Cross-frequency coupling

交叉频率耦合
  • 文章类型: Journal Article
    与年龄相关的神经退行性疾病和相关医疗保健成本的全球负担不断升级,需要创新的干预措施来稳定或增强认知功能。工作记忆(WM)的缺陷与前额叶theta-gamma交叉频率耦合的变化有关。低强度经颅交流电刺激(tACS)已成为一种非侵入性,低成本的方法,能够通过夹带调节目标大脑区域的持续振荡。这项研究调查了对背外侧前额叶皮层(DLPFC)给予多会话峰值耦合θ-γ交叉频率tACS对老年人WM表现的影响。在一个随机的,假控制,三盲设计,77名参与者在执行n-back任务的同时,在六周内接受了16次刺激训练。信号检测措施显示增加的2-back灵敏度和响应偏差的鲁棒调制,表明改进的WM和决策适应性,分别。在1-back条件下没有观察到影响,强调对认知负荷的依赖。重复的tACS加强了行为变化,通过增加效果大小来表示。这项研究支持将前额叶theta-gamma偶联与WM过程相关的先前研究,并提供了对重复tACS干预的神经认知益处的独特见解。老年人中耐受性良好且高度有效的多阶段tACS干预措施强调了其在脆弱人群中的治疗潜力。
    The escalating global burden of age-related neurodegenerative diseases and associated healthcare costs necessitates innovative interventions to stabilize or enhance cognitive functions. Deficits in working memory (WM) are linked to alterations in prefrontal theta-gamma cross-frequency coupling. Low-intensity transcranial alternating current stimulation (tACS) has emerged as a non-invasive, low-cost approach capable of modulating ongoing oscillations in targeted brain areas through entrainment. This study investigates the impact of multi-session peak-coupled theta-gamma cross-frequency tACS administered to the dorsolateral prefrontal cortex (DLPFC) on WM performance in older adults. In a randomized, sham-controlled, triple-blinded design, 77 participants underwent 16 stimulation sessions over six weeks while performing n-back tasks. Signal detection measures revealed increased 2-back sensitivity and robust modulations of response bias, indicating improved WM and decision-making adaptations, respectively. No effects were observed in the 1-back condition, emphasizing dependencies on cognitive load. Repeated tACS reinforces behavioral changes, indicated by increasing effect sizes. This study supports prior research correlating prefrontal theta-gamma coupling with WM processes and provides unique insights into the neurocognitive benefits of repeated tACS intervention. The well-tolerated and highly effective multi-session tACS intervention among the elderly underscores its therapeutic potential in vulnerable populations.
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  • 文章类型: Journal Article
    参与认知功能的神经过程的电活动被捕获在EEG信号中,允许探索跨多个时空尺度的神经元振荡的整合和协调。我们提出了一种新颖的方法,将EEG信号转换为图像序列,考虑涉及低级听觉处理的交叉频率相位同步(CFS)动力学,随着用于检测发展性阅读障碍(DD)的两阶段深度学习模型的发展。这种深度学习模型利用图像序列中保存的空间和时间信息来发现相位同步随时间变化的判别模式,达到高达83%的平衡精度。该结果支持了典型和诵读困难的7岁读者之间存在差异的大脑同步动力学。此外,我们使用一种新的特征掩模获得了可解释的表示,将分类过程中最相关的区域与正常阅读的认知过程以及与阅读障碍中发现的代偿机制相对应的认知过程联系起来.
    The electrical activity of the neural processes involved in cognitive functions is captured in EEG signals, allowing the exploration of the integration and coordination of neuronal oscillations across multiple spatiotemporal scales. We have proposed a novel approach that combines the transformation of EEG signal into image sequences, considering cross-frequency phase synchronisation (CFS) dynamics involved in low-level auditory processing, with the development of a two-stage deep learning model for the detection of developmental dyslexia (DD). This deep learning model exploits spatial and temporal information preserved in the image sequences to find discriminative patterns of phase synchronisation over time achieving a balanced accuracy of up to 83%. This result supports the existence of differential brain synchronisation dynamics between typical and dyslexic seven-year-old readers. Furthermore, we have obtained interpretable representations using a novel feature mask to link the most relevant regions during classification with the cognitive processes attributed to normal reading and those corresponding to compensatory mechanisms found in dyslexia.
