (protein CycD) as the control input node. In this model, one of the state nodes, protein Cdc20, plays a central role in the separation of sister chromatids. Hence, if any Cdc20 sequence can be obtained, fully controlling the mammalian cell cycle is feasible. Motivated by this fact, we study whether any Cdc20 sequence can be obtained theoretically. We formulate the foregoing problem as the invertibility of BCNs, that is, whether one can obtain any Cdc20 sequence by designing input (i.e., protein CycD) sequences. We give an algorithm to verify the invertibility of any BCN, and find that the BCN model for the core network regulating the mammalian cell cycle is not invertible, that is, one cannot obtain any Cdc20 sequence. We further present another algorithm to test whether a finite Cdc20 sequence can be generated by the BCN model, which leads to a series of periodic infinite Cdc20 sequences with alternately active and inactive Cdc20 segments. States of these sequences are alternated between the two attractors in the proposed model, which reproduces correctly how a cell exits the cell cycle to enter the quiescent state, or the opposite.