Elsevier

Sleep Medicine Reviews 睡眠医学评论

Volume 37, February 2018, Pages 85-93
第 37 卷，2018 年 2 月，第 85-93 页
Sleep Medicine Reviews

Technical Review 技术审查
Nonlinear dynamical analysis of sleep electroencephalography using fractal and entropy approaches
使用分形和熵方法对睡眠脑电图进行非线性动态分析

医学TOPCPU 医学学科一流ESI学科分类：神经科学与行为简介JCI 1.79IF(5) 12.7 如果(5) 12.7SCU 医学BSCI升级版 医学1区SCI基础版 医学1区SCI Q1 SCI第一季度IF 11.2 如果11.2CUG 医学T1XJU 一区SWJTU A++ 西南交通大学A++
https://doi.org/10.1016/j.smrv.2017.01.003 IF: 11.2 Q1
https://doi.org/10.1016/j.smrv.2017.01.003如果：11.2 Q1 Get rights and content 获取权利和内容

Summary 概括

The analysis of electroencephalography (EEG) recordings has attracted increasing interest in recent decades and provides the pivotal scientific tool for researchers to quantitatively study brain activity during sleep, and has extended our knowledge of the fundamental mechanisms of sleep physiology. Conventional EEG analyses are mostly based on Fourier transform technique which assumes linearity and stationarity of the signal being analyzed. However, due to the complex and dynamical characteristics of EEG, nonlinear approaches are more appropriate for assessing the intrinsic dynamics of EEG and exploring the physiological mechanisms of brain activity during sleep. Therefore, this article introduces the most commonly used nonlinear methods based on the concepts of fractals and entropy, and we review the novel findings from their clinical applications. We propose that nonlinear measures may provide extensive insights into brain activities during sleep. Further studies are proposed to mitigate the limitations and to expand the applications of nonlinear EEG analysis for a more comprehensive understanding of sleep dynamics.
近几十年来，脑电图（EEG）记录的分析引起了越来越多的兴趣，为研究人员定量研究睡眠期间的大脑活动提供了关键的科学工具，并扩展了我们对睡眠生理学基本机制的了解。传统的脑电图分析主要基于傅立叶变换技术，该技术假设被分析信号的线性和平稳性。然而，由于脑电图的复杂性和动态特性，非线性方法更适合评估脑电图的内在动力学并探索睡眠期间大脑活动的生理机制。因此，本文介绍了基于分形和熵概念的最常用的非线性方法，并回顾了它们在临床应用中的新发现。我们认为非线性测量可以提供对睡眠期间大脑活动的广泛了解。进一步的研究旨在减轻非线性脑电图分析的局限性并扩展其应用，以便更全面地了解睡眠动态。

Keywords 关键词

Electroencephalography
Brain activity
Nonlinear
Sleep medicine
Sleep stages
Fractal
Entropy
Complexity

脑电图
大脑活动
非线性
睡眠药
睡眠阶段
分形
熵
复杂

Abbreviations 缩写

Background 背景

Sleep, in contrast to wakefulness, is characterized by reduced awareness and responsiveness. A basic model of sleep homeostasis is based on the concept of sleep-wake transition [1]. Conventionally, sleep stages in humans are classified as wake, rapid eye movement (REM) sleep, and an approximate continuum of depth during non-REM (NREM) sleep based on electroencephalographic patterns, which comprises about 80% of the entire sleep [2]. This cycling model of wake/NREM/REM sleep switches has been the primary focus of sleep research for decades. However, this reductionist type of approach is over-simplified and has limitation in understanding pathophysiological mechanisms in sleep disorders.
与清醒相反，睡眠的特点是意识和反应能力降低。睡眠稳态的基本模型基于睡眠-觉醒转变的概念[1] 。传统上，人类的睡眠阶段根据脑电图模式分为清醒期、快速眼动 (REM) 睡眠和非快速眼动 (NREM) 睡眠期间的近似连续深度，约占整个睡眠的 80% [2] 。几十年来，这种觉醒/非快速眼动/快速眼动睡眠切换的循环模型一直是睡眠研究的主要焦点。然而，这种还原论类型的方法过于简单化，并且在理解睡眠障碍的病理生理机制方面存在局限性。

Sleep is not simply a succession of human invented stages, but a delicate and sophisticated nonlinear symphony played by the brain in a democratic and mutual interaction with the rest of the body [2]. Real sleep stages are dynamic transitions between multiple physiological states swinging between the dual condition of stability and instability to warrant environmental adaptations and achieve physical and mental restoration [3].
睡眠不仅仅是人类发明的一系列阶段，而是大脑在与身体其他部分民主且相互互动的过程中演奏的精致而复杂的非线性交响曲[2] 。真正的睡眠阶段是多种生理状态之间的动态转变，在稳定与不稳定的双重状态之间摇摆，以适应环境，实现身心恢复[3] 。

Quantification of sleep stages via the analysis of electroencephalography (EEG) signal has been a challenge for years. Conventional visual sleep stage scoring is arbitrary and does not fully capture intrinsic EEG activity [4]. Fourier-based spectral analysis can quantify frequency compositions in EEG signals and is the most commonly used EEG analysis; however, it has intrinsic limitations to capturing underlying dynamics of the brain oscillations. First, fast Fourier transform (FFT)-based analysis assumes that complex oscillations embedded in the EEG signal are comprised of sine waves with different frequencies [5]. In this context, EEG signal can be decomposed into frequency components such as beta, alpha, theta, or delta frequency bands. However, it has long been known that brain oscillation is not a linear combination of these arbitrary frequency components, a property called “nonlinearity” [6]. Second, FFT-based spectral analysis assumes that none of these frequency components change in amplitude or shape as time evolves, which is clearly against what has been observed in complex brain oscillations, a property called “nonstationarity” [5].
多年来，通过分析脑电图 (EEG) 信号来量化睡眠阶段一直是一个挑战。传统的视觉睡眠阶段评分是任意的，并且不能完全捕捉内在的脑电图活动[4] 。基于傅里叶的频谱分析可以量化脑电图信号中的频率成分，是最常用的脑电图分析；然而，它在捕捉大脑振荡的潜在动态方面具有内在的局限性。首先，基于快速傅立叶变换(FFT) 的分析假设 EEG 信号中嵌入的复杂振荡由不同频率的正弦波组成[5] 。在这种情况下，EEG 信号可以分解为频率分量，例如 beta、alpha、theta 或 delta 频带。然而，人们很早就知道，大脑振荡并不是这些任意频率分量的线性组合，这种特性被称为“非线性” [6] 。其次，基于 FFT 的频谱分析假设这些频率分量的幅度或形状不会随着时间的推移而变化，这显然与在复杂的大脑振荡中观察到的情况（一种称为“非平稳性”的属性） [5]相悖。

It has long been observed that physiologic output of human body is nonstationary and nonlinear. Controls of physiological systems and outputs such as heartbeat, respiration, and brain wave oscillations are extraordinary complex [7]. Such complexity is believed to arise from nonlinear interactions among multiple control nodes in different physiological body systems that operate at multiple time scales. It has been hypothesized that the complexity of a biological system should be related to the system's capacity to adapt and function in an ever changing environment [7]. Conventionally, scientists employ a reductionist approach to disassemble the complex system into constituent pieces, examine each component, and, finally, reassemble them to recreate the original entity. However, this approach is often unrealistic. In most circumstances, we can observe only the macroscopic output of physiological functions, such as an EEG, heart rate, or respiration. In the language of complex systems, the composite behavior cannot be fully understood by “adding up” the components. Instead, one needs new approaches to measuring a system's integrative behavior. Thus, the understanding of the complex dynamics of the physiologic output, such as changes in EEG dynamics observed across different sleep stages, will be improved by applying nonlinear dynamical approaches to the analysis of EEG signal.
人们很早就观察到人体的生理输出是非平稳和非线性的。生理系统和输出（例如心跳、呼吸和脑电波振荡）的控制非常复杂[7] 。这种复杂性被认为是由在多个时间尺度上运行的不同生理系统中的多个控制节点之间的非线性相互作用引起的。据推测，生物系统的复杂性应与系统在不断变化的环境中适应和发挥作用的能力有关[7] 。传统上，科学家采用还原论方法将复杂系统分解成组成部分，检查每个组成部分，最后重新组装它们以重新创建原始实体。然而，这种方法往往是不现实的。在大多数情况下，我们只能观察生理功能的宏观输出，例如脑电图、心率或呼吸。在复杂系统的语言中，复合行为不能通过“相加”组件来完全理解。相反，我们需要一种新的方法来衡量系统的综合行为。因此，通过应用非线性动力学方法来分析脑电图信号，将改善对生理输出的复杂动态的理解，例如在不同睡眠阶段观察到的脑电图动态变化。

