{"paper_id":"14c1eb4a-8a4d-470a-997c-1872d524fa91","body_text":"Emergent critical oscillations in motor cortex of Parkinson’s1\npatients2\nShort Title : Critical oscillations in Parkinson’s disease.3\nCheng Ly 1*, J. Sam Sooter 2, Andrea K. Barreiro 3, Woodrow L. Shew 2\n4\n1*Mathematics Department, Virginia Commonwealth University, 1015 Floyd Ave,5\nRichmond, 23284-2014, VA, U.S.A.6\n2Physics Department, University of Arkansas, 1 University of Arkansas, Fayetteville,7\n72701, AR, U.S.A.8\n3Mathematics Department, Southern Methodist University, P.O. Box 750156, Dallas,9\n75275-0156, TX, U.S.A.10\n*Corresponding author(s). E-mail(s): cly@vcu.edu;11\nContributing authors: sooter@uark.edu; abarreiro@smu.edu; shew@uark.edu;12\nAbstract13\nThe dynamical state of cortical neural activity constrains the complexity of functions it can perform.14\nA marginally stable dynamical state - called criticality - is thought to be beneficial for brain functions15\nthat require multiple time scales, broad dynamic range, and large information storage and transmis-16\nsion. A growing body of evidence suggests that criticality is a feature of healthy brain dynamics, but17\nbreaks down in certain brain disorders. Here we ask whether Parkinson’s disease incurs deviation from18\ncriticality compared to healthy controls. We analyze human resting state EEG activity in primary19\nmotor cortex. Parkinson’s patients exhibit prominent oscillatory brain activity in multiple frequency20\nbands (low delta and high theta) that is not present in controls. Surprisingly, we find that these21\nemergent oscillations are close to criticality, i.e., amplitude fluctuations with approximate temporal22\nscale invariance. We compare traditional signatures of criticality and more principled measurements23\nof proximity to criticality using our new approach based on temporal renormalization group theory24\nand information theory. Our new approach and traditional methods agree, demonstrating that critical25\ndynamics are not always healthy; Parkinson’s disease is associated with the emergence of near-critical26\noscillations in motor cortex.27\nKeywords: criticality, Parkinson’s, EEG, motor cortex, temporal renormalization group, detrended28\nfluctuation analysis29\nAuthor Summary30\nBrain function is thought to be optimal when its activity is near the border of order and chaos — a state31\ncalled criticality. This state is thought to help the brain stay flexible and process information efficiently.32\n1\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nWe investigate whether Parkinson’s disease disrupts this balance like in other diseases and pathologies.33\nUsing resting EEG brain activity, we found that people with Parkinson’s show strong rhythmic signals34\nnot seen in healthy brains, and surprisingly these rhythms are also near the critical state. Using both35\nestablished and new theoretical tools, we show that critical dynamics can accompany disease, suggesting36\nthat being closer to criticality is not always a sign of healthy brain function.37\nIntroduction38\nWhat are the properties of cortical neural activity that confer its ability to perform healthy functions?39\nOne long-standing hypothesis posits that a healthy brain operates in a dynamical state near criticality40\n- a special, marginally stable state imbued with a wide range of scale-invariant time scales and optimal41\ncomputation (Shew and Plenz, 2013; Hengen and Shew, 2025). Indeed, evidence for criticality is associated42\nwith improved cognitive performance in humans (M¨ uller et al., 2025; Xin et al., 2025; Ezaki et al., 2020)43\nand multiple beneficial computational properties including efficient coding (Safavi et al., 2024), large44\ndynamic range (Kinouchi and Copelli, 2006; Shew et al., 2009; Gautam et al., 2015), discrimination of45\nsensory input (Clawson et al., 2017; Gautam et al., 2015), and more. If these properties of criticality46\nare needed for healthy brain function, it stands to reason that unhealthy dysfunction may be associated47\nwith deviation from criticality. This has indeed been reported in multiple studies (Zimmern, 2020). For48\ninstance, Alzheimer’s disease (Montez et al., 2009; Ghassemkhani et al., 2025; McGregor et al., 2024),49\nschizophrenia (Nikulin et al., 2012; Moran et al., 2019), depression (Linkenkaer-Hansen et al., 2005),50\nand epilepsy (Fusc` a et al., 2023) are associated with deviation from criticality compared to controls.51\nHowever, the notion that healthy brain function requires closeness to criticality is challenged by some52\nstudies. For instance, sustained, focused attention seem to cause deviation from criticality (Irrmischer53\net al., 2018; Fagerholm et al., 2015). Here our primary goal was to determine how Parkinson’s disease54\nimpacts criticality in motor cortex. Motor cortex is a crucial area for voluntary movement and muscle55\ncontrol, functions that are severely impaired in Parkinson’s disease. We analyze a publicly available EEG56\ndataset (Jackson et al., 2019; Swann et al., 2015; George et al., 2013) and ask whether motor cortex57\ndynamics are closer to criticality for healthy control subjects or Parkinson’s patients.58\nTo rigorously measure proximity to criticality, we use our newly-developed approach based on infor-59\nmation theory and Gaussian autoregressive processes that we term temporal Renormalization Group60\n(tRG) (Sooter et al., 2025). This approach measures distance from criticality (d 2) from time series data61\nbased