Oscillations in the beta/low gamma range (10–45 Hz) are recorded in diverse neural structures. They have successfully been modeled as sparsely synchronized oscillations arising from reciprocal interactions between randomly connected excitatory (E) pyramidal cells and local interneurons (I). The synchronization of spatially distant oscillatory spiking E–I modules has been well-studied in the rate model framework but less so for modules of spiking neurons. Here, we first show that previously proposed modifications of rate models provide a quantitative description of spiking E–I modules of Exponential Integrate-and-Fire (EIF) neurons. This allows us to analyze the dynamical regimes of sparsely synchronized oscillatory E–I modules connected by long-range excitatory interactions, for two modules, as well as for a chain of such modules. For modules with a large number of neurons (> 105), we obtain results similar to previously obtained ones based on the classic deterministic Wilson-Cowan rate model, with the added bonus that the results quantitatively describe simulations of spiking EIF neurons. However, for modules with a moderate (~ 104) number of neurons, stochastic variations in the spike emission of neurons are important and need to be taken into account. On the one hand, they modify the oscillations in a way that tends to promote synchronization between different modules. On the other hand, independent fluctuations on different modules tend to disrupt synchronization. The correlations between distant oscillatory modules can be described by stochastic equations for the oscillator phases that have been intensely studied in other contexts. On shorter distances, we develop a description that also takes into account amplitude modes and that quantitatively accounts for our simulation data. Stochastic dephasing of neighboring modules produces transient phase gradients and the transient appearance of phase waves. We propose that these stochastically-induced phase waves provide an explanative framework for the observations of traveling waves in the cortex during beta oscillations.

中文翻译：

---

具有空间结构连接的神经元网络中的同步、随机性和相位波

β/低伽马范围 (10–45 Hz) 的振荡记录在不同的神经结构中。它们已成功地被建模为由随机连接的兴奋性 (E) 锥体细胞和局部中间神经元 (I) 之间的相互相互作用引起的稀疏同步振荡。空间遥远的振荡尖峰 E-I 模块的同步在速率模型框架中得到了很好的研究，但对尖峰神经元模块的研究较少。在这里，我们首先表明，先前提出的速率模型修改提供了指数积分触发 (EIF) 神经元的尖峰 E-I 模块的定量描述。这使我们能够分析由长程兴奋性相互作用连接的稀疏同步振荡 E-I 模块的动力学机制，用于两个模块以及一连串这样的模块。对于具有大量神经元 (> 105) 的模块，我们获得的结果与之前基于经典确定性 Wilson-Cowan 速率模型获得的结果相似，另外的好处是结果定量描述了尖峰 EIF 神经元的模拟。然而，对于具有中等（~104）数量神经元的模块，神经元尖峰发射的随机变化很重要，需要考虑在内。一方面，它们以一种倾向于促进不同模块之间同步的方式修改振荡。另一方面，不同模块上的独立波动往往会破坏同步。远距离振荡模块之间的相关性可以通过振荡相位的随机方程来描述，这些方程在其他上下文中得到了深入研究。在较短的距离上，我们开发了一种描述，该描述也考虑了振幅模式，并定量地解释了我们的模拟数据。相邻模块的随机去相产生瞬态相位梯度和相位波的瞬态外观。我们建议这些随机诱导的相位波为在 β 振荡期间观察皮层中的行波提供了一个解释性框架。