Nonlinear analysis of sleep EEG
睡眠脑电图的非线性分析

The term nonlinear applies to systems in which components interact non-additively [8]. One example of the non-additivity is brain electrical activity that reflects combinations of excitatory and inhibitory postsynaptic potentials in apical dendrites of pyramidal neurons in the superficial layers of the cortex [9]. To understand the nonlinear feature of the EEG signal, in 1985, Rapp et al. pioneeringly performed a chaos analysis of spontaneous neural activity in monkeys [10], and Babloyantz et al. examined the correlation dimension (CD) of human sleep EEG [11]. At the early stages (1985–1990) and since 1990, nonlinear EEG analysis was mainly referred to low-dimensional chaotic dynamics and surrogate data testing, respectively [3]. In the late 1990s, phase synchronization and generalized synchronization became widely used. Recently, the concepts and methods originated from the chaos theory or complexity science has attracted considerable attention. Nonlinear approaches are suggested to be superior to traditional linear methods in understanding EEG dynamics [12]. In addition, the nonlinear approaches provide clearer insights into dynamical nature and variability of the brain signal and have shown their ability to surpass traditional spectral techniques, such as tracing epileptic changes in the EEG signal [13].
术语“非线性”适用于其中组件以非相加方式相互作用的系统[8] 。非相加性的一个例子是脑电活动，它反映了皮质浅层锥体神经元顶端树突的兴奋性和抑制性突触后电位的组合[9] 。为了理解脑电图信号的非线性特征，1985年，Rapp等人。开创性地对猴子的自发神经活动进行了混沌分析[10] ，Babloyantz 等人。检查了人类睡眠脑电图的相关维度（CD） [11] 。早期（1985-1990）和1990年以来，非线性脑电分析主要分别指低维混沌动力学和替代数据检验[3] 。 20世纪90年代末，相位同步和广义同步得到广泛应用。近年来，源自混沌理论或复杂性科学的概念和方法引起了人们的广泛关注。在理解脑电图动力学方面，非线性方法被认为优于传统的线性方法[12] 。 此外，非线性方法可以更清晰地了解大脑信号的动态性质和变异性，并显示出超越传统频谱技术的能力，例如追踪脑电图信号中的癫痫变化[13] 。

Quantification of the nonlinear features of sleep physiology is very important in two aspects: 1) for evaluating dynamical models of sleep homeostasis and 2) for clinically monitoring alteration/degradation of normal sleep physiology with aging and pathological conditions [8]. In the last two decades, novel approaches derived from concepts of nonlinear dynamics and statistical physics theories have been developed and applied to probe generic features of complex systems. These approaches revealed that the fluctuations in the outputs of these systems contain important information about the underlying mechanisms of system controls. For instance, robust fractal/scale-invariant, multifractal, and time irreversibility were observed in healthy physiologic systems (indicating complex physiological control) and the alterations of these nonlinear statistical properties are associated with aging and pathological states ∗[8], ∗[14]. Moreover, certain generic features exist in a various number of physiological systems (e.g., similar scale-invariant correlations in heartbeat fluctuations, motor activity, gait, respiration, and brain wave oscillations), indicating a universal “rule” in the underlying mechanisms of nonlinear interactions in physiological systems. These universal features of different physiological systems provide an important guidance for building physiologically meaningful models of integrative physiological systems.
睡眠生理学非线性特征的量化在两个方面非常重要：1）评估睡眠稳态的动态模型；2）临床监测正常睡眠生理学随衰老和病理条件的改变/退化[8] 。在过去的二十年中，人们开发了源自​​非线性动力学和统计物理理论概念的新方法，并将其应用于探测复杂系统的一般特征。这些方法揭示了这些系统输出的波动包含有关系统控制的基本机制的重要信息。例如，在健康的生理系统中观察到稳健的分形/尺度不变、多重分形和时间不可逆性（表明复杂的生理控制），并且这些非线性统计特性的改变与衰老和病理状态相关*[8] ， *[14 ] 。此外，某些通用特征存在于各种生理系统中（例如，心跳波动、运动活动、步态、呼吸和脑电波振荡中类似的尺度不变相关性），表明非线性潜在机制中存在普遍的“规则”生理系统中的相互作用。不同生理系统的这些普遍特征为构建具有生理意义的综合生理系统模型提供了重要指导。

Nonlinear dynamics theory provides new opportunities for the understanding of sleep EEG behavior [15], and increasing amount of studies have used nonlinear methods to investigate the characteristics of brain activities during sleep. In this article, we therefore present the most commonly used nonlinear analyses of sleep EEG signal, such as fractal or entropy methods, and review their applications to sleep studies. This review intends to inspire future sleep studies to understand the complex nature of sleep physiology, particular the dynamical changes in brain activity during sleep.
非线性动力学理论为理解睡眠脑电行为提供了新的机会[15] ，越来越多的研究利用非线性方法来研究睡眠期间大脑活动的特征。因此，在本文中，我们提出了最常用的睡眠脑电图信号非线性分析，例如分形或熵方法，并回顾了它们在睡眠研究中的应用。这篇综述旨在启发未来的睡眠研究，以了解睡眠生理学的复杂本质，特别是睡眠期间大脑活动的动态变化。

Fractal-based methods 基于分形的方法

Mandelbrot [16] introduced the fractal theory, which can be expressed by two phenomenon: self-similarity and fractional dimensionality. Fractal theory has been used for explaining natural landscapes, modeling temporal dynamics of a time series, and predicting extreme events or human behavior [17]. Therefore, fractal analysis has potential to describing the dynamics of brain electrical activities under physiological and pathological conditions. To quantify fractal scaling behavior in a time series, several methods have been developed to quantify the fractal dimension, including the CD, Hurst exponent (H), and detrended fluctuation analysis (DFA).
Mandelbrot [16]引入了分形理论，它可以用两种现象来表达：自相似性和分数维数。分形理论已用于解释自然景观、对时间序列的时间动态进行建模以及预测极端事件或人类行为[17] 。因此，分形分析具有描述生理和病理条件下脑电活动动态的潜力。为了量化时间序列中的分形尺度行为，已经开发了几种量化分形维数的方法，包括 CD、赫斯特指数( H ) 和去趋势波动分析 (DFA)。

Correlation dimension 相关维度

The correlation dimension (CD, or D2 in certain literature) [18] describes the fractional dimensionality of an underlying process in relation to its geometrical reconstruction in embedded phase space. The values of the CD range between zero and the value of embedding dimension and can be used to quantify the complex dynamics of the brain activity [19]. In a chaotic system, the CD usually shows a non-integer value larger than 1, indicating an increased complexity of system dimensionality.
相关维数（CD，或某些文献中的D 2 ） [18]描述了与其在嵌入相空间中的几何重建相关的基础过程的分数维数。 CD 的值介于零和嵌入维数之间，可用于量化大脑活动的复杂动态[19] 。在混沌系统中，CD通常呈现大于1的非整数值，表明系统维数的复杂性增加。

In the analysis of sleep EEG recorded from healthy adults, the CD has been consistently reported to decrease from wake to sleep stages N1–N3 and increase during REM [11], [15], [20], [21], [22], [23], [24]. Furthermore, studies of neonatal EEG suggested that CD during active sleep tends to be higher than that during quiet sleep, whereas CD during indeterminate sleep is at the midpoint of the range between active and quiet sleep states [25]. This trend of changes in the values of the CD between different sleep stages may be associated with the sleep depth [15], [20]. Studies have reported that the CD of the sleep EEG in N2 stages was significantly lower in the first half of the night than that in the second half of the night, indicating an association of sleep EEG complexity with the restoration effect of the sleep [26], [27]. For healthy subjects under total sleep deprivation, EEG in sleep-deprived states yields lower CD values than that in normal states, suggesting that sleep deprivation results in the decrease of the brain signal complexity [28].
在对健康成人记录的睡眠脑电图的分析中，一致报道 CD 从清醒到睡眠阶段 N1-N3 减少，并在REM期间增加[11] , [15] , [20] , [21] , [22] ， [23] ， [24] 。此外，新生儿脑电图研究表明，主动睡眠期间的CD往往高于安静睡眠期间的CD，而不确定睡眠期间的CD则处于主动睡眠状态和安静睡眠状态之间范围的中点[25] 。不同睡眠阶段之间CD值的这种变化趋势可能与睡眠深度有关[15] 、 [20] 。有研究报道，N2阶段睡眠脑电图的CD在前半夜显着低于后半夜，表明睡眠脑电图复杂性与睡眠的恢复效果存在关联[26] ， [27] 。 对于完全睡眠剥夺的健康受试者，睡眠剥夺状态下的脑电图CD值低于正常状态，表明睡眠剥夺导致大脑信号复杂性降低[28] 。

Compared to healthy subjects, CD was lower during stage 2 and REM sleep in unmedicated schizophrenic patients [29] while CD was found to be decreased during slow-wave sleep (SWS) in depression [30]. In contrast, CD derived from resting-state awake EEG of schizophrenic patients was not distinguishable from healthy volunteers while CD was found to be lower in schizophrenic patients under arithmetic test [31]. These alterations in the nonlinear dynamical properties of EEG in sleep/wake and task conditions indicate that brain's ability to process information may be disturbed in depression and schizophrenia [29], [30]. In the studies of neurological disorders, CD in SWS was lower in narcoleptic patients than that in healthy subjects, indicating a lower degree of complexity in the sleep-wake regulation among narcoleptic patients [32].
与健康受试者相比，未接受药物治疗的精神分裂症患者在第 2 阶段和快速眼动睡眠期间 CD 较低[29] ，而抑郁症患者在慢波睡眠 (SWS) 期间 CD 下降[30] 。相比之下，精神分裂症患者静息态清醒脑电图得出的 CD 与健康志愿者无法区分，而在算术测试中发现精神分裂症患者的 CD 较低[31] 。睡眠/清醒和任务条件下脑电图非线性动力学特性的这些变化表明，大脑处理信息的能力可能在抑郁症和精神分裂症中受到干扰[29] , [30] 。在神经系统疾病的研究中，发作性睡病患者的SWS CD低于健康受试者，表明发作性睡病患者睡眠-觉醒调节的复杂程度较低[32] 。

