Standard multiscale entropy reflects spectral power at mismatched temporal scales: What’s signal irregularity got to do with it?

The (ir)regularity of neural time series patterns as assessed via Multiscale Sample Entropy (MSE; e.g., Costa et al., 2002) has been proposed as a complementary measure to signal variance, but the con- and divergence between these measures often remains unclear in applications. Importantly, the estimation of sample entropy is referenced to the magnitude of fluctuations, leading to a trade-off between variance and entropy that questions unique entropy modulations. This problem deepens in multi-scale implementations that aim to characterize signal irregularity at distinct timescales. Here, the normalization parameter is traditionally estimated in a scale-invariant manner that is dominated by slow fluctuations. These issues question the validity of the assumption that entropy estimated at finer/coarser time scales reflects signal irregularity at those same scales. While accurate scale-wise mapping is critical for valid inference regarding signal entropy, systematic analyses have been largely absent to date. Here, we first simulate the relations between spectral power (i.e., frequency-specific signal variance) and MSE, highlighting a diffuse reflection of rhythms in entropy time scales. Second, we replicate known cross-sectional age differences in EEG data, while highlighting how timescale-specific results depend on the spectral content of the analyzed signal. In particular, we note that the presence of both low- and high-frequency dynamics leads to the reflection of power spectral density slopes in finer time scales. This association co-occurs with previously reported age differences in both measures, suggesting a common, power-based origin. Furthermore, we highlight that age differences in high frequency power can account for observed entropy differences at coarser scales via the traditional normalization procedure. By systematically assessing the impact of spectral signal content and normalization choice, our findings highlight fundamental biases in traditional MSE implementations. We make multiple recommendations for future work to validly interpret estimates of signal irregularity at time scales of interest. Highlights Multiscale sample entropy (MSE) links to spectral power via an internal similarity criterion. Counterintuitively, traditional MSE implementations lead to slow-frequency reflections in fine-scale entropy, and high-frequency biases on coarse-scale entropy. Fine-scale entropy reflects power spectral density slopes, a multi-scale property. Narrowband sample entropy indexes (non-stationary) rhythm (ir)regularity at matching time scales.

MSE, highlighting a diffuse reflection of rhythms in entropy time scales. Second, we replicate known 23 cross-sectional age differences in EEG data, while highlighting how timescale-specific results depend 24 on the spectral content of the analyzed signal. In particular, we note that the presence of both low-and 25 high-frequency dynamics leads to the reflection of power spectral density slopes in finer time scales.

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This association co-occurs with previously reported age differences in both measures, suggesting a 27 common, power-based origin. Furthermore, we highlight that age differences in high frequency power 28 can account for observed entropy differences at coarser scales via the traditional normalization 29 procedure. By systematically assessing the impact of spectral signal content and normalization choice, Neural times series exhibit a wealth of dynamic patterns that may be tightly linked to neural 47 computations. While some of these patterns consist of stereotypical deflections (e.g., periodic 48 neural rhythms; Buzsaki & Draguhn, 2004; X. J. Wang, 2010), others have a more complex 49 appearance that may still be equally relevant for characterizing neural function (S. R. Cole & 50 Voytek, 2017; Diaz, Bassi, Coolen, Vivaldi, & Letelier, 2018). Multiscale entropy (MSE) 51 (Costa, Goldberger, & Peng, 2002, 2005 has been proposed as an information-theoretic metric 52 that estimates the temporal irregularity in a signal (in theory providing information above and 53 beyond traditional spectral metrics), while accommodating that neural dynamics occur across 54 multiple spatiotemporal scales. In tandem, dynamic perspectives on brain function in the 55 framework of nonlinear dynamics and complex systems have gained traction (Breakspear,56 2017; Stam, 2005;Vakorin & McIntosh, 2012), suggesting that optimal computations in the 57 brain may be characterized by metastable states that afford flexible movement between distinct 58 attractor states. Following this conceptual framework, MSE has been increasingly applied to 59 characterize the apparent "irregularity" (or non-linearity) of neural dynamics of different brain 60 states, across the lifespan and in relation to health and disease ( Sample entropy is an information theoretic metric that indexes the pattern irregularity (or 82 "complexity") of time series as the conditional probability that two sequences remain similar 83 when another sample is included in the sequence (for a visual example see Figure 1A). Hence, 84 sample entropy compares the relative rate of similar to dissimilar time domain patterns. 85 Whereas signals with a similar/repetitive structure (like rhythmic fluctuations) are assigned low 86 entropy, less predictable/dissimilar (or random) signals are characterized as having higher 87 entropy. We presume that a necessary condition for valid non-linear interpretations of sample 88 entropy is that "the degree of irregularity of a complex signal [ is traditionally assessed relative to the standard deviation of the broadband signal to intuitively 92 normalize the estimation of irregularity for overall distributional width (Richman & Moorman, 93 2000). In particular, the similarity parameter r directly reflects the tolerance against which 94 temporal patterns are labelled as being similar or different (for an example, see Figure 1A; for 95 Matches are detected when m consecutive samples fall within the templates' similarity bounds as indicated by the grey shading. Entropy is based on the ratio of m+1 vs. m target matches and increases with a disproportional number of patterns of length m that do not remain similar at length m+1 (non-matches). This procedure is iteratively repeated across samples, deriving the entropy for each template in time. (A) Sample entropy varies as a function of the variance-dependent similarity criterion r that in turn relies on the signal's spectral variance. Empirical example of fine-scale entropy estimation in identical high-frequency (A1) and broadband (A2) signals. The superimposed formula exemplifies the sample entropy calculation for the current template. When the same signal is constrained to high frequency content (A1), its variance and the associated similarity criterion reflect a conservative criterion for pattern similarity. This results in high sample entropy estimates that accurately reflect high frequency pattern irregularity. (A2), In contrast, broadband signals are typically characterized by strong low-frequency fluctuations that lead to large similarity criteria at fine scales (A2), which are more appropriate for characterizing the largeamplitude fluctuations of slow dynamics (B; note different x-axis scaling). (C) Scale-wise estimates may not reflect the irregularity of spectral events at matching time scales depending on filter choices. In addition to influencing the similarity criterion, added spectral systematicity also modulates entropy estimates at varying time scales as a function of filter choice. The schematic shows an exemplary power spectrum with a characteristic 1/f shape, i.e., dominance of power/variance at low frequencies and a prominent alpha frequency peak. Low-pass filtering leads to slow dynamics dominating fine time scales, whereas high-pass filtering leads to reflections of rhythmicity at coarse time scales. details see methods). In particular, for each point in the time series, a repeating pattern is 96 identified by falling within a range that is defined by the standard deviation of the signal (see 97 Figure A1). However, contrary to the assumption that "[d]efining r as a fraction of the standard 98 deviation eliminates the dependence of [sample entropy] on signal amplitude" (Bruce et al.,99 2009, p. 259; see also Costa et al., 2004), it is rather plausible that this procedure in itself 100 introduces dependencies between signal variance and entropy. Specifically, as the magnitude 101 of signal fluctuations increases, the threshold for pattern similarity becomes more liberal as 102 more pattern are identified as similar (see Figure A2), thereby reducing estimated entropy and 103 leading to a general anti-correlation between signal variance and entropy (Nikulin & Brismar,104 2004; Richman & Moorman, 2000; Shafiei et al., 2019). Hence, contrary to common belief, the 105 use of a variance-based normalization criterion may invoke rather than remove dependencies 106 between entropy estimates and signal variance (see Hypothesis A in section 1.5). 107 This problem is compounded in the case of multiscale sample entropy (MSE), which aims 108 to describe entropy at different time scales -from fast dynamics at fine (also referred to as 109 'short) time scales to slow fluctuations at coarse (or 'long) time scales. referenced to signal variance dominated by slow fluctuations (see Figure 1B). In principle, this 176 may reliably manifest as an association between spectral slopes and fine-scale entropy that has 177 been observed both across subjects and wakefulness states ( Hypothesis C (see section 1.5). In worst-case scenarios, a conjunction of the mechanisms 183 described above may thus lead to a reflection of fast dynamics at coarse scales and a reflection 184 of slow dynamics at fine time scales, potentially inverting the interpretation of MSE time scales 185 in general.

