Limits...
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.


Relationship between edge control strength and minimal control time.For the T-cell network, (a) an inverted rectangular control signal of duration  and amplitude , where μ0 is the original parameter value and μn is the control parameter value. A saddle-node bifurcation occurs for μ=μc, so Δe=μc−μn is the excessive amount of the parameter change over the critical value μc. (b,c) Minimal control time  versus μn, where parameter control is applied to the activation edge from node ‘S1P' to node ‘PDGFR' and to the inhibitory edge from ‘DISC' to ‘MCL1', respectively. These four nodes are indicated with the solid black circles in Fig. 1a. The corresponding plots on a logarithmic scale in the insets of (b,c) suggest a power-law scaling behaviour between  and Δe (equation (2)). The fitted power-law scaling exponents are β≈−0.44 and −0.55, respectively, for (b,c).
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f2: Relationship between edge control strength and minimal control time.For the T-cell network, (a) an inverted rectangular control signal of duration and amplitude , where μ0 is the original parameter value and μn is the control parameter value. A saddle-node bifurcation occurs for μ=μc, so Δe=μc−μn is the excessive amount of the parameter change over the critical value μc. (b,c) Minimal control time versus μn, where parameter control is applied to the activation edge from node ‘S1P' to node ‘PDGFR' and to the inhibitory edge from ‘DISC' to ‘MCL1', respectively. These four nodes are indicated with the solid black circles in Fig. 1a. The corresponding plots on a logarithmic scale in the insets of (b,c) suggest a power-law scaling behaviour between and Δe (equation (2)). The fitted power-law scaling exponents are β≈−0.44 and −0.55, respectively, for (b,c).

Mentions: We detail how actual control can be implemented based on the attractor network for the T-cell system. To be concrete, we assume that the control signal has the shape of a rectangular pulse in the plot of a parameter versus time, as shown in Fig. 2a, where the control parameter is μ and the rectangular pulse has duration and amplitude , with μ0 denoting the nominal parameter value and μn being the value during the time interval when control is on. For the T-cell network, we set μ0=1.0. As μ is reduced, the system approaches a bifurcation point. (In other examples, a bifurcation can be reached by increasing a control parameter, as in the low-dimensional GRNs detailed in the Methods.) Extensive numerical simulations show that, to control the T-cell network from a cancerous state (C1 or C2) to the normal state N, there are wide ranges of choices for Δμ and . In fact, once μn is decreased through the bifurcation point μc at which the initial attractor loses its stability, the goal of control can be realized. The critical value μc for each parameter can be identified from the bifurcation analysis. In addition, for μn<μc, there exists a required minimum control time , over which the system will move into the original basin of the target attractor before control is activated. Insofar as , one does not need longer duration of control as the system will evolve into the target attractor following its natural dynamical evolution with the nominal parameter μ0. The value of increases as μn is closer to μc, where if μn=μc, is infinite due to the critical slowing down at the bifurcation point μc. Figure 2b,c show, respectively, for the T-cell network, the relationship between and μn when controlling the strength of the activation edge from the node ‘S1P' to the node ‘PDGFR', and that of the inhibitory edge from the node ‘DISC' to the node ‘MCL1' (cf., Fig. 1a, the nodes denoted as black circles and connected by bold coupling edges). The critical value μc (indicated by the dotted line) can be estimated accordingly. The insets in Fig. 2b and c show the corresponding plots of the relationships on a double logarithmic scale, with the horizontal axis to be Δe=μc−μn, the exceeded value of μn over the critical point μc. We observe the following power-law scaling behaviour:


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)

Relationship between edge control strength and minimal control time.For the T-cell network, (a) an inverted rectangular control signal of duration  and amplitude , where μ0 is the original parameter value and μn is the control parameter value. A saddle-node bifurcation occurs for μ=μc, so Δe=μc−μn is the excessive amount of the parameter change over the critical value μc. (b,c) Minimal control time  versus μn, where parameter control is applied to the activation edge from node ‘S1P' to node ‘PDGFR' and to the inhibitory edge from ‘DISC' to ‘MCL1', respectively. These four nodes are indicated with the solid black circles in Fig. 1a. The corresponding plots on a logarithmic scale in the insets of (b,c) suggest a power-law scaling behaviour between  and Δe (equation (2)). The fitted power-law scaling exponents are β≈−0.44 and −0.55, respectively, for (b,c).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Relationship between edge control strength and minimal control time.For the T-cell network, (a) an inverted rectangular control signal of duration and amplitude , where μ0 is the original parameter value and μn is the control parameter value. A saddle-node bifurcation occurs for μ=μc, so Δe=μc−μn is the excessive amount of the parameter change over the critical value μc. (b,c) Minimal control time versus μn, where parameter control is applied to the activation edge from node ‘S1P' to node ‘PDGFR' and to the inhibitory edge from ‘DISC' to ‘MCL1', respectively. These four nodes are indicated with the solid black circles in Fig. 1a. The corresponding plots on a logarithmic scale in the insets of (b,c) suggest a power-law scaling behaviour between and Δe (equation (2)). The fitted power-law scaling exponents are β≈−0.44 and −0.55, respectively, for (b,c).
Mentions: We detail how actual control can be implemented based on the attractor network for the T-cell system. To be concrete, we assume that the control signal has the shape of a rectangular pulse in the plot of a parameter versus time, as shown in Fig. 2a, where the control parameter is μ and the rectangular pulse has duration and amplitude , with μ0 denoting the nominal parameter value and μn being the value during the time interval when control is on. For the T-cell network, we set μ0=1.0. As μ is reduced, the system approaches a bifurcation point. (In other examples, a bifurcation can be reached by increasing a control parameter, as in the low-dimensional GRNs detailed in the Methods.) Extensive numerical simulations show that, to control the T-cell network from a cancerous state (C1 or C2) to the normal state N, there are wide ranges of choices for Δμ and . In fact, once μn is decreased through the bifurcation point μc at which the initial attractor loses its stability, the goal of control can be realized. The critical value μc for each parameter can be identified from the bifurcation analysis. In addition, for μn<μc, there exists a required minimum control time , over which the system will move into the original basin of the target attractor before control is activated. Insofar as , one does not need longer duration of control as the system will evolve into the target attractor following its natural dynamical evolution with the nominal parameter μ0. The value of increases as μn is closer to μc, where if μn=μc, is infinite due to the critical slowing down at the bifurcation point μc. Figure 2b,c show, respectively, for the T-cell network, the relationship between and μn when controlling the strength of the activation edge from the node ‘S1P' to the node ‘PDGFR', and that of the inhibitory edge from the node ‘DISC' to the node ‘MCL1' (cf., Fig. 1a, the nodes denoted as black circles and connected by bold coupling edges). The critical value μc (indicated by the dotted line) can be estimated accordingly. The insets in Fig. 2b and c show the corresponding plots of the relationships on a double logarithmic scale, with the horizontal axis to be Δe=μc−μn, the exceeded value of μn over the critical point μc. We observe the following power-law scaling behaviour:

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.