Hurst exponent 赫斯特指数

The Hurst analysis is an important method derived from chaos theory to quantify the long-term memory processes of a time series [33]. The biological time series often exhibits long-range dependence which refers to dependence structures that decay slowly with increasing distance, such as a power-law decay [34]. The value of the Hurst exponent (H) ranges between 0 and 1. Based on H, a time series can be classified into three categories: a) H = 0.5, indicating the presence of uncorrelated randomness; b) 0 < H < 0.5, indicating an anticorrelated process; and c) 0.5 < H < 1, indicating a long-range correlation process with characteristics of 1/f power spectrum.
Hurst分析是源自混沌理论的一种重要方法，用于量化时间序列的长期记忆过程[33] 。生物时间序列通常表现出长程依赖性，这是指随着距离的增加而缓慢衰减的依赖性结构，例如幂律衰减[34] 。赫斯特指数 ( H ) 的值介于 0 和 1 之间。根据H ，时间序列可以分为三类： a) H = 0.5，表示存在不相关的随机性； b) 0 < H < 0.5，表示反相关过程； c) 0.5 < H < 1，表示具有1/f功率谱特征的长程相关过程。

Although H has been applied in many studies to distinguish brain signal dynamics from different states or pathological conditions [35], it was less frequently used in sleep EEG studies compared with other nonlinear approaches. Results reported from existing studies are inconsistent. In Acharya's et al. study, H values were higher in wake and REM sleep, but lowest in stage 3 and 4 [15]. While in Weiss's et al. studies, H was higher in NREM stage 4 than stage 2 and REM period [36], [37], suggesting that sleep EEG signals may be less fractal during SWS [36]. However, the discrepancy in studies using Hurst exponent indicates a need for a comprehensive comparison of different approaches and careful evaluation of the fractal property.
尽管H已在许多研究中用于区分不同状态或病理条件下的大脑信号动态[35] ，但与其他非线性方法相比，它在睡眠脑电图研究中的使用频率较低。现有研究报告的结果不一致。在 Acharya 等人中。研究表明， H值在清醒和快速眼动睡眠中较高，但在第 3 和第 4 阶段最低[15] 。而在韦斯等人。研究表明，NREM 4 阶段的H高于 2 阶段和 REM 阶段[36] 、 [37] ，这表明 SWS 期间睡眠 EEG 信号的分形可能较少[36] 。然而，使用赫斯特指数的研究中的差异表明需要对不同方法进行全面比较并仔细评估分形特性。

Detrended fluctuation analysis
去趋势波动分析

DFA, originally introduced by Peng et al. [38], is a widely used method for quantifying the long-range correlation of the physiological time series, such as heart beat time series, respiratory signals, and EEG recordings [39], [40], [41]. The scaling exponent (α) derived from the DFA method represents the long-range correlation properties of the signal: α = 0.5 indicates the presence of uncorrelated randomness; α < 0.5 suggests the presence of anti-correlated process; 0.5 < α < 1 represents the existence of long-range correlations in the time series and α = 1 resembles 1/f noise. When α > 1 and approaches to 1.5, a Brownian noise is indicated, which is the integration of the white noise.
DFA，最初由 Peng 等人提出。 [38] ，是一种广泛使用的量化生理时间序列的长程相关性的方法，例如心跳时间序列、呼吸信号和脑电图记录[39] 、 [40] 、 [41] 。 DFA 方法导出的标度指数( α ) 表示信号的长程相关性： α = 0.5 表示存在不相关的随机性； α <0.5表明存在反相关过程； 0.5 < α < 1 表示时间序列中存在长程相关性， α = 1 类似于 1/f 噪声。当α > 1并且接近1.5时，表明布朗噪声，这是白噪声的积分。

In healthy subjects, the values of DFA scaling exponent increase beyond 1 with increased depth of sleep stages [39], [42], suggesting that the dynamics of the sleep EEG is more like a Brownian noise process in deeper sleep stages [39]. Studies of obstructive sleep apnea (OSA) [39], [43] have reported that patients with OSA exhibited a similar trend of the DFA exponent in different stages of NREM sleep, but their DFA exponent values were lower than those from healthy subjects [39].
在健康受试者中，DFA 标度指数的值随着睡眠阶段深度的增加而增加超过 1 [39] 、 [42] ，这表明睡眠 EEG 的动态更像是较深睡眠阶段的布朗噪声过程[39] 。阻塞性睡眠呼吸暂停（OSA）研究[39] 、 [43]报道，OSA患者在NREM睡眠的不同阶段表现出相似的DFA指数趋势，但其DFA指数值低于健康受试者[39] ] 。

In depression-related sleep disturbance, a study reported that the DFA scaling exponents were lower in patients with major depression during stage 2 and SWS; the authors suggested that these findings might be related to the sleep fragmentation and instability observed in major depressive disorders. Another study found that DFA exponents of sleep EEG in depressed patients were higher than those from healthy controls [37]. This study also identified a significant correlation between the severity of depression and DFA exponent [40], suggesting that the sleep EEG under the depressive state exhibits a slower decay of long-range temporal correlations which is associated with the severity of depression [40].
在抑郁症相关的睡眠障碍中，一项研究报告称，2 期和 SWS 期间重度抑郁症患者的 DFA 标度指数较低；作者认为，这些发现可能与重度抑郁症中观察到的睡眠碎片化和不稳定有关。另一项研究发现，抑郁症患者睡眠脑电图的 DFA 指数高于健康对照者[37] 。这项研究还发现抑郁的严重程度与 DFA 指数之间存在显着相关性[40] ，这表明抑郁状态下的睡眠脑电图表现出与抑郁严重程度相关的长程时间相关性的较慢衰减[40] 。

Implications of fractal-based methods
基于分形的方法的含义

Based on aforementioned literature (Table 1), both the values of the CD and DFA exponent change in accordance with the advancement of NREM sleep stages (Fig. 1a). Consistent across three different fractal-based methods, sleep EEG in awake status is more compatible with 1/f noise, an important characteristic of nonlinear dynamics, and EEG signals in NREM stages become more ordered with reduced fractal dimensional complexity. There has been increasing evidence that the activity of brain circuit becomes more coherent and ordered as sleep stage goes deeper [44], [45], which may contribute to the reduced complexity found in EEG signals during NREM sleep. Collectively, this evidence suggests that fractal-based EEG markers may be of use to track sleep stages, and the degree of changes in fractal-based EEG markers between awake and NREM stages may be of help to distinguish between healthy sleep and pathologic sleep conditions, such as insomnia. In addition, fractal-based EEG markers show a significant difference between wake and REM period [46], despite these two conditions involve intense brain activities, the importance of this finding remains elusive but worth further studies.
根据上述文献(表1 )，CD和DFA指数的值均随着NREM睡眠阶段的进展而变化(图1a )。与三种不同的基于分形的方法一致，清醒状态下的睡眠脑电图与 1/f 噪声（非线性动力学的一个重要特征）更兼容，并且 NREM 阶段的脑电图信号变得更加有序，分形维数复杂性降低。越来越多的证据表明，随着睡眠阶段的深入，大脑回路的活动变得更加连贯和有序[44] ， [45] ，这可能有助于降低 NREM 睡眠期间脑电图信号的复杂性。总的来说，这一证据表明，基于分形的脑电图标记可能可用于跟踪睡眠阶段，并且清醒阶段和非快速眼动阶段之间基于分形的脑电图标记的变化程度可能有助于区分健康睡眠和病理性睡眠状况。比如失眠。此外，基于分形的脑电图标记显示清醒期和快速眼动期之间存在显着差异[46] ，尽管这两种情况涉及剧烈的大脑活动，但这一发现的重要性仍然难以捉摸，但值得进一步研究。

Table 1. Applications of fractal-based methods*.
表格1 。基于分形的方法的应用*。

Citations 引文Study subjects 学习科目Main finding(s) 主要发现）
Correlation Dimension 相关维度
Roschke, et al. 1992 [21]
罗施克等人。 1992年[21]		12 healthy males, 20-31 y; sleep under lorazepam versus placebo
12名健康男性，20-31岁；服用劳拉西泮与安慰剂相比睡眠		SWS depicts a much smaller dimensionality than light or REM sleep; lorazepam does not alter the EEG's dimensionality except in stage 2 and REM sleep.
SWS 描述的维度比浅睡眠或快速眼动睡眠小得多；除第二阶段和快速眼动睡眠外，劳拉西泮不会改变脑电图的维度。

Roschke, et al. 1994 [30]
罗施克等人。 1994年[30]		9 depressive and 11 schizophrenic inpatients compared to healthy controls
9 名抑郁症住院患者和 11 名精神分裂症住院患者与健康对照者相比		Altered nonlinear brain dynamics mainly during slow wave sleep in depression and during REM sleep in schizophrenia.
非线性大脑动力学的改变主要发生在抑郁症的慢波睡眠期间和精神分裂症的快速眼动睡眠期间。

Achermann, et al. 1994 [24]
阿赫曼等人。 1994年[24]		11 healthy males, 23-32 y
11名健康男性，23-32岁		CD was high in REM sleep, declined progressively within each NREM sleep episode, and reached a low level at times when SWS was dominant.
CD 在 REM 睡眠中较高，在每次 NREM 睡眠期间逐渐下降，并在 SWS 占主导地位时达到较低水平。