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We argue that narrowband rhythms provide an optimal test case to assess a proper mapping 187 of neural irregularity to specific time scales (see Figure 1C), given that they are a well-188 researched characteristic of brain function, and given their specific definition of the time scale 189 of events (i.e., period = inverse of frequency for both regarding fast dynamics), this correspondence is surprising upon closer inspection 220 given the presumed anticorrelation between the magnitude of stereotypic rhythm dynamics and 221 their estimated entropy. Given uncertainty regarding the unique information offered by entropy 222 modulations, as well as concerns regarding the valid interpretation of time scales of entropy 223 effects, we attempted to reconcile these various issues by investigating the relation between 224 cross-sectional age effects on both MSE and spectral power. 225 226

Hypotheses and current study 227 228
We used simulations and empirical EEG data to probe the relationship between spectral 229 power and multiscale sample entropy (MSE), with a specific focus on the relation between 230 rhythmic frequencies and entropy time scales. We formulated the following general hypotheses 231 regarding the link between spectral variance and MSE: 232 233 A. The magnitude of the variance-based similarity criterion is negatively correlated with 234 entropy estimates.

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B. 'Original' scale-invariant similarity criteria produce increasingly biased thresholds for the 236 detection of time series pattern similarity towards coarser time scales. The magnitude of 237 this bias scales with the amount of excluded high frequency variance. This produces scale-238 to-frequency mismatches, wherein power differences at high frequencies manifest as 239 differences in coarse-scale entropy. 240 C. When fine time scales characterize signals that include both fast and slow fluctuations, fine-241 scale entropy estimates (and age differences therein) will relate to PSD slopes. Such an 242 association will be absent when slow fluctuations are removed.

