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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

Illustration of pseudo potential landscape.‘Pseudo' potential  of the two-node GRN system (a) for a1=1.0 (Δd≈0.3549), σ=0.05 and (b) for a1=1.3 (Δd≈0.0549), σ=0.05. Regions of warm and cold colours indicate the states with large and small pseudo energies, respectively. (c) For fixed σ=0.02, two-dimensional representation of  for a number of values of a1. (d) For fixed a1=1.3, two-dimensional representation of  for a number of values of σ.
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f6: Illustration of pseudo potential landscape.‘Pseudo' potential of the two-node GRN system (a) for a1=1.0 (Δd≈0.3549), σ=0.05 and (b) for a1=1.3 (Δd≈0.0549), σ=0.05. Regions of warm and cold colours indicate the states with large and small pseudo energies, respectively. (c) For fixed σ=0.02, two-dimensional representation of for a number of values of a1. (d) For fixed a1=1.3, two-dimensional representation of for a number of values of σ.

Mentions: The potential landscape for a GRN under Gaussian noise can be constructed from the dynamical equations of the system using the concept of ‘pseudo' energy57. For the two-node GRN system (equation (3)) subject to stochastic disturbance N(0, σ2), we can compute the potential landscape for any combination of some system parameter (say a1) and the noise amplitude σ. Figure 6a,b shows two examples of the landscape (in three-dimensional representation) for a1=1.0 and a1=1.3, where the noise amplitude is σ=0.05. We see that, for example, for a1=1.0, there are four valleys (attractors). Figure 6c shows, for σ=0.02, a two-dimensional representation of the pseudo-energy for a number of values of a1. We observe that, for a same noise amplitude, as a1 is increased, the transition rate from A to B becomes higher (the colour becomes warmer).


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)

Illustration of pseudo potential landscape.‘Pseudo' potential  of the two-node GRN system (a) for a1=1.0 (Δd≈0.3549), σ=0.05 and (b) for a1=1.3 (Δd≈0.0549), σ=0.05. Regions of warm and cold colours indicate the states with large and small pseudo energies, respectively. (c) For fixed σ=0.02, two-dimensional representation of  for a number of values of a1. (d) For fixed a1=1.3, two-dimensional representation of  for a number of values of σ.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f6: Illustration of pseudo potential landscape.‘Pseudo' potential of the two-node GRN system (a) for a1=1.0 (Δd≈0.3549), σ=0.05 and (b) for a1=1.3 (Δd≈0.0549), σ=0.05. Regions of warm and cold colours indicate the states with large and small pseudo energies, respectively. (c) For fixed σ=0.02, two-dimensional representation of for a number of values of a1. (d) For fixed a1=1.3, two-dimensional representation of for a number of values of σ.
Mentions: The potential landscape for a GRN under Gaussian noise can be constructed from the dynamical equations of the system using the concept of ‘pseudo' energy57. For the two-node GRN system (equation (3)) subject to stochastic disturbance N(0, σ2), we can compute the potential landscape for any combination of some system parameter (say a1) and the noise amplitude σ. Figure 6a,b shows two examples of the landscape (in three-dimensional representation) for a1=1.0 and a1=1.3, where the noise amplitude is σ=0.05. We see that, for example, for a1=1.0, there are four valleys (attractors). Figure 6c shows, for σ=0.02, a two-dimensional representation of the pseudo-energy for a number of values of a1. We observe that, for a same noise amplitude, as a1 is increased, the transition rate from A to B becomes higher (the colour becomes warmer).

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