Fell, et al. 1996 [22]
菲尔等人。 1996年[22]		12 healthy males, 23-36 y
12名健康男性，23-36岁		Nonlinear measures yield additional information, which improves the ability to discriminate sleep stages and which may in general improve the ability to distinguish different psychophysiological states.
非线性测量产生额外的信息，这提高了区分睡眠阶段的能力，并且通常可以提高区分不同心理生理状态的能力。

Pereda, et al. 1998 [20]
佩雷达等人。 1998年[20]		9 healthy males and females, mean age 27.3 y
健康男女各9名，平均年龄27.3岁		EEG exhibits random fractal structure with 1/f -β (1 < β < 3) and a negative linear correlation between CD and fractal exponent (β) in all states except during SWS.
EEG 表现出具有 1/f -β (1 < β < 3) 的随机分形结构，并且在除 SWS 期间之外的所有状态下 CD 和分形指数 (β) 之间呈负线性相关。


Ferri, et al. 1999 [32]
费里等人。 1999 [32]		9 narcoleptic patients, male and female, 20–55 y
9 名发作性睡病患者，男性和女性，20-55 岁		CD was higher in normal controls than in narcoleptic patients.
正常对照者的 CD 高于发作性睡病患者。

Kobayashi, et al. 2000 [27]
小林等人。 2000年[27]		1 healthy male, 22 y
1名健康男性，22岁		CD decreased from wake to sleep stage 1 to 3, and increased for REM sleep.
CD 从清醒到睡眠阶段 1 至 3 下降，而在快速眼动睡眠阶段则增加。

Kobayashi, et al. 2000 [26]
小林等人。 2000年[26]		10 healthy males, mean age 23.6 y
10名健康男性，平均年龄23.6岁		CD significantly decreased from wake to sleep stage 1, 2, 3 and increased during REM sleep. The mean CD of the sleep EEG in the second half of the night was significantly higher than those in the first half of the night.
CD 从清醒到睡眠阶段 1、2、3 显着下降，而在 REM 睡眠期间增加。后半夜睡眠脑电图平均CD显着高于前半夜。

Jeong, et al. 2001 [28]
郑等人。 2001年[28]		20 healthy male volunteers, 23.4±1.9 y. A 24-hour schedule of sleep deprivation began on morning awakening following a normal sleep night.
20名健康男性志愿者，23.4±1.9岁。 24小时睡眠剥夺计划从正常睡眠后早上醒来开始。		The sleep-deprived states had lower CD values at three channels (P4, O2, and C3) than normal states.
睡眠剥夺状态下三个通道（P4、O2 和 C3）的 CD 值低于正常状态。

Kobayashi, et al. 2001 [90]
小林等人。 2001年[90]		9 male subjects, 21–24 y, in good health and with no history of alcoholism. PSG was recorded on baseline night (no ethanol) and on study night when ethanol (0.8 g/kg) was given 15 minutes prior to sleep study.
9 名男性受试者，21-24 岁，身体健康，无酗酒史。在基线晚上（无乙醇）和研究晚上（在睡眠研究前 15 分钟给予乙醇（0.8 g/kg））记录 PSG。		The mean CD of EEG during sleep stage 2 and those for the second sleep cycle on the ethanol night were significantly higher than those on the baseline night (no ethanol). The changes in CD between sleep cycles were reduced on ethanol night as compared to baseline night.
第 2 阶段睡眠期间的脑电图平均 CD 以及乙醇夜间第二个睡眠周期的脑电图平均 CD 显着高于基线夜间（无乙醇）的脑电图平均 CD。与基线夜间相比，乙醇夜间睡眠周期之间 CD 的变化减少。

Ferri, et al. 2001 [89]
费里等人。 2001年[89]		5 children with ESES, males and female, 6.5-10 y
5 名 ESES 儿童，男性和女性，6.5-10 岁		In NREM sleep, the possible presence of low-dimensional chaos could be suspected. EEG without ESES could not be distinguished from linearly filtered noise.
在非快速眼动睡眠中，可以怀疑可能存在低维混沌。没有 ESES 的 EEG 无法与线性过滤的噪声区分开来。

Acharya, et al. 2005 [15]
阿查里亚等人。 2005年[15]		8 healthy Caucasian, males and females, 21-35 y
8 名健康白人，男性和女性，21-35 岁		CD decreases from wake to sleep stages 1-4 and then increases during REM sleep.
CD 从清醒到睡眠阶段 1-4 降低，然后在快速眼动睡眠期间增加。

Scher, et al. 2005 [91]
谢尔等人。 2005年[91]		116 EEG recordings in 55 neonatal subjects (28-43 wk gestational age)
55 名新生儿受试者（胎龄 28-43 周）的 116 条脑电图记录		For full-term infants, CD between AS and QS was significantly different. A positive correlation between CD and increasing conceptional age was noted.
对于足月婴儿，AS 和 QS 之间的 CD 显着不同。注意到 CD 与受孕年龄增加之间呈正相关。

Janjarasjitt, et al. 2008 [25]
Janjarasjit等人。 2008年[25]		50 healthy neonates (22 male) with postmenstrual age of 28-42 wk.
50 名健康新生儿（22 名男性），月经后年龄为 28-42 周。		CD during AS is higher than during QS, and CD during indeterminate sleep is virtually at the midpoint between them. The birth status (preterm or full-term) of the neonate has an influence on CD.
AS 期间的 CD 高于 QS 期间的 CD，而不确定睡眠期间的 CD 实际上处于两者之间的中点。新生儿的出生状态（早产或足月）对 CD 有影响。

Bell, et al. 2012 [88]
贝尔等人。 2012年[88]		54 subjects with histories of coffee-induced insomnia, male and female, mean age 20 y
54 名有咖啡引起失眠史的受试者，男性和女性，平均年龄 20 岁		Both Coffea cruda 30c and Nux vomic 30c increased CD in SWS in the post-remedy night.
Coffea cruda 30c 和Nux vomic 30c 都增加了治疗后晚上 SWS 中的 CD。

Hurst Exponent 赫斯特指数
Acharya, et al. 2005 [15]
阿查里亚等人。 2005年[15]		8 healthy Caucasian, males and females, 21-35 y
8 名健康白人，男性和女性，21-35 岁		H values were higher in wake and REM sleep, indicating higher self-similarity, but were lowest in stage 3 and 4.
H值在清醒和快速眼动睡眠中较高，表明自相似性较高，但在第 3 和第 4 阶段最低。

Weiss, et al. 2009 [36]
韦斯等人。 2009年[36]		10 healthy subjects, male and female, 17–53 y
10 名健康受试者，男性和女性，17-53 岁		Higher H values during stage 4 compared to stage 2 and REM sleep in all electrodes.
与所有电极中的第 2 阶段和 REM 睡眠相比，第 4 阶段的H值更高。

Weiss, et al. 2011 [37]
韦斯等人。 2011年[37]		22 healthy subjects 22名健康受试者Highest H values emerged frontally during all sleep stages, while the minimum was found during REM in the central zone.
最高的H值出现在所有睡眠阶段的正面，而最低的 H 值出现在中央区域的快速眼动期间。

Detrended Fluctuation Analysis
去趋势波动分析

Lee, et al. 2002 [43]
李等人。 2002年[43]		17 PSG data in MIT/BIH database
17 MIT/BIH 数据库中的 PSG 数据		The mean scaling exponents of EEG is discriminated according to NREM, REM and wake, and gradually increased from stage 1 to stage 2, 3 and 4.
脑电图的平均标度指数根据 NREM、REM 和唤醒进行区分，并从阶段 1 逐渐增加到阶段 2、3 和 4。

Lee, et al. 2004 [39]
李等人。 2004年[39]		The sleep EEGs of six healthy males (30–35 y) and six sleep apnea EEGs (slp02a, slp14, slp16, slp37, slp61, and slp66; aged 32–39 years; all of them were males) from MIT/BIH PSG database.
来自 MIT/BIH PSG 数据库的 6 名健康男性（30-35 岁）的睡眠脑电图和 6 名睡眠呼吸暂停脑电图（slp02a、slp14、slp16、slp37、slp61 和 slp66；年龄 32-39 岁；均为男性） 。		The mean scaling exponents increased from wake to sleep stage 1, 2, 3 and 4, but decreased during REM sleep. The scaling exponents of the apnea were lower than those of the healthy subjects for all the stages.
从清醒到睡眠第 1、2、3 和 4 阶段，平均标度指数增加，但在快速眼动睡眠期间减少。在所有阶段，呼吸暂停的标度指数均低于健康受试者。

Ferri, et al. 2005 [95]
费里等人。 2005年[95]		5 healthy subjects, male and female, 20–32 y
5 名健康受试者，男性和女性，20-32 岁		Higher levels of interregional synchronization during CAP sleep than during non-CAP with a small but significant difference between its A and B phases. Only the first DFA exponent showed different values during the different sleep stages.
CAP 睡眠期间的区域间同步水平高于非 CAP 期间，其 A 阶段和 B 阶段之间存在微小但显着的差异。只有第一个 DFA 指数在不同睡眠阶段表现出不同的值。

Lee, et al. 2007 [40]
李等人。 2007年[40]		11 unmedicated unipolar depressed patients and 11 non-depressed, age-matched controls.
11 名未接受药物治疗的单相抑郁症患者和 11 名年龄匹配的非抑郁症对照者。		All the scaling exponents in depressed patients and healthy controls were between 0.5 and 1.0. The scaling exponents of depressed patients have relatively higher values in whole brain regions compared to healthy controls, with significant differences at F3, C3, T3, T4 and O1 channels. A significant linear correlation was observed between the severity of depression and the scaling exponent over most of the channels, except O2.
抑郁症患者和健康对照者的所有标度指数均在 0.5 至 1.0 之间。与健康对照相比，抑郁症患者整个大脑区域的标度指数值相对较高，其中F3、C3、T3、T4和O1通道差异显着。在除 O2 之外的大多数通道上，抑郁症的严重程度与标度指数之间都观察到显着的线性相关性。