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Extending these hypotheses to the domain of age-related differences in EEG-based MSE 245 and spectral power, we assessed the following hypotheses: To assess the influence of rhythmicity on entropy estimates, we simulated varying  To investigate the influence of similarity criteria and filter ranges in empirical data, we used 298 resting-state EEG data collected in the context of a larger assessment prior to task performance 299 and immediately following electrode preparation. 2.3 Calculation of standard and "modified" multiscale entropy 343 344 The calculation of standard MSE and the point averaging procedure followed (Costa et al., 345 2002(Costa et al., 345 , 2005. In short, sample entropy quantifies the irregularity of a time series of length N by 346 assessing the conditional probability that two sequences of m consecutive data points will 347 remain similar when another sample (m+1) is included in the sequence (for a visual example 348 see Figure 1A). The embedding dimension m was set to 2 in our applications. Sample entropy 349 is defined as the inverse natural logarithm of this conditional similarity: SampEn(+, -, .) = 350 − log 5 6 789 (:) 6 7 (:) ;. Crucially, the similarity criterion (r) defines the tolerance within which time 351 points are considered similar and is traditionally defined relative to the standard deviation (i.e., 352 square root of signal variance; here set to r = .5). Note that a larger, more liberal, similarity 353 criterion increases the likelihood of finding matching patterns, hence reducing entropy 354 estimates (see Figure 1A). Furthermore, in traditional applications (e.g., Costa  of scale-wise signal variance) and assessed the differences in resulting entropy estimates. 365 To assess entropy at coarser time scales, while the original MSE method coarse-grains the 366 data by averaging time points within discrete time bins (i.e., 'point averaging'; equivalent to 367 applying a finite-impulse response (FIR) filter to the original time series followed by down- estimates were computed based on their summed counts. As down-sampling represents a form 380 of low-pass filter, it is not employed in the 'high-pass' case. Thus, estimates are based on the 381 original sampling rate (i.e., embedding dimension of 1) with an exclusive modulation of the 382 spectral content according to the high-pass filter. Hence, we dissociated the embedding 383 dimension from the frequency content of the signal. As entropy (re-)calculation at the original 384 sampling rate introduces higher computational demands, scales were sampled in step sizes of 3 385 for empirical data and later spline-interpolated. As the interpretation of time scales is bound to 386 the sampling rate of the data (to assess scale-wise sampling rates) as well as the remaining 387 spectral content, our figures indicate the Nyquist frequency at each scale, except for the high-388 pass case (see above). Note that the sampling rate of the simulated data was 250 Hz, whereas 389 the empirical data had a sampling rate of 500 Hz, which renders consideration of the Nyquist 390 frequency particularly important. We refer to a traditional implementation with scale-invariant 391 similarity criterion and time point averaging as 'Original' in both the main text and Figures Spectral power, even in the narrowband case, is unspecific to the occurrence of systematic 413 rhythmic events as it also characterizes periods of absent rhythmicity (e.g., Jones, 2016). 414 Dedicated rhythm detection alleviates this problem by specifically detecting rhythmic episodes 415 in the ongoing signal. To investigate the potential relation between the occurrence of stereotypic 416 spectral events and narrowband entropy, we detected single-trial spectral events using the entropy estimates. In short, this method identifies stereotypic 'rhythmic' events at the single-420 trial level, with the assumption that such events have significantly higher power than the 1/f 421 background and occur for a minimum number of cycles at a particular frequency. This 422 effectively dissociates narrowband spectral peaks from the arrhythmic background spectrum. 423 Here, we used a one cycle threshold during detection, while defining the power threshold as the 424 95 th percentile above the individual background power. A 5-cycle wavelet was used to provide 425 the time-frequency transformations for 49 logarithmically-spaced center frequencies between 426 1 and 64 Hz. Rhythmic episodes were detected as described in Kosciessa et al. (2019).

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Following the detection of spectral events, the rate of spectral episodes longer than 3 cycles 428 was computed by counting the number of episodes with a mean frequency that fell in a moving 429 window of 3 adjacent center frequencies. This produced a channel-by-frequency representation 430 of spectral event rates, which were the basis for subsequent significance testing. Event rates 431 and statistical results were averaged within frequency bins from 8-12 Hz (alpha) and 14-20 Hz 432 (beta) to assess relations to narrowband entropy and for the visualization of topographies. To 433 visualize the stereotypic depiction of single-trial alpha and beta events, the original time series 434 were time-locked to the trough of individual spectral episodes and averaged across events (c.f., 435 Sherman et al., 2016). More specifically, the trough was chosen to be the local minimum during 436 the spectral episode that was closest to the maximum power of the wavelet-transformed signal. 437 To better estimate the local minimum, the signal was low-pass filtered at 25 Hz for alpha and 438 bandpass-filtered between 10 and 25 Hz for beta using a 6 th order Butterworth filter. A post-439 hoc duration threshold of one cycle was used for the visualization of beta events, whereas a 440 three-cycle criterion was used to visualize alpha events. Alpha and beta events were visualized 441 at channels POz and Cz, respectively. 442 443 2.6 Statistical analyses 444 Spectral power and entropy were compared across age groups within condition by means 445 of independent samples t-tests; cluster-based permutation tests (Maris & Oostenveld, 2007) 446 were performed to control for multiple comparisons. Initially, a clustering algorithm formed 447 clusters based on significant t-tests of individual data points (p <.05, two-sided; cluster entry 448 threshold) with the spatial constraint of a cluster covering a minimum of three neighboring 449 channels. Then, the significance of the observed cluster-level statistic, based on the summed t-450 values within the cluster, was assessed by comparison to the distribution of all permutation-451 based cluster-level statistics. The final cluster p-value that we report in all figures was assessed 452 as the proportion of 1000 Monte Carlo iterations in which the cluster-level statistic was 453 exceeded. Cluster significance was indicated by p-values below .025 (two-sided cluster 454 significance threshold). Effect sizes for MSE age differences with different filter settings were 455 computed on the basis of the cluster results in the 'Original' version. This was also the case for 456 analyses of partial correlations. Raw MSE values were extracted from channels with indicated 457 age differences at the initial three scales 1-3 (>65 Hz) for fine MSE and scales 39-41 (<6.5 Hz) 458 for coarse MSE. R T was calculated based on the t-values of an unpaired t-test: (Lakens, 2013). The measure describes the variance in the age difference explained by the 460 measure of interest, with the square root being identical to Pearson's correlation coefficient 461 between continuous individual values and binary age group. Effect sizes were compared using 462 the r-to-z-transform and a successive comparison of the z-value difference against zero: 463 (Brandner, 1933). Unmasked t-values are presented in support of the 464 assessment of raw statistics in our data (Allen, Erhardt, & Calhoun, 2012). highly similar regardless of actual pattern fluctuations (see Figure 1AB). Low entropy values 477 could result at fast entropy scales simply for this reason. In principle, this problem could be 478 alleviated by using spectral filters to constrain signals to the frequency range of interest. In 479 particular, we expected that scale-dependent low-pass filters would lead to a low-frequency 480 representation also at finer time scales, whereas slow fluctuations would exclusively modulate 481 entropy at coarser time scales if high-pass filters were applied ( Figure 1C). 482 483