on the nature of temporal fluctuations. We apply this framework to EEG data for the first time,62\nto our knowledge, and compare it to traditional methods of quantifying timescales from time series (Zer-63\naati et al., 2024) including the decay times of autocorrelation function ( ACF) (M¨ uller and Meisel, 2023;64\n2\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nM¨ uller et al., 2025), and long-range temporal correlation via detrended fluctuation analysis (DF A) (Peng65\net al., 1994; Hardstone et al., 2012). However, we emphasize that our new approach rests on a math-66\nematically rigorous definition of proximity to criticality, which is lacking in traditional methods (Tian67\net al., 2022).68\nWe find that motor cortex EEG activity in Parkinson’s patients is marked by the emergence of near69\ncritical oscillations that are not present in healthy controls. Two recent studies are consistent with our70\nresults, although we measure distance to criticality directly. Calvo et al. (2024) found in whole-brain71\nhuman MEG data that control subjects were further from a chaotic point in more frequency bands than72\nParkinson’s, and Lee et al. (2024) found in whole-brain human EEG that control subjects had shorter73\ntimescales than Parkinson’s in the theta band in some regions. Our results indicate that critical dynamics74\nare not always beneficial; Parkinson’s disease seems to cause critical oscillations.75\nResults76\nWe use freely available resting state EEG data (Fig 1A) collected by a lab in UCSD (Jackson et al., 2019;77\nSwann et al., 2015; George et al., 2013) using a standardized format (Pernet et al., 2019; Appelhoff et al.,78\n2019). Following previous studies (Jackson et al., 2019; Swann et al., 2015), we analyze data from the79\ntwo electrodes positioned over left and right primary motor cortex (M1) labeled C3 and C4 (Fig 1A), an80\nimportant brain region for motor planning and voluntary movement, functions that are impaired in these81\nParkinson’s patients. The dataset consists of 3 minute recording sessions from 16 healhty control subjects82\n(control) and 15 subjects with Parkinson’s disease in two states: off drugs and on drugs to manage their83\nsymptoms. The Parkinson’s patients exhibited slight variability in severity of the disease as measured84\nby Unified Parkinson’s Disease Rating Scale (UPDRS) III, but otherwise were not cognitively impaired85\ncompared to control subjects via Mini-Mental Status Exam ( MMSE) or the North American Reading86\nTest (NAAR T) (George et al., 2013). We use a common approach of applying a band-pass filter to the87\nEEG data and subsequently extracted the amplitude envelope (Fig 1A right panel) to be used as the88\nsignal for all the analyses here (except Fig 1B). By studying fluctuations of the amplitude envelope, we can89\nassess whether the oscillations at particular frequency bands are near or far from criticality. By definition90\na critical oscillation will have amplitude fluctuations that are temporally scale invariant (Fontenele et al.,91\n2025; Palva and Palva, 2018). This approach follows the long tradition of studying critical oscillations,92\ntypically referred to as long range temporal correlations (LR TC) (Linkenkaer-Hansen et al., 2001, 2005;93\nJackson et al., 2019; Hohlefeld et al., 2012, 2015).94\nThis EEG data has power in select frequency bands (Fig 1B), a common observation in other resting95\nstate EEG data (Newson and Thiagarajan, 2019). Importantly, the timescales in the broadband signal96\n3\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\n10 0 10 110 \n0\n10 \n2\n10 \n4\n10 \n6\n10 \n8\nδ\n[1,4]\nθ\n[4,8]\nα \n[8,13]\nβ \n[13,30]\n10 0 10 110 \n0\n10 \n2\n10 \n4\n10 \n6\n10 \n8\nδ\n[1,4]\nθ\n[4,8]\nα \n[8,13]\nβ \n[13,30]\n10 0 10 110 \n0\n10 \n2\n10 \n4\n10 \n6\n10 \n8\nδ\n[1,4]\nθ\n[4,8]\nα \n[8,13]\nβ \n[13,30]\nB\nA\nEEG\nC3 C4 \nFrequency (Hz)\nPower Spectrum \nHealthy\nParkinson’s (Off)\nParkinson’s (On)\n0.4 s\n200 μV\nBandpass Filter ( δ,θ,α,β)\n0.5 s\na.u. \nExtract Envelope \nC3 (right impairment) C4 (left impairment) Average \nFrequency (Hz)\nPower Spectrum \nFrequency (Hz)\nPower Spectrum \nFig. 1 Emergent δ and θ oscillations in Parkinson’s patients. A) EEG time-series from two electrodes near the\nprimary motor cortex on the left (C3) and right side (C4); right panel illustrates extracting the envelope of the band-passed\nEEG recording. B) The population-averaged power spectrum of the envelope of the EEG (without any band-pass filtering)\nfor respectively the average of C3 and C4, as well as C3 and C4 individually, all exhibit peaks in the lower delta-band for\nParkinson’s patients off medication (red), peaks in upper theta- to lower alpha-bands for Parkinson’s patients (on and off\ndrugs), and a peak in the alpha-band for control (black). The shaded region above the curve corresponds to one standard\ndeviation across the subjects.