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A geometrical approach to control and controllability of nonlinear dynamical networks.

Wang LZ, Su RQ, Huang ZG, Wang X, Wang WX, Grebogi C, Lai YC - Nat Commun (2016)

Bottom Line: To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily.We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected.We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

View Article: PubMed Central - PubMed

Affiliation: School of Electrical, Computer and Energy Engineering, Arizona State University, 650 E. Tyler Mall, Tempe, Arizona 85287-5706, USA.

ABSTRACT
In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains an outstanding problem. Here we develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily. We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected. We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

No MeSH data available.


Related in: MedlinePlus

Control of a GRN of two nodes.(a) Simplified model of the two-node GRN, where the arrowhead and bar-head edges represent activation and inhibition regulations, respectively, and the sawtooth lines denote the strength of the tunable edge. (b) Bifurcation diagram with respect to the control parameter a1, where the red and grey solid lines denote the stable and unstable steady states, respectively. In the two parallel cross-sections (with dashed line boundaries) for  and , the yellow and brown dots represent the corresponding stable attractors, respectively. (c) Control signals required to drive the system from attractor A to attractor B. In d–f, grey dashed lines represent the basin boundaries; black solid circles and green crosses denote attractors and unstable steady states, respectively. (d) For the initial parameter setting, , the system is at a low concentration state A, and the target state is B. (e) By changing a1 from  to , attractor A is destabilized and its original basin is absorbed into that of the intermediate attractor B′, so the system approaches B′. (f) When control perturbation upon a1 is released, the landscape recovers to that in d. Once the system has entered the basin of the target state B during the process in e it will evolve spontaneously towards B. Parameters in simulation are , , t0=0, t1=23 and t2=40.
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f4: Control of a GRN of two nodes.(a) Simplified model of the two-node GRN, where the arrowhead and bar-head edges represent activation and inhibition regulations, respectively, and the sawtooth lines denote the strength of the tunable edge. (b) Bifurcation diagram with respect to the control parameter a1, where the red and grey solid lines denote the stable and unstable steady states, respectively. In the two parallel cross-sections (with dashed line boundaries) for and , the yellow and brown dots represent the corresponding stable attractors, respectively. (c) Control signals required to drive the system from attractor A to attractor B. In d–f, grey dashed lines represent the basin boundaries; black solid circles and green crosses denote attractors and unstable steady states, respectively. (d) For the initial parameter setting, , the system is at a low concentration state A, and the target state is B. (e) By changing a1 from to , attractor A is destabilized and its original basin is absorbed into that of the intermediate attractor B′, so the system approaches B′. (f) When control perturbation upon a1 is released, the landscape recovers to that in d. Once the system has entered the basin of the target state B during the process in e it will evolve spontaneously towards B. Parameters in simulation are , , t0=0, t1=23 and t2=40.

Mentions: We use a two-node GRN to understand the dynamical underpinning of the attractor network. As shown in Fig. 4a, the network contains two auto-activation nodes (genes) and together they form a mutual inhibitory circuit. Such a topology was shown to be responsible for the regulation of blood stem cell differentiation53. In addition, it is conceivable that such topologies can be constructed with tunable inputs using synthetic biology approaches29.


A geometrical approach to control and controllability of nonlinear dynamical networks.

Wang LZ, Su RQ, Huang ZG, Wang X, Wang WX, Grebogi C, Lai YC - Nat Commun (2016)

Control of a GRN of two nodes.(a) Simplified model of the two-node GRN, where the arrowhead and bar-head edges represent activation and inhibition regulations, respectively, and the sawtooth lines denote the strength of the tunable edge. (b) Bifurcation diagram with respect to the control parameter a1, where the red and grey solid lines denote the stable and unstable steady states, respectively. In the two parallel cross-sections (with dashed line boundaries) for  and , the yellow and brown dots represent the corresponding stable attractors, respectively. (c) Control signals required to drive the system from attractor A to attractor B. In d–f, grey dashed lines represent the basin boundaries; black solid circles and green crosses denote attractors and unstable steady states, respectively. (d) For the initial parameter setting, , the system is at a low concentration state A, and the target state is B. (e) By changing a1 from  to , attractor A is destabilized and its original basin is absorbed into that of the intermediate attractor B′, so the system approaches B′. (f) When control perturbation upon a1 is released, the landscape recovers to that in d. Once the system has entered the basin of the target state B during the process in e it will evolve spontaneously towards B. Parameters in simulation are , , t0=0, t1=23 and t2=40.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4834635&req=5

f4: Control of a GRN of two nodes.(a) Simplified model of the two-node GRN, where the arrowhead and bar-head edges represent activation and inhibition regulations, respectively, and the sawtooth lines denote the strength of the tunable edge. (b) Bifurcation diagram with respect to the control parameter a1, where the red and grey solid lines denote the stable and unstable steady states, respectively. In the two parallel cross-sections (with dashed line boundaries) for and , the yellow and brown dots represent the corresponding stable attractors, respectively. (c) Control signals required to drive the system from attractor A to attractor B. In d–f, grey dashed lines represent the basin boundaries; black solid circles and green crosses denote attractors and unstable steady states, respectively. (d) For the initial parameter setting, , the system is at a low concentration state A, and the target state is B. (e) By changing a1 from to , attractor A is destabilized and its original basin is absorbed into that of the intermediate attractor B′, so the system approaches B′. (f) When control perturbation upon a1 is released, the landscape recovers to that in d. Once the system has entered the basin of the target state B during the process in e it will evolve spontaneously towards B. Parameters in simulation are , , t0=0, t1=23 and t2=40.
Mentions: We use a two-node GRN to understand the dynamical underpinning of the attractor network. As shown in Fig. 4a, the network contains two auto-activation nodes (genes) and together they form a mutual inhibitory circuit. Such a topology was shown to be responsible for the regulation of blood stem cell differentiation53. In addition, it is conceivable that such topologies can be constructed with tunable inputs using synthetic biology approaches29.

Bottom Line: To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily.We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected.We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

View Article: PubMed Central - PubMed

Affiliation: School of Electrical, Computer and Energy Engineering, Arizona State University, 650 E. Tyler Mall, Tempe, Arizona 85287-5706, USA.

ABSTRACT
In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains an outstanding problem. Here we develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily. We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected. We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

No MeSH data available.


Related in: MedlinePlus