Emergent critical oscillations in motor cortex of Parkinson’s patients

criticality, Parkinson’s, EEG, motor cortex, temporal renormalization group, detrended28 fluctuation analysis29 Author Summary30 Brain function is thought to be optimal when its activity is near the border of order and chaos — a state31 called criticality. This state is thought to help the brain stay flexible and process information efficiently.32 1 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint We investigate whether Parkinson’s disease disrupts this balance like in other diseases and pathologies.33 Using resting EEG brain activity, we found that people with Parkinson’s show strong rhythmic signals34 not seen in healthy brains, and surprisingly these rhythms are also near the critical state. Using both35 established and new theoretical tools, we show that critical dynamics can accompany disease, suggesting36 that being closer to criticality is not always a sign of healthy brain function.37 Introduction38 What are the properties of cortical neural activity that confer its ability to perform healthy functions?39 One long-standing hypothesis posits that a healthy brain operates in a dynamical state near criticality40 - a special, marginally stable state imbued with a wide range of scale-invariant time scales and optimal41 computation (Shew and Plenz, 2013; Hengen and Shew, 2025). Indeed, evidence for criticality is associated42 with improved cognitive performance in humans (M¨ uller et al., 2025; Xin et al., 2025; Ezaki et al., 2020)43 and multiple beneficial computational properties including efficient coding (Safavi et al., 2024), large44 dynamic range (Kinouchi and Copelli, 2006; Shew et al., 2009; Gautam et al., 2015), discrimination of45 sensory input (Clawson et al., 2017; Gautam et al., 2015), and more. If these properties of criticality46 are needed for healthy brain function, it stands to reason that unhealthy dysfunction may be associated47 with deviation from criticality. This has indeed been reported in multiple studies (Zimmern, 2020). For48 instance, Alzheimer’s disease (Montez et al., 2009; Ghassemkhani et al., 2025; McGregor et al., 2024),49 schizophrenia (Nikulin et al., 2012; Moran et al., 2019), depression (Linkenkaer-Hansen et al., 2005),50 and epilepsy (Fusc` a et al., 2023) are associated with deviation from criticality compared to controls.51 However, the notion that healthy brain function requires closeness to criticality is challenged by some52 studies. For instance, sustained, focused attention seem to cause deviation from criticality (Irrmischer53 et al., 2018; Fagerholm et al., 2015). Here our primary goal was to determine how Parkinson’s disease54 impacts criticality in motor cortex. Motor cortex is a crucial area for voluntary movement and muscle55 control, functions that are severely impaired in Parkinson’s disease. We analyze a publicly available EEG56 dataset (Jackson et al., 2019; Swann et al., 2015; George et al., 2013) and ask whether motor cortex57 dynamics are closer to criticality for healthy control subjects or Parkinson’s patients.58 To rigorously measure proximity to criticality, we use our newly-developed approach based on infor-59 mation theory and Gaussian autoregressive processes that we term temporal Renormalization Group60 (tRG) (Sooter et al., 2025). This approach measures distance from criticality (d 2) from time series data61 based on the nature of temporal fluctuations. We apply this framework to EEG data for the first time,62 to our knowledge, and compare it to traditional methods of quantifying timescales from time series (Zer-63 aati et al., 2024) including the decay times of autocorrelation function ( ACF) (M¨ uller and Meisel, 2023;64 2 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint M¨ uller et al., 2025), and long-range temporal correlation via detrended fluctuation analysis (DF A) (Peng65 et al., 1994; Hardstone et al., 2012). However, we emphasize that our new approach rests on a math-66 ematically rigorous definition of proximity to criticality, which is lacking in traditional methods (Tian67 et al., 2022).68 We find that motor cortex EEG activity in Parkinson’s patients is marked by the emergence of near69 critical oscillations that are not present in healthy controls. Two recent studies are consistent with our70 results, although we measure distance to criticality directly. Calvo et al. (2024) found in whole-brain71 human MEG data that control subjects were further from a chaotic point in more frequency bands than72 Parkinson’s, and Lee et al. (2024) found in whole-brain human EEG that control subjects had shorter73 timescales than Parkinson’s in the theta band in some regions. Our results indicate that critical dynamics74 are not always beneficial; Parkinson’s disease seems to cause critical oscillations.75 Results76 We use freely available resting state EEG data (Fig 1A) collected by a lab in UCSD (Jackson et al., 2019;77 Swann et al., 2015; George et al., 2013) using a standardized format (Pernet et al., 2019; Appelhoff et al.,78 2019). Following previous studies (Jackson et al., 2019; Swann et al., 2015), we analyze data from the79 two electrodes positioned over left and right primary motor cortex (M1) labeled C3 and C4 (Fig 1A), an80 important brain region for motor planning and voluntary movement, functions that are impaired in these81 Parkinson’s patients. The dataset consists of 3 minute recording sessions from 16 healhty control subjects82 (control) and 15 subjects with Parkinson’s disease in two states: off drugs and on drugs to manage their83 symptoms. The Parkinson’s patients exhibited slight variability in severity of the disease as measured84 by Unified Parkinson’s Disease Rating Scale (UPDRS) III, but otherwise were not cognitively impaired85 compared to control subjects via Mini-Mental Status Exam ( MMSE) or the North American Reading86 Test (NAAR T) (George et al., 2013). We use a common approach of applying a band-pass filter to the87 EEG data and subsequently extracted the amplitude envelope (Fig 1A right panel) to be used as the88 signal for all the analyses here (except Fig 1B). By studying fluctuations of the amplitude envelope, we can89 assess whether the oscillations at particular frequency bands are near or far from criticality. By definition90 a critical oscillation will have amplitude fluctuations that are temporally scale invariant (Fontenele et al.,91 2025; Palva and Palva, 2018). This approach follows the long tradition of studying critical oscillations,92 typically referred to as long range temporal correlations (LR TC) (Linkenkaer-Hansen et al., 2001, 2005;93 