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  • 文章类型: Journal Article
    韦斯特综合征(WS)是一种破坏性的癫痫性脑病,发病于婴儿期和幼儿期。它的特点是聚集性癫痫痉挛,发展性逮捕,脑电图(EEG)和发作间性心律失常。心律失常被认为是WS的标志,但由于其广泛的可变性和缺乏可量化的定义,其视觉评估具有挑战性。本研究旨在分析WS中的EEG模式,并确定该疾病的计算诊断生物标志物。
    比较了31例WS患者和20例年龄匹配对照的EEG记录的线性和非线性特征。随后,研究了已识别特征与结构和遗传异常的相关性。
    WS患者的α带活性显着升高(0.2516vs.0.1914,p<0.001)和降低的δ带活性(0.5117vs.0.5479,p<0.001),特别是在枕骨区域,以及全球增强的θ带活性(0.2145vs.0.1655,p<0.001)在功率谱分析中。此外,小波-双相干分析显示,WS患者的交叉频率耦合明显衰减。此外,双通道相干性分析显示WS患者的连通性改变较小。在EEG数据的四个非线性特征中(即,近似熵,样本熵,排列熵,和小波熵),排列熵显示,与对照组相比,WS患者的脑电图总体减少最显著(1.4411vs.1.5544,p<0.001)。多元回归结果表明,遗传病因可能会影响WS的脑电图,而结构性因素则不行。
    θ活性的整体增强和排列熵的整体降低的组合可以用作WS的计算EEG生物标志物。在临床实践中实施这些生物标志物可以加快WS的诊断和治疗。从而改善长期结果。
    UNASSIGNED: West syndrome (WS) is a devastating epileptic encephalopathy with onset in infancy and early childhood. It is characterized by clustered epileptic spasms, developmental arrest, and interictal hypsarrhythmia on electroencephalogram (EEG). Hypsarrhythmia is considered the hallmark of WS, but its visual assessment is challenging due to its wide variability and lack of a quantifiable definition. This study aims to analyze the EEG patterns in WS and identify computational diagnostic biomarkers of the disease.
    UNASSIGNED: Linear and non-linear features derived from EEG recordings of 31 WS patients and 20 age-matched controls were compared. Subsequently, the correlation of the identified features with structural and genetic abnormalities was investigated.
    UNASSIGNED: WS patients showed significantly elevated alpha-band activity (0.2516 vs. 0.1914, p < 0.001) and decreased delta-band activity (0.5117 vs. 0.5479, p < 0.001), particularly in the occipital region, as well as globally strengthened theta-band activity (0.2145 vs. 0.1655, p < 0.001) in power spectrum analysis. Moreover, wavelet-bicoherence analysis revealed significantly attenuated cross-frequency coupling in WS patients. Additionally, bi-channel coherence analysis indicated minor connectivity alterations in WS patients. Among the four non-linear characteristics of the EEG data (i.e., approximate entropy, sample entropy, permutation entropy, and wavelet entropy), permutation entropy showed the most prominent global reduction in the EEG of WS patients compared to controls (1.4411 vs. 1.5544, p < 0.001). Multivariate regression results suggested that genetic etiologies could influence the EEG profiles of WS, whereas structural factors could not.
    UNASSIGNED: A combined global strengthening of theta activity and global reduction of permutation entropy can serve as computational EEG biomarkers for WS. Implementing these biomarkers in clinical practice may expedite diagnosis and treatment in WS, thereby improving long-term outcomes.
