Regularity of boundary quasi-potentials for planar systems

Day, Martin V.

doi:10.1007/bf01261992

Cited by 22 publications

(12 citation statements)

References 6 publications

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“…From [15, 16], we know that the Freidlin-Wentzell quasi-potential function W ( x ) has high regularity in the neighborhood of stable nodes and limit cycles. Thus the example discussed below in 3.2 satisfies the conditions of Theorem 1.…”

Section: Proofmentioning

confidence: 99%

Quantification of degeneracy in biological systems for characterization of functional interactions between modules

Dwivedi

Huang

et al. 2012

Journal of Theoretical Biology

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There is an evolutionary advantage in having multiple components with overlapping functionality (i.e degeneracy) in organisms. While theoretical considerations of degeneracy have been well established in neural networks using information theory, the same concepts have not been developed for differential systems, which form the basis of many biochemical reaction network descriptions in systems biology. Here we establish mathematical definitions of degeneracy, complexity and robustness that allow for the quantification of these properties in a system. By exciting a dynamical system with noise, the mutual information associated with a selected observable output and the interacting subspaces of input components can be used to define both complexity and degeneracy. The calculation of degeneracy in a biological network is a useful metric for evaluating features such as the sensitivity of a biological network to environmental evolutionary pressure. Using a two-receptor signal transduction network, we find that redundant components will not yield high degeneracy whereas compensatory mechanisms established by pathway crosstalk will. This form of analysis permits interrogation of large-scale differential systems for non-identical, functionally equivalent features that have evolved to maintain homeostasis during disruption of individual components.

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Section: Proofmentioning

confidence: 99%

Quantification of degeneracy in biological systems for characterization of functional interactions between modules

Dwivedi

Huang

et al. 2012

Journal of Theoretical Biology

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“…(1) For any > 0 there exists 0 < w 0 < so that V is constant over the W level set fx 2 D : W(x) = w 0 g: (2) V is smooth in D. This can be derived from condition (4) without much trouble, but it will come out directly in our proof of Theorem 2. We believe that M 0 = M @ D is actually equivalent to the conditions of the theorem, but we have not proved it.…”

Section: X5: Proof Of the Sufficiency Theoremmentioning

confidence: 90%

Exit cycling for the Van der Pol oscillator and quasipotential calculations

Day

1996

J Dyn Diff Equat

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Abstract. We discuss the phenomenon of cycling for noise induced escape to a unstable periodic orbit. The presence of cycling is shown to follow from qualitative properties of two quasipotential functions. A method of numerically evaluating these quasipotential functions is described, and applied to the Van der Pol oscillator as an example. Figures resulting from these calculations reveal that nonconvergent cycling of exit measures does occur for the Van der Pol example. x1: IntroductionThe Van der Pol equation We are going to consider < 0 (which is equivalent t o r e v ersing time for > 0). This makes (1.2) a system with a stable critical point at the origin, surrounded by an unstable limit cycle. The e ects of adding an asymptotically small random perturbation to such a system have been discussed in recent w ork, such as 2] and other references cited there. Speci cally we compare the solution x( ) o f ( 1 . 2 ) , with initial condition x(0) = x 0 , to the solution x (t) of the following Itô equation:The parameter > 0 is viewed as the strength of the random perturbation. Our interest is in asymptotic behavior as # 0. It can be shown in several ways that x (t) ! x(t) uniformly (in probability) over t in any xed time interval 0 T ]. The story is di erent however when we l o o k at the whole time axis, t 2 0 1).Let D be the region enclosed by the limit cycle of (1.2), with @ Dbeing the limit cycle itself. For any x 0 2 D the solution x(t) of (1.2) remains forever in D and converges to 0 as t ! 1 . In contrast to this, x (t) w i l l e v entually (with probability 1 ) w ander out to @ D , r e a c hing it rst at some random time < 1.A particularly interesting phenomena occurs in the study of the distribution of this point x ( ) o f r s t \noise induced" escape to the limit cycle @ D :(1.4) (A) = P x ( ) 2 A] A @ D :1991 Mathematics Subject Classi cation. 60H30, 60H10, 60J70, 60J60.

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“…This provides an alternative way of verifying H 0 ). For results regarding high regularity of the quasi-potential function, see [8,9].…”

Section: Concentration Of Stationary Measuresmentioning

confidence: 99%

“…Without loss of generality, we assume that A = {0}. It follows from H 1 ) and the WKB expansion (see [8,9,33]) that there is a function W ∈ C 2 (R n ), called quasi-potential function, such that the density function of µ for each ∈ (0, * ) has the form…”

Section: Entropy-dimension Relationshipmentioning

confidence: 99%

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Systematic Measures of Biological Networks I: Invariant Measures and Entropy

2016

Comm Pure Appl Math

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This paper is part I of a two‐part series devoted to the study of systematic measures in a complex biological network modeled by a system of ordinary differential equations. As the mathematical complement to our previous work with collaborators, the series aims at establishing a mathematical foundation for characterizing three important systematic measures: degeneracy, complexity, and robustness, in such a biological network and studying connections among them. To do so, we consider in part I stationary measures of a Fokker‐Planck equation generated from small white noise perturbations of a dissipative system of ordinary differential equations. Some estimations of concentration of stationary measures of the Fokker‐Planck equation in the vicinity of the global attractor are presented. The relationship between the differential entropy of stationary measures and the dimension of the global attractor is also given.© 2016 Wiley Periodicals, Inc.

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Regularity of boundary quasi-potentials for planar systems

Cited by 22 publications

References 6 publications

Quantification of degeneracy in biological systems for characterization of functional interactions between modules

Quantification of degeneracy in biological systems for characterization of functional interactions between modules

Exit cycling for the Van der Pol oscillator and quasipotential calculations

Systematic Measures of Biological Networks I: Invariant Measures and Entropy

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