Leistedt, et al. 2007 [41]
莱斯特等人。 2007年[41]		10 unmedicated inpatients with acute major depression, and 14 normal controls
10 名未接受药物治疗的急性重度抑郁症住院患者和 14 名正常对照		The median values of alpha were lower in patients during sleep stage 2 and SWS.
睡眠阶段 2 和 SWS 期间患者的 α 中位值较低。

Leistedt, et al. 2007 [94]
莱斯特等人。 2007年[94]		10 untreated depressed men in full to partial remission (42.43+/-5.62 y) and 14 healthy subjects (42.8+/-8.55 y)
10 名未经治疗的抑郁症男性完全缓解至部分缓解 (42.43+/-5.62 岁) 和 14 名健康受试者 (42.8+/-8.55 岁)		Significant difference and deviation of the scaling exponents between the two groups were not observed during targeted three sleep stages (stage 2, SWS and REM).
在目标三个睡眠阶段（第 2 阶段、SWS 和 REM）期间，未观察到两组间标度指数的显着差异和偏差。

Dumont, et al. 2007 [93]
杜蒙等人。 2007年[93]		24 patients with sleep apnea-hypopnea syndrome (12 moderate-to-severe and 12 severe subjects respectively), and 12 normal controls; mean age 44 y
24名睡眠呼吸暂停低通气综合征患者（分别为12名中重度和12名重度受试者）和12名正常对照；平均年龄 44 岁		For all sleep bands, the fluctuations of the synchronization between sleep EEG and heart activity appear scale free and the scaling exponent is close to one as for 1/f noise. We could not detect any effect due to sleep apnea-hypopnea syndrome.
对于所有睡眠带，睡眠脑电图和心脏活动之间的同步波动似乎是无标度的，并且对于 1/f 噪声来说，标度指数接近于 1。我们无法检测到睡眠呼吸暂停低通气综合征造成的任何影响。

D'Rozario, et al. 2013 [92]
德罗萨里奥等人。 2013年[92]		8 untreated OSA patients and 13 non-OSA controls
8 名未经治疗的 OSA 患者和 13 名非 OSA 对照		DFA scaling exponent and power spectra biomarkers significantly correlated with simultaneously tested performance and self-rated sleepiness across the testing period in OSA patients and controls. Baseline DFA scaling exponent were markers of impaired simulated driving after 24-h extended wakefulness in OSA. OSA patients had a higher scaling exponent and delta power during wakefulness than controls.
DFA 标度指数和功率谱生物标志物与 OSA 患者和对照测试期间同时测试的表现和自评困倦显着相关。基线 DFA 缩放指数是 OSA 中 24 小时长时间清醒后模拟驾驶受损的标志。 OSA 患者在清醒期间比对照组具有更高的标度指数和增量功率。

* Only methods introduced in the review are listed in the table. Some studies included more than one method.
* 表中仅列出了综述中介绍的方法。一些研究包括不止一种方法。

Abbreviations: AS, active sleep; BIH, Beth Israel Hospital; CAP, cyclic alternating pattern; CD, correlation dimension; DFA, detrended fluctuation analysis; EEG, electroencephalography; ESES, electrical status epilepticus in sleep; H, Hurst exponent; MIT, Massachusetts institute of technology; NREM, non-rapid eye movement; OSA, obstructive sleep apnea; PSG, polysomnography; QS, quiet sleep; REM, rapid eye movement; SWS, slow wave sleep.
缩写：AS，主动睡眠； BIH，贝斯以色列医院； CAP，循环交替模式； CD，相关维数； DFA，去趋势波动分析；脑电图、脑电图； ESES，睡眠中癫痫持续状态； H ，赫斯特指数； MIT，麻省理工学院； NREM，非快速眼球运动； OSA，阻塞性睡眠呼吸暂停； PSG，多导睡眠图； QS，安静睡眠； REM，快速眼球运动； SWS，慢波睡眠。

Fig. 1
  1. Download: Download high-res image (146KB)
    下载：下载高分辨率图像 (146KB)
  2. Download: Download full-size image
    下载：下载全尺寸图像

Fig. 1. Reported trends of fractal- and entropy-based outcomes for different sleep stages.
图。1 。报告了不同睡眠阶段基于分形和熵的结果的趋势。

Future studies with incorporation of latest neuroimaging techniques, such as functional magnetic resonance imaging (fMRI), may also help address several unsolved questions, including: which brain circuit or network (i.e., measured by brain connectivity) is associated with loss of fractal complexity of the sleep EEG in NREM stage? Is loss of sleep EEG fractal complexity associated with memory or cognitive functions? Is brain circuit involved in REM sleep different with those in awake period? Importantly, findings in these fractal-based literature may be useful in developing and testing mathematical models of sleep wake regulations [39].
未来的研究结合最新的神经影像技术，例如功能磁共振成像（fMRI），也可能有助于解决一些未解决的问题，包括：哪个大脑回路或网络（即通过大脑连接性测量）与分形复杂性的丧失相关。 NREM阶段的睡眠脑电图？睡眠不足脑电图分形复杂性与记忆或认知功能相关吗？快速眼动睡眠期的大脑回路与清醒期的大脑回路有什么不同吗？重要的是，这些基于分形的文献中的发现可能有助于开发和测试睡眠觉醒调节的数学模型[39] 。

Entropy-based methods 基于熵的方法

In information theory, entropy measures the uncertainty about the information source and the probability distribution of the samples drawn from it, thus the estimates of entropy can be an indicator of a system's complexity. Complexity analysis of the physiological time series has revealed the fundamental mechanisms of the human physiologic system [47]. Aging and illness have been shown to exhibit a progressive loss of physiologic complexity ∗[8], ∗[14], reflecting the reduced adaptability of physiologic system to intrinsic or extrinsic stimuli. Several entropy analyses have been introduced in sleep literature, including Shannon entropy [48], permutation entropy [49], spectrum entropy [50], approximate entropy (ApEn) [51], sample entropy (SampEn) [52], and multiscale entropy (MSE) ∗[53], [54]. Three of commonly used analyses (ApEn, SampEn, and MSE) are detailed below. In contrast with fractal-based measures, these entropic-based analyses tend to measure similar physical properties (e.g., irregularity), thus we outline their applications to sleep EEG signals in a single section “Applications of entropy-based methods”.
在信息论中，熵衡量信息源的不确定性以及从中抽取的样本的概率分布，因此熵的估计可以作为系统复杂性的指标。生理时间序列的复杂性分析揭示了人体生理系统的基本机制[47] 。衰老和疾病已被证明表现出生理复杂性的逐渐丧失*[8] ， *[14] ，反映了生理系统对内在或外在刺激的适应性降低。睡眠文献中引入了几种熵分析，包括香农熵[48] 、排列熵[49] 、谱熵[50] 、近似熵 (ApEn) [51] 、样本熵 (SampEn) [52]和多尺度熵(MSE) *[53] ， [54] 。下面详细介绍了三种常用的分析（ApEn、SampEn 和 MSE）。与基于分形的测量相比，这些基于熵的分析倾向于测量类似的物理特性（例如，不规则性），因此我们在“基于熵的方法的应用”一节中概述了它们在睡眠脑电图信号中的应用。

Approximate entropy 近似熵

ApEn was introduced to assess the irregularity of the time sequence data [51]. Two input parameters, a pattern length m and tolerance factor r, are specified to compute ApEn. Increased value of ApEn indicates increased irregularity of the time series data and has been used extensively to characterize the degree of randomness in studies of physiologic time series [55], [56].
ApEn被引入来评估时间序列数据的不规则性[51] 。指定两个输入参数，即图案长度m和容差因子r来计算 ApEn。 ApEn 值的增加表明时间序列数据的不规则性增加，并已广泛用于表征生理时间序列研究中的随机程度[55] 、 [56] 。

Sample entropy 样本熵

SampEn was developed to solve the shortcomings of ApEn, in which the precise estimate of ApEn requires substantial lengths of the data and lack of relative consistency (e.g., if the ApEn of one data set is higher than that of another, then it should, but often does not, remain higher under all parameters tested) [52]. Similar to ApEn, a higher value of SampEn indicates increased irregularity in the time series. SampEn has been found to be more consistent and less vulnerable to the constraint of time series length than the ApEn algorithm [57].
SampEn的开发是为了解决ApEn的缺点，其中ApEn的精确估计需要大量数据并且缺乏相对一致性（例如，如果一个数据集的ApEn高于另一数据集的ApEn，那么它应该，但是通常不会，在所有测试参数下仍保持较高水平） [52] 。与 ApEn 类似，SampEn 值较高表示时间序列的不规则性增加。人们发现 SampEn 比 ApEn 算法更一致，并且更不易受时间序列长度的约束[57] 。