484
To probe the relationship between low-frequency rhythmic power and estimated multiscale 485 sample entropy, we systematically varied the magnitude of simulated alpha power and assessed 486 its influence on estimated MSE using different filter settings. Our first aim was to establish an 487 inversion between similarity criteria and MSE estimates. In line with Hypothesis A, variations 488 in the similarity criterion as a function of rhythmic power tightly covaried with entropy 489 estimates; increased rhythmic power rendered the higher similarity criterion easier to surpass, 490 in turn decreasing entropy estimates by increasing pattern matches (see Figure 1A, Figure 2). 491 Importantly for scale-dependent inferences, with 'Original' settings, the effect of alpha power 492 on r and MSE estimates was not specific to the time scale corresponding to the simulated 493 frequency (Figure 2A). This can be attributed to the broadband similarity criterion, which by 494 definition prohibits scale-specific allocations of the added signal variance. In contrast, when 495 scale-dependent similarity criteria were used ( Figure 2BC), strong alpha rhythmicity 496 systematically decreased entropy at finer time scales than the simulated frequency (decreases 497 from baseline to the left of the vertical line in Figure 2C). Hence, the presence of the low 498 frequency rhythm diffusely affected fine-scale MSE estimates. This results from the low-pass 499 filter (LPF) characteristics of the scale-wise estimation procedure for which the low-frequency 500 rhythm is removed by LPFs < 10 Hz (see schematic in Figure 1C). As in previous work 501 (Valencia et al., 2009), dedicated low-pass filtering provided a better spectral suppression 502 compared with 'Original' point-averaging ( Figure 2B), but with otherwise comparable results. 503  In contrast to low-pass filter results, when high-pass filters were used, rhythmicity reduced 504 entropy at time scales below 10 Hz, hence leading to estimates of high frequency entropy that 505 were independent of low frequency power ( Figure 2D). Finally, when band-pass filters were 506 used ( Figure 2E), rhythmicity modulated entropy at the target frequency (although they also 507 produced edge artifacts surrounding the time scale of rhythmicity). In sum, these analyses 508 highlight that power increases of narrowband rhythms can diffusely modulate diverging 509 temporal scales as a function of the MSE implementation. In addition, these analyses highlight 510 that decreases in estimated entropy are often accompanied by comparable increases in the 511 liberality of similarity criteria. rhythms of higher frequency increased entropy at slightly finer time scales than the simulated 529 frequency (see increases in entropy above baseline to the left of the dotted vertical lines in 530 Figure 3A-C). Importantly, such sharp entropy increases were only observed with low-pass 531 implementations ( Figure 3A-C). Moreover, with scale-invariant r parameters ( Figure 3A), these 532 increases were paralleled by decreasing entropy at coarser time scales (i.e., to the right of the 533 dotted lines in Figure 3A). This is in line with our observation of relatively broadband, 534 amplitude-dependent, entropy decreases (cf., Figure 2A). Crucially, increased entropy relative 535 to baseline is counterintuitive to the idea that the addition of a stereotypic pattern should 536 decrease rather than increase pattern irregularity. Moreover, the results suggest that 537 combinations of amplitude-varying contributions of spectral content can induce ambiguous 538 scale-dependent results. In sum, our simulations highlight that the choice of similarity criterion 539 and the signal's spectral content grossly affect the interpretation of entropy time scales. 540 Furthermore, our frequency-resolved simulations suggest that a previously observed linear 541 frequency-to-scale mapping does not provide sufficient evidence that entropy towards finer 542 time scales dominantly represents the pattern irregularity of faster neural dynamics. Rather, 543 such assumptions rely on puzzling entropy increases with the addition of faint rhythmic 544 regularity that are counteracted by more dominant, and expected, decreases in entropy when 545 the signal contains strong rhythmic predictability. in the entropy of neural activity patterns or whether such effects can alternatively be accounted 556 for by differences in spectral power (see Hypothesis D). To assess the relations between age 557 differences in spectral power and multiscale entropy during eyes open rest, we used the 558 following strategies: (1) we statistically compared spectral power and MSE between two age 559 groups of younger and older adults; (2) we assessed the impact of scale-wise similarity criteria 560 and different filtering procedures on age differences in MSE and (3) we probed the relationship 561 between the r parameter and MSE. 562 563 564 565 Using traditional ('Original') settings, we replicated previous observations of scale-566 dependent entropy differences between younger and older adults ( Figure 4A1, Figure 5A). 567 Specifically, compared with younger adults, older adults exhibited lower entropy at coarse 568 scales, while they showed higher entropy at fine scales (Hypothesis D; Figure 4A1). Mirroring 569 these results in spectral power, older adults had lower parieto-occipital alpha power and 570 increased frontal high frequency power ( Figure 4A2) compared to younger adults. This was 571 globally associated with a shift from steeper to shallower PSD slopes with increasing age 572 ( Figure 4D). At face value, this suggests joint shifts of both power and entropy, in the same 573 direction and at matching time scales. Crucially, however, the spatial topography of differences 574 in entropy inversely mirrored differences in power between fast and slow dynamics ( Figure 4B  575 & C; cf., upper and lower topographies), such that frontal high frequency power differences 576 appeared inversely represented in coarse entropy scales ( Figure 4B), while parieto-occipital age 577 differences in slow frequency power more closely resembled fine-scale entropy differences 578 ( Figure 4D). This rather suggests scale-mismatched associations between entropy and power in 579 line with our simulations and theoretical expectations (Hypothesis D1 & D2). We investigated 580 their potential relationships more closely in the following sections regarding the potential 581 mechanistic associations proposed in Hypotheses B and C. 584 Importantly, as suggested by our simulations, filter choice affected the estimation of age 585 differences in entropy alongside differences in similarity thresholds ( Figure 5). As described 586 above, 'Original' settings indicated increased fine-scale and decreased coarse-scale entropy for 587 older compared to younger adults, whereas no group differences in the global r parameter were 588 indicated ( Figure 5A). In contrast, scale-wise similarity criteria Figure 5B) abolished age 589 differences in coarse-scale entropy (effect size was significantly reduced from r = .58 to r = .07; 590 p=6.8*10^-5), while fine-scale entropy differences remained unchanged when low-pass filters 591 were used ( Figure 5B/C) (from r = .44 to r = .45; p=.934). However, when constraining the 592 signal at fine scales to high frequency content (via high-pass filters; Figure 5D), no fine-scale 593 age differences were observed and the age effect was significantly reduced (r = .09; p = .008. 594 An age effect was only indicated once low-frequency dynamics contributed to the entropy 595 estimation again at coarse scales. Both of these effects were in line with our Hypotheses D1 596 and D2 regarding the influence of spectral filtering on entropy estimates. Interestingly, we 597 observed inverted age differences in the bandpass version ( Figure 5E), with larger 'narrowband' 598 entropy indicated in the alpha range and lower entropy in the beta range for older adults 599 compared with younger adults. In the following sections, we investigate these results more 600 closely with regard to the putative mechanisms linking spectral power and entropy.  Figure 5: Average multiscale entropy and similarity criterion by age depend on the specifics of the estimation method. Grand average traces of entropy (1 st row) and similarity criteria (3 rd row) alongside t-maps from statistical contrasts of age differences (2 nd + 4 th row). 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 invariant similarity criteria, 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).

