Probabilistic Boolean Control Networks Research Articles

Finite-time observability of probabilistic Boolean multiplex control networks

Finite-time observability of probabilistic Boolean multiplex control networks

Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data

Article Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data Hongwei Chen 1,*, Qi Chen 1, Bo Shen 1, and Yang Liu 2 1 College of Information Science and Technology, Donghua University, Shanghai 201620, China 2 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China * Correspondence: hongwei@dhu.edu.cn Received: 23 September 2023 Accepted: 27 November 2023 Published: 26 March 2024 Abstract: This paper investigates the parameter learning problem for the probabilistic Boolean control networks (PBCNs) with input-output data. Firstly, an algebraic expression of the PBCNs is obtained by taking advantage of the semi-tensor product technique, and then, the parameter learning problem is transformed into an optimal problem to reveal the parameter matrices of a linear system in a computationally efficient way. Secondly, two recursive semi-tensor product based algorithms are designed to calculate the forward and backward probabilities. Thirdly, the expectation maximization algorithm is proposed as an elaborate technique to address the parameter learning problem. In addition, a useful index is introduced to describe the performance of the proposed parameter learning algorithm. Finally, two numerical examples are employed to demonstrate the reliability of the proposed parameter learning approach.

Set stabilization of logical control networks: A minimum node control approach

In network systems, control using minimum nodes or pinning control can be effectively used for stabilization problems to cut down the cost of control. In this paper, we investigate the set stabilization problem of logical control networks. In particular, we study the set stabilization problem of probabilistic Boolean networks (PBNs) and probabilistic Boolean control networks (PBCNs) via controlling minimal nodes. Firstly, an algorithm is given to search for the minimum index set of pinning nodes. Then, based on the analysis of its high computational complexity, we present optimized algorithms with lower computational complexity to ascertain the network control using minimum node sets. Moreover, some sufficient and necessary conditions are proposed to ensure the feasibility and effectiveness of the proposed algorithms. Furthermore, a theorem is presented for PBCNs to devise all state-feedback controllers corresponding to the set of pinning nodes. Finally, two models of gene regulatory networks are considered to show the efficacy of obtained results.

Finite-time set stabilization of probabilistic Boolean control networks via output-feedback control

This paper investigates the finite-time output-feedback set stabilization problem for probabilistic Boolean control networks (PBCNs), a crucial aspect of set convergence. Firstly, an algorithm is presented to effectively identify the largest output-feedback control invariant subset in a given set. Next, a sufficient condition is put forward to bridge set stabilization and the question at hand via the minimum transient period. Additionally, sufficient and necessary criteria are proposed, addressing the solvability of finite-time output-feedback set stabilization, followed by two corresponding algorithms for effectively designing all feasible output-feedback stabilizers. Finally, a biological example is successfully explored to demonstrate our theoretical results.

Pinning Stabilizer Design for Probabilistic Boolean Control Networks via Condensation Digraph.

This article investigates the design of pinning controllers for state feedback stabilization of probabilistic Boolean control networks (PBCNs), based on the condensation digraph method. First, two effective algorithms are presented to achieve state feedback stabilization of the considered system from the perspective of condensation digraph. One is to find the desired matrix, and the other is to search for the minimum number of pinned nodes and specific pinned nodes. Then, all the mode-independent pinning controllers can be designed based on the desired matrix and pinned nodes. Several examples are delineated to illustrate the validity of the main results.

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.

A unified transform method for general robust property of probabilistic Boolean control networks

This article proposes a unified transform method for general robust property of probabilistic Boolean control networks (PBCNs). Here, the general robust property of PBCNs means any given robust property of any given PBCN with probability one. Specifically, the method is that the investigation on any given robust property of any given PBCN with probability one can be equivalently transformed into the investigation on the given robust property of the corresponding Boolean control network (BCN). Note that the PBCN has time-varying probability distribution and the property can be set according to the needs of specific problems. As an application of the method, we study the robust set stabilization with probability one of PBCNs with positive time-varying probability distribution. We derive the results through a discussion of the robust set stabilization of the corresponding BCNs. Also, we characterize all of the time-optimal robust set stabilizers of the PBCNs. A reduced disturbed PBCN model of the lac operon in the Escherichia coli is provided to show the effectiveness of the achieved results.

Maximum-Likelihood State Estimators in Probabilistic Boolean Control Networks.

This study addresses state estimation problems for probabilistic Boolean control networks (PBCNs). Compared with deterministic Boolean networks, PBCNs have the stochastic switching in logical update functions in the state equation. Consequently, statistical analysis is required to estimate unavailable states, which induces an optimization problem called maximum-likelihood estimation. This article mainly focuses on two scenarios: 1) state estimation from partially measured state and 2) state estimation from output data, meaning observer design. The resulting optimization problems are solved using efficient algorithms based on dynamic programming. Concurrently, Dijkstra-type algorithms, which solve equivalent shortest path problems, are also proposed using best-first search. Furthermore, both the proposed algorithms derive novel observer design methods for PBCNs. The proposed algorithms are evaluated with practical estimation problems aiming to the sensor reduction and applied to gene regulatory networks of apoptosis and Lac operon.

