Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Gibo, Shingo; Ito, Hiroshi

doi:10.1016/j.jtbi.2015.04.024

Cited by 8 publications

(4 citation statements)

References 27 publications

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“…Many studies have been conducted on the dynamical properties of ultradiscrete equations derived from continuous differential equations for non-integrable systems, such as reaction-diffusion systems [1][2][3], an inflammatory response system [4,5], and a biological negativefeedback system [6]. For these derivations, it is necessary to adopt a difference method that preserves the positivity of the continuous equations in order to apply the ultradiscrete limit [7].…”

Section: Introductionmentioning

confidence: 99%

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Ohmori,

Yamazaki

2023

JSIAM Letters

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Relationship for dynamical properties in the vicinity of fixed points between two-dimensional continuous and its positivity-preserving discretized dynamical systems is studied. Based on linear stability analysis, we reveal the conditions under which the dynamical structures of the original continuous dynamical systems are retained in their discretized dynamical systems, and the types of fixed points are identified if they change due to discretization. We also discuss stability of the fixed points in the discrete dynamical systems. The obtained general results are applied to Sel'kov model and Lengyel-Epstein model.

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Section: Introductionmentioning

confidence: 99%

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Ohmori,

Yamazaki

2023

JSIAM Letters

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“…The ultradiscretization has also been applied to nonintegrable dynamical systems, such as SIR model [3], a model for an inflammatory response [4], Fisher-KPP equation [5], Allen-Cahn equation [5], Gray-Scott model [6], a model for biological rhythms [7], a reactiondiffusion model [8], normal forms in one dimensional dynamical systems [9], Sel'kov model [10,11], and van der Pol equation [12]. These studies also show how the ultradiscretized equations retain the dynamical features of the original equations, as well as how new dynamical features are introduced by the ultradiscretization.…”

Section: Introductionmentioning

confidence: 99%

Poincaré map approach to limit cycles of a simplified ultradiscrete Sel'kov model

Ohmori

Yamazaki

2022

JSIAM Letters

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Dynamical properties of the two limit cycles in a ultradiscrete Sel'kov model are analytically investigated. We construct a Poincaré map for the limit cycles and reveal their stabilities; one is attracting and the other is repelling. Basins for the limit cycles are identified. It is found that the basin of the repelling limit cycle has self-similar structure. We also review the Poincaré map from the viewpoint of integrable system theory.

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“…Non-integrable dissipative systems and reaction-diffusion systems have also been target for application of ultradiscretization [3,4,5,6,7,8,9,10]. Recently, we have applied ultradiscretization to bifurcation phenomena in one-dimensional dynamical systems [10].…”

mentioning

confidence: 99%

Dynamical properties of max-plus equations obtained from tropically discretized Sel'kov model

Ohmori

Yamazaki²

2021

Preprint

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Max-plus equations are derived from tropically discretized Sel'kov model via ultradiscretization. These max-plus equations possess common dynamical structures with the discretized model: Neimark-Sacker bifurcation and limit cycles. The limit cycles of the ultradiscrete maxplus equations have seven discrete states. Furthermore, these max-plus equations exhibit excitability. Relationship between the tropically discretized model and the derived max-plus equations is also discussed based on numerical results. It is found that the derived max-plus equations correspond to a limiting case of the tropically discretized model.Ultradiscretization is known to be a method to derive max-plus equations from difference equations. Actually for example, ultradiscretization has been applied to a discretized KdV equation. As a result, a max-plus equation has been obtained, which is often referred as the boxand-ball system [1,2]. The box-and-ball system (ultradiscrete system) exhibits soliton behaviors which are also confirmed in the original KdV equation (continuous system). The important point is that ultradiscretization successfully retains and elucidates the essence of dynamical structures in the original nonlinear systems (soliton behaviors in this example), although the ultradiscrete system does not reproduce all dynamical properties of the original system.

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Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Cited by 8 publications

References 27 publications

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Poincaré map approach to limit cycles of a simplified ultradiscrete Sel'kov model

Dynamical properties of max-plus equations obtained from tropically discretized Sel'kov model

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