\n(Fig 1B) is distinct from the timescales in the band-passed power-envelope signal that is the main focus97\nof our study. Figure 1B shows the population average (across 16 control subjects and 15 Parkinson’s98\npatients) power spectrum of the envelope of the EEG data (y-axes on a log-scale) without band-pass99\nfiltering. Whether considering the average of both C3 and C4 (left panel), or a single electrode alone (C3100\nin middle, C4 in right panel), it is evident that control subjects on average (black curve) have peaks in101\ntheir power spectrum in the upper theta-band (4 to 8 Hz) to lower alpha-band (8 to 13 Hz). Parkinson’s102\npatients exhibit similar power spectra to controls, except for the emergence of oscillations in the lower103\nδ-band (1 to 4 Hz) for patients off medication and upper θ- to α-bands for all patients.104\nFirst, we analyze the timescales of fluctuations in oscillation amplitudes using two traditional tools:105\nautocorrelation ( ACF, see Fig 2A and Eq (1)), detrended fluctuation analysis ( DF A, see Fig 3A and106\nMethods: Detrended Fluctuation Analysis). Then, we compare to our new information theoretic107\nmethod for measuring distance to criticality based on tRG theory (d2) that goes beyond simple timescales.108\nFor the control subjects, we simply use the average of both electrodes, but for Parkinson’s patients we109\nuse the electrode that corresponds to the side that subjects are reported to have physical limitations,110\nsee Table 3. (The results reported in the main text (Figs 2–4) also hold when we use both electrodes111\nin Parkinson’s patients, see Supplementary Text S1 and Figures S1, S2.) Figure 2A shows the average112\nautocorrelation function (over the number of subjects) in the four frequency bands (the alpha- and113\n4\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\ntheta- bands zoomed-in to the right of the main axes) where the control subjects (black curves) have114\nconsistently faster decay than Parkinson’s patients (blue and red, x-axis at specific time lags is a log-scale).115\nA more direct measure of timescales from the ACF is to calculate the time a subject’s ACF falls below116\na chosen threshold (Fig 2B). The population ACF timescales exhibit statistically significant differences117\nbetween healthy control and Parkinson’s patients, specifically that the control subjects generally have118\nfaster timescales than Parkinson’s patients (both on and off drugs) in the delta- and theta- frequency119\nbands (population summarized with box plots, Fig 2C), statistical significance was assessed with the120\nWilcoxon rank-sum test (see Methods: Wilcoxon Rank-Sum T est for details). When the differences121\nwere significant, the effect sizes are medium to large (see Table 1). There are no statistical differences122\nbetween Parkinson’s patients on drugs versus off drugs (see Table 1; the population means are included123\nin Fig 2D for completeness). Although commonly used, the autocorrelation function simply measures124\nstatistical correlation at a specific time lag averaged over the entire time series, in contrast to other125\nmeasures that account for fluctuation trends as window sizes vary (DFA). Note that there are other126\nmethods for extracting timescales from the ACF, such as fitting an exponential function (Siegle et al.,127\n2021; Li and Wang, 2022; Zeraati et al., 2022, 2024) or its variants (Zeraati et al., 2022; van Meegen and128\nvan Albada, 2021); but in this data, neither the population averages nor individual ACFs are well-fit129\nexponential functions.130\nTable 1 Statistics to show that control subjects have shorter time\nscales than Parkinson’s using ACF timescale measure; see Figure\n2C. Using Wilcoxon rank-sum test where the null hypothesis is that both data\nsamples are drawn from the same distribution. Top shows p−values, bottom\nshows effect size (see Methods: Wilcoxon Rank-Sum Test).\nRelationship / p−value δ band θ band α band β band\nCntrl. vs. Park. (Off drugs) 1.5 × 10−2 4.6 × 10−5 0.33 0.18\nCntrl. vs. Park. (On drugs) 6.6 × 10−2 2.2 × 10−3 0.42 0.24\n(Park.) On vs. Off 0.43 0.27 0.91 0.88\nRelationship / Effect Size δ band θ band α band β band\nCntrl. vs. Park. (Off drugs) 0.44 (med) 0.73 (lrg) 0.41 (n/a) 0.24 (n/a)\nCntrl. vs. Park. (On drugs) 0.33 (med) 0.55 (lrg) 0.15 (n/a) 0.21 (n/a)\n(Park.) On vs. Off 0.14 (n/a) 0.13 (n/a) 2.3 × 10−2 (n/a) 2.7 × 10−2 (n/a)\nNext we perform DFA analysis, which characterizes how flucuations vary across different timescale,131\nalso known as long range temporal correlation ( LR TC) analysis. In the DFA analysis we find that132\ncontrol subjects have on average shorter range temporal correlation than Parkinson’s patients, consistent133\nwith the lower frequency bands in the ACF timescale analysis. A demonstration of the DFA method is134\ndepicted in Figure 3A on a control subject’s resting state EEG in the delta-band where the fluctuation135\namplitude as a function of time window length in log-log coordinates requires 2 lines at a manually chosen136\ndividing point; such a dividing point is required in about 72% to 85% of the time (counting all frequency137\nbands, subject type, and electrode combinations). In such cases, we use the slope of the best fit line for138\n5\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\n0\n0.5\n1\n1.5\n2\n10 -1 \n10 0\n*** \n** \nδ\n[1,4]Hz\nθ\n[4,8]Hz\nα \n[8,13]Hz\nβ \n[13,30]Hz\n*** \n***\n10 \n-2 10 \n-1 \n10 \n0\nTime (s)\nδ\nθα\nβ\n(das\nhed) \n0\n0.2\n0.4\n0.6\n0.8\n1\nα\nA\nAutocorrelation Funct. \nHealthy\nParkinson’s (Off Drugs)\nParkinson’s (On Drugs)\nC\nθ\nZoomed-in to see black \ncurve below\nD\nPop. Avg. \nACF Time-scale (s) \nδ\n[1,4]Hz\nθ\n[4,8]Hz\nα \n[8,13]Hz\nβ \n[13,30]Hz\nHealthy\nParkinson’s (Off Drugs)\nParkinson’s (On Drugs)\nB\nACF Time-scale (s) \n10 -2 10 0-0.2\n0\n0.2\n0.4\n0.6\n0.8\n1\nTime (s)\nAutocorrelation Funct. \nChosen Threshold\nACF Time-scale= \nTime first cross \nbelow threshold\nFig. 2 Emergent Parkinson’s oscillations have large autocorrelation time. A) The population-averaged ACF in\nParkinson’s patients has longer timescales (red, blue: slower decay) in motor cortex EEG activity across all 4 frequency\nbands than healthy (control). ACF in alpha- and theta-band are zoomed-in and shifted for clarity. B) Example calculation\nof ACF timescale, i.e., a measure of ACF decay time, for 2 subjects; the first time where a subject’s ACF falls below a\nchosen threshold of 0.1\n. C) Summary of ACF timescale with box plots in different frequency bands shows that control\nsubjects on average have faster ACF time decay (smaller timescale) than Parkinson’s patients for the lower frequency bands\n(delta and theta). The horizontal lines in the boxes represent inter-quartiles: 25 th percentile, median, and 75 th percentile.