论文标题
多习惯霍奇金 - 赫克斯利方程的几何奇异扰动分析
Geometric singular perturbation analysis of the multiple-timescale Hodgkin-Huxley equations
论文作者
论文摘要
我们提出了一个新颖的全球三维减少，对四维Hodgkin-Huxley方程的非二维版本[J。 Rubin和M. Wechselberger，《巨型鱿鱼》 - 隐藏的卡纳德：Hodgkin-Huxley模型的3D几何学，《生物控制论》，97（2007），第5---32页]，基于几何学奇异触发理论（GSPT）。我们以两个参数状态研究了所得的三维系统的动力学，其中流在三个不同的时间尺度上演变。具体而言，我们证明该系统根据[P. Kaklamanos，N。Popović和K. U. Kristiansen，三段尺度系统中混合模式振荡的分叉：一个扩展的原型例子，混乱：非线性科学跨学科杂志，32（2022），p。 013108]，我们根据外部应用电流对各种射击模式进行了分类。虽然[S. doi，S。nabetani和S. kumagai，由时间尺度变化引起的霍奇金 - 赫克斯利方程的复杂非线性动力学，生物学控制论，85（2001），第51---64页]，对于多态霍奇金 - 赫克斯利方程，我们都在介绍了几何形式，以前彼此之间的几何形式均不在那些彼此之间的变化。
We present a novel and global three-dimensional reduction of a non-dimensionalised version of the four-dimensional Hodgkin-Huxley equations [J. Rubin and M. Wechselberger, Giant squid--hidden canard: the 3D geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), pp. 5--32] that is based on geometric singular perturbation theory (GSPT). We investigate the dynamics of the resulting three-dimensional system in two parameter regimes in which the flow evolves on three distinct timescales. Specifically, we demonstrate that the system exhibits bifurcations of oscillatory dynamics and complex mixed-mode oscillations (MMOs), in accordance with the geometric mechanisms introduced in [P. Kaklamanos, N. Popović, and K. U. Kristiansen, Bifurcations of mixed--mode oscillations in three--timescale systems: An extended prototypical example, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32 (2022), p. 013108], and we classify the various firing patterns in terms of the external applied current. While such patterns have been documented in [S. Doi, S. Nabetani, and S. Kumagai, Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes, Biological Cybernetics, 85 (2001), pp. 51--64] for the multiple-timescale Hodgkin-Huxley equations, we elucidate the geometry that underlies the transitions between them, which had not been previously emphasised.