Scale-invariant similarity criteria increasingly bias entropy towards coarser scales 605 606 607 608
Scale-dependent entropy effects in the face of scale-invariant similarity criteria (as observed 609 in the 'Original' implementation; Figure 5A) may intuitively suggest scale-wise variations in 610 pattern irregularity in the absence of variance differences. However, a fixed similarity criterion 611 is an artificial constraint that does not reflect the spectral shape of the broadband signal, leading 612 to potentially profound mismatches between the scale-dependent signal variance and the 613 invariant similarity criterion. That is, the total broadband variance may be similar while spectral 614 slopes and/or narrow-band frequency content differ. This is true for the case of aging as can be 615 appreciated by comparing the global r parameter with the age-specific frequency spectra 616 ( Figure 6A & B). As this scale-invariant criterion thresholds a successively low-pass filtered 617 signal, this induces a relative mismatch between the scale-specific variance and the global 618 Figure 6: Mismatches between scale-specific signal variance and global similarity criteria (r parameters) can account for age differences in coarse-scale entropy. (A, B) A global similarity criterion does not reflect the spectral shape, thus leading to disproportionally liberal criteria at coarse scales following the successive removal of highfrequency variance. Scale-dependent variance (as reflected in r) is more quickly reduced in older compared to younger adults (A) due to their removal of more high-frequency variance (B). This leads to a differential bias, as reflected in the increasingly mismatched distance between the two invariant and scale-dependent similarity criteria towards coarser scales. This mismatch, in turn, should scale with the amount of variance removed up to a particular scale. Letter insets refer to the relevant subplots. (C) The r adjustment in the rescaled version is associated with a comparable increase in coarse-scale entropy. This shift is more pronounced in older adults. (D) With global similarity criteria, coarse-scale entropy strongly reflects high frequency power due to the proportionally more liberal similarity threshold associated. Data in A and B are global averages, data in C and D are averages from frontal Original effect cluster (see Figure 4B) at time scales below 6 Hz. similarity criterion that successively increases towards coarser scales ( Figure 6A). Importantly, 619 the same broadband variance will pose a relatively higher (i.e. liberal) similarity threshold if 620 low-pass filtering removes more high-frequency variance. In turn, the coarse-scale MSE 621 estimate would be modulated as a function of high frequency power (i.e., Hypothesis B). To 622 assess this hypothesis, we probed the link between the change in r and MSE between the use 623 of a global and a scale-varying similarity criterion. As expected, we observed a strong anti-624 correlation between inter-individual differences in r and MSE ( Figure 6C). That is, the more 625 individual thresholds were re-adjusted to the lower scale-wise variance, the more entropy 626 estimates increased. Crucially, this difference was more pronounced for the older adults (paired 627 t-test; r: p = 5e-6; MSE: p = 3e-4). That is, due to their increased high frequency power, low-628 pass filtering decreased older adults' variance proportionally more than younger adults' 629 variance. Thus, in 'Original' settings, older adults' global criterion presented a more liberal 630 threshold at coarser scales than the threshold of younger adults, which can account for the 631 'lower' MSE estimates observed for older adults with 'Original' settings. In line with this 632 assumption, individual high frequency power at frontal channels was inversely related to 633 coarse-scale entropy estimates when a scale-invariant similarity criterion was applied ( Figure  634 6C), but not when the similarity criterion was recomputed for each scale (YA: r = -0.15; p = 635 .302; OA: r = .2, p = .146). This is further in line with the observation that coarse-scale age 636 differences ( Figure 5A) disappeared when a scale-wise similarity criterion was used ( Figure  637 5B). Taken together, this indicates that the observed age difference at coarse entropy scales can 638 be largely accounted for by high frequency power differences between young and old adults 639 and provides an explanation for the inverse group differences between high frequency power 640 and coarse-scale entropy (Hypothesis D1). 641 642 3.4 Low-frequency contributions render fine-scale entropy a proxy measure of PSD slope 643 644 A common observation in the MSE literature is a high sensitivity to task and behavioral 645 differences even at the original sampling rates (i.e., fine scales), which are commonly assumed 646 to reflect fast dynamics. This sensitivity is surprising given that little power generally exists in 647 high-frequency ranges in humans or animals (Hipp & Siegel, 2013). Interestingly, multiple 648 previous studies suggest that fine-scale entropy reflects the slope of power spectral density (e.g., 649 Bruce suggests that in low-pass scenarios, in which the target signal is dominated by low frequency 660 fluctuations, fine-scale entropy is sensitive to the ratio of high-to-low frequency variance, as 661 captured by PSD slopes. To highlight that fine-scale entropy does not exclusively relate to the 662 signal irregularity of high-frequency activity, we observed that following a high-pass filter to 663 the signal, the positive relation of fine-scale entropy to PSD slopes disappeared in both age 664 groups ( Figure 7B, dotted lines), and turned negative in older adults (see Supplementary Figure  665 3), alongside age differences in fine-scale entropy ( Figure 5D). In turn, relations between PSD 666 slopes and age differences re-emerged once low-frequency content was included in the entropy 667 estimation ( Figure 7C, dotted lines). Hence, the positive relation of fine-scale entropy to PSD 668 slopes was conditional on the presence of both low-and high-frequency dynamics. 669 In line with the hypothesis that fine-scale age differences are dependent on the presence of 670 slow fluctuations, we observed no age differences in fine-scale entropy when signals 671 exclusively contained high-frequency content (see section 3.2). To assess whether age 672 differences in PSD slope could account for fine-scale age differences in 'Original' entropy, we 673 computed partial correlations between the measures. In line with fine-scale entropy primarily 674 reflecting PSD slope variations, no significant prediction of age group status by fine-scale 675 entropy was observed when controlling for the high collinearity with PSD slopes (r = -.06, p = 676 .59). In contrast, PSD slopes significantly predicted age group status when controlling for MSE 677 (r = .38, p <.001), suggesting that differences in PSD slopes primarily account for observed age 678 differences in MSE, but not vice-versa (in line with Hypothesis D2). 679 On a side note, spectral slopes were anticorrelated with coarse-scale entropy when global 680 similarity criteria were used ( Figure 7C, continuous lines), but not when criteria were scale-681 wise re-estimated ( Figure 7C, broken lines). This likely reflects the bias described in section 682 3.2. That is, subjects with shallower slopes (more high frequency power) had increasingly 683 liberal-biased thresholds towards coarse scales, thereby resulting in decreased entropy 684 estimates. 685 Jointly, these empirical examples indicate that the use of global similarity criteria, as well 686 as the presence of large amplitude low frequency dynamics can severely bias scale-wise MSE. 687 Hence, differences in the spectral power and the r parameter (typically neglected as measures 688 of interest when estimating MSE) may actually account for a large proportion of reported MSE 689 effects; in this scenario, the pattern irregularity of fast dynamics per se may do little to drive 690 MSE estimates. 691 692 3.5 Narrowband MSE indicates age differences in signal irregularity in alpha and beta band 693 694