\nDifference in distributions are statistically significant measured by Wilcoxon rank-sum test (see Table 1 for details). D)\nThe population means of ACF timescale are plotted for completeness.\nlarger time windows (second segment to the right), and call this the DFA coefficient. When there are no139\ntemporal correlations (i.e., white noise) the DFA coefficient is 0.5. In a random walk, temporal memory140\nis infinite and the DFA exponent is 1.5. DFA coefficients between 0.5 and 1.5 indicate intermediate141\ntemporal correlations. A summary of all DFA coefficients is shown with box plots in Fig 3B with four142\nfrequency bands: on average control subjects have a much shorter range of temporal correlation, i.e.,143\ntimescales, than Parkinson’s patients (on or off drugs). The trend that control subjects have shorter144\ntimescales than Parkinson’s patients is robust across all four frequency bands we consider, with the145\nalpha-band results having comparatively weaker results with larger p−values using Wilcoxon rank-sum146\ntest. The DFA coefficients shown include all 16 control subjects, but for Parkinson’s 1 or 2 patients were147\nexcluded (depending on frequency band) because the fluctuation amplitudes for a few subjects were too148\nvariable to be well fit by a line (see Fig S3 and Supplementary Text S1). Figure 3C clearly demonstrates149\nhow different the population averages are; control subjects have much smaller average DFA coefficients150\n6\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nthan Parkinson’s, and Parkinson’s patients have similar DFA coefficients regardless of whether on or off151\ndrug treatments.152\n0.5\n1\n0.75\nWhite noise\n*** \n*\n** \n  *\n*** \n** \nδ \n[1,4]Hz\nθ \n[4,8]Hz\nα \n[8,13]Hz\nβ \n[13,30]Hz\n*** \n***\n*     0.05 < p < 0.1 \n**   0.01 < p < 0.05\n***  p<0.01 \nA\nHealthy\nParkinson’s (Off Drugs)\nParkinson’s (On Drugs)\nB\nDFA Coeff. \n0.3 0.5 1 2 4 6\n6\n7\n8\n9\nTime Window Length (s)\nFlucuation Amplitude F Data\nBest fit line(s)\nSlope of line = 0.72\nSlope of line  in smaller \nwindows= 1.31 (not used)\nDFA Method C\n0.5\n0.55\n0.6\n0.65\n0.7\n0.75\n0.8\nPop. Avg. DFA Coeff .\nδ \n[1,4]Hz\nθ \n[4,8]Hz\nα \n[8,13]Hz\nβ \n[13,30]Hz\nHealthy\nParkinson’s (Off Drugs)\nParkinson’s (On Drugs)\nFig. 3 Emergent Parkinson’s oscillations have larger DFA exponents. A) Example DFA coefficient calculation\n(control subject 1 in delta-band) well-fit with 2 line segments, where a choice for the time window for where to segment\nthe data has to be made. When 2 lines are used, the slope of the right line segment for larger time windows is reported. B)\nSummary of DFA coefficients with box plots in different frequency bands is largely consistent with the ACF results (Fig\n2C,D). Box plot convention are the same as in Figure 2C. The results are not as strong in the alpha-band. Difference in\ndistributions are statistically significant measured by Wilcoxon rank-sum test (see Table 2 for details). C) The population\nmeans of DFA coefficients are plotted to clearly illustrate that control subjects are further from criticality/scale-invariance\nthan Parkinson’s patients.\nTable 2 Statistics to show that control subjects have shorter time scales\nthan Parkinson’s using DFA coefficient; see Figure 3B . Using Wilcoxon\nrank-sum test where the null hypothesis is that both data samples are drawn from the\nsame distribution. Top shows p−values, bottom shows effect size (see Methods:\nWilcoxon Rank-Sum Test).\nRelationship / p−value δ band θ band α band β band\nCntrl. vs. Park. (Off drugs) 9.4 × 10−3 9.4 × 10−3 8.6 × 10−2 2.8 × 10−2\nCntrl. vs. Park. (On drugs) 7.7 × 10−2 5.7 × 10−3 2.4 × 10−2 9.1 × 10−3\n(Park.) On vs. Off 0.28 0.51 0.89 0.96\nRelationship / Effect Size δ band θ band α band β band\nCntrl. vs. Park. (Off drugs) 0.47 (med) 0.47 (med) 0.31 (med) 0.11 (sm)\nCntrl. vs. Park. (On drugs) 0.32 (med) 0.51 (lrg) 0.42 (med) 0.14 (sm)\n(Park.) On vs. Off 0.2 (n/a) 0.13 (n/a) 0.26 (n/a) 1.5 × 10−3 (n/a)\nAlthough both ACF and DFA results provide evidence that control subjects’ EEG motor cortex153\nactivity is further from criticality than Parkinson’s patients, these analyses do not directly measure154\ndistance to criticality. To this end, we developed a rigorous tRG theory and implemented pragmatic155\ncomputational tools to directly calculate distance to criticality ( d2). The d2 measure has several specific156\nadvantages : i) unlike DFA, it does not require specifically choosing a time window segment and assessing157\nquality of linear fits (Fig S3), ii) unlike ACF, it does not require a prescribed threshold to find timescale,158\niii) the distance to criticality d2 (Fig 4A) is a precise quantification of distance independent of model159\nparameterization, calculated in units of bits/sec (the bits/sec