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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.


The benefit of noise in controlling the T-cell network.(a) Success rate to control the T-cell network from the cancerous state C1 to the normal state N using a combination of parameter perturbation and external noise (of amplitude) σ, where Δd≡μn−μc is the parameter deficiency. Warm colours indicate higher probability values of successful control. The perturbation duration is . The results are averaged over 1,000 realizations. (b) A three-dimensional plot: success rate versus Δd and σ.
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f3: The benefit of noise in controlling the T-cell network.(a) Success rate to control the T-cell network from the cancerous state C1 to the normal state N using a combination of parameter perturbation and external noise (of amplitude) σ, where Δd≡μn−μc is the parameter deficiency. Warm colours indicate higher probability values of successful control. The perturbation duration is . The results are averaged over 1,000 realizations. (b) A three-dimensional plot: success rate versus Δd and σ.

Mentions: Suppose the control parameter is set to the value μn, which is insufficient to induce escape from the undesired attractor in the absence of noise. When noise is present, the system dynamics is stochastic. To characterize the control performance, we carry out independent simulations starting from one cancerous state, for example, C1, but with insufficient control strength as characterized by the deficiency parameter Δd≡μn−μc, and calculate the probability P of control success through the number of trials that the system can be successfully driven to the normal state N. Figure 3a shows, on a double logarithmic scale, the values of P in the parameter plane of σ and Δd, where the control parameter is the strength of the activation edge from node ‘S1P' to node ‘PDGFR' in the T-cell network. A three-dimensional plot of P versus σ and Δd is shown in Fig. 3b. We see that, for fixed σ, P decreases with Δd but, for any fixed value of Δd, the probability P increases with σ, indicating the beneficial role of noise in facilitating control. In the parameter plane, there exists a well-defined boundary, below which the control probability assumes large values but above which the probability is near zero. Testing alternative control parameters yields essentially the same results, because of the simplicity of the attractor network for the T-cell system and the multiple directed edges from each cancerous state to the normal state.


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)

The benefit of noise in controlling the T-cell network.(a) Success rate to control the T-cell network from the cancerous state C1 to the normal state N using a combination of parameter perturbation and external noise (of amplitude) σ, where Δd≡μn−μc is the parameter deficiency. Warm colours indicate higher probability values of successful control. The perturbation duration is . The results are averaged over 1,000 realizations. (b) A three-dimensional plot: success rate versus Δd and σ.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: The benefit of noise in controlling the T-cell network.(a) Success rate to control the T-cell network from the cancerous state C1 to the normal state N using a combination of parameter perturbation and external noise (of amplitude) σ, where Δd≡μn−μc is the parameter deficiency. Warm colours indicate higher probability values of successful control. The perturbation duration is . The results are averaged over 1,000 realizations. (b) A three-dimensional plot: success rate versus Δd and σ.
Mentions: Suppose the control parameter is set to the value μn, which is insufficient to induce escape from the undesired attractor in the absence of noise. When noise is present, the system dynamics is stochastic. To characterize the control performance, we carry out independent simulations starting from one cancerous state, for example, C1, but with insufficient control strength as characterized by the deficiency parameter Δd≡μn−μc, and calculate the probability P of control success through the number of trials that the system can be successfully driven to the normal state N. Figure 3a shows, on a double logarithmic scale, the values of P in the parameter plane of σ and Δd, where the control parameter is the strength of the activation edge from node ‘S1P' to node ‘PDGFR' in the T-cell network. A three-dimensional plot of P versus σ and Δd is shown in Fig. 3b. We see that, for fixed σ, P decreases with Δd but, for any fixed value of Δd, the probability P increases with σ, indicating the beneficial role of noise in facilitating control. In the parameter plane, there exists a well-defined boundary, below which the control probability assumes large values but above which the probability is near zero. Testing alternative control parameters yields essentially the same results, because of the simplicity of the attractor network for the T-cell system and the multiple directed edges from each cancerous state to the normal state.

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.