大脑在各种频带中产生节律。有些可能是神经元过程的副产品;其他人被认为是自上而下的。完全自然产生,这些节奏有清晰可识别的节拍,但它们远非数学意义上的周期性。信号是宽带的,情节,在振幅和频率上徘徊;节奏来来去去,退化和再生。伽玛节奏,特别是,已经被许多计算神经科学的作者研究过,使用简化的模型以及数百到数千个集成和激发神经元的网络。所有这些模型都成功捕获了伽马节律的振荡性质,但伽玛在简化模型中的不规则特征尚未得到彻底研究。在这篇文章中,我们解决了一个数学问题,即是否可以从低维动力系统中产生具有大脑节律特性的信号。我们发现,虽然在单个周期周期中添加白噪声可以在某种程度上模拟伽马动力学,这样的模型往往是有限的,在他们的能力,以捕捉范围的行为观察。使用具有受FitzHugh-Nagumo和Leslie-Gower模型启发的两个变量的ODE,随机变化的系数设计为独立控制振幅,频率,和退化程度,我们能够复制自然大脑节律的定性特征。为了展示模型的多功能性,我们模拟了实验中记录的各种大脑状态中伽马节律的功率谱密度。
The brain produces rhythms in a variety of frequency bands. Some are likely by-products of neuronal processes; others are thought to be top-down. Produced entirely naturally, these rhythms have clearly recognizable beats, but they are very far from periodic in the sense of mathematics. The signals are broad-band, episodic, wandering in amplitude and frequency; the rhythm comes and goes, degrading and regenerating. Gamma rhythms, in particular, have been studied by many authors in computational neuroscience, using reduced models as well as networks of hundreds to thousands of integrate-and-fire neurons. All of these models captured successfully the oscillatory nature of gamma rhythms, but the irregular character of gamma in reduced models has not been investigated thoroughly. In this article, we tackle the mathematical question of whether signals with the properties of brain rhythms can be generated from low dimensional dynamical systems. We found that while adding white noise to single periodic cycles can to some degree simulate gamma dynamics, such models tend to be limited in their ability to capture the range of behaviors observed. Using an ODE with two variables inspired by the FitzHugh-Nagumo and Leslie-Gower models, with stochastically varying coefficients designed to control independently amplitude, frequency, and degree of degeneracy, we were able to replicate the qualitative characteristics of natural brain rhythms. To demonstrate model versatility, we simulate the power spectral densities of gamma rhythms in various brain states recorded in experiments.