Jackson et al., 2019; Hohlefeld et al., 2012, 2015).94 This EEG data has power in select frequency bands (Fig 1B), a common observation in other resting95 state EEG data (Newson and Thiagarajan, 2019). Importantly, the timescales in the broadband signal96 3 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint 10 0 10 110 0 10 2 10 4 10 6 10 8 δ [1,4] θ [4,8] α [8,13] β [13,30] 10 0 10 110 0 10 2 10 4 10 6 10 8 δ [1,4] θ [4,8] α [8,13] β [13,30] 10 0 10 110 0 10 2 10 4 10 6 10 8 δ [1,4] θ [4,8] α [8,13] β [13,30] B A EEG C3 C4 Frequency (Hz) Power Spectrum Healthy Parkinson’s (Off) Parkinson’s (On) 0.4 s 200 μV Bandpass Filter ( δ,θ,α,β) 0.5 s a.u. Extract Envelope C3 (right impairment) C4 (left impairment) Average Frequency (Hz) Power Spectrum Frequency (Hz) Power Spectrum Fig. 1 Emergent δ and θ oscillations in Parkinson’s patients. A) EEG time-series from two electrodes near the primary motor cortex on the left (C3) and right side (C4); right panel illustrates extracting the envelope of the band-passed EEG recording. B) The population-averaged power spectrum of the envelope of the EEG (without any band-pass filtering) for respectively the average of C3 and C4, as well as C3 and C4 individually, all exhibit peaks in the lower delta-band for Parkinson’s patients off medication (red), peaks in upper theta- to lower alpha-bands for Parkinson’s patients (on and off drugs), and a peak in the alpha-band for control (black). The shaded region above the curve corresponds to one standard deviation across the subjects. (Fig 1B) is distinct from the timescales in the band-passed power-envelope signal that is the main focus97 of our study. Figure 1B shows the population average (across 16 control subjects and 15 Parkinson’s98 patients) power spectrum of the envelope of the EEG data (y-axes on a log-scale) without band-pass99 filtering. Whether considering the average of both C3 and C4 (left panel), or a single electrode alone (C3100 in middle, C4 in right panel), it is evident that control subjects on average (black curve) have peaks in101 their power spectrum in the upper theta-band (4 to 8 Hz) to lower alpha-band (8 to 13 Hz). Parkinson’s102 patients exhibit similar power spectra to controls, except for the emergence of oscillations in the lower103 δ-band (1 to 4 Hz) for patients off medication and upper θ- to α-bands for all patients.104 First, we analyze the timescales of fluctuations in oscillation amplitudes using two traditional tools:105 autocorrelation ( ACF, see Fig 2A and Eq (1)), detrended fluctuation analysis ( DF A, see Fig 3A and106

Detrended Fluctuation Analysis). Then, we compare to our new information theoretic107

for measuring distance to criticality based on tRG theory (d2) that goes beyond simple timescales.108 For the control subjects, we simply use the average of both electrodes, but for Parkinson’s patients we109 use the electrode that corresponds to the side that subjects are reported to have physical limitations,110 see Table 3. (The results reported in the main text (Figs 2–4) also hold when we use both electrodes111 in Parkinson’s patients, see Supplementary Text S1 and Figures S1, S2.) Figure 2A shows the average112 autocorrelation function (over the number of subjects) in the four frequency bands (the alpha- and113 4 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint theta- bands zoomed-in to the right of the main axes) where the control subjects (black curves) have114 consistently faster decay than Parkinson’s patients (blue and red, x-axis at specific time lags is a log-scale).115 A more direct measure of timescales from the ACF is to calculate the time a subject’s ACF falls below116 a chosen threshold (Fig 2B). The population ACF timescales exhibit statistically significant differences117 between healthy control and Parkinson’s patients, specifically that the control subjects generally have118 faster timescales than Parkinson’s patients (both on and off drugs) in the delta- and theta- frequency119 bands (population summarized with box plots, Fig 2C), statistical significance was assessed with the120 Wilcoxon rank-sum test (see Methods: Wilcoxon Rank-Sum T est for details). When the differences121 were significant, the effect sizes are medium to large (see Table 1). There are no statistical differences122 between Parkinson’s patients on drugs versus off drugs (see Table 1; the population means are included123 in Fig 2D for completeness). Although commonly used, the autocorrelation function simply measures124 statistical correlation at a specific time lag averaged over the entire time series, in contrast to other125 measures that account for fluctuation trends as window sizes vary (DFA). Note that there are other126

for extracting timescales from the ACF, such as fitting an exponential function (Siegle et al.,127 2021; Li and Wang, 2022; Zeraati et al., 2022, 2024) or its variants (Zeraati et al., 2022; van Meegen and128 van Albada, 2021); but in this data, neither the population averages nor individual ACFs are well-fit129 exponential functions.130 Table 1 Statistics to show that control subjects have shorter time scales than Parkinson’s using ACF timescale measure; see Figure 2C. Using Wilcoxon rank-sum test where the null hypothesis is that both data samples are drawn from the same distribution. Top shows p−values, bottom shows effect size (see Methods: Wilcoxon Rank-Sum Test). Relationship / p−value δ band θ band α band β band Cntrl. vs. Park. (Off drugs) 1.5 × 10−2 4.6 × 10−5 0.33 0.18 Cntrl. vs. Park. (On drugs) 6.6 × 10−2 2.2 × 10−3 0.42 0.24 (Park.) On vs. Off 0.43 0.27 0.91 0.88 Relationship / Effect Size δ band θ band α band β band Cntrl. vs. Park. (Off drugs) 0.44 (med) 0.73 (lrg) 0.41 (n/a) 0.24 (n/a) Cntrl. vs. Park. (On drugs) 0.33 (med) 0.55 (lrg) 0.15 (n/a) 0.21 (n/a) (Park.) On vs. Off 0.14 (n/a) 0.13 (n/a) 2.3 × 10−2 (n/a) 2.7 × 10−2 (n/a) Next we perform DFA analysis, which characterizes how flucuations vary across different timescale,131 also known as long range temporal correlation ( LR TC) analysis. In the DFA analysis we find that132 control subjects have on average shorter range temporal correlation than Parkinson’s patients, consistent133 with the lower frequency bands in the ACF timescale analysis. A demonstration of the DFA method is134 depicted in Figure 3A on a control subject’s resting state EEG in the delta-band where the fluctuation135 amplitude as a function of time window length in log-log coordinates requires 2 lines at a manually chosen136 dividing point; such a dividing point is required in about 72% to 85% of the time (counting all frequency137 bands, subject type, and electrode