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  • 文章类型: Journal Article
    大脑中的电生理信号之间的交叉频率耦合(CFC)是一种长期研究的现象,在帕金森氏病和癫痫等疾病中已观察到其异常。最近,在胃-脑电生理研究中已观察到CFC,因此成为涉及肠-脑轴畸变的疾病的诱人可能目标。然而,当前检测耦合的方法,特别是相位-振幅耦合(PAC),不要试图捕获相位和振幅的统计关系。
    在本文中,我们首先演示了一种使用灵活的参数方法对这些联合统计进行建模的方法,其中,我们使用带有傅立叶回归函数的伽马分布广义线性模型(GLM)对给定相位的振幅条件分布进行建模。我们用最小描述长度(MDL)原则进行模型选择,演示一种评估拟合优度(GOF)的方法,并在多个脑电图(EEG)数据集中展示这种方法的有效性。其次,我们展示了我们如何利用互信息,在联合分发上运作,作为耦合的规范度量,因为当且仅当相位和幅度在统计上不独立时,它是非零和非负的。此外,我们建立了Martinez-Cancino等人以前的工作。,和Voytek等人。,并显示信息密度,使用我们的方法沿着给定的样本路径进行评估,是一种有前途的时间分辨PAC度量。
    使用合成生成的肠-脑耦合信号,我们证明,我们的方法优于现有的黄金标准方法,可通过接收器工作特性(ROC)曲线分析检测到低水平的相位幅度耦合。为了验证我们的方法,我们通过生成comodulograms来测试侵入性脑电图记录,并将我们的方法与黄金标准PAC措施进行比较,调制指数,在探索性分析中表现可比。此外,为了展示其在联合肠脑电生理数据中的应用,我们生成了同时高密度脑电图和胃电图记录的拓扑图,并重现了Richter等人的开创性工作。证明了肠脑PAC的存在。使用模拟数据,我们针对不同类型的时变耦合验证了我们的方法,然后证明了其在睡眠纺锤脑电图和失配负(MMN)数据集中跟踪时变PAC的性能.
    我们使用GammaGLM和互信息对PAC进行的新度量展示了一种有前途的新方法,可以使用振幅和相位上的完全联合分布来计算PAC值。我们的措施优于PAC最常见的现有措施,并在识别电生理数据集中随时间变化的PAC方面显示出有希望的结果。此外,我们提供了使用我们的方法进行多重比较,并表明我们的测量在使用同时的肠-脑数据集的电生理记录中可能具有更多的统计功效.
    UNASSIGNED: Cross frequency coupling (CFC) between electrophysiological signals in the brain is a long-studied phenomenon and its abnormalities have been observed in conditions such as Parkinson\'s disease and epilepsy. More recently, CFC has been observed in stomach-brain electrophysiologic studies and thus becomes an enticing possible target for diseases involving aberrations of the gut-brain axis. However, current methods of detecting coupling, specifically phase-amplitude coupling (PAC), do not attempt to capture the phase and amplitude statistical relationships.
    UNASSIGNED: In this paper, we first demonstrate a method of modeling these joint statistics with a flexible parametric approach, where we model the conditional distribution of amplitude given phase using a gamma distributed generalized linear model (GLM) with a Fourier basis of regressors. We perform model selection with minimum description length (MDL) principle, demonstrate a method for assessing goodness-of-fit (GOF), and showcase the efficacy of this approach in multiple electroencephalography (EEG) datasets. Secondly, we showcase how we can utilize the mutual information, which operates on the joint distribution, as a canonical measure of coupling, as it is non-zero and non-negative if and only if the phase and amplitude are not statistically independent. In addition, we build off of previous work by Martinez-Cancino et al., and Voytek et al., and show that the information density, evaluated using our method along the given sample path, is a promising measure of time-resolved PAC.
    UNASSIGNED: Using synthetically generated gut-brain coupled signals, we demonstrate that our method outperforms the existing gold-standard methods for detectable low-levels of phase-amplitude coupling through receiver operating characteristic (ROC) curve analysis. To validate our method, we test on invasive EEG recordings by generating comodulograms, and compare our method to the gold standard PAC measure, Modulation Index, demonstrating comparable performance in exploratory analysis. Furthermore, to showcase its use in joint gut-brain electrophysiology data, we generate topoplots of simultaneous high-density EEG and electrgastrography recordings and reproduce seminal work by Richter et al. that demonstrated the existence of gut-brain PAC. Using simulated data, we validate our method for different types of time-varying coupling and then demonstrate its performance to track time-varying PAC in sleep spindle EEG and mismatch negativity (MMN) datasets.
    UNASSIGNED: Our new measure of PAC using Gamma GLMs and mutual information demonstrates a promising new way to compute PAC values using the full joint distribution on amplitude and phase. Our measure outperforms the most common existing measures of PAC, and show promising results in identifying time varying PAC in electrophysiological datasets. In addition, we provide for using our method with multiple comparisons and show that our measure potentially has more statistical power in electrophysiologic recordings using simultaneous gut-brain datasets.
