In this paper, we study the dynamics of a quadratic integrate-and-fire neuron, spiking in the gamma (30-100 Hz) range, coupled to a delta/theta frequency (1-8 Hz) neural oscillator. Using analytical and semianalytical methods, we were able to derive characteristic spiking times for the system in two distinct regimes (depending on parameter values): one regime where the gamma neuron is intrinsically oscillating in the absence of theta input, and a second one in which gamma spiking is directly gated by theta input, i.e., windows of gamma activity alternate with silence periods depending on the underlying theta phase. In the former case, we transform the equations such that the system becomes analogous to the Mathieu differential equation. By solving this equation, we can compute numerically the time to the first gamma spike, and then use singular perturbation theory to find successive spike times. On the other hand, in the excitable condition, we make direct use of singular perturbation theory to obtain an approximation of the time to first gamma spike, and then extend the result to calculate ensuing gamma spikes in a recursive fashion. We thereby give explicit formulas for the onset and offset of gamma spike burst during a theta cycle, and provide an estimation of the total number of spikes per theta cycle both for excitable and oscillator regimes.

中文翻译：

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关于 theta-gamma 耦合神经振荡器的分析见解。

在本文中，我们研究了二次积分和激发神经元的动力学，在 gamma (30-100 Hz) 范围内尖峰，耦合到 delta/theta 频率 (1-8 Hz) 神经振荡器。使用分析和半分析方法，我们能够在两种不同的方式（取决于参数值）中推导出系统的特征尖峰时间：一种方式是伽马神经元在没有 theta 输入的情况下本质上振荡，第二种方式是伽马尖峰由θ输入直接门控，即伽马活动窗口与静默期交替，取决于基础θ相位。在前一种情况下，我们变换方程，使系统变得类似于 Mathieu 微分方程。通过求解这个方程，我们可以数值计算到第一个伽马峰值的时间，然后使用奇异微扰理论找到连续的尖峰时间。另一方面，在可激发条件下，我们直接使用奇异微扰理论来获得第一个伽马尖峰时间的近似值，然后将结果扩展到以递归方式计算随后的伽马尖峰。因此，我们给出了在 θ 周期期间伽马尖峰爆发的开始和偏移的明确公式，并提供了对可激发和振荡器机制每个 θ 周期的尖峰总数的估计。