Cerf R, Oumarrakchi M, Ben Maati M L, Sefrioui M
Laboratoire d'Ultrasons et de Dynamique des Fluides Complexes, Unité de Recherche Associée au C.N.R.S. No. 851, Université Louis Pasteur, Strasbourg, France.
Biol Cybern. 1992;68(2):115-24. doi: 10.1007/BF00201433.
The search for the low-dimensional attractor behaviour and the dynamic self-organization of neuronal systems, from an analysis of electroencephalographic (EEG) signals, must be carried out under conditions in which the signals are not stationary for more than a few seconds. We employ a technique that we have introduced for analyzing short signals obeying a differential equation and develop it further. The technique uses the fact that in plots of "slope curves" of d log C(r)/d log r against log C(r), C(r) = correlation integral, for short time sequences, the dynamics may be "trans-embedding-scaled", i.e. a horizontal power-law structure builds up, that is constructed from different slope curves (different embeddings), and appears at the right value of the correlation dimension, although no single slope curve exhibits scaling. Patterns of the family of slope curves are described exhibiting the "doublet-split-scaling" of the correlation integrals. Examples include a solution of the Mackey and Glass delay differential equation and EEG signals. The two components of a doublet differ in the dimensions of the embeddings of which they are formed, i.e. low- and high-dimensions, respectively. The advantages subsequent to recognizing trans-embedding-scaled correlation integrals and doublet-split-scaling are illustrated for EEG delta sleep signals, with emphasis on ideal doublet-split-scaling. Unambiguous evidence of attractor behaviour in delta sleep is presented.
从脑电图（EEG）信号分析中寻找神经元系统的低维吸引子行为和动态自组织，必须在信号不超过几秒保持静止的条件下进行。我们采用一种我们已引入的用于分析服从微分方程的短信号的技术，并对其进一步发展。该技术利用了这样一个事实：对于短时间序列，在d log C(r)/d log r 相对于 log C(r) 的“斜率曲线”图中（C(r) = 关联积分），动力学可能是“跨嵌入缩放”的，即形成一个水平幂律结构，它由不同的斜率曲线（不同的嵌入）构建而成，并且在关联维数的正确值处出现，尽管没有单个斜率曲线表现出缩放。描述了斜率曲线族的模式，展示了关联积分的“双峰分裂缩放”。示例包括Mackey和Glass延迟微分方程的一个解以及EEG信号。双峰的两个分量在其形成的嵌入维数上不同，分别为低维和高维。针对EEGδ睡眠信号说明了识别跨嵌入缩放关联积分和双峰分裂缩放后的优势，重点是理想的双峰分裂缩放。给出了δ睡眠中吸引子行为的明确证据。
