Rubin Jonathan, Wechselberger Martin
Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, PA, USA.
Biol Cybern. 2007 Jul;97(1):5-32. doi: 10.1007/s00422-007-0153-5. Epub 2007 Apr 26.
This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin-Huxley model, depending on parameters tau( h ), tau( n ) that scale the rates of change of the gating variables. After reducing the model to an appropriate nondimensionalized form featuring one fast and two slow variables, we use geometric singular perturbation theory to analyze the model's dynamics under systematic variation of the parameters tau( h ), tau( n ), and applied current I. As expected, we find that for fixed (tau( h ), tau( n )), the model undergoes a transition from excitable, with a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO's), featuring slow action potential generation, arise for an intermediate range of I values, if tau( h ) or tau( n ) is sufficiently large. Our analysis explains in detail the geometric mechanisms underlying these results, which depend crucially on the presence of two slow variables, and allows for the quantitative estimation of transitional parameter values, in the singular limit. In particular, we show that the subthreshold oscillations in the observed MMO patterns arise through a generalized canard phenomenon. Finally, we discuss the relation of results obtained in the singular limit to the behavior observed away from, but near, this limit.
这项工作的动机源于对未耦合的霍奇金 - 赫胥黎模型中显著缓慢放电的观察，其取决于缩放门控变量变化率的参数τ(h)和τ(n)。在将模型简化为具有一个快速变量和两个缓慢变量的适当无量纲形式后，我们使用几何奇异摄动理论来分析在参数τ(h)、τ(n)和施加电流I的系统变化下模型的动力学。正如预期的那样，我们发现对于固定的(τ(h)，τ(n))，随着I的增加，模型经历从具有稳定静息平衡态的可兴奋状态到具有经典弛豫振荡的振荡状态的转变。有趣的是，如果τ(h)或τ(n)足够大，对于中间范围的I值会出现以缓慢动作电位生成为特征的混合模式振荡（MMO）。我们的分析详细解释了这些结果背后的几何机制，这关键取决于两个缓慢变量的存在，并允许在奇异极限下对过渡参数值进行定量估计。特别是，我们表明观察到的MMO模式中的阈下振荡是通过广义鸭现象产生的。最后，我们讨论了在奇异极限下获得的结果与在远离但接近此极限处观察到的行为之间的关系。
