Time irreversibility (TIR) refers to the manifestation of nonequilibrium brain activity influenced by various physiological conditions; however, the influence of sleep on electroencephalogram (EEG) TIR has not been sufficiently investigated. In this paper, a comprehensive study on permutation TIR (pTIR) of EEG data under different sleep stages is conducted. Two basic ordinal patterns (i.e., the original and amplitude permutations) are distinguished to simplify sleep EEGs, and then the influences of equal values and forbidden permutation on pTIR are elucidated. To detect pTIR of brain electric signals, 5 groups of EEGs in the awake, stages I, II, III, and rapid eye movement (REM) stages are collected from the public Polysomnographic Database in PhysioNet. Test results suggested that the pTIR of sleep EEGs significantly decreases as the sleep stage increases (p<0.001), with the awake and REM EEGs, demonstrating greater differences than others. Comparative analysis and numerical simulations support the importance of equal values. Distribution of equal states, a simple quantification of amplitude fluctuations, significantly increases with the sleep stage (p<0.001). If these equalities are ignored, incorrect probabilistic differences may arise in the forward-backward and symmetric permutations of TIR, leading to contradictory results; moreover, the ascending and descending orders for symmetric permutations also lead different outcomes in sleep EEGs. Overall, pTIR in sleep EEGs contributes to our understanding of quantitative TIR and classification of sleep EEGs.

翻译：时间不可逆性（TIR）是受多种生理条件影响的非平衡脑活动的表现，然而睡眠对脑电图（EEG）TIR的影响尚未得到充分研究。本文对不同睡眠阶段下EEG数据的排列时间不可逆性（pTIR）进行了全面研究。首先区分两种基本序模式（即原始排列和振幅排列）以简化睡眠脑电图，进而阐明等值与禁忌排列对pTIR的影响。为检测脑电信号的pTIR，从PhysioNet公共多导睡眠图数据库中收集了清醒期、I期、II期、III期和快速眼动（REM）期共5组EEG数据。测试结果表明，随着睡眠阶段加深，睡眠EEG的pTIR显著降低（p<0.001），其中清醒期和REM睡眠期的EEG显示出比其他阶段更大的差异。对比分析与数值模拟支持等值的重要性。等值状态分布——振幅波动的简单量化指标——随睡眠阶段加深而显著增加（p<0.001）。若忽略这些等值，TIR的前后向排列和对称排列中可能产生错误的概率差异，从而导出矛盾结果；此外，对称排列中的升序与降序规则在睡眠EEG中也会导致不同结果。总体而言，睡眠EEG中的pTIR有助于我们理解定量TIR及睡眠EEG的分类。