Lattice structures for attractors  I

Kalies, William D.; Mischaikow, Konstantin; Vandervorst, Robert

doi:10.3934/jcd.2014.1.307

Cited by 32 publications

(96 citation statements)

References 15 publications

(21 reference statements)

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“…To address the lack of robustness of invariant sets with respect to parameters, another change in perspective is needed. Initiated by C. Conley [37] and developed over the last 40 years [38][39][40][41] the emphasis shifts from invariant sets to positively attracting sets.…”

Section: Discussionmentioning

confidence: 99%

Multistability in the epithelial-mesenchymal transition network

2020

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Background: The transitions between epithelial (E) and mesenchymal (M) cell phenotypes are essential in many biological processes like tissue development and cancer metastasis. Previous studies, both modeling and experimental, suggested that in addition to E and M states, the network responsible for these phenotypes exhibits intermediate phenotypes between E and M states. The number and importance of such states is subject to intense discussion in the epithelial-mesenchymal transition (EMT) community. Results: Previous modeling efforts used traditional bifurcation analysis to explore the number of the steady states that correspond to E, M and intermediate states by varying one or two parameters at a time. Since the system has dozens of parameters that are largely unknown, it remains a challenging problem to fully describe the potential set of states and their relationship across all parameters. We use the computational tool DSGRN (Dynamic Signatures Generated by Regulatory Networks) to explore the intermediate states of an EMT model network by computing summaries of the dynamics across all of parameter space. We find that the only attractors in the system are equilibria, that E and M states dominate across parameter space, but that bistability and multistability are common. Even at extreme levels of some of the known inducers of the transition, there is a certain proportion of the parameter space at which an E or an M state co-exists with other stable steady states. Conclusions: Our results suggest that the multistability is broadly present in the EMT network across parameters and thus response of cells to signals may strongly depend on the particular cell line and genetic background.

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Section: Discussionmentioning

confidence: 99%

Multistability in the epithelial-mesenchymal transition network

2020

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“…This process can be made fairly general, see e.g. [KMV14,KMV16]. For illustration, we now outline one constructive approach that we used to produce the sample results throughout the paper.…”

Section: Computational Conley Indexmentioning

confidence: 99%

Sofic Shifts via Conley Index Theory: Computing Lower Bounds on Recurrent Dynamics for Maps

Day¹,

Frongillo²

2019

SIAM J. Appl. Dyn. Syst.

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We extend and demonstrate the applicability of computational Conley index techniques for computing symbolic dynamics and corresponding lower bounds on topological entropy for discrete-time systems governed by maps. In particular, we describe an algorithm that uses Conley index information to construct sofic shifts that are topologically semi-conjugate to the system under study. As illustration, we present results for the two-dimensional Hénon map, the three-dimensional LPA map, and the infinite-dimensional Kot-Schaffer map. This approach significantly builds on methods first presented in [DFT08] and is related to work in [Kwa00, Kwa04].

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“…Further development of these methods relies on understanding the way in which lattices and order naturally play a role in dynamics on the fundamental level of attractors, repellers, and invariant sets, cf. [19]. Analogues of these basic concepts in dynamical systems theory also exist in directed graphs and are used to analyze combinatorial representations of dynamical systems.…”

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confidence: 99%

“…A subset N Ă X is an attracting block if f pcl Nq Ă int N, and ωpNq " A is the associated attractor, cf. [19,20]. In many cases attracting blocks are readily computable, while the lattice of attractors is not directly computable in general.…”

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confidence: 99%

“…The latter is a sublattice of the Boolean algebra RpXq of regular closed sets in X with the binary operations _ " Y and^" cl pint p¨X¨qq. The definitions of all of these structures and their properties are given in detail in [19,20] to which we refer the reader, see also Section 1.2. We emphasize that a fundamental consideration when working with the above structures is that, while the lattice operations on ABlock R p f q are join and meet in RpXq, the lattice operations on Attp f q are _ " Y and^" ωp¨X¨q, where ω denotes the operation of taking the ω-limit set.…”

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confidence: 99%

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An Algorithmic Approach to Lattices and Order in Dynamics

Kalies

Kasti²,

Vandervorst

2018

SIAM J. Appl. Dyn. Syst.

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Recurrent versus gradient-like behavior in global dynamics can be characterized via a surjective lattice homomorphism between certain bounded, distributive lattices, that is, between attracting blocks (or neighborhoods) and attractors. Using this characterization, we build finite, combinatorial models in terms of surjective lattice homomorphisms, which lay a foundation for a computational theory for dynamical systems that focuses on Morse decompositions and index lattices. In particular we present an algorithm that builds a combinatorial model that represents a Morse decomposition for the underlying dynamics. We give computational examples that illustrate the theory for both maps and flows.

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Lattice structures for attractors I

Cited by 32 publications

References 15 publications

Multistability in the epithelial-mesenchymal transition network

Multistability in the epithelial-mesenchymal transition network

Sofic Shifts via Conley Index Theory: Computing Lower Bounds on Recurrent Dynamics for Maps

An Algorithmic Approach to Lattices and Order in Dynamics

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