Cooperative Boolean systems with generically long attractors I

Just, Winfried; Malicki, Maciej

doi:10.1080/10236198.2012.691167

Cited by 4 publications

(1 citation statement)

References 15 publications

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“…For instance, what happens for the family of SIDs with positives arcs only? It is an interesting problem, because the BN having such SIDs are monotone, and since Tarski [39] monotone networks received great attention [8,35,3,5,26,15,16,17]. We hope to be able to adapt our constructions in order to fit this new constraint.…”

Section: Minimum Fixed Point Problemmentioning

confidence: 99%

Complexity of Maximum Fixed Point Problem in Boolean Networks

Bridoux¹,

Durbec²,

Perrot³

et al. 2019

Computing With Foresight and Industry

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A Boolean network (BN) with n components is a discrete dynamical system described by the successive iterations of a function f : {0, 1} n → {0, 1} n. This model finds applications in biology, where fixed points play a central role. For example, in genetic regulations, they correspond to cell phenotypes. In this context, experiments reveal the existence of positive or negative influences among components: component i has a positive (resp. negative) influence on component j meaning that j tends to mimic (resp. negate) i. The digraph of influences is called signed interaction digraph (SID), and one SID may correspond to a large number of BNs (which is, in average, doubly exponential according to n). The present work opens a new perspective on the well-established study of fixed points in BNs. When biologists discover the SID of a BN they do not know, they may ask: given that SID, can it correspond to a BN having at least/at most k fixed points? Depending on the input, we prove that these problems are in P or complete for NP, NP NP , NP #P or NEXPTIME. In particular, we prove that it is NP-complete (resp. NEXPTIME-complete) to decide if a given SID can correspond to a BN having at least two fixed points (resp. no fixed point).

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Section: Minimum Fixed Point Problemmentioning

confidence: 99%

Complexity of Maximum Fixed Point Problem in Boolean Networks

Bridoux¹,

Durbec²,

Perrot³

et al. 2019

Computing With Foresight and Industry

View full text Add to dashboard Cite

show abstract

Complexity of fixed point counting problems in Boolean networks

Bridoux

Durbec

Perrot

et al. 2022

Journal of Computer and System Sciences

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Cooperative Boolean systems with generically long attractors II

Just

Malicki

2013

Adv Differ Equ

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The purpose of this paper is to investigate to what extent cooperativity, that is, the absence of negative interactions, in Boolean networks with synchronous updating, imposes limits on chaos-like properties that are possible in such systems. Our focus is on notions of sensitive dependence on initial conditions, or a combination of sensitive dependence and large basins of attraction of exponentially long attractors, both of which are well-recognized hallmarks of chaotic dynamics in the Boolean context.We prove that a strong notion of sensitive dependence on initial conditions that formalizes decoherence along the attractor is precluded by cooperativity. Weaker notions of sensitive dependence that formalize decoherence at some time during the trajectory and sensitive dependence of the basin of attraction on initial conditions, respectively, are shown to be consistent with cooperativity, but if each regulatory function is binary AND or binary OR, in N-dimensional networks they impose an upper bound of ≈ √ 3 N on the lengths of attractors that can be reached from a fraction p ≈ 1 of initial conditions. The upper bound is shown to be optimal. These results indicate that the transfer of analogous results for differential equations models crucially depends on the precise conceptualization of chaos in the Boolean context. MSC: 34C12; 39A33; 94C10

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Cooperative Boolean systems with generically long attractors I

Cited by 4 publications

References 15 publications

Complexity of Maximum Fixed Point Problem in Boolean Networks

Complexity of Maximum Fixed Point Problem in Boolean Networks

Complexity of fixed point counting problems in Boolean networks

Cooperative Boolean systems with generically long attractors II

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