用随机变系数的ODE模拟脑节律
Simulating brain rhythms using an ODE with stochastically varying coefficients
摘要：
大脑在不同的频段产生节奏。有些可能是神经过程的副产品;另一些被认为是自上而下的。这些节奏完全是自然产生的，有清晰可识别的节拍，但它们远非数学意义上的周期性。它们产生的信号是宽带的、间歇性的，在幅度、频率和相位上都是漂移的;节奏来来去去，退化和再生。具有这些特征的节奏与存在布朗噪声时的周期性、准周期性或周期性运动的标准动力系统范例不匹配。到目前为止，只有使用数百个整合和激发神经元的网络才能令人满意地复制它们。在本文中，我们解决了具有这些特性的信号是否可以从更简单的动力系统中产生的数学问题。使用受FitzHugh-Nagumo模型启发的具有两个变量的ODE，并随机改变控制退化幅度，频率和程度的三个参数，我们能够复制这些自然脑节律的定性特征。将这两个变量视为局部人群中典型神经元的兴奋性和抑制性电导，我们的模型产生的结果非常类似于真实皮层中的伽马带活动，包括实验中看到的E和I电流的时刻平衡。
英文摘要：
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. They produce signals that are broad-band, episodic, wandering in magnitude, in frequency and in phase; the rhythm comes and goes, degrading and regenerating. Rhythms with these characteristics do not match standard dynamical systems paradigms of periodicity, quasi-periodicity, or periodic motion in the presence of a Brownian noise. Thus far they have been satisfactorily reproduced only using networks of hundreds of integrate-and-fire neurons. In this paper, we tackle the mathematical question of whether signals with these properties can be generated from simpler dynamical systems. Using an ODE with two variables inspired by the FitzHugh-Nagumo model, and varying randomly three parameters that control the magnitude, frequency and degree of degradation, we were able to replicate the qualitative characteristics of these natural brain rhythms. Viewing the two variables as Excitatory and Inhibitory conductances of a typical neuron in a local population, our model produces results that closely resemble gamma-band activity in real cortex, including the moment-to-moment balancing of E and I-currents seen in experiments.
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