695
The previous analyses highlighted how the interpretation of scale-dependent results 696 critically depends on the spectral content of the signal, in some cases giving rise to mismatching 697 time scales. However, our simulations also suggest an accurate mapping between entropy and 698 power when scale-wise bandpass filters are used ( Figure 3A). Concurrently, the empirical band-699 pass results indicate a partial decoupling between entropy and variance age differences as 700 reflected in the r parameter ( Figure 5E). Specifically, older adults exhibited higher parieto-701 occipital entropy at alpha time scales (˜8-12 Hz) and lower central entropy at beta time scales 702 (˜12-20 Hz) than in younger adults ( Figure 5; Figure 8AB). Whereas alpha-band entropy was 703 moderately and inversely correlated with alpha power ( Figure 8C) and the age difference was 704 inversely reflected in the similarity criterion in a topographically similar fashion ( Figure 8E), 705 the same was not observed for entropy in the beta range for both age groups ( Figure 8DF). a divergence of entropy estimates from spectral power as it should be the rate of stereotypic 713 spectral events that reduces pattern irregularity rather than the overall power within a frequency 714 band. To test this hypothesis, we applied single-trial rhythm detection to extract the individual 715 rate of alpha (8-12 Hz) and beta (14-20 Hz) events. As predicted, individual alpha events had a 716 more sustained appearance compared with beta events as shown in Figure  rate and entropy within the two frequencies bands. This is important, as our simulations suggest 724 increased entropy estimates around narrow-band filtered rhythmicity (see Figure 2A). 725 Furthermore, a permutation test indicated age differences in beta rate that were opposite in sign 726 to the entropy age difference (see Figure 8L). In particular, older adults had a higher number of 727 central beta events during the resting state compared with younger adults, thus rendering their 728 beta-band dynamics more stereotypic. In sum, these results suggest that narrowband MSE 729 estimates approximate the irregularity of spectral events at matching time scales. 730 731

733
For entropy to be a practical and non-redundant measure in cognitive neuroscience, both its 734 convergent and discriminant validity to known signal characteristics has to be established. 735 Spectral features have a long history in cognitive electrophysiology and many procedures and 736 theoretical work are available for their interpretation. In the face of this existing literature, it 737 has been proposed that entropy is sensitive to non-linear time series characteristics that can 738 complement linear spectral information. If and to what extent these measures are independent 739 is however often not assessed, but tacitly inferred from applying a variance-based 740 'normalization' during the entropy calculation. Contrary to orthogonality assumptions, our 741 analyses suggest that differences in the similarity criterion may account for a significant 742 proportion of entropy effects in the literature, and thereby fundamentally affect the 743 interpretation of observed effects. In traditional applications, these effects can be differentiated 744 into separable effects of (a) biases arising from scale-invariant similarity criteria and (b) 745 challenges in the presence of broadband, low-frequency dominated, signals (see Figure 9A for 746 a schematic summary). In the following, we discuss these effects and how they can affect 747 traditional inferences regarding signal irregularity. 748 749 4.1 Narrowband rhythmicity diffusely affects entropy scales 750 751 The use of MSE is often motivated by its sensitivity to non-linear properties of brain 752 dynamics, that are assumed to reflect phenomena such as spontaneous network reconfigurations 753 and brain state transitions (e.g., Deco    highlights that low-pass filters render multiscale entropy especially sensitive to variance at low 774 frequencies, while further suggesting that slow events (e.g. event-related potentials) will be 775 reflected in a broad-scale manner. In contrast, we verified that the manipulation of spectral 776 content via high-or band-pass filters controlled the reflection of rhythms in MSE time scales. 777 The diffuse reflection of rhythms across many entropy time scales may initially seem at odds 778 with previous simulations that suggested a linear mapping of increasing frequencies onto 779 coarse-to-fine 'Original' MSE scales (Park et al., 2007;Takahashi et al., 2010;Vakorin & 780 McIntosh, 2012). Curiously, such previous simulations indicated the frequency-to-scale 781 mapping by considering the reflection of rhythms in positive entropy peaks. While we replicate 782 such increases, we highlight their dependence on low rhythm strength. Specifically, whereas 783 strong rhythmicity led to a sizeable reduction in entropy, fainter rhythmicity increased entropy 784 at slightly finer time scales above baseline. However, increases in entropy contrast with our 785 expectations that the addition of a more stereotypic pattern would decrease sample entropy and 786 were quickly counteracted by more diffuse entropy decreases once rhythm magnitude 787 increased. While the mechanistic origin of entropy increases with faint regularity remains 788 unclear, previous conclusions may thus have overemphasized the scale-specificity of rhythmic 789 influences. Hence, while rhythms of different frequencies modulate entropy at appropriate time 790 scales, they also induce broadband effects, thereby leading to potential scale-to-frequency 791 mismatches.

