In the present study, we attempted to characterize two characteristic features within the dynamic behavior of wideband electrocorticography data, which were recorded as the brain waves of epilepsy, comprising high-frequency oscillations (HFOs) and interictal epileptic slow (red slow). The results of power spectrum and nonlinear time series analysis indicate that, on one hand, HFOs at epileptic focus are characterized by one-dimensional dynamical systems in ictal onset time segments at an epileptic focus for two patients' datasets; on the other hand, an interictal epileptic slow is characterized by the residue of power spectrum. The results suggest that the degree of freedom of the brain dynamics during epileptic seizure with HFO degenerates to low-dimensional dynamics; hence, the interictal epileptic slow as the precursors of the seizure onset can be detected simply from interictal brain wave data for the dataset of one patient. Therefore, our results are essential to understand the brain dynamics in epilepsy.

中文翻译：

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癫痫的数学结构：高频振荡和间歇性癫痫慢（红色慢）

在本研究中，我们试图表征宽带脑电图数据的动态行为中的两个特征，这些特征被记录为癫痫的脑电波，包括高频振荡 (HFO) 和发作间期癫痫缓慢（红色缓慢）。功率谱和非线性时间序列分析结果表明，一方面，对于两个患者的数据集，癫痫灶处的HFOs在癫痫灶处的发作时间段具有一维动力系统特征；另一方面，发作间期癫痫缓慢的特征是功率谱的残留。结果表明，HFO 癫痫发作期间脑动力学的自由度退化为低维动力学；因此，可以简单地从一位患者的数据集的发作间期脑电波数据中检测到发作间期癫痫缓慢作为癫痫发作的前兆。因此，我们的结果对于了解癫痫的大脑动力学至关重要。