Automatic decomposition of electrophysiological data into distinct nonsinusoidal oscillatory modes

Figures

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  Figure 1.. Limitations of empirical mode decomposition (EMD). A: standard EMD sifting applied on a pure 4 Hz iterated sine function. With no noise, EMD can accurately identify an intrinsic mode function (IMF) that represents the nonsinusoidal signal. B: in the presence of white noise and a 30-Hz burst (arrow), standard EMD shows heavy mode mixing. C: EMD with an appropriate dyadic mask sift will recover most of the iterated sine (IMF-3) and the intermittent burst (IMF-2) signals. D: masked EMD and ensemble EMD can better reconstruct nonsinusoidal wave shape in signal with low noise unlike standard EMD. The figure shows the phase-aligned instantaneous frequency calculated from 100 runs of B and C. Means ± standard error (shaded) shown.

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  Figure 2.. Iterated masking empirical mode decomposition (EMD) (itEMD) on simulated data. A: example endpoint of itEMD, i.e. intrinsic mode functions (IMFs) from last of 15 iterations on the same signal as in Fig. 1B. B: itEMD convergence to equilibrium starting with a mask drawn randomly from a uniform distribution of 1–128 Hz. Thin gray lines are all 100 individual runs, colored lines are median trajectories with line thickness scaled to maximum value of that IMF. itEMD converges rapidly, adapts mask frequencies to the signal content, and correctly finds both the nonsinusoidal 4 Hz oscillation and intermittent 30 Hz burst with no prior knowledge about the frequencies contained in the signal.

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  Figure 3.. Influence of noise on empirical mode decomposition (EMD) performance on simulated data. A: Pearson correlation coefficient between reconstructed and ground truth instantaneous frequency against increasing white noise amplitude. B: example 5 s of iterated masking EMD (itEMD) sift results for σnoise = 0.5, frequency distortion (FD) = 68%. Iterated sine function is captured by IMF-4. C: pseudo-mode splitting index (PMSI) against white noise amplitude, with higher PMSI values indicating higher mode mixing. Means ± standard error across n = 100 noise realizations shown. Black line indicates regions where itEMD performs significantly better than the best of the other techniques with P < 0.01 (multiple comparisons Bonferroni corrected). Red shaded region shows noise levels where itEMD reached maximum number of iterations in >20% of noise realizations. The novel itEMD performs significantly better for highly noisy data in the region between σnoise ≈ 1 and σnoise ≈ 2 with reduced mode mixing and accurate waveshape reconstruction. D: example dyadic mask sift results for σnoise = 0.5, FD = 68%. Intrinsic mode function (IMF)-4 shows significantly more mode splitting than itEMD results. RMS,…

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  Figure 4.. Influence of frequency distortion on empirical mode decomposition (EMD) performance in simulated data. A: Pearson correlation coefficient between reconstructed and ground truth instantaneous frequency against increasing frequency distortion. B: example 5 s of itEMD sift results for σnoise = 1, frequency distortion (FD) = 80%. C: pseudo-mode splitting index (PMSI) against frequency distortion, with higher PMSI values indicating higher mode mixing. Means ± standard error across n = 100 noise realizations (shaded) shown. Black line indicates regions where itEMD performs significantly better than the best of the other techniques with P < 0.01 (multiple comparisons Bonferroni corrected). The novel iterated masking EMD (itEMD) performs significantly better for highly nonsinusoidal data in the region FD > 50% with reduced mode mixing and accurate waveshape reconstruction. D: example 5 s of dyadic mask sift results for σnoise = 1, FD = 80%. Intrinsic mode function (IMF)-4 shows significantly more mode mixing with existing methods than with our novel itEMD.