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  • 文章类型: Journal Article
    最近,神经科学已经看到了从地方主义方法到全网络脑功能研究的转变。跨不同空间和时间尺度的神经生理信号提供对神经通信的洞察。然而,在调查全网络的脑动力学而不是局部效应时,会出现其他方法学考虑。具体来说,大量的数据,在更高维度的空间中进行研究,是必要的。这里,我们介绍了FINN(寻找神经生理学网络),一个新颖的工具箱,用于分析神经生理数据,重点是功能和有效的连通性。FINN提供广泛的数据处理方法以及统计和可视化工具,以促进对连接性估计和由此产生的脑动力学指标的检查。Python工具箱及其文档可作为支持信息免费提供。我们在概念和实施层面上根据许多已建立的框架评估了FNN。我们发现FINN比其他工具箱需要更少的处理时间和内存。此外,FINN坚持易于访问和可修改的设计理念,同时提供高效的数据处理实现。由于对网络级神经动力学的研究越来越感兴趣,我们将FINN作为开源软件置于神经科学社区的支配之下。
    Recently, neuroscience has seen a shift from localist approaches to network-wide investigations of brain function. Neurophysiological signals across different spatial and temporal scales provide insight into neural communication. However, additional methodological considerations arise when investigating network-wide brain dynamics rather than local effects. Specifically, larger amounts of data, investigated across a higher dimensional space, are necessary. Here, we present FiNN (Find Neurophysiological Networks), a novel toolbox for the analysis of neurophysiological data with a focus on functional and effective connectivity. FiNN provides a wide range of data processing methods and statistical and visualization tools to facilitate inspection of connectivity estimates and the resulting metrics of brain dynamics. The Python toolbox and its documentation are freely available as Supporting Information. We evaluated FiNN against a number of established frameworks on both a conceptual and an implementation level. We found FiNN to require much less processing time and memory than other toolboxes. In addition, FiNN adheres to a design philosophy of easy access and modifiability, while providing efficient data processing implementations. Since the investigation of network-level neural dynamics is experiencing increasing interest, we place FiNN at the disposal of the neuroscientific community as open-source software.
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  • 文章类型: Journal Article
    尽管人们可以通过用视觉代替听觉信号来识别环境,受试者是否可以将音景感知为视觉或类似视觉的感觉受到质疑。在这项研究中,我们研究了分层过程,以阐明被蒙住眼睛的受试者通过音景刺激对视觉区域的募集机制。对22名健康受试者进行了反复训练,以识别由字母的视觉形状信息转换的音景刺激。采用一种称为动态因果模型(DCM)的有效连接方法来揭示大脑是如何分层组织以识别音景刺激的。视觉心理意象模型自下而上地生成五个感兴趣区域的皮层源信号,跨模态感知,混合模型。分析了视觉心理意象模型中大脑区域之间的光谱耦合。虽然在传递感官信息的自下而上处理中,频率内耦合是显而易见的,在自上而下的处理中,交叉频率耦合是突出的,与信息的期望和解释相对应。蒙住眼睛的受试者大脑中的感觉替代通过结合自下而上和自上而下的处理来获得视觉心理意象。
    Although one can recognize the environment by soundscape substituting vision to auditory signal, whether subjects could perceive the soundscape as visual or visual-like sensation has been questioned. In this study, we investigated hierarchical process to elucidate the recruitment mechanism of visual areas by soundscape stimuli in blindfolded subjects. Twenty-two healthy subjects were repeatedly trained to recognize soundscape stimuli converted by visual shape information of letters. An effective connectivity method called dynamic causal modeling (DCM) was employed to reveal how the brain was hierarchically organized to recognize soundscape stimuli. The visual mental imagery model generated cortical source signals of five regions of interest better than auditory bottom-up, cross-modal perception, and mixed models. Spectral couplings between brain areas in the visual mental imagery model were analyzed. While within-frequency coupling is apparent in bottom-up processing where sensory information is transmitted, cross-frequency coupling is prominent in top-down processing, corresponding to the expectation and interpretation of information. Sensory substitution in the brain of blindfolded subjects derived visual mental imagery by combining bottom-up and top-down processing.
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  • 文章类型: Journal Article
    在阿尔茨海默病(AD)患者中观察到的认知障碍与海马活动改变有关。尽管该机制仍在广泛研究中,已提出神经原纤维缠结和淀粉样斑块是脑活动改变的原因。旨在揭示机制,研究人员已经开发了几种AD的转基因模型。然而,在不同的遗传背景和年龄中发现的海马振荡改变的变异性尚不清楚。
    为了评估与动物发育年龄和蛋白质内含物相关的振荡变化,淀粉样蛋白β(Aβ)负荷,和异常磷酸化的tau(pTau),我们回顾并分析了已发表的峰值功率数据,频率,和θ-γ交叉频率耦合(调制指数值)的量化。
    为了确保搜索尽可能及时,对2000年1月至2023年2月发表的涉及AD转基因小鼠模型体内海马局部场电位记录的所有研究进行了系统综述,并对其进行了摘录.
