Fig 1.
Traditional MSE estimation procedure.
(A) Multi-scale entropy is an extension of sample entropy, an information-theoretic metric intended to describe the temporal irregularity of time series data. To estimate entropy for different time scales, the original signal is traditionally ‘coarse-grained’ using low-pass filters, followed by the calculation of the sample entropy. (B) Sample entropy estimation procedure. Sample entropy measures the conditional probability that two amplitude patterns of sequence length m (here, 2) remain similar (or matching) when the next sample m + 1 is included in the sequence. Hence, sample entropy increases with temporal irregularity, i.e., with the number of m-length patterns that do not remain similar at length m+1 (non-matches). To discretize temporal patterns from continuous amplitudes, similarity bounds (defined as a proportion r, here .5, of the signal’s standard deviation [SD]) define amplitude ranges around each sample in a given template sequence, within which matching samples are identified in the rest of the time series. These are indicated by horizontal grey and green bars around the first three template samples. This procedure is applied to each template sequence in time, and the pattern counts are summed to estimate the signal’s entropy. The exemplary time series is a selected empirical EEG signal that was 40-Hz high-pass filtered with a 6th order Butterworth filter.
Fig 2.
Issue 1: Global similarity bounds systematically confound the entropy of coarse-scale signals with removed spectral power.
(A, B) Similarity bounds constrain sample entropy as shown schematically for entropy estimation using narrower (A) and wider (B) similarity bounds. For clarity, only a subset of pattern matches (green ticks) and mismatches (red cross) are indicated for a sequence length m = 1(cf. Fig 1B). Wider, more liberal similarity bounds indicate more pattern matches than narrow, conservative bounds, thereby decreasing entropy. S2 Fig shows the empirical link between liberal similarity bounds and sample entropy estimates. (C-E) Divergence between global similarity bounds and scale-wise signal SD biases coarse-scale entropy. (C) Coarse-graining (see Fig 1A) progressively reduces variance from the original broadband signal (as shown in panel E). (D) At original sampling rates (i.e., time scale 1; marked red in panels DE and F), neural signal variance is usually composed of broadband 1/f content and narrowband rhythmic peaks. Note that the x-axis plots decreasing frequencies to align with the traditional MSE low-pass filter direction. Towards coarser scales (e.g., scale 30; marked blue in CD and E), signal variance progressively decreases, as the signal becomes more specific to low frequencies. (E) Due to the systematic and cumulative reduction of variance in scale-wise signals, global similarity bounds become liberally biased (‘broad’). Critically, systematic differences in the magnitude of this bias (e.g., due to different spectral slopes) introduce systematic entropy differences at coarser scales.
Fig 3.
Issue 2: Traditional scale derivation leads to diffuse time-scale reflections of spectral power.
(A) Exemplary sample entropy estimation in the same empirical EEG signal shown in Fig 1B, but without application of a high-pass filter, thus including dominant slow dynamics. See Fig 1B for a legend of the Figure elements. In brief, green elements indicate pattern matches at m+1, whereas red elements indicate pattern mismatches at m+1. In the presence of large low-frequency fluctuations, sample entropy at fine scales (here scale 1) may to a large extent characterize the temporal regularity of slow dynamics. Note that this is not a case of biased similarity bounds, but a desired adjustment to the large amplitude of slow fluctuations. The inset shows an extended segment (800 ms) of the same signal, allowing for an assessment of the slower signal dynamics. The red box indicates the 100 ms signal shown in the main plot. (B) A scale-wise filter implementation controls the scale-wise spectral content, as schematically shown here for the filter-dependent representation of spectral content at a time scale of approximately 10 Hz (for a note on the x-axis labeling, see methods: Calculation of multi-scale sample entropy). Traditionally, low-pass filters are used to derive coarser scales, which introduces a sensitivity to slower fluctuations. However, other filter implementations can be used to e.g., investigate the pattern irregularity of fast signal variations. No matter whether low or high pass filters are used, the spectral content influencing entropy estimates is by definition not specific to any particular time scale; band-pass filters provide one viable solution permitting such specificity.
Fig 4.
Rhythmic power manifests at different time scales depending on filter choice and similarity bound.