Multiscale entropy 多尺度熵

MSE analysis was introduced based on SampEn to quantify the entropy over different time scales ∗[53], [54]. The motivation of MSE is that increased entropy is not always associated with increased dynamical complexity. For example, an uncorrelated randomness (such as irregular heart rate seen in atrial fibrillation) has high entropy but did not convey physiologically meaningful complexity [8]. MSE method measures the entropy of a time series over different time scales, enabling a dimensional view of complexity that differentiates true complex dynamics from regularity and uncorrelated randomness. For example, a regular time series would exhibit low entropy in all time scales, whereas a random time series could show a high entropy in short time scale and the entropy decades as scale factor increases [53].
基于SampEn引入MSE分析来量化不同时间尺度上的熵*[53] ， [54] 。 MSE 的动机是熵的增加并不总是与动态复杂性的增加相关。例如，不相关的随机性（例如心房颤动中看到的不规则心率）具有高熵，但没有传达生理上有意义的复杂性[8] 。 MSE 方法测量不同时间尺度上时间序列的熵，从而实现复杂性的维度视图，将真正的复杂动态与规律性和不相关的随机性区分开来。例如，规则时间序列在所有时间尺度上都会表现出低熵，而随机时间序列可能会在短时间尺度上表现出高熵，并且随着尺度因子的增加而呈现数十年的熵[53] 。

Applications of entropy-based methods
基于熵的方法的应用

For healthy adults, the entropy of sleep EEG signals gradually decreases from wake to sleep stages N1, N2, to N3 (or stage 3 + 4), and increases during REM [50], [58], [59], [60], [61], [62] (Fig. 1b). The results are consistent regardless of the different entropic methods used in the literature. Nonetheless, inconsistency was found in the comparison between the entropy of REM and other sleep stages. Some studies found entropy of REM EEG was between that of wake and N1 [50], [59], whereas others reported to be between N1 and N2 [50], [58], [61], [62]. The similar trend of entropy change is also present in children (wake > REM sleep > NREM sleep) [60] and in newborns (active sleep > quiet sleep) [63].
对于健康成年人来说，睡眠脑电信号的熵从清醒到睡眠阶段N1、N2、N3（或3+4阶段）逐渐减少，并在快速眼动期间增加[50] 、 [58] 、 [59] 、 [60] ， [61] ， [62] （图1b ）。无论文献中使用不同的熵方法，结果都是一致的。尽管如此，快速眼动睡眠阶段的熵与其他睡眠阶段的熵的比较却发现不一致。一些研究发现 REM 脑电图的熵介于唤醒和 N1 之间[50] 、 [59] ，而其他研究则报道介于 N1 和 N2 之间[50] 、 [58] 、 [61] 、 [62] 。类似的熵变化趋势也存在于儿童（清醒> REM睡眠> NREM睡眠） [60]和新生儿（主动睡眠>安静睡眠） [63] 。

The entropy of sleep EEG signals may be of help to assess the trajectory of brain maturation in newborns. For example, a study in newborns (age 25–60 wk) reported that entropy of EEG increased during both active sleep and quiet sleep before approximately 42 wk in age but ceased to increase in quiet sleep and even decreased in active sleep after newborns reached term age [63].
睡眠脑电图信号的熵可能有助于评估新生儿大脑成熟的轨迹。例如，一项针对新生儿（25-60 周）的研究报告称，在大约 42 周之前，脑电图熵在主动睡眠和安静睡眠期间都会增加，但在新生儿足月后，安静睡眠中不再增加，甚至在主动睡眠中下降年龄[63] 。

In pathological conditions such as Parkinson's disease (PD), sleep stage-specific increased MSE was observed during NREM sleep in PD patients, compared with non-PD controls, and the difference was significant at high time-scale factors, which might reflect a compensatory mechanism for early defects in neuronal network control machinery in PD [64].
在帕金森病(PD) 等病理情况下，与非 PD 对照相比，PD 患者在NREM 睡眠期间观察到睡眠阶段特异性 MSE 增加，并且在高时间尺度因素下差异显着，这可能反映了代偿性PD 神经元网络控制机制早期缺陷的机制[64] 。

Implications of entropy-based methods
基于熵的方法的含义

In general, findings from the sleep EEG studies using entropy-based methods are relatively consistent (Table 2), compared to those using fractal-based analyses. Although entropy and fractal-based methods have very different physical definitions, the ubiquitous principle among different complexity analyses is to quantify how the variability (or temporal dynamics) of the time series changes with different time scales [65], and the properties of regularity and randomness captured by these methods is interchangeable. Using methods of nonlinear dynamics, sleep EEG signals become less complex (i.e., toward regularity) as sleep stage advances, suggesting that brain activity during sleep becomes more coherent and periodic compared to the wake or REM period.
一般来说，与使用基于分形的分析相比，使用基于熵的方法进行的睡眠脑电图研究的结果相对一致（表2 ）。尽管基于熵和分形的方法具有非常不同的物理定义，但不同复杂性分析之间普遍存在的原则是量化时间序列的变异性（或时间动态）如何随不同时间尺度变化[65] ，以及规律性和这些方法捕获的随机性是可以互换的。使用非线性动力学方法，随着睡眠阶段的进展，睡眠脑电图信号变得不那么复杂（即趋向规律性），这表明睡眠期间的大脑活动与清醒或快速眼动时期相比变得更加连贯和周期性。

Table 2. Applications of entropy-based methods*.
表 2 .基于熵的方法的应用*。

Citations 引文Study subjects 学习科目Main finding(s) 主要发现）
Approximate Entropy 近似熵
Acharya, et al. 2005 [15]
阿查里亚等人。 2005年[15]		8 healthy Caucasian, males and females, 21-35 y
8 名健康白人，男性和女性，21-35 岁		In sleep stage 4, the ApEn was the lowest due to the very low variation in the EEG signal. In REM sleep, the variation is slightly more and as a result the ApEn increases.
在睡眠阶段 4，由于 EEG 信号的变化非常小，ApEn 最低。在快速眼动睡眠中，变化稍大，因此 ApEn 增加。

He, et al. 2005 [50]
他，等人。 2005年[50]		8 healthy subjects, 21-35 y
8 名健康受试者，21-35 岁		ApEn declined from wake to each NREM sleep episode progressively, and reached a low level at times when SWS was dominant. ApEn in REM sleep is higher than in than SWS.
ApEn 从醒来到每个 NREM 睡眠阶段逐渐下降，并在 SWS 占主导地位时达到较低水平。 REM 睡眠中的 ApEn 高于 SWS 睡眠中的 ApEn。

Burioka, et al. 2005 [59]
布里奥卡等人。 2005年[59]		8 healthy males, 23-26 y
8名健康男性，23-26岁		ApEn of EEG was significantly lower during sleep stage 4 and higher during wake and REM sleep.
脑电图的 ApEn 在睡眠第 4 阶段显着较低，而在清醒和快速眼动睡眠期间则较高。

Lee, et al. 2013 [60]
李等人。 2013年[60]		6 adults, 19-25 y; 6 children, 11-13 y.
6名成人，19-25岁； 6 名儿童，11-13 岁。		ApEn trends for both age groups: wake > REM sleep > NREM sleep. Adults had significantly larger ApEn values than children during wakefulness.
两个年龄组的 ApEn 趋势：清醒 > 快速眼动睡眠 > 非快速眼动睡眠。成人在清醒状态下的 ApEn 值显着高于儿童。

Sample Entropy 样本熵
Zhang, et al. 2009 [63]
张，等人。 2009年[63]		168 newborns with postmenstrual age ranging from 25 to 60 wk
168 名月经后年龄为 25 至 60 周的新生儿		SampEn of EEG during AS is higher than that during QS. SampEn increases during both AS and QS before about 42 wk in PMA while it ceases its increase in QS and even decreases in AS after newborns reaching term age. A distinct decrease in the interquartile range of SampEn is found with increasing PMA (25-50 wk), followed by maintenance of low fluctuation in SampEn curves.
AS期间EEG的SampEn高于QS期间。在 PMA 中约 42 周之前，SampEn 在 AS 和 QS 期间均增加，而在 QS 中停止增加，甚至在新生儿达到足月后在 AS 中减少。随着 PMA 的增加（25-50 周），SampEn 的四分位数范围明显减小，随后 SampEn 曲线维持低波动。

Chouvarda, et al. 2010 [62]
乔瓦尔达等人。 2010年[62]		10 healthy subjects, male and female, 25-45 y
10 名健康受试者，男性和女性，25-45 岁		SampEn values are related to both the sleep stages and the subtype of CAP. Complexity features can serve as consistent descriptors of sleep dynamics and can potentially assist in the classification of sleep stages
SampEn 值与睡眠阶段和 CAP 子类型相关。复杂性特征可以作为睡眠动态的一致描述符，并可能有助于睡眠阶段的分类

Chouvarda, et al. 2011 [58]
乔瓦尔达等人。 2011年[58]		10 healthy subjects, male and female, 25-45 y
10 名健康受试者，男性和女性，25-45 岁		SampEn declined from wake to each NREM sleep episode. SampEn in REM sleep is higher than in SWS. For CAP sleep, A3 presented a quite similar complexity independently of the sleep stage, while A1 and A2 showed higher complexity in light sleep than during deep sleep.
SampEn 从清醒到每个 NREM 睡眠阶段都会出现下降。 REM 睡眠中的 SampEn 高于 SWS 睡眠中的 SampEn。对于 CAP 睡眠，A3 表现出非常相似的复杂性，与睡眠阶段无关，而 A1 和 A2 在浅度睡眠中表现出比深度睡眠更高的复杂性。