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”

Sampled-data Control of Probabilistic Boolean Control Networks: A Deep Reinforcement Learning Approach

The rise of reinforcement learning (RL) has guided a new paradigm: unraveling the intervention strategies to control systems with unknown dynamics. Model-free RL provides an exhaustive framework to devise therapeutic methods to alter the regulatory dynamics of gene regulatory networks (GRNs). This paper presents an RL-based technique to control GRNs modeled as probabilistic Boolean control networks (PBCNs). In particular, a double deep-Q network (DDQN) approach is proposed to address the sampled-data control (SDC) problem of PBCNs, and optimal state feedback controllers are obtained, rendering the PBCNs stabilized at a given equilibrium point. Our approach is based on options, i.e., the temporal abstractions of control actions in the Markov decision processes (MDPs) framework. First, we define options and hierarchical options and give their properties. Then, we introduce multi-time models to compute the optimal policies leveraging the options framework. Furthermore, we present a DDQN algorithm: i) to concurrently design the feedback controller and the sampling period; ii) wherein the controller intelligently decides the sampled period to update the control actions under the SDC scheme. The presented method is model-free and offers scalability, thereby providing an efficient way to control large-scale PBCNs. Finally, we compare our control policy with state-of-the-art control techniques and validate the presented results.

Self-triggered control of probabilistic Boolean control networks: A reinforcement learning approach

In this work, strategies to devise an optimal feedback control of probabilistic Boolean control networks (PBCNs) are discussed. Reinforcement learning (RL) based control is explored in order to minimize model design efforts and regulate PBCNs with high complexities. A Q-learning random forest (QLRF) algorithm is proposed; by making use of the algorithm, state feedback controllers are designed to stabilize the PBCNs at a given equilibrium point. Further, by adopting QLRF stabilized closed-loop PBCNs, a Lyapunov function is defined, and a method to construct the same is presented. By utilizing such Lyapunov functions, a novel self-triggered control (STC) strategy is proposed, whereby the controller is recomputed according to a triggering schedule, resulting in an optimal control strategy while retaining the closed-loop PBCN stability. Finally, the results are verified using computer simulations.

Finite-time sampled-data set stabilisation of delayed probabilistic Boolean control networks

In this paper, the finite-time constrained set stabilisation of delayed probabilistic Boolean control networks (DPBCNs) is considered via sampled-data control (SDC). The main mathematical tool is semi-tensor product of matrices, which is used to convert the dynamics of DPBCNs into an equivalent algebraic state space representation (ASSR). Based on the ASSR of DPBCNs, the constrained sampled-data transition probability matrix is constructed. Then, new criteria are proposed for the finite-time constrained reachability of DPBCNs under uniform SDC. By virtue of finite-time constrained reachability, a procedure is established to design state feedback sampled-data stabilisers for DPBCNs. The obtained results are applied to the ‘minimal’ model of lactose operon in E. coli.

On robust set stability and set stabilization of probabilistic Boolean control networks

Under the semi-tensor product method, this paper investigates the robust set stability and robust set stabilization problems for a class of probabilistic Boolean control networks (PBCNs) with disturbances. First, an algorithm to determine the largest robust invariant set (LRIS) of a given set for a PBN with probability one is proposed, and the necessary and sufficient conditions to detect whether the PBN is globally finite-time stable to this invariant set with probability one are established. Second, the PBNs with control inputs are considered, and an algorithm for the largest robust control invariant set (LRCIS) with probability one is provided, based on which, some necessary and sufficient conditions for finite-time robust set stabilization with probability one of PBCNs are presented. Furthermore, the design scheme of time-optimal state feedback stabilizers via antecedence solution technique is derived. The study of illustrative examples shows the effectiveness of the obtained new results.

On state feedback asymptotical stabilization of probabilistic Boolean control networks

This article devotes to the design of all state feedback controllers for the asymptotical stabilization of probabilistic Boolean control networks (PBCNs). Firstly, a class of complete family of reachable sets (CFRSs) is constructed. Based on the CFRSs, all possible state feedback controllers are designed to stabilize PBCNs to a given control-fixed point in distribution. Secondly, by removing the finite-time stabilizers from the controllers that make PBCNs stabilizable in distribution, all state feedback controllers which asymptotically stabilize PBCNs with probability one are further obtained. Thirdly, the CFRSs are applied to solve the problem of asymptotical set stabilization. Finally, an example of lac operon in the Escherichia coli is given to illustrate the proposed methods.

