There are typically two distinct types of single‐neuron dynamics, known as Class I and Class II. Fast‐spiking (FS) neurons are a type of Class II neuron, which tend to exhibit periodic spikes in the gamma frequency. Although the population dynamics of Class I neurons are well elucidated with the Fokker‐Planck equation (FPE), much remains unknown about the population dynamics of Class II neurons. This is because the population model of Class II neurons typically leads to a high‐dimensional FPE that is hard to analyze. The modified theta (MT) model is one of the simplest mathematical models of Class I neurons, which possesses voltage‐dependent dynamics and has advantages in the analysis of population dynamics. In this study, we propose two approximation methods to derive a one‐dimensional FPE for FS interneurons using the framework of MT transformation. One method is mean‐field approximation of the variable related to adaptation; the other is semi‐adiabatic approximation of the variable. We confirm the firing characteristics of a single neuron in these approximated models match well to that of the original model proposed by Izhikevich. For the populational dynamics of the gamma oscillation, FPEs with these approximations make it possible to identify the region where Class II neuron firings synchronize. Bifurcation analyses of two FPEs show that semi‐adiabatic approximation fits better to the numerical results than mean‐field approximation. FPE with semi‐adiabatic approximation better illustrates the fact that a population of FS interneurons has a narrower region of gamma oscillation than that of Class I neurons. Moreover, analysis with semi‐adiabatic approximation of a population composed of both Class I and FS interneurons has a narrower region of synchronization with an increase in the FS interneuron ratio. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

中文翻译：

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宏观伽马振荡下快速加标中子失步的计算研究

通常有两种不同的单神经元动力学类型，称为I类和II类。快发（FS）神经元是II类神经元的一种，倾向于在γ频率上表现出周期性的尖峰。尽管可以通过Fokker-Planck方程（FPE）很好地阐明了I类神经元的种群动态，但是对于II类神经元的种群动态仍知之甚少。这是因为II类神经元的种群模型通常会导致难以分析的高维FPE。修正theta（MT）模型是I类神经元最简单的数学模型之一，它具有电压依赖的动力学特性，并且在分析种群动态特性方面具有优势。在这个研究中，我们提出了两种近似方法，利用MT变换的框架为FS中间神经元导出一维FPE。一种方法是与适应相关的变量的均值场逼近。另一个是变量的半绝热近似。我们确认在这些近似模型中单个神经元的放电特性与Izhikevich提出的原始模型的匹配良好。对于伽玛振荡的总体动力学，具有这些近似值的FPE使得可以识别II类神经元放电同步的区域。对两个FPE的分叉分析表明，半绝热逼近比均值场逼近更适合数值结果。具有半绝热近似的FPE更好地说明了这样一个事实，即FS中间神经元群体比I类神经元具有更窄的伽马振荡区域。此外，对由I类和FS中神经元组成的总体进行半绝热逼近分析，随着FS中神经元比率的增加，同步区域变窄。©2020日本电气工程师学会。由Wiley Periodicals LLC发布。