School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA.
Bull Math Biol. 2018 Jan;80(1):130-150. doi: 10.1007/s11538-017-0366-2. Epub 2017 Nov 17.
It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction-diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction-diffusion system is Baer and Rinzel's continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon "complete accommodation."
已经证实，在具有缓慢通过 Hopf 分岔（动态 Hopf 分岔）的问题中，直到缓慢变化的参数大大超过静态 Hopf 分岔分析（时间延迟效应）预测的值，才可能发生到大振幅振荡的转变，延迟的长度取决于缓慢变化的参数的初始值（时间记忆效应）。在本文中，我们使用分析（WKB 方法）和数值方法介绍了新的延迟和记忆效应现象。我们提出了一个反应扩散系统，对于该系统，通过 Hopf 分岔缓慢地升高刺激参数（注入电流）会引起大振幅振荡，这些振荡局限于与注入点有很大距离的位置（空间延迟效应）。此外，如果初始电流值发生变化，则该位置可能会发生变化（空间记忆效应）。我们的反应扩散系统是 Baer 和 Rinzel 的棘突树突电缆的连续体模型；该系统由与可兴奋的树突棘弱耦合的被动树突电缆组成。我们将该系统的结果与系统中反应和扩散部分之间具有更强耦合的神经电缆模型的结果进行了比较。最后，我们从数学上证明了 Hodgkin 和 Huxley 的观点是正确的，即对于足够缓慢的电流斜坡和足够大的电缆长度，没有注入电流的值会导致他们的可兴奋电缆模型发生点火；我们称这种现象为“完全适应”。