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  Figure 5.. Example 2 s of sifting results for 4th-order iterated sine with white noise σnoise = 0.1 in simulated data. A: intrinsic mode functions (IMFs) for the novel iterated masking empirical mode decomposition (itEMD). Iterated sine is in IMF-3 with very little mode mixing. B: IMFs for dyadic mask sifting. Signal is split between IMF-3 and IMF-5. C: IMFs for ensemble sifting. Iterated sine is mostly in IMF-4, but mode mixing is present. D, top: average reconstructed waveform, bottom: reconstructed phase-aligned instantaneous frequency (IF); means (line) ± standard error across cycles (shaded) shown. Dyadic mask sift waveform (orange) fails to reconstruct nonsinusoidality. Ensemble sift recovers more shape detail but suffers from high mode mixing. itEMD is able to reconstruct more of the waveform shape than either existing method while lowering mode mixing. EEMD, ensemble empirical mode decomposition.

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  Figure 6.. Influence of signal sparsity on empirical mode decomposition (EMD) performance on simulated data. A: Pearson correlation coefficient between reconstructed and ground truth instantaneous frequency against increasing burst length. B: example iterated masking EMD (itEMD) sift results for σnoise = 1, FD = 68%, 10 cycles. C: pseudo-mode splitting index (PMSI) against burst length. Means ± standard error across n = 100 noise realizations (shaded) shown. Black line indicates regions where itEMD performs significantly better than the best existing technique with P < 0.01 (multiple comparisons Bonferroni corrected). The novel itEMD performs significantly better for a wide range of burst durations. D: example dyadic mask sift results for σnoise = 1, FD = 60%, 10 cycles in burst. Iterated sine function is captured by intrinsic mode function (IMF)-4 and IMF-5 due to mode mixing. FD, frequency distortion.

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  Figure 7.. Rat hippocampal local field potential (LFP) results. A: power spectrum of the full recording showing a θ peak harmonic. B: phase-aligned instantaneous frequency of θ cycles (means ± standard error across all cycles shown). Existing methods including dyadic mask sift and ensemble sift fail to capture high nonsinusoidality of θ oscillations unlike iterated masking empirical mode decomposition (itEMD). C: violin plots of the pseudo-mode splitting index (PMSI, a measure of mode mixing) across n = 20 segments of the 1,000 s recording. Iterated masking had significantly lower PMSI than both dyadic mask sift and ensemble sift (P < 10−6, Bonferroni-corrected across methods). D, top: example itEMD sift results from 2 s of the LFP recording, bottom: Hilbert–Huang Transform (HHT) for the same data. θ Oscillations are well-captured by intrinsic mode function (IMF)-4 with minimal mode mixing. E and F: same as D but for the dyadic mask sift and ensemble EMD (EEMD). Significant mode mixing is present. G: wavelet transform of the same data as in D–F. Similar dynamics to the HHT in D are present but with lower resolution. Compared with E and F, we see the artifacts in poor sifts (red arrows). H…

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  Figure 8.. Human magnetoencephalography (MEG) occipital α results. A: group-level violin plots of the pseudo-mode splitting index (PMSI, a measure of mode mixing). Iterated masking had significantly lower PMSI than both dyadic mask sift (P = 6e−5), and ensemble sift (P = 0.0062, both Bonferroni-corrected). B: group-level phase-aligned instantaneous frequency (means ± standard error shaded). Both iterated masking empirical mode decomposition (itEMD) and ensemble empirical mode decomposition (EEMD) detect the 10 Hz occipital α oscillation with no significant nonsinusoidality. Masked sift fails to capture the oscillation well due to mode mixing. C: one-dimensional (1-D) Hilbert–Huang Transform (HHT) for intrinsic mode functions (IMFs) from an example subject. D: mean subject phase-aligned instantaneous frequency against peak α frequency from the Fourier power spectral density. Iterated masking linearly reproduces between-subject variability in α frequency (P = 0.00739, F test against constant null hypothesis). E: example 5 s of raw itEMD sift results. α Oscillations are in IMF-3 with sharp features in IMF-2 and minimal mode mixing. **P < 0.01, ***P < 0.001 (Bonferroni-corrected).