Simulations indicate at which time scales the addition of varying magnitudes of stereotypic narrowband 10 Hz rhythms (red-to-white line color gradient) modulate entropy compared to the baseline 1/f signal (black line). Simulations indicate that increases in rhythmicity strongly reduce entropy estimates alongside increases in the similarity bound. The affected scales vary as a function of global vs. scale-dependent similarity bounds and the spectral filtering used to derive coarser time scales. Crucially, in ‘Original’ implementations, added narrowband rhythmicity decreased entropy with low scale-specificity, in line with global increases in the similarity bound (A). In contrast, the use of scale-varying thresholds (B) and dedicated filtering (C-E) increased specificity regarding the time scales at which rhythmicity was reflected. Note that timescales are presented in Hz to facilitate the visual assessment of rhythmic modulation. For all versions except high pass, the scale represents the upper Nyquist bound of the embedding dimension. For the high pass variant, the scale represents the high pass frequency (see methods). Time scales are log-scaled. Spectral attenuation properties of the Butterworth filters are shown in S4 Fig.
Fig 5.
Influence of rhythmic frequency on MSE estimates and similarity bounds across different MSE variants.
Simulations of different frequencies indicate a linear frequency-to-scale mapping of simulated sinusoids. Broken vertical lines indicate the simulated frequency. The original MSE variant (A) shows increased entropy at time scales finer than the simulated frequency in combination with a global entropy decrease. Low-, high- and band-pass variants exhibit the properties observed in the alpha case, with a reduction above (B, C), below (D) or at the simulated frequency (E). Time scales are log-scaled.
Fig 6.
Timescale-dependent age differences in spectral power and entropy during eyes open rest.
(A) MSE (A1) and power (A2) spectra for the two age groups. Error bars show standard errors of the mean. Note that in contrast to standard presentations of power, the log-scaled x-axis in A2 is sorted by decreasing frequency to enable a better visual comparison with entropy time scales (see also Fig 2D). Similarly, the x-axis in A1 has been log-scaled to allow easier visual comparison with log-scaled values in A2 and emphasize fine-scale differences (cf. Fig 7A1). Inset labels refer to the approximate time scales across which topographies are plotted in B & C. T-values of power age contrast are shown in S5 Fig. (B, C) Topographies of age differences indicate mirrored age differences in fast entropy and low frequency power, as well as coarse entropy and high frequency power. Significant differences are indicated by yellow dots. P-values correspond to the two/sided significance test of the cluster-level statistic. (D1) Spectral slopes across age groups. (D2) Age differences in spectral slopes.
Fig 7.
Multiscale entropy age differences depend on the specifics of the estimation method.
Grand average traces of entropy (1st row) and similarity bounds (3rd row) alongside t-maps from statistical contrasts of age group differences (2nd + 4th row: younger minus older adults for entropy and bounds, respectively), shown by channel on the y-axis. Age differences were assessed by means of cluster-based permutation tests and are indicated via opacity. Original MSE (A) replicated reported scale-dependent age differences, with older adults exhibiting higher entropy at fine scales and lower entropy at coarse scales, compared with younger adults. The coarse-scale difference was exclusively observed when using global similarity bounds, whereas the fine-scale age difference was indicated with all low-pass versions (A, B, C), but not when signals were constrained to high-frequency or narrow-band ranges (D, E). In contrast, narrowband MSE indicated inverted age differences within the alpha and beta band (E).
Fig 8.
Divergence of scale-specific signal variance from global similarity bounds accounts for age differences in coarse-scale entropy.
(A, B) A global similarity bound does not reflect the spectral shape, thus leading to disproportionally liberal criteria at coarse scales following the successive removal of high-frequency variance (see Fig 2C–2E for the schematic example). Scale-dependent variance is more quickly reduced in older compared to younger adults (A) due to the removal of more prevalent high-frequency variance in the older group (B). This leads to a differential bias across age groups, as reflected in the differentially mismatched distance between global and scale-dependent similarity bounds at coarser scales. (C) Removing this bias by adjusting the similarity bounds to the scale-dependent signal is associated with increases in coarse-scale entropy. This shift is more pronounced in older adults following the removal of a more prevalent bias. (D) With global similarity bounds, coarse-scale entropy strongly reflects high frequency power due to the proportionally more liberal similarity threshold associated. Low frequency power < 8 Hz was not consistently related to coarse-scale entropy (log10-power as in D; YA: r = .12; p = .419; OA: r = .36, p = .009). Data in A and B are global averages, data in C and D are averages from frontal ‘Original’ effect cluster (see Fig 7A) at entropy time scales below 8 Hz.
Fig 9.
The presence of low- and high-frequency content renders fine entropy slopes sensitive to PSD slopes.