combinations). In such cases, we use the slope of the best fit line for138 5 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint 0 0.5 1 1.5 2 10 -1 10 0 *** ** δ [1,4]Hz θ [4,8]Hz α [8,13]Hz β [13,30]Hz *** *** 10 -2 10 -1 10 0 Time (s) δ θα β (das hed) 0 0.2 0.4 0.6 0.8 1 α A Autocorrelation Funct. Healthy Parkinson’s (Off Drugs) Parkinson’s (On Drugs) C θ Zoomed-in to see black curve below D Pop. Avg. ACF Time-scale (s) δ [1,4]Hz θ [4,8]Hz α [8,13]Hz β [13,30]Hz Healthy Parkinson’s (Off Drugs) Parkinson’s (On Drugs) B ACF Time-scale (s) 10 -2 10 0-0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Autocorrelation Funct. Chosen Threshold ACF Time-scale= Time first cross below threshold Fig. 2 Emergent Parkinson’s oscillations have large autocorrelation time. A) The population-averaged ACF in Parkinson’s patients has longer timescales (red, blue: slower decay) in motor cortex EEG activity across all 4 frequency bands than healthy (control). ACF in alpha- and theta-band are zoomed-in and shifted for clarity. B) Example calculation of ACF timescale, i.e., a measure of ACF decay time, for 2 subjects; the first time where a subject’s ACF falls below a chosen threshold of 0.1 . C) Summary of ACF timescale with box plots in different frequency bands shows that control subjects on average have faster ACF time decay (smaller timescale) than Parkinson’s patients for the lower frequency bands (delta and theta). The horizontal lines in the boxes represent inter-quartiles: 25 th percentile, median, and 75 th percentile. Difference in distributions are statistically significant measured by Wilcoxon rank-sum test (see Table 1 for details). D) The population means of ACF timescale are plotted for completeness. larger time windows (second segment to the right), and call this the DFA coefficient. When there are no139 temporal correlations (i.e., white noise) the DFA coefficient is 0.5. In a random walk, temporal memory140 is infinite and the DFA exponent is 1.5. DFA coefficients between 0.5 and 1.5 indicate intermediate141 temporal correlations. A summary of all DFA coefficients is shown with box plots in Fig 3B with four142 frequency bands: on average control subjects have a much shorter range of temporal correlation, i.e.,143 timescales, than Parkinson’s patients (on or off drugs). The trend that control subjects have shorter144 timescales than Parkinson’s patients is robust across all four frequency bands we consider, with the145 alpha-band results having comparatively weaker results with larger p−values using Wilcoxon rank-sum146 test. The DFA coefficients shown include all 16 control subjects, but for Parkinson’s 1 or 2 patients were147 excluded (depending on frequency band) because the fluctuation amplitudes for a few subjects were too148 variable to be well fit by a line (see Fig S3 and Supplementary Text S1). Figure 3C clearly demonstrates149 how different the population averages are; control subjects have much smaller average DFA coefficients150 6 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint than Parkinson’s, and Parkinson’s patients have similar DFA coefficients regardless of whether on or off151 drug treatments.152 0.5 1 0.75 White noise *** * ** * *** ** δ [1,4]Hz θ [4,8]Hz α [8,13]Hz β [13,30]Hz *** *** * 0.05 < p < 0.1 ** 0.01 < p < 0.05 *** p<0.01 A Healthy Parkinson’s (Off Drugs) Parkinson’s (On Drugs) B DFA Coeff. 0.3 0.5 1 2 4 6 6 7 8 9 Time Window Length (s) Flucuation Amplitude F Data Best fit line(s) Slope of line = 0.72 Slope of line in smaller windows= 1.31 (not used) DFA Method C 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Pop. Avg. DFA Coeff . δ [1,4]Hz θ [4,8]Hz α [8,13]Hz β [13,30]Hz Healthy Parkinson’s (Off Drugs) Parkinson’s (On Drugs) Fig. 3 Emergent Parkinson’s oscillations have larger DFA exponents. A) Example DFA coefficient calculation (control subject 1 in delta-band) well-fit with 2 line segments, where a choice for the time window for where to segment the data has to be made. When 2 lines are used, the slope of the right line segment for larger time windows is reported. B) Summary of DFA coefficients with box plots in different frequency bands is largely consistent with the ACF results (Fig 2C,D). Box plot convention are the same as in Figure 2C. The results are not as strong in the alpha-band. Difference in distributions are statistically significant measured by Wilcoxon rank-sum test (see Table 2 for details). C) The population means of DFA coefficients are plotted to clearly illustrate that control subjects are further from criticality/scale-invariance than Parkinson’s patients. Table 2 Statistics to show that control subjects have shorter time scales than Parkinson’s using DFA coefficient; see Figure 3B . Using Wilcoxon rank-sum test where the null hypothesis is that both data samples are drawn from the same distribution. Top shows p−values, bottom shows effect size (see Methods: Wilcoxon Rank-Sum Test). Relationship / p−value δ band θ band α band β band Cntrl. vs. Park. (Off drugs) 9.4 × 10−3 9.4 × 10−3 8.6 × 10−2 2.8 × 10−2 Cntrl. vs. Park. (On drugs) 7.7 × 10−2 5.7 × 10−3 2.4 × 10−2 9.1 × 10−3 (Park.) On vs. Off 0.28 0.51 0.89 0.96 Relationship / Effect Size δ band θ band α band β band Cntrl. vs. Park. (Off drugs) 0.47 (med) 0.47 (med) 0.31 (med) 0.11 (sm) Cntrl. vs. Park. (On drugs) 0.32 (med) 0.51 (lrg) 0.42 (med) 0.14 (sm) (Park.) On vs. Off 0.2 (n/a) 0.13 (n/a) 0.26 (n/a) 1.5 × 10−3 (n/a) Although both ACF and DFA results provide evidence that control subjects’ EEG motor cortex153 activity is further from criticality than Parkinson’s patients, these analyses do not directly measure154 distance to criticality. To this end, we developed a rigorous tRG theory and implemented pragmatic155 computational tools to directly calculate distance to criticality ( d2). The d2 measure has several specific156 advantages : i) unlike DFA, it does not require specifically choosing a time window segment and assessing157 quality of linear fits (Fig S3), ii) unlike ACF, it does not require a prescribed threshold to find timescale,158 iii) the distance to criticality d2 (Fig 4A) is a precise quantification of distance independent of model159 parameterization, calculated in units of bits/sec (the bits/sec quantifies accumulation of evidence for160 ruling out being at criticality). Our framework requires first fitting an auto-regressive (AR) model to the161 data, then calculating the distance of the fitted model to the critical state; see Methods: T emporal162 Renormalization Group Theory and Figs S5–S7 for further details.163 7 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint A BSpace of all AR