    Aβ的存在与电生理改变有关,电生理改变主要反映在功率增加上,频率降低,和较低的调制指数值。同时,pTau积累与电生理改变有关,电生理改变主要反映在功率下降,频率降低,调制指数值没有显著变化。
    在这项研究中,我们表明,从病理的前驱阶段到晚期,电生理参数都发生了变化。因此,我们发现Aβ沉积与大脑网络过度兴奋有关,而pTau沉积主要导致转基因模型中脑网络兴奋性低下。
    UNASSIGNED: Cognitive deficits observed in Alzheimer\'s disease (AD) patients have been correlated with altered hippocampal activity. Although the mechanism remains under extensive study, neurofibrillary tangles and amyloid plaques have been proposed as responsible for brain activity alterations. Aiming to unveil the mechanism, researchers have developed several transgenic models of AD. Nevertheless, the variability in hippocampal oscillatory alterations found in different genetic backgrounds and ages remains unclear.
    UNASSIGNED: To assess the oscillatory alterations in relation to animal developmental age and protein inclusion, amyloid-β (Aβ) load, and abnormally phosphorylated tau (pTau), we reviewed and analyzed the published data on peak power, frequency, and quantification of theta-gamma cross-frequency coupling (modulation index values).
    UNASSIGNED: To ensure that the search was as current as possible, a systematic review was conducted to locate and abstract all studies published from January 2000 to February 2023 that involved in vivo hippocampal local field potential recording in transgenic mouse models of AD.
    UNASSIGNED: The presence of Aβ was associated with electrophysiological alterations that are mainly reflected in power increases, frequency decreases, and lower modulation index values. Concomitantly, pTau accumulation was associated with electrophysiological alterations that are mainly reflected in power decreases, frequency decreases, and no significant alterations in modulation index values.
    UNASSIGNED: In this study, we showed that electrophysiological parameters are altered from prodromal stages to the late stages of pathology. Thus, we found that Aβ deposition is associated with brain network hyperexcitability, whereas pTau deposition mainly leads to brain network hypoexcitability in transgenic models.
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  • 文章类型: Preprint
    高频振荡是癫痫组织的重要新型生物标志物。不同时间尺度上的振荡相互作用被揭示为交叉频率耦合(CFC),代表了大脑节律功能组织中的高阶结构。深度学习神经网络等新的人工智能方法可以为脑电图的自动分析提供强大的工具。在这里,我们介绍了一种堆叠稀疏自动编码器(SSAE),该编码器经过训练,可以根据头皮EEG中的交叉频率模式来识别缺勤癫痫发作活动。我们使用了天普大学医院数据库中的脑电图记录。对12例患者的癫痫发作(n=94)以及背景活动片段进行了分析。一半的记录是随机选择的,用于网络训练,另一半用于测试。使用EEGLAB工具箱在2-120Hz的所有频率之间成对地计算功率-功率耦合。所得CFC矩阵用作自动编码器的训练或测试输入。受训网络能够识别背景和癫痫发作段(训练中未使用),灵敏度为96.3%,特异性为99.8%,总体准确率为98.5%。我们的结果提供了证据,表明SSAE神经网络可用于自动检测头皮脑电图中的失神发作。
    High frequency oscillations are important novel biomarkers of epileptogenic tissue. The interaction of oscillations across different time scales is revealed as cross-frequency coupling (CFC) representing a high-order structure in the functional organization of brain rhythms. New artificial intelligence methods such as deep learning neural networks can provide powerful tools for automated analysis of EEG. Here we present a Stacked Sparse Autoencoder (SSAE) trained to recognize absence seizure activity based on the cross-frequency patterns within scalp EEG. We used EEG records from the Temple University Hospital database. Absence seizures (n = 94) from 12 patients were taken into analysis along with segments of background activity. Half of the records were selected randomly for network training and the second half were used for testing. Power-to-power coupling was calculated between all frequencies 2-120 Hz pairwise using the EEGLAB toolbox. The resulting CFC matrices were used as training or testing inputs to the autoencoder. The trained network was able to recognize background and seizure segments (not used in training) with a sensitivity of 96.3%, specificity of 99.8% and overall accuracy of 98.5%. Our results provide evidence that the SSAE neural networks can be used for automated detection of absence seizures within scalp EEG.