A) Sample entropy at fine time scales represents the slope of power spectral density across age groups. The 7–13 Hz range was excluded prior to the PSD slope fit to exclude the rhythmic alpha peak (see Fig 8B). (B) The presence of both slow and fast dynamics is required for positive associations with PSD slopes to emerge. The direction and magnitude of correlations of scale-wise entropy with PSD slopes depends on the choice of global vs. rescaled similarity bounds, as well as the choice of filtering. Original entropy inverts from a positive correlation with PSD slope at fine scales to a negative association at coarse scales. Rescaling of the similarity bound abolishes the negative correlation of coarse-scale entropy with PSD slopes. S6 Fig presents scatter plots of these relationships. The x-axis indicates the upper frequency bounds for the low-pass version.
Fig 10.
Narrowband MSE reflects age differences in alpha- and beta-specific event (ir)regularity.
(A, B) Narrowband MSE indicates age differences in the pattern complexity at alpha (A) and beta (B) frequencies. (C, D) Alpha, but not beta power consistently correlates negatively with individual narrowband entropy within clusters of age differences. (E, F) Similarly, alpha but not beta similarity bounds show an inverted age effect with similar topography. (G, H) Single-trial rhythm detection highlights a more transient appearance of beta compared with alpha events. Data are collapsed across age groups. (I, J) The rate of stereotypical single-trial alpha and beta events is anticorrelated with individual narrowband entropy. (K, L) The rate of spectral events exhibits age differences that mirror those observed for entropy. Note that the same color range, plotted in the lower row, was plotted for all topographies.
Fig 11.
Nonstationary alpha events transiently reduce broadband irregularity.
(A) Testing for transient broadband changes during alpha events requires control for narrowband circularity. We simulated 250 ms signals consisting of varying slope coefficients (plotted on the x-axis) in the presence or absence of alpha rhythms. Bars indicate first-scale entropy estimates (i.e., sample entropy; SampEn) for these signals, as well as bandstop-filtered versions. Left: Valid slope shallowing in the presence of alpha events was indicated both when alpha was included in estimates (red background), as well as when band-stop filters removed the influence of alpha regularity (green background). Right: Crucially, when no bandstop filters were applied, sample entropy decreased also in the absence of slope variations due to the added alpha regularity (red background). This effectively represents narrowband circularity in the analysis. In contrast, bandstop filters removed the influence of alpha regularity and permitted estimation of valid reductions in broadband irregularity (green background). (B, C, D) Empirical analysis of transient entropy decreases during alpha events. (B Alpha events were selected across channels with high amounts of detected events (black dots). Lower: Broadband entropy was calculated for 250 ms segments preceding and following the on- and offset of alpha events. (C1) During eyes open rest, nonstationary alpha events of high strength transiently reduce broadband irregularity, also after accounting for alpha circularity. Raincloud plots (RCPs) indicate the intervals schematically plotted in the bottom panel of B. For visualization, RCPs display estimates that are centered within-subject (condition-wise data minus individual across-condition average plus global average); statistics were calculated on uncentered estimates. ***1: d’ = -1.91; p ~ 0. ***2: d’ = -1.61; p ~ 0. ***3: d’ = -0.63; p = 1e-8. ***4 d’ = -0.54; p = 6e-7 [d' = ()/STD(Xalpha-Xpre/post)]. (C2) Slope fits indicate a shallowing of slopes during alpha events. The inset bar plot indicates mean slopes estimates with within-subject standard errors. (D1) In contrast, irregularity decreases were indicated for low-amplitude alpha events only when circularity was not accounted for, but not after alpha was removed. This indicates that bandstop filtering successfully avoids circularity in empirical use cases. ***1: d’ = -0.52; p = 1e-6. ***2: d’ = -0.75; p = 3e-11. n.s.3: d’ = -0.05; p = 0.63. n.s.4 d’ = -0.04; p = 0.67. (D2) No significant slope changes were observed during low-amplitude alpha events. Note that black dotted line is covered here. Error bars reflect within-subject standard errors.
Fig 12.
Summary of the identified time-scale mismatches and recommendations for future studies.
(A) We highlight two scale-dependent mismatches that run counter to the intuition that entropy at fine scales primarily refers to fast dynamics, and vice-versa: (1) Coarse-scale entropy is biased towards reflecting high-frequency content when signals of decreasing variance are compared to a global, and increasingly inadequate, similarity bound. (2) Fine-scale entropy characterizes scale-free 1/f slopes when broadband signals include slow frequency content. Dashed colored arrows indicate the mismatched relations observed in the current study. (B) Beyond time-scale mismatches, brain signal entropy and variance/power can often be collinear, in part due to their shared description of linear signal characteristics, such as rhythmicity. To identify complementary and unique relations of pattern complexity compared to more established measures of variance, explicit statistical control is required for the latter. (C) We propose multiple strategies to safeguard future applications against the highlighted issues.