models AR models at criticality Best fit AR d2 C Healthy Parkinson’s (Off Drugs) Parkinson’s (On Drugs) Time bin (s) Healthy Parkinson’s (On Drugs) Parkinson’s (Off Drugs) Pop. d2 (bits/s) δ [1,4]Hz θ [4,8]Hz α [8,13]Hz β [13,30]Hz 0 0.05 0.1 10 0 0 0.05 0.1 10 0 0 0.05 0.1 10 0 0 1 2 Temporal Reach (s) 10 0 KL distance δ [1,4]Hz θ [4,8]Hz α [8,13]Hz β [13,30]Hz 0 1 2 Temporal Reach (s) 10 0 0 1 2 Temporal Reach (s) 10 0 Pop. d2 (bits/s) 0 1 2 Temporal Reach (s) 10 0 D 1 p- value 0.05 0.01 Healthy = Park. (Off) Healthy = Park. (On) Effect Size 0 0.2 0.4 0.6 0.8 Large Medium Small 0 1 2 Temporal Reach (s) 0 1 2 Temporal Reach (s) 0 1 2 Temporal Reach (s) 0 1 2 Temporal Reach (s) Fig. 4 Emergent Parkinson’s oscillations are closer to criticality. A) Using tRG theory to quantify differences in distance to criticality between controls and Parkinson’s patients after data is fit with an AR model. B) The population d2 (bits/s) values (log-scale) grouped by control and two Parkinson’s state as a function of coarse-grained time bin with AR model order 20 shows little difference across different band-passed frequencies. C) Summary of population d2 values (log-scale) for many time bins and model orders; the x-axis represents the ‘temporal reach’, i.e., model order multiplied by time bin length (varies from 2 ms to 100 ms). The control subjects consistently had larger d2 and were thus further from criticality than Parkinson’s patients, independent of model order, time bin length, or frequency band. D) Quantifying the statistical significance of our results using Wilcoxon rank-sum test, showing the p−values (log-scale) and effect sizes (see

Wilcoxon Rank-Sum Test ). The different shades of colors in C) and D) correspond to AR model fits of different orders ranging from 16 to 24. We perform a detailed comparison of Parkinson’s and control subjects using d2 and find control164 subjects’ EEG in primary motor cortex are generally further from criticality than Parkinson’s patients.165 We specifically vary the AR model order (16 to 24) as well as the time bin length (2 ms up to 100 ms) – we166 previously showed that increasing the time bin can unveil critical dynamics (Fontenele et al., 2024) and167 that d2 is expected to decrease monotonically with increasing model order and with increasing time bin168 length (Sooter et al., 2025). Figure 4B shows, within a given subject group (healthy, Parkinson’s on/off169 drugs) for a fixed AR model order 20, that d2 tends to decrease as time bin length increases, except170 occasionally in the delta-band (light blue), and that there are minor differences in population d2 across171 8 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint frequency bands with a given time bin. Varying AR model order and time bin length simultaneously172

in a wide variety of ‘ temporal reach ’ (x-axis of Fig 4C,D) defined as the the AR model order173 multiplied by the specific time bin, i.e., the maximal time in the past that can influence present AR model174 value. The temporal reach values we consider have a large range from 32 ms to 2.4 s. The population175 averaged d2 for many temporal reach values is shown in Figure 4C (y-axis is log-scale, different color176 shades correspond to different AR model order), where it is evident that Parkinson’s patients (red and177 blue dots) are closer to criticality (d 2 below) than controls (black/gray) in the delta- and theta-bands.178 We use Wilcoxon rank-sum test to analyze whether differences between Parkinson’s and control are179 statistically significant under the null hypothesis that the values were generated from the same probability180 distribution (p−values in top row of Fig 4D). The differences are clear in the delta- and theta-bands for a181 wide range of temporal reach values, there is no differences in the alpha band, and differences in the beta-182 band are only evident with small temporal reach values. The effect size and a qualitative characterization183 of effect size (small, medium, large (Cohen, 2013; Tomczak and Tomczak, 2014)) is shown in the bottom184 row of Figure 4D.185 Discussion186 Here we have shown that the prominentδ andθ band oscillations that emerge in Parkinson’s disease are,187 in fact, near-critical oscillations. Although each of these oscillations is defined by particular timescales188 (the oscillation periods), the power (amplitude) of these oscillations exhibits fluctuations across a wide189 range of time scales. These amplitude fluctuations are approximately scale invariant, which is how critical190 oscillations are defined (Fontenele et al., 2025; Palva and Palva, 2018). In contrast, in healthy controls,191 the same frequency bands have amplitude fluctuations that are further from criticality.192 The distance measure d2 enables a fair comparison of different time series, and is a rigorous193 information-theoretic entity in units of bits/s that measures the amount of evidence for ruling out the194 hypothesis that the data are at criticality (Sooter et al., 2025). Although the ACF and DFA analy-195 ses yielded similar results, d2 is a direct measure for distance to temporal scale-invariance, and proved196 to be cleaner for delineating differences (control vs. Parkinson’s), and did not require making specific197 choices regarding threshold cut-offs, which time window segments to use, etc. Unlike traditional methods,198 the analysis with d2 goes beyond just measuring timescales, and also clearly shows how the differences199 depend on the ‘temporal reach’, and that distances to criticality tend to decrease with increasing tempo-200 ral reach. For both DFA andd2, the strongest separation between control and Parkinson’s patients are in201 the delta- and theta-bands, followed by the beta-band, with the weakest results in the alpha-band. The202 ACF timescales only had significant differences in the delta- and theta- band. The relative consistency203 9 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint of these results suggests that there are real and surprising differences between control and Parkinson’s204 patients in motor cortex EEG.205 Our results in motor cortex are at odds with the idea that, in general, healthy brains operate closer206 to criticality than pathological ones (Hengen and Shew, 2025; O’Byrne and Jerbi, 2022), as discussed in207 the Introduction. However, our results are in line with two recent publications where it was reported that208 Parkinson’s patients i) have more frequency bands closer to ‘edge of chaos’ (a point related to criticality)209 than control subjects in whole brain MEG (Calvo et al., 2024), and ii) can have longer timescales