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FIGURE A1. More examples on harmonic structures. Top -signal and bottom -instantaneous frequency (IF) for (A) the triangular wave, (B) the saw-tooth wave, (C) the square wave (all formed from first 1000 harmonics). We see IF spikes near sharp edges but is otherwise well-defined. In (D), we take similar signals as in Fig. 1 but with HF 4x the amplitude and 3x the frequency of LF. It is clear the signal properties (amplitude, frequency) are largely determined by HF and not the LF base. We thus do not classify this as a harmonic structure.
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The analysis of harmonics and non-sinusoidal waveform shape in time-series data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an integer frequency ratio, constant phase, and a wel...
Citations
... Although PAPTO does not address harmonics, it detects oscillatory events in the presence of 1/f background activity. Lastly, empirical mode decomposition (EMD) approaches have been studied in the context of neural harmonics and non-sinusoidal activity (Quinn 2021, Fabus 2022. EMD has an intrinsic relationship with extrema finding, in contrast with the present technique. ...
Detecting temporal and spectral features of neural oscillations is essential to understanding dynamic brain function. Traditionally, the presence and frequency of neural oscillations are determined by identifying peaks over 1/f noise within the power spectrum. However, this approach solely operates within the frequency domain and thus cannot adequately distinguish between the fundamental frequency of a non-sinusoidal oscillation and its harmonics. Non-sinusoidal signals generate harmonics, significantly increasing the false-positive detection rate — a confounding factor in the analysis of neural oscillations. To overcome these limitations, we define the fundamental criteria that characterize a neural oscillation and introduce the Cyclic Homogeneous Oscillation (CHO) detection method that implements these criteria based on an auto-correlation approach that determines the oscillation’s periodicity and fundamental frequency. We evaluated CHO by verifying its performance on simulated sinusoidal and non-sinusoidal oscillatory bursts convolved with 1/f noise. Our results demonstrate that CHO outperforms conventional techniques in accurately detecting oscillations. Specifically, we determined the sensitivity and specificity of CHO as a function of signal-to-noise ratio (SNR). We further assessed CHO by testing it on electrocorticographic (ECoG, 8 subjects) and electroencephalographic (EEG, 7 subjects) signals recorded during the pre-stimulus period of an auditory reaction time task and on electrocorticographic signals (6 SEEG subjects and 6 ECoG subjects) collected during resting state. In the reaction time task, the CHO method detected auditory alpha and pre-motor beta oscillations in ECoG signals and occipital alpha and pre-motor beta oscillations in EEG signals. Moreover, CHO determined the fundamental frequency of hippocampal oscillations in the human hippocampus during the resting state (6 SEEG subjects). In summary, CHO demonstrates high precision and specificity in detecting neural oscillations in time and frequency domains. The method’s specificity enables the detailed study of non-sinusoidal characteristics of oscillations, such as the degree of asymmetry and waveform of an oscillation. Furthermore, CHO can be applied to identify how neural oscillations govern interactions throughout the brain and to determine oscillatory biomarkers that index abnormal brain function.
... Note that the existence of power dependencies by itself does not constitute harmonics all by itself, but strong narrow-band peaks warrant further careful checks. Recently, identification of harmonics via the instantaneous frequency has been put forward (Fabus et al., 2022). The bispectrum and bicoherence as a measure derived from that enables to identify the joint distribution of power in a frequency-resolved manner, with harmonics appearing as localized peaks (Kovach et al., 2018;Shahbazi Avarvand et al., 2018). ...
Beta-band activity in the human cortex as recorded with non-invasive electrophysiology is of diverse origin. In addition to genuine beta-rhythms, there are numerous non-sinusoidal alpha-band rhythms present in the human brain, which will result in harmonic beta-band peaks. This type of activity has different temporal and response dynamics than genuine beta-rhythms. Here it is argued that in the analysis of higher frequency rhythms the relationship to lower frequency rhythms needs to be clarified. Only in that way we can arrive at strong, methodologically valid interpretations of potential functional roles and generative mechanisms of neural oscillations.
Objective: The study aims to investigate the relationship between amplitude modulation (AM) of EEG and anesthesia depth during general anesthesia. Methods: In this study, Holo-Hilbert spectrum analysis (HHSA) was used to decompose the multichannel EEG signals of 15 patients to obtain the spatial distribution of AM in the brain. Subsequently, HHSA was applied to the prefrontal EEG (Fp1) obtained during general anesthesia surgery in 15 and 34 patients, and the α-θ and α-δ regions of feature (ROFs) were defined in Holo-Hilbert spectrum (HHS) and three features were derived to quantify AM in ROFs. Results: During anesthetized phase, an anteriorization of the spatial distribution of AMs of α-carrier in brain was observed, as well as AMs of α-θ and α-θ in the EEG of Fp1. The total power ( P
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), mean carrier frequency ( MF
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) and mean amplitude frequency ( MF
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) of AMs changed during different anesthesia states. Conclusion: HHSA can effectively analyze the cross-frequency coupling of EEG during anesthesia and the AM features may be applied to anesthesia monitoring. Significance: The study provides a new perspective for the characterization of brain states during general anesthesia, which is of great significance for exploring new features of anesthesia monitoring.
