Periodic Orbits and Equilibria in Glass Models for Gene Regulatory Networks

Zinovik, Igor; Chebiryak, Yury; Kroening, Daniel

doi:10.1109/tit.2009.2037078

Cited by 10 publications

(3 citation statements)

References 40 publications

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“…Rigorous investigations of Glass networks have considered them within the framework of differential inclusions [1,39]. To facilitate construction of networks with customized dynamics [64], systematically classified cyclic attractors on Glass networks with up to six switching units. Walsh and colleagues studied periodic orbits in a discontinuous vector field as a model of cycling phenomena in glacial dynamics [63].…”

Section: Further Applicationsmentioning

confidence: 99%

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Park

Shaw

Chiel³

et al. 2018

Eur. J. Appl. Math

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The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

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Section: Further Applicationsmentioning

confidence: 99%

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Park

Shaw

Chiel³

et al. 2018

Eur. J. Appl. Math

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“…The choice of edge orientation in turn is linked to the specification of focal points of the PLDE. Therefore, the presence of nodes that are not dominated indicates that the phase flow along the attractor is robust to any variations of the coefficients that define the equations in the orthant corresponding to that node [20]. We say that a node that is not dominated by the cycle is shunned by the cycle.…”

Section: Introductionmentioning

confidence: 99%

Finding Lean Induced Cycles in Binary Hypercubes

Chebiryak

Wahl

Kroening

et al. 2009

Lecture Notes in Computer Science

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Abstract. Induced (chord-free) cycles in binary hypercubes have many applications in computer science. The state of the art for computing such cycles relies on genetic algorithms, which are, however, unable to perform a complete search. In this paper, we propose an approach to finding a special class of induced cycles we call lean, based on an efficient propositional SAT encoding. Lean induced cycles dominate a minimum number of hypercube nodes. Such cycles have been identified in Systems Biology as candidates for stable trajectories of gene regulatory networks. The encoding enabled us to compute lean induced cycles for hypercubes up to dimension 7. We also classify the induced cycles by the number of nodes they fail to dominate, using a custom-built All-SAT solver. We demonstrate how clause filtering can reduce the number of blocking clauses by two orders of magnitude.

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“…Intuitively, C "preserves distances" up to k. Circuit codes were first introduced in [15] as a type of error-correcting code, but have since been employed in a variety of applications, including: rank-modulation schemes for flash memory [28,13,26], constructing worst-case examples for the analysis of combinatorial algorithms [3], and analyzing the behavior of models for gene regulatory networks [30]. Circuit codes have been extensively studied in the case k = 2, where they are called 'snakes in the box' or 'coils in the box' [8,1,27,25,29,10,18,19,21,2].…”

mentioning

confidence: 99%

The Maximum Length and Isomorphism of Circuit Codes with Long Bit Runs

Byrnes

2020

Preprint

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Recently, Byrnes [5] presented a formula for the maximum length of a symmetric circuit code that has a long bit run and odd spread. Here we show that the formula is also valid when the spread is even. We also establish that all maximum length symmetric circuit codes with long bit runs are isomorphic for an infinite and nontrivial family of circuit codes, extending a previous result of Douglas [9].

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Periodic Orbits and Equilibria in Glass Models for Gene Regulatory Networks

Cited by 10 publications

References 40 publications

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Finding Lean Induced Cycles in Binary Hypercubes

The Maximum Length and Isomorphism of Circuit Codes with Long Bit Runs

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