as210 measured with DFA coefficients in whole brain EEG in the theta-band (Lee et al., 2024). These studies211 are different than ours because they focused on whole brain imaging and included many more subjects212 with which they aggregated/averaged. The reasonable number of subjects enabled detailed analysis, for213 example to assess the quality of model fits for each subject.214 Along these lines, another recent study showed that a measure of ‘intrinsic neural timescale’ using215 fMRI was longer in late stage Parkinson’s patients than in healthy controls in the anterior cortical region216 (Wei et al., 2024). This study is in line with our results, but unlike ourd2 their measure of intrinsic neural217 timescale is indirect because it involves calculating when various ACFs first cross below a threshold,218 smoothing the maximum of those values over space and applying a z-transform. The timescales of fMRI219 measurements are coarser than those of EEG, with resolution on the order of seconds, so we cannot make220 any direct comparisons with our results.221 Interestingly, Parkinson patients on versus off drugs to treat motor symptoms did not ever have222 statistically significant differences in their motor cortex EEG, independent of the methods (ACF, DFA,223 tRG). Presumably, these drugs helped mitigate their motor symptoms to some extent, but the motor224 cortex activity that is responsible for voluntary movement planning and muscle control did not exhibit225 any changes in timescales. Thus, it stands to reason that the timescales of the EEG in motor cortex226 might not be a direct reflection of mitigated motor symptoms, but rather a wholesale difference between227 Parkinson’s disease and control is manifested in these timescales.228 Methods229 Ethics statement230 This article presents an accurate account of the work performed by the stated authors, and all underlying231 data are represented accurately with consent from the owners. To the best of our knowledge, this work232 is original and is not under consideration for publication elsewhere. The study used publicly available233 data with accurate citation, and all methods were performed in accordance with relevant guidelines and234 regulations. The authors declare no conflicts of interest related to this research.235 10 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint This study uses third party human EEG data that is publicly available (George et al., 2013; Swann236 et al., 2015; Jackson et al., 2019) (see Data and code availability section). The Materials and methods237 section in their papers explicitly state that ‘All the participants provided written informed consent238 according to an Institutional Review Board Protocol at the University of California, San Diego and the239 Declaration of Helsinki’. We have also obtained written approval from the authors to use their data in240 this study.241 Parkinson’s patient characteristics242 Table 3 shows side of physical impairment in the Parkinson’s patients.243 Table 3 The side of reported physical impairment in Parkinson’s patients (Appelhoff et al., 2019; Rockhill et al., 2021) and thus corresponding electrode used. Electrode C3 is on the left motor cortex, C4 on the right motor cortex. Note that all subjects had the same side for physical impairment on and off drug treatment except for subject 14 who switched to Right side (C3) while on drug treatment. Subject Impairment Side Electrode 1 Right C3 2 Both Both 3 Right C3 4 Right C3 5 Left C4 6 Right C3 7 Left C4 8 Left C4 9 Left C4 10 Right C3 11 Right C3 12 Right C3 13 Left C4 14 Left* C4* 15 Right C3 Autocorrelation function244 The autocorrelation function is a common tool used to characterize how related (correlated) a time series of data x(t) is with specific time lags τ. The autocovariance function of a time series x(t) is: ˜A(τ) := Et[x(t)x(t +τ)]− ( Et[x(t)] )2 , (1) and the autocorrelation function is simply: A(τ) = ˜A(τ)/σ2 X (2) 11 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint where σ2 X := Et [( x(t)−µX )2] is the point-wise variance of the time series. We calculate the auto-245 correlation function of a particular time series of EEG via the Matlab function autocorr on centered246 data x(t)− Et[x(t)] with 10,000 lags (recall the time bins are 2 ms) and 1.96 standard deviations:247 autocorr(X-mean(X),‘NumLags’,10000,‘NumStd’,1.96). The results are in Figures 2, S1.248 Detrended Fluctuation Analysis (DF A)249 DFA is a common method for quantifying the degree of long-range temporal correlations (Peng et al.,250 1994). For a given time-series, the DFA coefficient was calculated by assessing the correlation of fluc-251 tuation amplitudes in various time window lengths. Start with a time-series xj, then calculate the252 cumulative sum:Yt := t∑ j=1 xj. The ‘entire’ Yt time-series is divided into n equal lengths for the duration253 of the specified time window τx of length (τ x/dt + 1) – if the length of Yt cannot be evenly divided,254 the end of the time-series is truncated, so n = ⌊N/(τx/dt + 1)⌋. Then for each segment of length255 (τx/dt + 1) the local trend (least squares linear fit Lk) is calculated. After which the mean-squared256 deviation is calculated: G(n,i ) = 1 n (i−1)n+n∑ t=(i−1)n+1 (Yt−Lk(t))2, then the mean fluctuation amplitude is:257 F (n) =  √ 1 ⌊N/n⌋ ⌊N/n⌋∑ i=1 G(n,i ). Finally, the least squares linear fit between log( n) (horizontal axis) and258 log(F (n)) (vertical axis) is calculated – the slope of this line is called the DFA coefficient.259 In cases where log( n) versus log(F (n)) is not well-fit by a single line, the time windows are split260 in two segments determined manually, then two least squares linear fits are calculated with the larger261 windows (right half) determining the DFA coefficient (Gu et al., 2015).262 Temporal renormalization group (tRG) theory263 A system is at criticality if (1) it lies at a boundary between qualitatively different operating regimes264 and (2) it exhibits scale-invariance, i.e. the lack of a characteristic spatial or temporal scale (Hengen265 and Shew, 2025). The renormalization group ( RG), which was originally developed to study critical266 phenomena in condensed matter systems, brings mathematical precision to these statements. The core267 idea of RG is to gradually remove the fine-scale details of a model to generate new, effective models at268 coarser scales. Fixed points of the RG operation therefore correspond to models that are scale-invariant,269 and all of the models in the basin of attraction of such a fixed point share the same coarse-scale behavior270 - this is the fundamental reason why, for example, water