quantifies accumulation of evidence for160\nruling out being at criticality). Our framework requires first fitting an auto-regressive (AR) model to the161\ndata, then calculating the distance of the fitted model to the critical state; see Methods: T emporal162\nRenormalization Group Theory and Figs S5–S7 for further details.163\n7\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nA BSpace of all AR models\nAR models at criticality \nBest fit AR \nd2\nC\nHealthy\nParkinson’s (Off Drugs)\nParkinson’s (On Drugs)\nTime bin (s)\nHealthy Parkinson’s (On Drugs) Parkinson’s (Off Drugs)\nPop. d2  (bits/s) \nδ  [1,4]Hz\nθ  [4,8]Hz\nα  [8,13]Hz\nβ  [13,30]Hz\n0 0.05 0.1 \n10 0\n0 0.05 0.1 \n10 0\n0 0.05 0.1 \n10 0\n0 1 2\nTemporal Reach (s)\n10 0\nKL distance \nδ  [1,4]Hz θ  [4,8]Hz α  [8,13]Hz β  [13,30]Hz\n0 1 2\nTemporal Reach (s)\n10 0\n0 1 2\nTemporal Reach (s)\n10 0\nPop. d2  (bits/s) \n0 1 2\nTemporal Reach (s)\n10 0\nD\n1\np- value \n0.05\n0.01\nHealthy = Park. (Off) \nHealthy = Park. (On)\nEffect Size \n0\n0.2\n0.4\n0.6\n0.8\nLarge\nMedium\nSmall\n0 1 2\nTemporal Reach (s)\n0 1 2\nTemporal Reach (s)\n0 1 2\nTemporal Reach (s)\n0 1 2\nTemporal Reach (s)\nFig. 4 Emergent Parkinson’s oscillations are closer to criticality. A) Using tRG theory to quantify differences in\ndistance to criticality between controls and Parkinson’s patients after data is fit with an AR model. B) The population\nd2 (bits/s) values (log-scale) grouped by control and two Parkinson’s state as a function of coarse-grained time bin with\nAR model order 20 shows little difference across different band-passed frequencies. C) Summary of population d2 values\n(log-scale) for many time bins and model orders; the x-axis represents the ‘temporal reach’, i.e., model order multiplied by\ntime bin length (varies from 2 ms to 100 ms). The control subjects consistently had larger d2 and were thus further from\ncriticality than Parkinson’s patients, independent of model order, time bin length, or frequency band. D) Quantifying the\nstatistical significance of our results using Wilcoxon rank-sum test, showing the p−values (log-scale) and effect sizes (see\nMethods: Wilcoxon Rank-Sum Test ). The different shades of colors in C) and D) correspond to AR model fits of\ndifferent orders ranging from 16 to 24.\nWe perform a detailed comparison of Parkinson’s and control subjects using d2 and find control164\nsubjects’ EEG in primary motor cortex are generally further from criticality than Parkinson’s patients.165\nWe specifically vary the AR model order (16 to 24) as well as the time bin length (2 ms up to 100 ms) – we166\npreviously showed that increasing the time bin can unveil critical dynamics (Fontenele et al., 2024) and167\nthat d2 is expected to decrease monotonically with increasing model order and with increasing time bin168\nlength (Sooter et al., 2025). Figure 4B shows, within a given subject group (healthy, Parkinson’s on/off169\ndrugs) for a fixed AR model order 20, that d2 tends to decrease as time bin length increases, except170\noccasionally in the delta-band (light blue), and that there are minor differences in population d2 across171\n8\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nfrequency bands with a given time bin. Varying AR model order and time bin length simultaneously172\nresults in a wide variety of ‘ temporal reach ’ (x-axis of Fig 4C,D) defined as the the AR model order173\nmultiplied by the specific time bin, i.e., the maximal time in the past that can influence present AR model174\nvalue. The temporal reach values we consider have a large range from 32 ms to 2.4 s. The population175\naveraged d2 for many temporal reach values is shown in Figure 4C (y-axis is log-scale, different color176\nshades correspond to different AR model order), where it is evident that Parkinson’s patients (red and177\nblue dots) are closer to criticality (d 2 below) than controls (black/gray) in the delta- and theta-bands.178\nWe use Wilcoxon rank-sum test to analyze whether differences between Parkinson’s and control are179\nstatistically significant under the null hypothesis that the values were generated from the same probability180\ndistribution (p−values in top row of Fig 4D). The differences are clear in the delta- and theta-bands for a181\nwide range of temporal reach values, there is no differences in the alpha band, and differences in the beta-182\nband are only evident with small temporal reach values. The effect size and a qualitative characterization183\nof effect size (small, medium, large (Cohen, 2013; Tomczak and Tomczak, 2014)) is shown in the bottom184\nrow of Figure 4D.185\nDiscussion186\nHere we have shown that the prominentδ andθ band oscillations that emerge in Parkinson’s disease are,187\nin fact, near-critical oscillations. Although each of these oscillations is defined by particular timescales188\n(the oscillation periods), the power (amplitude) of these oscillations exhibits fluctuations across a wide189\nrange of time scales. These amplitude fluctuations are approximately scale invariant, which is how critical190\noscillations are defined (Fontenele et al., 2025; Palva and Palva, 2018). In contrast, in healthy controls,191\nthe same frequency bands have amplitude fluctuations that are further from criticality.192\nThe distance measure d2 enables a fair comparison of different time series, and is a rigorous193\ninformation-theoretic entity in units of bits/s that measures the amount of evidence for ruling out the194\nhypothesis that