Pinned Solutions in a Heterogeneous Three-Component FitzHugh–Nagumo Model

Heijster, Peter van; Chen, Chao-Nien; Nishiura, Yasumasa; Teramoto, Takashi

doi:10.1007/s10884-018-9694-7

Cited by 25 publications

(12 citation statements)

References 58 publications

(239 reference statements)

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“…[27], [31]) and for the Fitzhugh-Nagumo model (cf. [2], [10]), while a plant hormone (auxin) gradient is predicted to control the spatial locations of root formation in plant cells [1]. In other contexts, a spatial heterogeneity can trigger a self-replication loop consisting of spike formation, propagation, and annihilation against a domain boundary [19].…”

Section: Introductionmentioning

confidence: 99%

Spike density distribution for the Gierer–Meinhardt model with precursor

Kolokolnikov

Xie

2020

Physica D: Nonlinear Phenomena

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Section: Introductionmentioning

confidence: 99%

Spike density distribution for the Gierer–Meinhardt model with precursor

Kolokolnikov

Xie

2020

Physica D: Nonlinear Phenomena

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“…Now we are ready to compute the time derivatives of p and α by using the relations ( 38), ( 40), ( 41), (43), and (47).…”

Section: Discussionmentioning

confidence: 99%

“…Heterogeneities or defects in the media are extremely common [37][38][39] and influence the dynamics of moving patterns through pinning (or blocking), rebound, splitting, and annihilation depending on the strength of heterogeneity [40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Remarkably, even chaotic motion is observed in periodic media in case the heterogeneity period is comparable to the size of the traveling pulse [42].…”

Section: Introductionmentioning

confidence: 99%

Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media

Nishiura,

Watanabe

2021

Preprint

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The interplay between 1D traveling pulses with oscillatory tails (TPO) and heterogeneities of bump type is studied for a generalized three-component FitzHugh-Nagumo equation. First, we present that stationary pulses with oscillatory tails (SPO) form a "snaky" structure in homogeneous spaces, after which TPO branches form a "figure-eight-like stack of isolas" located adjacent to the snaky structure of SPO. Herein, we adopted input resources such as voltage-difference as a bifurcation parameter. A drift bifurcation from SPO to TPO was observed by introducing another parameter at which these two solution sheets merged. In contrast to the monotone tail case, a nonlocal interaction appeared in the heterogeneous problem between the TPO and heterogeneity, which created infinitely many saddle solutions and finitely many stable stationary solutions distributed across the entire line. The response of TPO displayed a variety of dynamics including pinning and depinning processes along with penetration (PEN) and rebound (REB). The stable/unstable manifolds of these saddles interacted with the TPO in a complex manner, which created a subtle dependence on the initial condition, and a difficulty to predict the behavior after collision even in 1D space. Nevertheless, for the 1D case, a systematic global exploration of solution branches induced by heterogeneities (heterogeneity-induced-ordered patterns; HIOP) revealed that HIOP contained all the asymptotic states after collision to predict the solution results without solving the PDEs. The reduction method of finite-dimensional ODE (ordinary differential equation) system allowed us to clarify the detailed mechanism of the transitions from PEN to pinning and pinning to REB based on a dynamical system perspective. Consequently, the basin boundary between two distinct outputs against the heterogeneities yielded an infinitely many successive reconnection of heteroclinic orbits among those saddles, because the strength of heterogeneity increased and caused the aforementioned subtle dependence of initial condition.

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“…Heterogeneity is the most important and ubiquitous type of external perturbation observed in natural environments. When a reaction-diffusion system is perturbed by some small heterogeneity, some new interesting phenomena [7,28] have been observed. With zero being an eigenvalue due to the translation free mode by periodic boundary conditions [12], we study the associated eigenfunctions for the linearization and the adjoint operator in Section 6.…”

Section: Shin-ichiro Ei and Shyuh-yaur Tzengmentioning

confidence: 99%

Spike solutions for a mass conservation reaction-diffusion system

Ichiro

Tzeng²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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This article deals with a mass conservation reaction-diffusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The profile of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived.

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Pinned Solutions in a Heterogeneous Three-Component FitzHugh–Nagumo Model

Cited by 25 publications

References 58 publications

Spike density distribution for the Gierer–Meinhardt model with precursor

Spike density distribution for the Gierer–Meinhardt model with precursor

Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media

Spike solutions for a mass conservation reaction-diffusion system

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