and ferromagnets poised near their respective271 phase transitions have quantitatively identical scaling exponents despite their drastic dissimilarities at272 microscopic length scales. Some RG fixed points are stable, meaning that there is an extended region in273 model space surrounding the fixed point such that every model in the region flows into the fixed point.274 12 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint Such fixed points fail to satisfy condition (1) in the definition of criticality. (For example, the RG fixed275 point corresponding to the disordered phase of an Ising model is stable.) Models lying in the basins of276 attraction of unstable fixed points, on the other hand, are both scale-invariant and poised at a boundary277 between different operating regimes, and hence are at criticality.278 In traditional applications of RG to spatially organized systems (e.g. Ising-type models), coarse- graining is implemented in space. Neural systems, on the other hand, can have rich temporal dynamics in an measurable entity (i.e., population EEG) independent of whether there is weak or crucial spatial structure. In Sooter et al. (2025), we argued that the appropriate way to define criticality in such systems is with a temporal RG ( tRG), wherein high-frequency features of a model are gradually removed to reveal its asymptotic behavior at low frequencies. We applied this procedure to a fundamental class of univariate discrete-time stochastic dynamical systems, Gaussian autoregressive ( AR) models: xt = n∑ j=1 φjxt−j +ξt. (3) We chose AR models both because they are analytically tractable, and because they are the optimal choice in a precise max ent sense. Specifically, if an observed time series is short enough that we can only confidently estimate its second-order (autocovariance) statistics, then AR models are the maximum entropy (i.e. minimally presumptive) way to model those statistics (Choi and Cover, 1984). In an AR model, the state xt at time t is a linear, Gaussian readout of the recent history (up to some maximum lag n, called the model order): xt∼N ( n∑ k=1 φkxt−k,σ 2 ) . In the space of order-n AR models, there are n+1 tRG fixed points, which we can label using the power-279 law exponents of their respective power spectra, β = 0, 2,..., 2n. The β = 0 fixed point is stable and280 corresponds to white noise - this is the ”trivial” fixed point that any AR model with a finite characteristic281 timescale flows into. The basins of attraction of the β≥ 2 fixed points constitute the AR models that282 are at criticality.283 Next, we asked how we should quantify proximity to these basins. That is, having determined which AR models are at criticality, can we say which ones are close to criticality? Naively, we could measure the Euclidean distance (in the parameter space defined by the AR model history kernel φ) from an AR model to each of the basins of attraction. However, there is no principled reason to use Euclidean distance rather than some other metric. To resolve this ambiguity, we turned to information theory and define proximity to criticality as distinguishability (per unit time) from a system at criticality. Specifically, for 13 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint a given order-n AR model B, let: dβ(B) := inf A∈A(n) β lim T→∞ 1 TKL (PB(x1,...,x T )||PA(x1,...,x T )) (4) where A(n) β is the set of order- n AR models that flow into the β fixed point, PA(x1,...,x T ) is the284 probability distribution for a T−step draw from the AR model A (and similarly for PB), and KL(·||·)285 is the Kullback-Leibler divergence. The structure of basins of attraction is such that that infimum taken286 over the set of all critical models ∪β≥2A(n) β is equal to the infimum taken over the β = 2 basin of287 attraction (Sooter et al., 2025); hence we only report d2 in this paper.288 To estimated2 from EEG data after bandpass filtering and extracting the envelope, we: 1) fit an AR289 model to the data using the Yule-Walker method, and 2) compute d2 for this model using Eq (4).290 Wilcoxon Rank-Sum Test291 We use the Wilcoxon rank-sum test (WRST) because it is ideal for the EEG. It is a nonparametric test of the null hypothesis that two groups of data are generated from the same distribution. The p−values of this test correspond to the probability that the null hypothesis holds. In addition, we report the Effect Size of the WRST: Effect Size := |z|√n1 +n2 (5) where z is the z−score of the U−statistic, z = (U−µU)/σU and nj are the sample sizes for the two292 populations. Effect sizes fort−test and Wilcoxon rank-sum test with values: (0, 0.2] are considered small,293 (0.2, 0.5] are medium, (0.5, 0.8] and above are large (Cohen, 2013; Tomczak and Tomczak, 2014); these294 labels are simply a qualitative assessment.295 EEG Data296 We used freely available EEG collected years ago that have appeared in many studies (Jackson et al., 2019;297 Swann et al., 2015; George et al., 2013) and was made widely applicable following common standards298 (Pernet et al., 2019; Appelhoff et al., 2019). We used all 16 control (control) subjects and all 15 Parkinson’s299 patients except for some of the DFA coefficients (see Figure S3). We used an EEG reader function by300 Tcheslavski (2025).301 Frequency Band Limits302 The frequency bands of interest were limited with an upper bound in the beta band (30 Hz) because the303 60 Hz grounding frequency (inferred by the power spectrum of all electrodes have a large peak at 60 Hz);304 14 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint any signals close to 60 Hz are considered artifacts perhaps due to electrical interference. For completeness305 and since some EEG studies report results in the gamma band frequency (George et al., 2013; Swann306 et al., 2015), we repeated our analysis on a lower gamma band frequency between 30 and 50 Hz (as was307 done in George et al. (2013), and found that the general trend of results we observed did not hold (see Fig308 S4, except for the population averaged ACF decay Fig S4A). We note however that in this lower gamma309 frequency band, the DFA analysis was messier than the other 4 lower frequency bands; in particular for310 the control subjects where 5 were excluded (see GitHub page), and in one of the ACF timescales was311 unusually long (longer than 20 s). Thus, these results should be taken with caution.312 Data and code availability313 Declarations314 Data availability . The raw EEG dataset was collected at UC San Diego from a team of researchers,315 it is freely available (Rockhill et al., 2021) at https://openneuro.org/datasets/ds002778/versions/1.0.2.316 