the data are at criticality (Sooter et al., 2025). Although the ACF and DFA analy-195\nses yielded similar results, d2 is a direct measure for distance to temporal scale-invariance, and proved196\nto be cleaner for delineating differences (control vs. Parkinson’s), and did not require making specific197\nchoices regarding threshold cut-offs, which time window segments to use, etc. Unlike traditional methods,198\nthe analysis with d2 goes beyond just measuring timescales, and also clearly shows how the differences199\ndepend on the ‘temporal reach’, and that distances to criticality tend to decrease with increasing tempo-200\nral reach. For both DFA andd2, the strongest separation between control and Parkinson’s patients are in201\nthe delta- and theta-bands, followed by the beta-band, with the weakest results in the alpha-band. The202\nACF timescales only had significant differences in the delta- and theta- band. The relative consistency203\n9\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nof these results suggests that there are real and surprising differences between control and Parkinson’s204\npatients in motor cortex EEG.205\nOur results in motor cortex are at odds with the idea that, in general, healthy brains operate closer206\nto criticality than pathological ones (Hengen and Shew, 2025; O’Byrne and Jerbi, 2022), as discussed in207\nthe Introduction. However, our results are in line with two recent publications where it was reported that208\nParkinson’s patients i) have more frequency bands closer to ‘edge of chaos’ (a point related to criticality)209\nthan control subjects in whole brain MEG (Calvo et al., 2024), and ii) can have longer timescales as210\nmeasured with DFA coefficients in whole brain EEG in the theta-band (Lee et al., 2024). These studies211\nare different than ours because they focused on whole brain imaging and included many more subjects212\nwith which they aggregated/averaged. The reasonable number of subjects enabled detailed analysis, for213\nexample to assess the quality of model fits for each subject.214\nAlong these lines, another recent study showed that a measure of ‘intrinsic neural timescale’ using215\nfMRI was longer in late stage Parkinson’s patients than in healthy controls in the anterior cortical region216\n(Wei et al., 2024). This study is in line with our results, but unlike ourd2 their measure of intrinsic neural217\ntimescale is indirect because it involves calculating when various ACFs first cross below a threshold,218\nsmoothing the maximum of those values over space and applying a z-transform. The timescales of fMRI219\nmeasurements are coarser than those of EEG, with resolution on the order of seconds, so we cannot make220\nany direct comparisons with our results.221\nInterestingly, Parkinson patients on versus off drugs to treat motor symptoms did not ever have222\nstatistically significant differences in their motor cortex EEG, independent of the methods (ACF, DFA,223\ntRG). Presumably, these drugs helped mitigate their motor symptoms to some extent, but the motor224\ncortex activity that is responsible for voluntary movement planning and muscle control did not exhibit225\nany changes in timescales. Thus, it stands to reason that the timescales of the EEG in motor cortex226\nmight not be a direct reflection of mitigated motor symptoms, but rather a wholesale difference between227\nParkinson’s disease and control is manifested in these timescales.228\nMethods229\nEthics statement230\nThis article presents an accurate account of the work performed by the stated authors, and all underlying231\ndata are represented accurately with consent from the owners. To the best of our knowledge, this work232\nis original and is not under consideration for publication elsewhere. The study used publicly available233\ndata with accurate citation, and all methods were performed in accordance with relevant guidelines and234\nregulations. The authors declare no conflicts of interest related to this research.235\n10\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nThis study uses third party human EEG data that is publicly available (George et al., 2013; Swann236\net al., 2015; Jackson et al., 2019) (see Data and code availability section). The Materials and methods237\nsection in their papers explicitly state that ‘All the participants provided written informed consent238\naccording to an Institutional Review Board Protocol at the University of California, San Diego and the239\nDeclaration of Helsinki’. We have also obtained written approval from the authors to use their data in240\nthis study.241\nParkinson’s patient characteristics242\nTable 3 shows side of physical impairment in the Parkinson’s patients.243\nTable 3 The side of reported physical\nimpairment in Parkinson’s patients (Appelhoff\net al., 2019; Rockhill et al., 2021) and thus\ncorresponding electrode used. Electrode C3 is on\nthe left motor cortex, C4 on the right motor\ncortex. Note that all subjects had the same side\nfor physical impairment on and off drug\ntreatment except for subject 14 who switched to\nRight side (C3) while on drug treatment.