Input-State Approach to Boolean Networks

Cheng, Daizhan

doi:10.1109/tnn.2008.2011359

Cited by 149 publications

(15 citation statements)

References 12 publications

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“…Ref. [20] shows the structure of cycles in BCNs as compounded cycles. Fixed points and cycles in input space, state space and input-state space are all considered.…”

Section: Introductionmentioning

confidence: 99%

Further results on dynamic-algebraic Boolean control networks

Wang

Feng

et al. 2018

Sci. China Inf. Sci.

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Restricted coordinate transformation, controllability, observability and topological structures of dynamic-algebraic Boolean control networks are investigated under an assumption. Specifically, given the input-state at some point, assume that the subsequent state is certain or does not exist. First, the system can be expressed in a new form after numbering the elements in admissible set. Then, restricted coordinate transformation is raised, which allows the dimension of new coordinate frame to be different from that of the original one. The system after restricted coordinate transformation is derived in the proposed form. Afterwards, three types of incidence matrices are constructed and the results of controllability, observability and topological structures are obtained. Finally, two practical examples are shown to demonstrate the theory in this paper.

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“…Ref. [20] shows the structure of cycles in BCNs as compounded cycles. Fixed points and cycles in input space, state space and input-state space are all considered.…”

Section: Introductionmentioning

confidence: 99%

Further results on dynamic-algebraic Boolean control networks

Wang

Feng

et al. 2018

Sci. China Inf. Sci.

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“…We summarize the mathematical tool of semi-tensor product in Cheng’s papers as follows. [14, 21] Cheng’s result 1: Any logical function f ( x 1 , x 2 ,⋯, x n ) with logical states can be expressed in a multi-linear form as where M is a 2×2 n logical matrix. Cheng’s result 2: Consider a Boolean network with states and denote integrated state , there exists a unique matrix such that L is the transition matrix of this Boolean network.…”

Section: Methodsmentioning

confidence: 99%

“…where V i and u j take value from the set {0,1} [14]. The representation of each Boolean function is defined as B i :{0,1} n + m →{0,1}, i =1,…, n , which is preassigned Boolean logical functions determined by the biological process.…”

Section: Introductionmentioning

confidence: 99%

“…If there is no control input u j , B i is a 2×2 n matrix because each logical value 0 or 1 is expressed as a vector (0,1) T or (1,0) T , respectively. The algebraic state-space representation of the Boolean control network is set up based on the semi-tensor product of matrices which will be introduced in our method part [10, 14, 15]. …”

Section: Introductionmentioning

confidence: 99%

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State feedback control design for Boolean networks

et al. 2016

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BackgroundDriving Boolean networks to desired states is of paramount significance toward our ultimate goal of controlling the progression of biological pathways and regulatory networks. Despite recent computational development of controllability of general complex networks and structural controllability of Boolean networks, there is still a lack of bridging the mathematical condition on controllability to real boolean operations in a network. Further, no realtime control strategy has been proposed to drive a Boolean network.ResultsIn this study, we applied semi-tensor product to represent boolean functions in a network and explored controllability of a boolean network based on the transition matrix and time transition diagram. We determined the necessary and sufficient condition for a controllable Boolean network and mapped this requirement in transition matrix to real boolean functions and structure property of a network. An efficient tool is offered to assess controllability of an arbitrary Boolean network and to determine all reachable and non-reachable states. We found six simplest forms of controllable 2-node Boolean networks and explored the consistency of transition matrices while extending these six forms to controllable networks with more nodes. Importantly, we proposed the first state feedback control strategy to drive the network based on the status of all nodes in the network. Finally, we applied our reachability condition to the major switch of P53 pathway to predict the progression of the pathway and validate the prediction with published experimental results.ConclusionsThis control strategy allowed us to apply realtime control to drive Boolean networks, which could not be achieved by the current control strategy for Boolean networks. Our results enabled a more comprehensive understanding of the evolution of Boolean networks and might be extended to output feedback control design.

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“…This means that all possible trajectories of the network consist of either cycles (loops or attractors) of any length from size one (a fixed point) to a maximum of 2 n , or transient states leading eventually to a cycle. An ideal total description of the network (in which one accounts for all 2 n states) can be achieved by matrix methods (Brown, 2003;Cheng, 2009;Cheng and Qi, 2010a;2010b;Cheng et al, 2011;Cull, 1971;Rushdi and Al-Otaibi, 2007), but can be realized only for small n and would be unfeasible for most networks of interest that usually have 100 or more nodes. Zhao (2005) showed that even the determination of the number of fixed points (cycles of length 1) for monotone Boolean networks and the determination of the existence of fixed points for general Boolean networks are both strong NP-complete problems, which means among other things that both problems are highly intractable and that the best algorithms that can ever be devised for them are highly inefficient.…”

Section: Introductionmentioning

confidence: 99%

On Reduced Scalar Equations for Synchronous Boolean Networks

Rushdi¹,

Alsogati²

2013

Journal of Mathematics and Statistics

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A total description of a synchronous Boolean network is typically achieved by a matrix recurrence relation. A simpler alternative is to use a scalar equation which is a possibly nonlinear equation that involves two or more instances of a single scalar variable and some Boolean operator(s). Further simplification is possible in terms of a linear reduced scalar equation which is the simplest two-term scalar equation that includes no Boolean operators and equates the value of a scalar variable at a latter instance t₂ to its value at an earlier instance t₁. This equation remains valid when the times t₁ and t₂ are both augmented by any integral multiple of the underlying time period. In other words, there are infinitely many versions of a reduced scalar equation, any of which is useful for deducing information about the cyclic behavior of the network. However, to obtain correct information about the transient behavior of the network, one must find the true reduced scalar equation for which instances t₁ and t₂ are minimal. This study investigates the nature, derivation and utilization of reduced scalar equations. It relies on Boolean-algebraic manipulations for the derivation of such equations and suggests that this derivation can be facilitated by seeking certain orthogonality relations among certain successive (albeit not necessarily consecutive) instances of the same scalar variable. We demonstrate, contrary to previously published assumptions or assertions, that there is typically no common reduced scalar equation for all the scalar variables. Each variable usually satisfies its own distinct reduced scalar equation. We also demonstrate that the derivation of a reduced scalar equation is achieved not only by proving it but also by disproving an immediately preceding version of it when such a version might exist. We also demonstrate that, despite the useful insight supplied by the reduced scalar equations, they do not provide a total solution like the one offered by matrix methods and therefore they need be supported by other techniques of mathematical reasoning. We present three classical examples to illustrate our techniques. Two examples are tutorials on the necessary Boolean-algebraic techniques. They present corrections of previously published results and refute purported claims of discrepancies between scalar and matrix methods. The third example illustrates how the reduced scalar equations can be supplemented by techniques of number theory, Diophantine equations and Boolean equations in making subtle inferences about Boolean networks. We achieved a better understanding of the nature of reduced scalar equations, demonstrated Boolean-algebraic techniques for deriving them, presented other mathematical tools for utilizing them and finally reconciled them with the more encompassing but more complex and less insightful matrix methods

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Input-State Approach to Boolean Networks

Cited by 149 publications

References 12 publications

Further results on dynamic-algebraic Boolean control networks

Further results on dynamic-algebraic Boolean control networks

State feedback control design for Boolean networks

On Reduced Scalar Equations for Synchronous Boolean Networks

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