Code availability . See https://github.com/chengly70/parkeeg for MATLAB code implementing all317 computational components in this paper.318 Author Contributions. Conceptualization: CL, WLS. Methodology: CL, JSS, WLS. Sofware: JSS,319 CL. Validation: CL. Formal Analysis: CL. Investigation: JSS, CL. Resources: N/A. Data Curation:320 CL. Writing original draft: CL, WLS, JSS, AKB. Writing review and editing: CL, JSS, AKB, WLS.321 Visualization: CL. Supervision: CL. Project administration: CL. Funding acquisition: CL, JSS, AKB,322 WLS.323 F unding. This study was supported by the National Institute on Drug Abuse (NIDA), National Insti-324 tutes of Health (NIH) under grant 1R01DA060744, part of the BRAIN Initiative (CL, JSS, AKB,325 WLS).326 Declaration of Competing Interests. The authors declare that no competing interests exist. The327 funders had no role in study design, data collection and analysis, decision to publish, or preparation of328 the manuscript.329 15 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint References330 Appelhoff, S., M. Sanderson, T.L. Brooks, M. van Vliet, R. Quentin, C. Holdgraf, M. Chaumon, E. Miku-331 lan, K. Tavabi, R. H¨ ochenberger, et al. 2019. Mne-bids: Organizing electrophysiological data into the332 bids format and facilitating their analysis. Journal of Open Source Software 4 (44) .333 Calvo, R., C. Martorell, G.B. Morales, S. Di Santo, and M.A. Mu˜ noz. 2024. Frequency-dependent covari-334 ance reveals critical spatiotemporal patterns of synchronized activity in the human brain. Physical335 Review Letters 133 (20): 208401 .336 Choi, B. and T.M. Cover. 1984. An information-theoretic proof of burg’s maximum entropy spectrum.337 Proceedings of the IEEE 72 (8): 1094–1096 .338 Clawson, W.P., N.C. Wright, R. Wessel, and W.L. Shew. 2017. Adaptation towards scale-free dynam-339 ics improves cortical stimulus discrimination at the cost of reduced detection. PLoS computational340 biology 13 (5): e1005574 .341 Cohen, J. 2013. Statistical power analysis for the behavioral sciences . Routledge.342 Ezaki, T., E.F.d. Reis, T. Watanabe, M. Sakaki, and N. Masuda. 2020. Closer to critical resting-343 state neural dynamics in individuals with higher fluid intelligence. Communications Biology 3 (1): 52.344 https://doi.org/10.1038/s42003-020-0774-y .345 Fagerholm, E.D., R. Lorenz, G. Scott, M. Dinov, P.J. Hellyer, N. Mirzaei, C. Leeson, D.W. Carmichael,346 D.J. Sharp, W.L. Shew, and R. Leech. 2015. Cascades and cognitive state: Focused attention incurs sub-347 critical dynamics. The Journal of Neuroscience 35 (11): 4626–4634. https://doi.org/10.1523/jneurosci.348 3694-14.2015 .349 Fontenele, A.J., J.S. Sooter, V.K. Norman, S.H. Gautam, and W.L. Shew. 2024. Low-dimensional350 criticality embedded in high-dimensional awake brain dynamics. Science Advances 10 (17): eadj9303 .351 Fontenele, A.J., J.S. Sooter, E. Ziarati, A.K. Barreiro, C. Ly, and W. Shew. 2025. Is critical brain352 dynamics more prevalent than previously thought? https://doi.org/10.1101/2025.09.02.673722 .353 Fusc` a, M., F. Siebenh¨ uhner, S.H. Wang, V. Myrov, G. Arnulfo, L. Nobili, J.M. Palva, and S. Palva. 2023.354 Brain criticality predicts individual levels of inter-areal synchronization in human electrophysiological355 data. Nature Communications 14 (1): 4736. https://doi.org/10.1038/s41467-023-40056-9 .356 16 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint Gautam, S.H., T.T. Hoang, K. McClanahan, S.K. Grady, and W.L. Shew. 2015. Maximizing sensory357 dynamic range by tuning the cortical state to criticality.PLoS Computational Biology 11(12): e1004576358 .359 George, J.S., J. Strunk, R. Mak-McCully, M. Houser, H. Poizner, and A.R. Aron. 2013. Dopaminergic360 therapy in parkinson’s disease decreases cortical beta band coherence in the resting state and increases361 cortical beta band power during executive control. NeuroImage: Clinical 3: 261–270 .362 Ghassemkhani, K., K.S. Saroka, and B.T. Dotta. 2025. Evaluating eeg complexity and spectral signa-363 tures in alzheimer’s disease and frontotemporal dementia: evidence for rostrocaudal asymmetry. npj364 Aging 11 (1): 50 .365 Gu, C., C.P. Coomans, K. Hu, F.A. Scheer, H.E. Stanley, and J.H. Meijer. 2015. Lack of exercise leads366 to significant and reversible loss of scale invariance in both aged and young mice. Proceedings of the367 National Academy of Sciences 112 (8): 2320–2324 .368 Hardstone, R., S.S. Poil, G. Schiavone, R. Jansen, V.V. Nikulin, H.D. Mansvelder, and K. Linkenkaer-369 Hansen. 2012. Detrended fluctuation analysis: A scale-free view on neuronal oscillations. Frontiers in370 Physiology 3: 450. https://doi.org/10.3389/fphys.2012.00450 .371 Hengen, K.B. and W.L. Shew. 2025. Is criticality a unified setpoint of brain function? Neuron 113: 1–17.372 https://doi.org/10.1016/j.neuron.2025.05.020 .373 Hohlefeld, F., F. Ehlen, H. Tiedt, L. Krugel, A. Horn, A. K¨ uhn, G. Curio, F. Klostermann, and V.V.374 Nikulin. 2015. Correlation between cortical and subcortical neural dynamics on multiple time scales375 in parkinson’s disease. Neuroscience 298: 145–160 .376 Hohlefeld, F., J. Huebl, C. Huchzermeyer, G.H. Schneider, T. Sch¨ onecker, A. K¨ uhn, G. Curio, and V.V.377 Nikulin. 2012. Long-range temporal correlations in the subthalamic nucleus of patients with parkinson’s378 disease. European Journal of Neuroscience 36 (6): 2812–2821 .379 Irrmischer, M., S.S. Poil, H.D. Mansvelder, F.S. Intra, and K. Linkenkaer-Hansen. 2018. Strong long-380 range temporal correlations of beta/gamma oscillations are associated with poor sustained visual381 attention performance. European Journal of Neuroscience 48 (8): 2674–2683 .382 Jackson, N., S.R. Cole, B. Voytek, and N.C. Swann. 2019. Characteristics of waveform shape in383 parkinson’s disease detected with scalp electroencephalography. eNeuro 6 (3) .384 17 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint Kinouchi, O. and M. Copelli. 2006. Optimal dynamical range of excitable networks at criticality. Nature385 Physics 2 (5): 348–351. https://doi.org/10.1038/nphys289. q-bio/0601037 .386 Lee, C.H., C.H. Juan, H.H. Chen, J.P. Hong, T.W. Liao, I. French, Y.S. Lo, Y.R. Wang, M.L. Cheng,387 H.C. Wu, et al. 2024. Long-range temporal correlations in electroencephalography for parkinson’s388 disease progression. Movement Disorders . https://doi.org/10.1002/mds.30074 .389 Li, S. and X.J. Wang. 2022. Hierarchical timescales in the neocortex: Mathematical mechanism and390 biological insights. Proceedings of the National Academy of Sciences 119 (6): e2110274119 .391 Linkenkaer-Hansen, K., S. Monto, H. Ryts¨ al¨ a, K. Suominen, E. Isomets¨ a, and S. K¨ ahk¨ onen. 2005.392 Breakdown of long-range temporal correlations in theta oscillations in patients with major depressive393 disorder. Journal of Neuroscience 25 (44): 10131–10137 .394 Linkenkaer-Hansen, K., V.V. Nikouline, J.M. Palva, and R.J. Ilmoniemi. 