\nSubject\nImpairment Side Electrode\n1 Right C3\n2 Both Both\n3 Right C3\n4 Right C3\n5 Left C4\n6 Right C3\n7 Left C4\n8 Left C4\n9 Left C4\n10 Right C3\n11 Right C3\n12 Right C3\n13 Left C4\n14 Left* C4*\n15 Right C3\nAutocorrelation function244\nThe autocorrelation function is a common tool used to characterize how related (correlated) a time series\nof data x(t) is with specific time lags τ. The autocovariance function of a time series x(t) is:\n˜A(τ) := Et[x(t)x(t +τ)]−\n(\nEt[x(t)]\n)2\n, (1)\nand the autocorrelation function is simply:\nA(τ) = ˜A(τ)/σ2\nX (2)\n11\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nwhere σ2\nX := Et\n[(\nx(t)−µX\n)2]\nis the point-wise variance of the time series. We calculate the auto-245\ncorrelation function of a particular time series of EEG via the Matlab function autocorr on centered246\ndata x(t)− Et[x(t)] with 10,000 lags (recall the time bins are 2 ms) and 1.96 standard deviations:247\nautocorr(X-mean(X),‘NumLags’,10000,‘NumStd’,1.96). The results are in Figures 2, S1.248\nDetrended Fluctuation Analysis (DF A)249\nDFA is a common method for quantifying the degree of long-range temporal correlations (Peng et al.,250\n1994). For a given time-series, the DFA coefficient was calculated by assessing the correlation of fluc-251\ntuation amplitudes in various time window lengths. Start with a time-series xj, then calculate the252\ncumulative sum:Yt :=\nt∑\nj=1\nxj. The ‘entire’ Yt time-series is divided into n equal lengths for the duration253\nof the specified time window τx of length (τ x/dt + 1) – if the length of Yt cannot be evenly divided,254\nthe end of the time-series is truncated, so n = ⌊N/(τx/dt + 1)⌋. Then for each segment of length255\n(τx/dt + 1) the local trend (least squares linear fit Lk) is calculated. After which the mean-squared256\ndeviation is calculated: G(n,i ) = 1\nn\n(i−1)n+n∑\nt=(i−1)n+1\n(Yt−Lk(t))2, then the mean fluctuation amplitude is:257\nF (n) =\n\n√\n1\n⌊N/n⌋\n⌊N/n⌋∑\ni=1\nG(n,i ). Finally, the least squares linear fit between log( n) (horizontal axis) and258\nlog(F (n)) (vertical axis) is calculated – the slope of this line is called the DFA coefficient.259\nIn cases where log( n) versus log(F (n)) is not well-fit by a single line, the time windows are split260\nin two segments determined manually, then two least squares linear fits are calculated with the larger261\nwindows (right half) determining the DFA coefficient (Gu et al., 2015).262\nTemporal renormalization group (tRG) theory263\nA system is at criticality if (1) it lies at a boundary between qualitatively different operating regimes264\nand (2) it exhibits scale-invariance, i.e. the lack of a characteristic spatial or temporal scale (Hengen265\nand Shew, 2025). The renormalization group ( RG), which was originally developed to study critical266\nphenomena in condensed matter systems, brings mathematical precision to these statements. The core267\nidea of RG is to gradually remove the fine-scale details of a model to generate new, effective models at268\ncoarser scales. Fixed points of the RG operation therefore correspond to models that are scale-invariant,269\nand all of the models in the basin of attraction of such a fixed point share the same coarse-scale behavior270\n- this is the fundamental reason why, for example, water and ferromagnets poised near their respective271\nphase transitions have quantitatively identical scaling exponents despite their drastic dissimilarities at272\nmicroscopic length scales. Some RG fixed points are stable, meaning that there is an extended region in273\nmodel space surrounding the fixed point such that every model in the region flows into the fixed point.274\n12\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nSuch fixed points fail to satisfy condition (1) in the definition of criticality. (For example, the RG fixed275\npoint corresponding to the disordered phase of an Ising model is stable.) Models lying in the basins of276\nattraction of unstable fixed points, on the other hand, are both scale-invariant and poised at a boundary277\nbetween different operating regimes, and hence are at criticality.278\nIn traditional applications of RG to spatially organized systems (e.g. Ising-type models), coarse-\ngraining is implemented in space. Neural systems, on the other hand, can have rich temporal dynamics\nin an measurable entity (i.e., population EEG) independent of whether there is weak or crucial spatial\nstructure. In Sooter et al. (2025), we argued that the appropriate way to define criticality in such systems\nis with a temporal RG ( tRG), wherein high-frequency features of a model are gradually removed to\nreveal its asymptotic behavior at low frequencies. We applied this procedure to a fundamental class of\nunivariate discrete-time stochastic dynamical systems, Gaussian autoregressive ( AR) models:\nxt =\nn∑\nj=1\nφjxt−j +ξt. (3)\nWe chose AR models both because they are analytically tractable, and because they are the optimal\nchoice in a precise max ent sense. Specifically, if an observed time series is short enough that we can\nonly confidently estimate its second-order (autocovariance) statistics, then AR models are the maximum\nentropy (i.e. minimally presumptive) way to model those statistics (Choi and Cover, 1984). In an AR\nmodel, the state xt at time t is a linear, Gaussian readout of the recent history (up to some maximum\nlag n, called the model order):\nxt∼N\n( n∑\nk=1\nφkxt−k,σ 2\n)\n.