Chouvarda, et al. 2012 [97]
乔瓦尔达等人。 2012年[97]		11 healthy subjects, male and female, 25-45 y
11 名健康受试者，男性和女性，25-45 岁		Based on the nonlinear properties of the EEG at transition points of the sequences that build the CAPs, EEG signal present significant differences between activations and non-activations in the SampEn.
基于构建 CAP 的序列转换点处 EEG 的非线性特性，EEG 信号在 SampEn 的激活和非激活之间呈现显着差异。

Chouvarda, et al. 2012 [98]
乔瓦尔达等人。 2012年[98]		11 healthy subjects, mean age 32.7 y, and 10 subjects diagnosed with primary insomnia, mean age 32.5 y, male and female
11 名健康受试者，平均年龄 32.7 岁，10 名诊断为原发性失眠的受试者，平均年龄 32.5 岁，男性和女性		As regards the deep sleep building phases defined by CAP, more irregular activation-deactivation patterns, with larger deactivation time, i.e., distance between consecutive activation events, and appearing with higher EEG complexity in deactivation. A longer duration of desynchronization phases, with increased EEG complexity and more irregular patterns.
对于CAP定义的深度睡眠建立阶段，激活-失活模式更加不规则，失活时间（即连续激活事件之间的距离）更大，并且失活中的脑电图复杂度更高。去同步阶段持续时间更长，脑电图复杂性增加，模式更不规则。

Mendez, et al. 2015 [96]
门德斯等人。 2015年[96]		10 healthy adult subjects (5 males), 25-45 y, mean age 32.7 y
10 名健康成年受试者（5 名男性），25-45 岁，平均年龄 32.7 岁		When define an onset window containing the first two seconds of the A-phase of CAP, SampEn showed statistical differences between the two consecutive no overlapped windows with duration of 2 seconds before the onset window, as well as between two windows after the onset window. On the other hand, the SampEn measure shows a different behavior during the onset.
当定义包含 CAP A 相前两秒的起始窗口时，SampEn 显示起始窗口之前持续时间为 2 秒的两个连续无重叠窗口之间以及起始窗口之后的两个窗口之间的统计差异。另一方面，SampEn 测量在发作期间显示出不同的行为。

Multiscale Entropy 多尺度熵
Bell, et al. 2012 [88]
贝尔等人。 2012年[88]		54 college students with histories of coffee-induced insomnia, male and female, mean age 20 y
54名有咖啡失眠史的大学生，男女，平均年龄20岁		MSE results indicate significant, remedy-specific directional effects, especially later in the night (Coffea cruda: remedy night increases and post-remedy night decreases in MSE at multiple sites for both stage 3 and 4 in both REM cycles; Nux vomica: remedy night decreases and post-remedy night increases).
MSE 结果表明显着的、特定于补救措施的方向性影响，尤其是在夜间（咖啡渣a：在两个 REM 周期的第 3 阶段和第 4 阶段的多个部位，补救措施夜间增加和补救后夜间 MSE 减少；马钱子：补救措施夜间减少，治疗后夜间增加）。

Chung, et al. 2013 [64]
钟，等人。 2013年[64]		9 patients with PD, (mean age 78.2 y) and 11 non-PD controls (mean age 61.2 y); male and female
9 名 PD 患者（平均年龄 78.2 岁）和 11 名非 PD 对照患者（平均年龄 61.2 岁）；男性和女性		Sleep stage-specific increased MSE was observed in patients with PD during NREM sleep. The difference was more marked and significant at higher time scale factors.
在 NREM 睡眠期间的 PD 患者中观察到睡眠阶段特异性 MSE 增加。在较高的时间尺度因素下，差异更加明显和显着。

Shi, et al. 2016 [66]
石，等人。 2016年[66]		4 healthy male subjects (27-38 yrs with mean age 32.0 ±4.6yrs)
4 名健康男性受试者（27-38 岁，平均年龄 32.0 ±4.6 岁）		Entropy is higher during wakefulness and increasing time scales at small scales (<0.04 s). In contrast, entropy is higher during deep sleep and lower with increasing time scales at large scales (0.25–2 s).
清醒时熵较高，并且小尺度（<0.04 s）的时间尺度增加。相比之下，深度睡眠期间熵较高，而在大尺度（0.25-2 s）下随着时间尺度的增加熵较低。

* Only methods introduced in the review are listed in the table. Some studies included more than one method.
* 表中仅列出了综述中介绍的方法。一些研究包括不止一种方法。

Abbreviations: ApEn, approximate entropy; AS, active sleep; CAP, cyclic alternating pattern; EEG, electroencephalography; MSE, multiscale entropy; NREM, non-rapid eye movement; PD, Parkinson’s disease; PMA, postmenstrual age; QS, quiet sleep; REM, rapid eye movement; SampEn, sample entropy; SWS, slow wave sleep.
缩写：ApEn，近似熵； AS，主动睡眠； CAP，循环交替模式；脑电图、脑电图； MSE，多尺度熵； NREM，非快速眼球运动； PD，帕金森病； PMA，月经后年龄； QS，安静睡眠； REM，快速眼球运动； SampEn，样本熵； SWS，慢波睡眠。

One limitation of the most existing studies is that complexity was not measured at multiple time scales. The EEG signal apparently contains multiple frequency components; each operates at distinct time scales [66]. Thus, investigation of complexity changes across different time scales in EEG signals may be of help to identify specific mode of brain activity that is related to sleep-wake regulation.
大多数现有研究的局限性之一是没有在多个时间尺度上测量复杂性。脑电图信号显然包含多个频率成分；每个都在不同的时间尺度上运行[66] 。因此，研究脑电图信号在不同时间尺度上的复杂性变化可能有助于识别与睡眠-觉醒调节相关的特定大脑活动模式。

Discussion 讨论

Our review of recent studies deduced that the findings of nonlinear approaches are somehow related and consistent regardless of the particular methods. From a fractal perspective, a fractal component decreases from wake to sleep stages N1–N3 and increases during REM sleep, and DFA scaling exponents increase from wake to stages N1–N3 and decrease during REM sleep. From the entropy perspective, the complexity of sleep EEG decreases from wake to sleep stages N1–N3 and increases during REM.
我们对最近研究的回顾表明，无论采用何种特定方法，非线性方法的研究结果在某种程度上都是相关且一致的。从分形角度来看，分形成分从清醒到睡眠阶段 N1-N3 减少，并在 REM 睡眠期间增加，DFA缩放指数从清醒到阶段 N1-N3 增加，并在 REM 睡眠期间减少。从熵的角度来看，睡眠脑电图的复杂性从清醒到睡眠阶段 N1-N3 降低，在 REM 期间增加。

Efficient sleep is supposed to be restful and restorative [2]. Nonlinear features exhibit a pattern of gradual change through sleep stages, which is consistent with NREM sleep stages approximately paralleling a depth-of-sleep continuum, with arousal thresholds generally lowest in N1 and highest in N3.
有效的睡眠应该是令人放松和恢复活力的[2] 。非线性特征表现出在整个睡眠阶段逐渐变化的模式，这与大约平行于睡眠深度连续体的 NREM 睡眠阶段一致，唤醒阈值通常在 N1 中最低，在 N3 中最高。

Sleep and wakefulness are influenced by different neurotransmitter signals in the brain. Neurons in the brainstem produce neurotransmitters such as serotonin and norepinephrine that keep some parts of the brain active while we are awake. Other neurons, which begin signaling when we fall asleep, appear to “switch off” the signals that keep us awake [67]. Neural models of information processing have suggested that both the degree of synchrony and time scale determine the maximum information transfer between neurons [68]. Overall, the cortex may become more inactive as a person proceeds through one stage to the next deeper stage until N3 [44]. As sleep is getting deeper, the reason why entropy, CD, and H decrease, could be that fewer neurons are available for processing information, or that the neurons are better synchronized to generate brain waves with less complexity [45]. In REM sleep, the brain becomes highly active again and can nearly attain the level of someone who is awake. In addition, cerebral blood flow and metabolism decreases in deep sleep, and remains about the same during REM sleep as in wakefulness, which reflects that cerebral synaptic activity levels are lower in deep sleep but higher in REM as in wake [69]. The cortex becomes more active and it is possible that additional neurons are available or neurons are desynchronized for processing information [70]; consequently, entropy, the CD, and H increase.
睡眠和觉醒受到大脑中不同神经递质信号的影响。脑干中的神经元会产生血清素和去甲肾上腺素等神经递质，使我们在清醒时大脑的某些部分保持活跃。当我们入睡时开始发出信号的其他神经元似乎“关闭”了让我们保持清醒的信号[67] 。信息处理的神经模型表明，同步程度和时间尺度决定了神经元之间的最大信息传递[68] 。总体而言，当一个人从一个阶段进入下一个更深的阶段直到 N3 时，皮层可能会变得更加不活跃[44] 。随着睡眠越来越深，熵、CD 和H减少的原因可能是可用于处理信息的神经元减少，或者神经元更好地同步以生成复杂性较低的脑电波[45] 。在快速眼动睡眠中，大脑再次变得高度活跃，几乎可以达到清醒时的水平。此外，深度睡眠时脑血流量和代谢减少，而快速眼动睡眠期间与清醒时大致相同，这反映出深度睡眠时大脑突触活动水平较低，但快速眼动睡眠时与清醒时相比较高[69] 。 皮质变得更加活跃，并且可能有额外的神经元可用或神经元不同步来处理信息[70] ；结果，熵、CD 和H增加。