2001. Long-range temporal395 correlations and scaling behavior in human brain oscillations. Journal of Neuroscience 21 (4): 1370–396 1377 .397 McGregor, J.N., C.A. Farris, S. Ensley, A. Schneider, L.J. Fosque, C. Wang, E.I. Tilden, Y. Liu, J. Tu,398 H. Elmore, K.D. Ronayne, R. Wessel, E.L. Dyer, K. Bhaskaran-Nair, D.M. Holtzman, and K.B. Hengen.399 2024. Failure in a population: Tauopathy disrupts homeostatic set-points in emergent dynamics despite400 stability in the constituent neurons. Neuron 112 (21): 3567–3584.e5. https://doi.org/10.1016/j.neuron.401 2024.08.006 .402 Montez, T., S.S. Poil, B.F. Jones, I. Manshanden, J.P.A. Verbunt, B.W.v. Dijk, A.B. Brussaard, A.v.403 Ooyen, C.J. Stam, P. Scheltens, and K. Linkenkaer-Hansen. 2009. Altered temporal correlations in404 parietal alpha and prefrontal theta oscillations in early-stage alzheimer disease. Proceedings of the405 National Academy of Sciences 106 (5): 1614–1619. https://doi.org/10.1073/pnas.0811699106 .406 Moran, J.K., G. Michail, A. Heinz, J. Keil, and D. Senkowski. 2019. Long-range temporal correlations407 in resting state beta oscillations are reduced in schizophrenia. Frontiers in Psychiatry 10: 517. https:408 //doi.org/10.3389/fpsyt.2019.00517 .409 M¨ uller, P.M. and C. Meisel. 2023. Spatial and temporal correlations in human cortex are inherently410 linked and predicted by functional hierarchy, vigilance state as well as antiepileptic drug load. PLOS411 Computational Biology 19 (3): e1010919. https://doi.org/10.1371/journal.pcbi.1010919 .412 18 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint M¨ uller, P.M., G. Miron, M. Holtkamp, and C. Meisel. 2025. Critical dynamics predicts cognitive per-413 formance and provides a common framework for heterogeneous mechanisms impacting cognition.414 Proceedings of the National Academy of Sciences 122(14): e2417117122. https://doi.org/10.1073/pnas.415 2417117122 .416 Newson, J.J. and T.C. Thiagarajan. 2019. Eeg frequency bands in psychiatric disorders: a review of417 resting state studies. Frontiers in human neuroscience 12: 521 .418 Nikulin, V.V., E.G. J¨ onsson, and T. Brismar. 2012. Attenuation of long-range temporal correlations419 in the amplitude dynamics of alpha and beta neuronal oscillations in patients with schizophrenia.420 NeuroImage 61 (1): 162–169. https://doi.org/10.1016/j.neuroimage.2012.03.008 .421 O’Byrne, J. and K. Jerbi. 2022. How critical is brain criticality?Trends in Neurosciences 45(11): 820–837422 .423 Palva, S. and J.M. Palva. 2018. Roles of brain criticality and multiscale oscillations in temporal predic-424 tions for sensorimotor processing. Trends in Neurosciences 41 (10): 729–743. https://doi.org/10.1016/425 j.tins.2018.08.008 .426 Peng, C.K., S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger. 1994. Mosaic427 organization of dna nucleotides. Physical review e 49 (2): 1685 .428 Pernet, C.R., S. Appelhoff, K.J. Gorgolewski, G. Flandin, C. Phillips, A. Delorme, and R. Oostenveld.429 2019. Eeg-bids, an extension to the brain imaging data structure for electroencephalography. Scientific430 data 6 (1): 103 .431 Rockhill, A.P., N. Jackson, J. George, A. Aron, and N.C. Swann. 2021. Uc san diego resting state eeg data432 from patients with parkinson’s disease. OpenNeuro. https://doi.org/10.18112/openneuro.ds002778.433 v1.0.5 .434 Safavi, S., M. Chalk, N.K. Logothetis, and A. Levina. 2024. Signatures of criticality in efficient coding435 networks. Proceedings of the National Academy of Sciences 121 (41): e2302730121. https://doi.org/436 10.1073/pnas.2302730121 .437 Shew, W.L. and D. Plenz. 2013. The functional benefits of criticality in the cortex. The neuroscien-438 tist 19 (1): 88–100 .439 19 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint Shew, W.L., H. Yang, T. Petermann, R. Roy, and D. Plenz. 2009. Neuronal avalanches imply maximum440 dynamic range in cortical networks at criticality. Journal of neuroscience 29 (49): 15595–15600 .441 Siegle, J.H., X. Jia, S. Durand, S. Gale, C. Bennett, N. Graddis, G. Heller, T.K. Ramirez, H. Choi,442 J.A. Luviano, et al. 2021. Survey of spiking in the mouse visual system reveals functional hierarchy.443 Nature 592 (7852): 86–92 .444 Sooter, J., A. Fontenele, A. Barreiro, C. Ly, K. Hengen, and W. Shew. 2025. Defining and measuring445 proximity to criticality. bioRxiv: 1–27. https://doi.org/10.1101/2025.08.03.668332 .446 Swann, N.C., C. De Hemptinne, A.R. Aron, J.L. Ostrem, R.T. Knight, and P.A. Starr. 2015. Elevated447 synchrony in p arkinson disease detected with electroencephalography. Annals of neurology 78 (5):448 742–750 .449 Tcheslavski, G. 2025. Eeg bdf reader. https://www.mathworks.com/matlabcentral/fileexchange/13070-450 eeg-bdf-reader. MATLAB Central File Exchange; Accessed October 2, 2024.451 Tian, Y., Z. Tan, H. Hou, G. Li, A. Cheng, Y. Qiu, K. Weng, C. Chen, and P. Sun. 2022. Theoretical452 foundations of studying criticality in the brain. Network Neuroscience 6 (4): 1148–1185 .453 Tomczak, M. and E. Tomczak. 2014. The need to report effect size estimates revisited. an overview of454 some recommended measures of effect size. Trends in sports sciences 21 .455 van Meegen, A. and S.J. van Albada. 2021. Microscopic theory of intrinsic timescales in spiking neural456 networks. Physical Review Research 3 (4): 043077 .457 Wei, Y., C. Zhang, Y. Peng, C. Chen, S. Han, W. Wang, Y. Zhang, H. Lu, and J. Cheng. 2024. Mri458 assessment of intrinsic neural timescale and gray matter volume in parkinson’s disease. Journal of459 Magnetic Resonance Imaging 59 (3): 987–995 .460 Xin, Y., Y. Cui, S. Yu, and N. Liu. 2025. Genetic contributions to brain criticality and its relationship with461 human cognitive functions. Proceedings of the National Academy of Sciences 122 (26): e2417010122.462 https://doi.org/10.1073/pnas.2417010122 .463 Zeraati, R., T.A. Engel, and A. Levina. 2022. A flexible bayesian framework for unbiased estimation of464 timescales. Nature computational science 2 (3): 193–204 .465 Zeraati, R., A. Levina, J.H. Macke, and R. Gao. 2024. Neural timescales from a computational466 perspective. arXiv preprint arXiv:2409.02684 .467 20 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint Zimmern, V. 2020. Why brain criticality is clinically relevant: A scoping review. Frontiers in Neural468 Circuits 14: 54. https://doi.org/10.3389/fncir.2020.00054 .469 21 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint