Bifurcation analysis of a normal form for excitable media: are stable dynamical alternans on a ring possible?
Bifurcation analysis of a normal form for excitable media: are stable dynamical alternans on a ring possible?
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可兴奋介质标准形式的分岔分析：环上的稳定动态交替是否可能？
作者：
G. Gottwald
本文对一维可激发介质中行波的规范形进行了分岔分析。最近提出的规范形式的唯象的理由是在微分延迟方程的形式。规范形式表现出一个保序的Hopf分支，可能与Bogdanov-Takens点中的鞍节点合并，以及一个保序的空间非均匀的干草叉分支。我们在这里研究的Hopf分支的传播的一个单一的脉冲在一个环的中心流形减少的装置，并为波列的多尺度分析，导致一个真实的Ginzburg-Landau方程作为相应的振幅方程。中心流形约化和多尺度分析都表明，Hopf分支总是亚临界的，与参数无关。这可能与心脏交替有关，到目前为止，心脏交替被认为是从超临界分叉产生的稳定振荡。我们讨论了心脏交替的影响，并重新审视一些可兴奋的介质中的振荡被认为是稳定的不稳定性。特别是，我们表明，我们的条件的发病的霍普夫分岔与众所周知的心脏交替的恢复条件相吻合。
We present a bifurcation analysis of a normal form for traveling waves in one-dimensional excitable media. The normal form that has been recently proposed on phenomenological grounds is given in the form of a differential delay equation. The normal form exhibits a symmetry-preserving Hopf bifurcation that may coalesce with a saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans, which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.
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DOI：
10.1103/physreve.62.4043
发表时间：
2000-09
期刊：
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
影响因子：
--
作者：
H. Hastings;F. Fenton;S. Evans;O. Hotomaroglu;J. Geetha;K. Gittelson;J. Nilson;A. Garfinkel
通讯作者：
H. Hastings;F. Fenton;S. Evans;O. Hotomaroglu;J. Geetha;K. Gittelson;J. Nilson;A. Garfinkel
影响因子：
2.4
作者：
Echebarria, Blas;Karma, Alain
通讯作者：
Karma, Alain
起搏心脏组织模型中传导阻滞的动力学。
DOI：
10.1103/physreve.71.051911
发表时间：
2005
期刊：
Physical review. E, Statistical, nonlinear, and soft matter physics .
影响因子：
--
作者：
Henry,Herve;Rappel,Wouter-Jan
通讯作者：
Rappel,Wouter-Jan
影响因子：
8.6
作者：
Echebarria, B;Karma, A
通讯作者：
Karma, A
DOI：
10.1152/ajpheart.00747.2003
发表时间：
2004-06-01
影响因子：
4.8
作者：
Cherry, EM;Fenton, FH
通讯作者：
Fenton, FH