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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.


Attractor network construction for a GRN of two nodes.(a–c) Bifurcation diagrams with respect to the coupling parameters a1, a2 and b2, respectively, where each bifurcation point can be exploited to design control. (d) The corresponding attractor network, in which a directed edge corresponds to an elementary control that is designed to steer the system from the original attractor to the directed one. The solid and dashed edges, respectively, denote the positive and negative changes in the corresponding control parameters. (e) Sequential control signals required to drive the system from attractor A to attractor C through the path A→B→C. In simulation, the original parameter values are  and . We set , followed by setting , and the respective durations of the parameter perturbation are  and .
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f5: Attractor network construction for a GRN of two nodes.(a–c) Bifurcation diagrams with respect to the coupling parameters a1, a2 and b2, respectively, where each bifurcation point can be exploited to design control. (d) The corresponding attractor network, in which a directed edge corresponds to an elementary control that is designed to steer the system from the original attractor to the directed one. The solid and dashed edges, respectively, denote the positive and negative changes in the corresponding control parameters. (e) Sequential control signals required to drive the system from attractor A to attractor C through the path A→B→C. In simulation, the original parameter values are and . We set , followed by setting , and the respective durations of the parameter perturbation are and .

Mentions: To obtain a global picture of all possible control outcomes, we construct the attractor network for the two-node GRN system, assuming that three parameters: a1, a2 and b2, are available for control. The corresponding bifurcation diagrams are shown in Fig. 5a–c, from which all saddle-node bifurcations can be identified for control design. When all the attractors are connected with directed and weighted edges through the control processes, that is, when none of the attractor is isolated, we obtain an attractor network, as shown in Fig. 5d. Specifically, the edge weight can be assigned by taking into account the key characteristics of control such as the critical parameter strength μc and the power-law scaling behaviour of the required minimum control time (see Supplementary Table 2 for details). From the attractor network, we can find all possible control paths for any given pair of original and desired states.


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)

Attractor network construction for a GRN of two nodes.(a–c) Bifurcation diagrams with respect to the coupling parameters a1, a2 and b2, respectively, where each bifurcation point can be exploited to design control. (d) The corresponding attractor network, in which a directed edge corresponds to an elementary control that is designed to steer the system from the original attractor to the directed one. The solid and dashed edges, respectively, denote the positive and negative changes in the corresponding control parameters. (e) Sequential control signals required to drive the system from attractor A to attractor C through the path A→B→C. In simulation, the original parameter values are  and . We set , followed by setting , and the respective durations of the parameter perturbation are  and .
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f5: Attractor network construction for a GRN of two nodes.(a–c) Bifurcation diagrams with respect to the coupling parameters a1, a2 and b2, respectively, where each bifurcation point can be exploited to design control. (d) The corresponding attractor network, in which a directed edge corresponds to an elementary control that is designed to steer the system from the original attractor to the directed one. The solid and dashed edges, respectively, denote the positive and negative changes in the corresponding control parameters. (e) Sequential control signals required to drive the system from attractor A to attractor C through the path A→B→C. In simulation, the original parameter values are and . We set , followed by setting , and the respective durations of the parameter perturbation are and .
Mentions: To obtain a global picture of all possible control outcomes, we construct the attractor network for the two-node GRN system, assuming that three parameters: a1, a2 and b2, are available for control. The corresponding bifurcation diagrams are shown in Fig. 5a–c, from which all saddle-node bifurcations can be identified for control design. When all the attractors are connected with directed and weighted edges through the control processes, that is, when none of the attractor is isolated, we obtain an attractor network, as shown in Fig. 5d. Specifically, the edge weight can be assigned by taking into account the key characteristics of control such as the critical parameter strength μc and the power-law scaling behaviour of the required minimum control time (see Supplementary Table 2 for details). From the attractor network, we can find all possible control paths for any given pair of original and desired states.

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.