Finite‐time observability of probabilistic Boolean control networks

AbstractIn this paper, the finite‐time observability (FO) of probabilistic Boolean control networks (PBCNs) based on set reachability and parallel extension is investigated. Under algebraic state space representation of PBCNs, finite‐time set reachability (FSR), finite‐time single input sequence set reachability (FSSSR), finite‐time arbitrary input sequence set reachability (FASSR), as well as finite‐time output feedback set reachability (FOSR) are addressed, where some efficient criteria are proposed. In the second part of this paper, several types of FO are provided, which can be converted to the corresponding FSR based on the parallel extension technique. Consequently, the obtained results for FSR are utilized to deal with FO. Finally, two examples including a biological network are given to demonstrate the theoretical results.

Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks

This study focuses on the finite-time set reachability of probabilistic Boolean multiplex control networks (PBMCNs). Firstly, based on the state transfer graph (STG) reconstruction technique, the PBMCNs are extended to random logic dynamical systems. Then, a necessary and sufficient condition for the finite-time set reachability of PBMCNs is obtained. Finally, the obtained results are effectively illustrated by an example.

Asymptotical Stabilization for Probabilistic Boolean Control Networks under Operators and Inputs Constraints

In this paper, asymptotical stabilization of probabilistic Boolean control networks (<monospace>PBCNs</monospace>) with a switching signal being an independent identically distributed process has been systematically studied. Here, the binary operators connecting logical functions of state nodes and pinning controllers are constrained to certain sets. By establishing matrix representation of operators and input constraints, several criteria for asymptotical stabilization have been established. The proposed asymptotical stabilization extends the results of Boolean networks (<monospace>BNs</monospace>) under static pinning control, which is more practical in real-word control design systems. Finally, simulations are given to verify the effectiveness of the obtained theoretical results.

Optimal control of probabilistic Boolean control networks: A scalable infinite horizon approach

AbstractOne of the major issues in systems biology is developing control theory for gene regulatory networks (GRNs). Particularly, an important objective is to develop therapeutic intervention strategies to alter regulatory dynamics of GRNs to avoid undesired or diseased states. Several optimal control strategies have been developed to find therapeutic intervention strategies for small (or medium) sized GRNs modeled as probabilistic Boolean control networks (PBCNs). However, due to the humongous nature of GRNs, we require an optimal control strategy that scales to large systems without posing any constraints on the network dynamics. In this article, we formulate the infinite horizon discounted cost optimal control problem by leveraging the Markov decision process (MDP) based PBCN structure to model GRNs. Further, we design a stationary optimal control strategy, thereby avoiding the states associated with a disease. An augmented state space is defined to facilitate the control design method. By exploiting Kullback–Leibler (KL) divergence and the log transformation of the cost function referred to as the desirability function, the optimal control is devised using the path integral (PI) approach. We propose a sampling‐based technique for the approximation of the PI and hence the optimal control of PBCNs. The sampling‐based strategy is amicable to parallel implementation, thereby addressing the optimal control problem of large‐scale PBCNs. Finally, the viability of the results in the article is shown by some illustrative examples.

Finite horizon tracking control of probabilistic Boolean control networks

In this paper, the finite horizon tracking control problem of probabilistic Boolean control networks (PBCNs) is studied. For a given reference output trajectory, two trackability definitions are introduced according to whether the tracking probability is 1. Under the framework of the semi-tensor product, some necessary and sufficient conditions are obtained to determine whether the reference output trajectory is trackable with probability (probability one) by a PBCN starting from a given initial state. Based on this, two algorithms are proposed to determine the maximum tracking probability and the corresponding optimal control policy sequence. By determining the tracking error of the reference output trajectory, two related optimal control problems are considered: one is to minimize the expected value of the total tracking error, and the other is to minimize the maximum tracking error. Inspired by dynamic programming, corresponding algorithms are given to solve these two problems. Finally, two examples are given to verify the validity and correctness of the results.

Asymptotical stability and stabilisation of probabilistic Boolean networks subject to function perturbation

This paper focuses on the effects of function perturbation for probabilistic Boolean networks (PBNs), including asymptotical stability and stabilisation. Firstly, by constructing a proper set, a necessary and sufficient condition is presented for the asymptotically stable PBN being stable in distribution after function perturbation. Secondly, by removing the case of changing to finite-time stability with probability one from the case of changing to stability in distribution, some criteria are given for the asymptotically stable PBN keeping asymptotically stable after function perturbation. These criteria need to calculate the number of positive-probability cycles. In order to simplify the calculation, two sufficient conditions are further established. Finally, as an application, the robustness of asymptotical feedback stabilisers is further studied for probabilistic Boolean control networks subject to function perturbation.