Both conventional spectral analyses and nonlinear measures have certain advantages. For example, nonlinear measures more effectively discriminated between N1 and N2, whereas the spectral measures were superior in separating N2 and SWS [22]. We believe that the dynamic complexity of sleep EEG is influenced by both linear and nonlinear features and can be effectively interpreted using comprehensive approaches including nonlinear measures of brain activity; furthermore, studies have proven that using nonlinear measures can yield valuable information compared to conventional linear measures such as Fourier transforms [71], [72].
传统的谱分析和非线性测量都具有一定的优点。例如，非线性测量可以更有效地区分 N1 和 N2，而谱测量在分离 N2 和SWS方面表现更佳[22] 。我们相信，睡眠脑电图的动态复杂性受到线性和非线性特征的影响，并且可以使用包括大脑活动的非线性测量在内的综合方法来有效解释；此外，研究证明，与傅里叶变换等传统线性测量相比，使用非线性测量可以产生有价值的信息[71] 、 [72] 。

In recent years, progresses have been made in advancing nonlinear methods. Examples of latest developments include 1) substituting a fuzzy membership function for the Heaviside function in SampEn to improve the consistency [73], 2) using symbolic series instead of the continuous variables to improve the robustness to outliers [74], and 3) measuring the distribution of inter-vector distances in order to mitigate the dependence on input parameters [75]. Other than the above reviewed fractal and entropy based methods, another widely-used methods, such as recurrence quantification analysis [76], are also capable of measuring complexity for non-stationary data including sleep EEG [77]. Of note, this review focuses mainly on the analysis of brain signal in temporal dimension using fractal or entropy methods. In recent years, there has been an increasing interest in applying graph theory to study the complexity of brain networks in spatial dimension [78], [79], [80] using various correlative methods to assess inter-dependency of brain signal among different brain regions [72], [81], [82], [83], [84], [85], [86].
近年来，非线性方法的研究取得了一定进展。最新进展的示例包括 1）用模糊隶属函数代替 SampEn 中的 Heaviside 函数以提高一致性[73] ，2）使用符号级数代替连续变量以提高对异常值的鲁棒性[74] ，以及 3）测量向量间距离的分布，以减轻对输入参数的依赖[75] 。除了上述基于分形和熵的方法之外，另一种广泛使用的方法，例如递归量化分析[76] ，也能够测量包括睡眠脑电图在内的非平稳数据的复杂性[77] 。值得注意的是，这篇综述主要关注使用分形或熵方法在时间维度上分析大脑信号。 近年来，人们越来越关注应用图论来研究空间维度上大脑网络的复杂性[78] ， [79] ， [80]，使用各种相关方法来评估不同大脑之间大脑信号的相互依赖性区域[72] 、 [81] 、 [82] 、 [83] 、 [84] 、 [85] 、 [86] 。

Nonlinear analysis of EEG signal can serve in the understanding or as consistent descriptors of sleep dynamics and potentially assist in automatic sleep classification. All existing findings encourage using nonlinear approaches as additional aids to visual or automated sleep staging. Furthermore, they may help in assessing pathologic conditions [87].
脑电图信号的非线性分析可以用于理解或作为睡眠动态的一致描述符，并可能有助于自动睡眠分类。所有现有的研究结果都鼓励使用非线性方法作为视觉或自动睡眠分期的额外辅助手段。此外，它们可能有助于评估病理状况[87] 。

Nonlinear approaches are promising and worth further investigation; however, some limitations must be mentioned. First, high-quality studies with well-designed experimental conditions and large samples are scant. Nearly all existing studies are based on the conventional definition of sleep stages. Therefore, the full advantages of nonlinear approaches have yet to be determined. Regarding the clinical use, it seems difficult to define the norm for sleep EEG by using nonlinear methods, because these methods themselves involve signal pre-processing and multiple parameters to be defined within the algorithm. In addition, the neurophysiologic mechanisms behind the complex oscillations of brain signals remain poorly understood. We suggest to evaluate a wide range of nonlinear measures in the large-scale sleep database, and to develop practical applications of nonlinear approaches to understanding the sleep EEG dynamics in healthy and pathological conditions.
非线性方法很有前途，值得进一步研究；然而，必须提及一些限制。首先，缺乏精心设计的实验条件和大样本的高质量研究。几乎所有现有的研究都是基于睡眠阶段的传统定义。因此，非线性方法的全部优点尚未确定。就临床使用而言，使用非线性方法来定义睡眠脑电图的标准似乎很困难，因为这些方法本身涉及信号预处理和算法中需要定义的多个参数。此外，人们对大脑信号复杂振荡背后的神经生理机制仍知之甚少。我们建议评估大规模睡眠数据库中的各种非线性测量，并开发非线性方法的实际应用来了解健康和病理条件下的睡眠脑电图动态。

Conclusion 结论

EEG is critical for extending the knowledge of sleep and revealing its fundamental mechanisms. Studies have shown that nonlinear analyses of sleep EEG signal can distinguish sleep stages as well as normal and pathological conditions. Both nonlinear and linear measures have certain advantages and disadvantages that are complementary to each other. Because of the nonlinear and nonstationary properties of brain activity, nonlinear approaches to sleep EEG are more appropriate for researching the physiologic and pathologic features of the brain activity during sleep. Nonlinear approaches using fractal or entropy methods may facilitate automatic sleep classification, but more importantly, additional studies are encouraged to mitigate the limitations toward expanding the application of nonlinear approaches to comprehensively understand sleep dynamics.

Practice points

  • 1)

    Sleep is not simply a succession of human invented stages, but a delicate and sophisticated nonlinear symphony played by the brain in a mutual interaction with the rest of the body.

  • 2)

    Nonlinear approaches are potentially promising because electrobiophysiological signals like EEG are typically nonlinear and non-stationary.

  • 3)

    Nonlinear analyses of sleep EEG signals are able to differentiate sleep states and distinguish pathological sleep conditions form healthy states.

  • 4)

    There are limitations in existing literature. Full advantages of nonlinear approaches have yet to be determined. Future studies are encouraged to investigate sleep neurophysiology by nonlinear approaches.

Research agenda

This review has addressed a number of important approaches and findings in nonlinear analyses for sleep EEG, and proposed some research questions warranting consideration in future studies for better understanding of sleep:

  • 1)

    Can nonlinear methods help to delineate critical and hidden dynamical properties of sleep?

  • 2)

    What are the biological mechanisms of changes in nonlinear indices during sleep?

  • 3)

    Can nonlinear approaches assist in automatic sleep scoring and classification?

  • 4)

    Can nonlinear approaches help to differentiate normal and pathologic conditions during sleep?


脑电图对于扩展睡眠知识和揭示其基本机制至关重要。研究表明，睡眠脑电图信号的非线性分析可以区分睡眠阶段以及正常和病理状态。非线性和线性测量都有一定的优点和缺点，并且是互补的。由于大脑活动的非线性和非平稳特性，睡眠脑电图的非线性方法更适合研究睡眠期间大脑活动的生理和病理特征。使用分形或熵方法的非线性方法可能有助于自动睡眠分类，但更重要的是，鼓励进行更多研究以减轻扩大非线性方法应用以全面了解睡眠动态的限制。

Practice points

  • 1)

    Sleep is not simply a succession of human invented stages, but a delicate and sophisticated nonlinear symphony played by the brain in a mutual interaction with the rest of the body.
    睡眠不仅仅是人类发明的一系列阶段，而是大脑与身体其他部位相互作用而演奏的一首精致而复杂的非线性交响曲。

  • 2)

    Nonlinear approaches are potentially promising because electrobiophysiological signals like EEG are typically nonlinear and non-stationary.
    非线性方法具有潜在的前景，因为像脑电图这样的电生物生理信号通常是非线性和非平稳的。

  • 3)

    Nonlinear analyses of sleep EEG signals are able to differentiate sleep states and distinguish pathological sleep conditions form healthy states.
    睡眠脑电图信号的非线性分析能够区分睡眠状态并将病理性睡眠状况与健康状态区分开来。

  • 4)

    There are limitations in existing literature. Full advantages of nonlinear approaches have yet to be determined. Future studies are encouraged to investigate sleep neurophysiology by nonlinear approaches.
    现有文献存在局限性。非线性方法的全部优点尚未确定。鼓励未来的研究通过非线性方法研究睡眠神经生理学。

Research agenda

This review has addressed a number of important approaches and findings in nonlinear analyses for sleep EEG, and proposed some research questions warranting consideration in future studies for better understanding of sleep:
这篇综述讨论了睡眠脑电图非线性分析的许多重要方法和发现，并提出了一些值得在未来研究中考虑的研究问题，以更好地理解睡眠：

  • 1)

    Can nonlinear methods help to delineate critical and hidden dynamical properties of sleep?
    非线性方法可以帮助描述睡眠的关键和隐藏的动态特性吗？

  • 2)

    What are the biological mechanisms of changes in nonlinear indices during sleep?
    睡眠期间非线性指标变化的生物学机制是什么？

  • 3)

    Can nonlinear approaches assist in automatic sleep scoring and classification?
    非线性方法可以帮助自动睡眠评分和分类吗？

  • 4)

    Can nonlinear approaches help to differentiate normal and pathologic conditions during sleep?
    非线性方法可以帮助区分睡眠期间的正常状况和病理状况吗？

Conflicts of interest 利益冲突

The authors report no conflict of interest in this review.
作者报告在本次审查中不存在利益冲突。

References∗

Cited by (0) 被引用 (0)

∗

The most important references are denoted by an asterisk.
最重要的参考文献用星号表示。

View Abstract 查看摘要
© 2017 Elsevier Ltd. All rights reserved.
© 2017 Elsevier Ltd. 保留所有权利。