792
In addition to diffuse scale effects, we observed that rhythm-induced changes in sample 793 entropy were strongly anti-correlated to changes in the r parameter, confirming Hypothesis A. 794 However, we note that in the case of simulated rhythmicity, increases in variance (and r) are 795 collinear with increases in signal regularity. Hence, entropy is not exclusively determined by 796 the similarity criterion, but also by the reduction in pattern irregularity due to the addition of a 797 predictable sinusoidal signal. This presents a challenge for dissociating valid differences in 798 pattern irregularity that covary with spectral power from erroneous entropy decreases due to 799 increased similarity criteria. To probe the main contributor to observed sample entropy effects, 800 we replicated our analyses using permutation entropy, a measure that does not use an intrinsic 801 similarity criterion (see Supplementary Materials). Crucially, we observed similar filter 802 influences on the scale-wise representation of rhythmicity, suggesting that an explicit similarity 803 criterion is not necessary to produce diffuse reflections of narrowband rhythms across multiple 804 temporal scales. Rather, when entropy is applied to broadband signals, low-frequencies with 805 high variance contribute in large part to fine-scale estimates (see also section 4.3). 806 807 4.2 Global similarity criteria bias coarse-scale entropy estimates 808 The global impact of frequency-specific events in 'Original' implementations is directly 809 coupled to the use of global similarity criteria and challenges the notion of an accurate 810 frequency-timescale mapping. The theoretical necessity of introducing scale-wise adaptations 811 of similarity criteria has previously been noted (Nikulin & Brismar, 2004;Valencia et al., 812 2009), and is highlighted here with a practical example. In particular, Nikulin and Brismar 813 (2004) discussed the ambiguity between variance and pattern irregularity that arises from using 814 scale-invariant criteria: "However, in the MSE approach the same r value is used for different 815 scales. Therefore, the changes in MSE on each scale will depend on both the regularity and 816 variation of the coarse-grained sequences.
[…] Therefore, the outcome of the MSE algorithm 817 does not allow one to make a clear conclusion as to what extent this separation is based on the 818 affected regularity or variation" (Nikulin & Brismar, 2004). In short, when the similarity 819 criterion is fixed in the presence of scale-dependent spectral content, the liberality of thresholds 820 systematically varies across scales. This introduces fundamental mismatches between the origin 821 of group differences (pattern irregularity vs. variance), and the time scales at which differences 822 manifest. These mismatches are independent of the values of the global similarity criterion -823 which did not differ across groups here -and rather depend on the slope of the power spectrum. 824 The rather than to indicate the presence of a systematic bias in estimation 1 . Importantly, the 832 dependence of such biases on the spectral shape of the signal also indicates that they cannot be 833 accounted for by choosing different constants of the similarity criterion. Importantly, this has 834 practical implications for functional inferences. In the current resting state EEG data, we 835 observed that an age-related increase in high frequency power manifested as a decrease in 836 coarse-scale entropy due to group differences in the scale-wise mismatch between the (low-837 passed) signal variance and the global r parameter. Specifically, older adults' increased high 838 frequency power strongly reduced variance with successive low-pass filtering towards coarser 839 scales. As the similarity criterion was fixed across time scales relative to the total variance, this 840 quickly invoked an increasingly liberal threshold. In comparison, less high-frequency variance 841 was removed for younger adults at coarse scales. Given comparable global similarity criteria 842 between groups, younger adults' criterion was thus more conservative, affording higher entropy 843 estimates at coarser time scales (see Figure 9A). Crucially, coarse-scale group differences were 844 not observed when scale-wise similarity criteria were applied, or when permutation entropy -845 a measure without a dedicated similarity threshold -was used (see Supplementary Materials), 846 therefore highlighting the dependence of the group difference on mismatched thresholds. Note 847 that we presume that this age difference arises from a relative bias. Pink noise signals, such as 848 those observed here, have a relatively low contribution from high compared to low frequencies, 849 rendering the absolute bias lower than for white noise signals with equal variance of these two 850 components (and therefore a quicker 'bias rate' towards coarser scales as more high frequency 851 variance is removed). However, variations in high-frequency variance (and thus the resulting 852 bias) suffice, even at low levels, to systematically impact coarse-scale estimates. This may be 853 independent from the main source of variance in course-scale entropy. Hence, the latter may be 854 dominated by slower fluctuations, while even a relatively low contribution of high-frequency 855 'bias' could specifically explain variance in a third variable of interest (e.g., age; see Figure  856 9B). Thus, beyond bias controls noted above, we argue for rigorous statistical controls when 857 evaluating the shared and unique predictive utility of power and multiscale entropy in neural 858 time series data. Crucially, our analyses suggest that fine-scale entropy does not specifically reflect the 896 pattern similarity of high frequency dynamics, but that the presence of both high-and low-897 frequency dynamics at fine time scales is necessary for a link between power spectral density 898 slopes and fine signal entropy to emerge. If low frequency information is removed and entropy 899 becomes specific to high-frequency content, the association with power spectral density fails to 900 persist. In this case, entropy may however provide a sensitive index of high frequency activity 901 (Werkle-Bergner et al., 2014). While there is a general relationship between the 1/f slope and 902 fine-scale entropy for broadband signals, it is also worth noting that our simulations suggest an 903 influence of band-limited power on fine entropy scales. This introduces ambiguities in the 904 interpretation of fine scales, as they appear sensitive to both arrhythmic and rhythmic content. scales and frequencies is not always readily apparent. Some of these discrepancies likely stem 930 from a combination of the reported effects: the global similarity criterion renders MSE sensitive 931 to the shape of the frequency spectrum across scales, whereas the low-pass procedure leads to 932 a strong sensitivity to low-frequency content. While many papers perform control analyses with 933 band-limited spectral power, such mechanisms may obscure key links between the two 934 measures.