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  • 文章类型: Journal Article
    作为基底前脑的脑胆碱能系统和Papez回路的重要组成部分,Meynert基底核(NBM)的功能障碍与各种神经退行性疾病有关。然而,没有毒品,包括现有的胆碱酯酶抑制剂,已经被证明可以逆转这种功能障碍。由于神经调节技术的进步,研究人员正在探索使用脑深部电刺激(DBS)治疗针对NBM(NBM-DBS)治疗精神和神经系统疾病以及相关机制。在这里,我们提供了有关认知相关神经网络振荡以及NBM与其他认知结构和电路之间复杂的解剖和投影关系的最新研究进展。此外,我们回顾了以前对NBM病变的动物研究,NBM-DBS模型,和临床案例研究,总结NBM在神经调节中的重要功能。除了阐明NBM神经网络的机制外,未来的研究应该集中在NBM中其他类型的神经元上,尽管胆碱能神经元仍然是DBS细胞类型特异性激活的关键靶标。
    As a crucial component of the cerebral cholinergic system and the Papez circuit in the basal forebrain, dysfunction of the nucleus basalis of Meynert (NBM) is associated with various neurodegenerative disorders. However, no drugs, including existing cholinesterase inhibitors, have been shown to reverse this dysfunction. Due to advancements in neuromodulation technology, researchers are exploring the use of deep brain stimulation (DBS) therapy targeting the NBM (NBM-DBS) to treat mental and neurological disorders as well as the related mechanisms. Herein, we provided an update on the research progress on cognition-related neural network oscillations and complex anatomical and projective relationships between the NBM and other cognitive structures and circuits. Furthermore, we reviewed previous animal studies of NBM lesions, NBM-DBS models, and clinical case studies to summarize the important functions of the NBM in neuromodulation. In addition to elucidating the mechanism of the NBM neural network, future research should focus on to other types of neurons in the NBM, despite the fact that cholinergic neurons are still the key target for cell type-specific activation by DBS.
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  • 文章类型: Journal Article
    过去几十年来,因果技术在神经时间序列中的应用已经广泛增加,包括广泛而多样的方法家族,专注于脑电图(EEG)分析。除了在定义的频带中推断的连通性,对交叉频率相互作用的分析越来越感兴趣,特别是相位和振幅耦合和方向性。一些研究表明,从高频到低频信号分量的耦合方向性结果矛盾,尽管通常认为低频相位调制了高频振幅。我们比较了两种广泛使用的方法来估计交叉频率耦合的方向性:条件互信息(CMI)和相位斜率指数(PSI)。后者,应用于从动物颅内记录推断交叉频率相位-振幅方向性,与CMI比较时给出相反的结果。这两个指标都在单向耦合Rössler系统的数值模拟示例中进行了测试,这有助于找到矛盾结果的解释:PSI正确估计了领先/滞后关系,然而,在非线性系统中耦合的方向性意义上,一般不等同于因果关系,通过使用CMI和替代数据测试来正确推断。
    Applications of causal techniques to neural time series have increased extensively over last decades, including a wide and diverse family of methods focusing on electroencephalogram (EEG) analysis. Besides connectivity inferred in defined frequency bands, there is a growing interest in the analysis of cross-frequency interactions, in particular phase and amplitude coupling and directionality. Some studies show contradicting results of coupling directionality from high frequency to low frequency signal components, in spite of generally considered modulation of a high-frequency amplitude by a low-frequency phase. We have compared two widely used methods to estimate the directionality in cross frequency coupling: conditional mutual information (CMI) and phase slope index (PSI). The latter, applied to infer cross-frequency phase-amplitude directionality from animal intracranial recordings, gives opposite results when comparing to CMI. Both metrics were tested in a numerically simulated example of unidirectionally coupled Rössler systems, which helped to find the explanation of the contradictory results: PSI correctly estimates the lead/lag relationship which, however, is not generally equivalent to causality in the sense of directionality of coupling in nonlinear systems, correctly inferred by using CMI with surrogate data testing.
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