\nIn the space of order-n AR models, there are n+1 tRG fixed points, which we can label using the power-279\nlaw exponents of their respective power spectra, β = 0, 2,..., 2n. The β = 0 fixed point is stable and280\ncorresponds to white noise - this is the ”trivial” fixed point that any AR model with a finite characteristic281\ntimescale flows into. The basins of attraction of the β≥ 2 fixed points constitute the AR models that282\nare at criticality.283\nNext, we asked how we should quantify proximity to these basins. That is, having determined which\nAR models are at criticality, can we say which ones are close to criticality? Naively, we could measure\nthe Euclidean distance (in the parameter space defined by the AR model history kernel φ) from an AR\nmodel to each of the basins of attraction. However, there is no principled reason to use Euclidean distance\nrather than some other metric. To resolve this ambiguity, we turned to information theory and define\nproximity to criticality as distinguishability (per unit time) from a system at criticality. Specifically, for\n13\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\na given order-n AR model B, let:\ndβ(B) := inf\nA∈A(n)\nβ\nlim\nT→∞\n1\nTKL (PB(x1,...,x T )||PA(x1,...,x T )) (4)\nwhere A(n)\nβ is the set of order- n AR models that flow into the β fixed point, PA(x1,...,x T ) is the284\nprobability distribution for a T−step draw from the AR model A (and similarly for PB), and KL(·||·)285\nis the Kullback-Leibler divergence. The structure of basins of attraction is such that that infimum taken286\nover the set of all critical models ∪β≥2A(n)\nβ is equal to the infimum taken over the β = 2 basin of287\nattraction (Sooter et al., 2025); hence we only report d2 in this paper.288\nTo estimated2 from EEG data after bandpass filtering and extracting the envelope, we: 1) fit an AR289\nmodel to the data using the Yule-Walker method, and 2) compute d2 for this model using Eq (4).290\nWilcoxon Rank-Sum Test291\nWe use the Wilcoxon rank-sum test (WRST) because it is ideal for the EEG. It is a nonparametric test\nof the null hypothesis that two groups of data are generated from the same distribution. The p−values\nof this test correspond to the probability that the null hypothesis holds. In addition, we report the Effect\nSize of the WRST:\nEffect Size := |z|√n1 +n2\n(5)\nwhere z is the z−score of the U−statistic, z = (U−µU)/σU and nj are the sample sizes for the two292\npopulations. Effect sizes fort−test and Wilcoxon rank-sum test with values: (0, 0.2] are considered small,293\n(0.2, 0.5] are medium, (0.5, 0.8] and above are large (Cohen, 2013; Tomczak and Tomczak, 2014); these294\nlabels are simply a qualitative assessment.295\nEEG Data296\nWe used freely available EEG collected years ago that have appeared in many studies (Jackson et al., 2019;297\nSwann et al., 2015; George et al., 2013) and was made widely applicable following common standards298\n(Pernet et al., 2019; Appelhoff et al., 2019). We used all 16 control (control) subjects and all 15 Parkinson’s299\npatients except for some of the DFA coefficients (see Figure S3). We used an EEG reader function by300\nTcheslavski (2025).301\nFrequency Band Limits302\nThe frequency bands of interest were limited with an upper bound in the beta band (30 Hz) because the303\n60 Hz grounding frequency (inferred by the power spectrum of all electrodes have a large peak at 60 Hz);304\n14\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint \n\nany signals close to 60 Hz are considered artifacts perhaps due to electrical interference. For completeness305\nand since some EEG studies report results in the gamma band frequency (George et al., 2013; Swann306\net al., 2015), we repeated our analysis on a lower gamma band frequency between 30 and 50 Hz (as was307\ndone in George et al. (2013), and found that the general trend of results we observed did not hold (see Fig308\nS4, except for the population averaged ACF decay Fig S4A). We note however that in this lower gamma309\nfrequency band, the DFA analysis was messier than the other 4 lower frequency bands; in particular for310\nthe control subjects where 5 were excluded (see GitHub page), and in one of the ACF timescales was311\nunusually long (longer than 20 s). Thus, these results should be taken with caution.312\nData and code availability313\nDeclarations314\nData availability . The raw EEG dataset was collected at UC San Diego from a team of researchers,315\nit is freely available (Rockhill et al., 2021) at https://openneuro.org/datasets/ds002778/versions/1.0.2.316\nCode availability . See https://github.com/chengly70/parkeeg for MATLAB code implementing all317\ncomputational components in this paper.318\nAuthor Contributions. Conceptualization: CL, WLS. Methodology: CL, JSS, WLS. Sofware: JSS,319\nCL. Validation: CL. Formal Analysis: CL. Investigation: JSS, CL. Resources: N/A. Data Curation:320\nCL. Writing original draft: CL, WLS, JSS, AKB. Writing review and editing: CL, JSS, AKB, WLS.321\nVisualization: CL. Supervision: CL. Project administration: CL. Funding acquisition: CL, JSS, AKB,322\nWLS.323\nF unding. This study was supported by the National Institute on Drug Abuse (NIDA), National Insti-324\ntutes of Health (NIH) under grant 1R01DA060744, part of the BRAIN Initiative (CL, JSS, AKB,325\nWLS).326\nDeclaration of Competing Interests. The authors declare that no competing interests exist. The327\nfunders had no role in study design, data collection and analysis, decision to publish, or preparation of328\nthe manuscript.329\n15\n.CC-BY 4.0 International licenseperpetuity. 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Frontiers in Neural468\nCircuits 14: 54. https://doi.org/10.3389/fncir.2020.00054 .469\n21\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 10, 2026. ; https://doi.org/10.64898/2026.01.09.698590doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}