935
Our results are particularly relevant for understanding multiscale entropy differences across 936 the lifespan, although our findings and suggestions presumably apply to any scenario in which 937 MSE is a measure of interest, such as for the assessment of clinical outcomes (e.g., Takahashi  differences in the entropy spectrum, with older adults exhibiting lower coarse-scale entropy and 942 higher entropy at fine time scales compared with younger adults. In the power spectrum, these 943 effects were inverted, with older subjects showing enhanced high-, and reduced low-frequency 944 power. This was previously taken as evidence that older adults' high-frequency dynamics were 945 not only enhanced in magnitude, but also more unpredictable compared with younger adults' 946 dynamics. While we replicate those results with relatively minimal resting-state data here, our 947 analyses question the validity of these intuitive previous interpretations. In particular, our 948 results suggest that an apparent age-related increase of coarse-scale entropy is not due to valid 949 group differences in pattern irregularity, but results from inadequate similarity criteria that 950 render coarse-scale entropy sensitive to high frequency power (Hypothesis D1). No coarse-951 scale age differences were observed with scale-varying thresholds or permutation entropy (see Similarly, our analyses indicate that differences in fine-scale 'pattern irregularity' rely on 954 variations in the magnitude of slow fluctuations, and describe age-related changes in PSD 955 slopes (Hypothesis D2). Taken together, our results thus fundamentally challenge mechanistic 956 inferences by suggesting that previously described age differences in entropy may be minimal 957 beyond a misattribution of traditional age differences in the magnitude of fluctuations (i.e., 958 signal variance). This is further in line with a previous application using surrogate data that 959 highlighted that age group differences were mainly captured by linear auto-correlative 960 properties (see appendix in Courtiol et al., 2016). 961 In contrast to existing 'broad-band' applications, our narrowband analyses suggested age-962 related entropy increases in the posterior-occipital alpha band and decreases in central beta 963 entropy. Whereas alpha power and MSE were inversely related and the similarity criterion 964 showed an inverted age effect, the situation was less clear for the beta band. One explanation 965 for such divergence is that many Fourier-based methods assume stationary sinusoidal rhythms, 966 whereas stereotypical spectral features, particularly in the beta band (Lundqvist, Herman, 967 Calculation of multiscale permutation entropy. Sample entropy's similarity criterion makes it difficult to differentiate between rhythmic modulations of MSE via added pattern regularity or the influence on similarity criteria. For this purpose, we extended our analyses to multiscale permutation entropy, a measure that assesses pattern irregularity independent of a similarity criterion. In particular, permutation entropy describes the randomness in the occurrence of symbolic sequences (rank-order permutations) (Bandt & Pompe, 2002;Riedl, Muller, & Wessel, 2013). To investigate the correspondence between sample entropy and permutation entropy, we repeated our analyses with identical settings as described for the MSE analyses. The calculation of permutation entropy followed previous implementations (e.g., Ouyang, Li, Liu, & Li, 2013). Specifically. for a given template length m (i.e., embedding dimension, here m = 4), all m! rank-order permutations were assessed with regard to their relative occurrence: ( ) = ( )/( − ( − 1) ), where N is the number of samples and l is the time delay/lag (here l = 1). The permutation entropy of a signal was defined as = − ∑ ( ) ln ( ) ! =1 . We calculated a normalized version of permutation entropy with bounds between zero and one. Specifically, complete randomness of permutation occurrence would result in values of one, whereas increasing regularity results in lower values. To assess the convergence between sample and permutation entropy, we repeated the simulations noted in the main text, and probed age differences in the traditional (i.e., low-pass) implementation.

SI Results
Dissociating between similarity criterion and spectral regularity using multiscale permutation entropy (MPE). In our MSE analyses, the intrinsic, variance-bound, similarity criterion makes it difficult to distinguish whether spectral events (e.g., narrowband rhythms) decrease entropy as a result of increasing the r parameter or via their contribution of added (sinusoidal) signal regularity. To probe this issue, we used multiscale permutation entropy (MPE) as a measure of signal complexity that does not use a variance-based threshold. In particular, permutation entropy assesses pattern complexity as the relative (im-)balance in the occurrence of symbolic patterns.
In simulations, rhythmicity modulated MPE in a similar fashion as MSE ( Figure S4A, B). Notably, MPE did not indicate rhythm-dependent increases in entropy, although it should be noted that permutation entropy was at ceiling even at baseline. Crucially, we observed a similar decrease of entropy at fine scales in the absence of variance normalization, suggesting that added rhythmicity decreased broadband 'fine-scale' estimates due to the added rhythmic regularity. We further assessed age effects in the traditional low-pass scenario. Most notably, permutation entropy in the low-pass implementation did not exhibit an age difference at coarse ( Figure S4C), in line with our suggestions that this MSE difference is exclusively induced by fixed similarity criteria. However, a fine-scale age difference was also observed in low-pass MPE ( Figure S4C), suggesting that this effect is not exclusively related to the similarity criterion. As in the MSE analysis, fine-scale estimates characterized individual PSD slopes, underlining the broadband origin of the effect.