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  • Open Access

Brain criticality through nonadditive entropic analysis of electroencephalograms

Henrique S. Lima1,*, Constantino Tsallis1,2,3,4, and Dimitri M. Abramov5

  • *Contact author: hslima94@cbpf.br
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APS Open Sci. 1, 000107 – Published 25 August, 2026

DOI: https://doi.org/10.1103/p73j-nq1w

Abstract

On the grounds of nonadditive entropies—appropriate for complex systems—we investigate the electroencephalogram amplitudes of typical and attention-deficit hyperactivity disorder (ADHD) children. The corresponding probability distributions are q-Gaussians, i.e., ρ(x)∝eq−βx2≡[1+(q−1)βx2]1/(1−q), where (q,β) are, respectively, the entropic index characterizing complexity and the inverse width. We show that q tends to monotonically vary with β for both typical and ADHD subjects, thus strongly suggesting a scalinglike organization of electroencephalogram amplitude statistics compatible with nearly critical or multiscale brain dynamics. Moreover, we verify that ADHD subjects have a higher complexity than the typical ones. Consistently, biomarkers for objective psychiatric diagnosis could emerge along this path.

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References (42)

  1. American Psychiatric Association, Diagnostic and Statistical Manual of Mental Disorders, 5th ed. (American Psychiatric Association, Washington, DC, 2022).
  2. D. M. Abramov, Complexity and psychopathology: From mechanistic science to a physical understanding of the mental conditions based on chaos and complexity, Front. Psychiatry 17, 1764422 (2026).
  3. J. O’Byrne and K. Jerbi, How critical is brain criticality? Trends Neurosci. 45, 820 (2022).
  4. Y. Tian, Z. Tan, H. Hou, G. Li, A. Cheng, Y. Qiu, K. Weng, C. Chen, and P. Sun, Theoretical foundations of studying criticality in the brain, Netw. Neurosci. 6, 1148 (2022).
  5. V. Zimmern, Why brain criticality is clinically relevant: A scoping review, Front. Neural Circuits 14, 54 (2020).
  6. B. S. Peterson, J. Trampush, M. Brown, M. Maglione, M. Bolshakova, M. Rozelle, J. Miles, S. Pakdaman, S. Yagyu, A. Motala, and S. Hempel, Tools for the diagnosis of ADHD in children and adolescents: A systematic review, Pediatrics 153, e2024065854 (2024).
  7. S. Koirala, G. Grimsrud, M. A. Mooney, B. Larsen, E. Feczko, J. T. Elison, S. M. Nelson, J. T. Nigg, B. Tervo-Clemmens, and D. A. Fair, Neurobiology of attention-deficit hyperactivity disorder: Historical challenges and emerging frontiers, Nat. Rev. Neurosci. 25, 759 (2024).
  8. A. L. Goldberger, C. K. Peng, and L. A. Lipsitz, What is physiologic complexity and how does it change with aging and disease? Neurobiol. Aging 23, 23 (2002).
  9. A. C. Yang and S. J. Tsai, Is mental illness complex? From behavior to brain, Prog. Neuro-Psychopharmacol. Biol. Psychiatry 45, 42 (2013).
  10. A. Chatterjee, G. Georgiev, and G. Iannacchione, Aging and efficiency in living systems: Complexity, adaptation and self-organization, Mech. Ageing Dev. 163, 2 (2017).
  11. H. Hadoush, M. Alafeef, and E. Abdulhay, Brain complexity in children with mild and severe autism spectrum disorders: Analysis of multiscale entropy in EEG, Brain Topogr. 32, 914 (2019).
  12. C. Gu, Z. X. Liu, and S. Woltering, Electroencephalography complexity in resting and task states in adults with attention-deficit/hyperactivity disorder, Brain Commun. 4, fcac054 (2022).
  13. Z. J. Lau, T. Pham, S. H. A. Chen, and D. Makowski, Brain entropy, fractal dimensions and predictability: A review of complexity measures for EEG in healthy and neuropsychiatric populations, Eur. J. Neurosci. 56, 5047 (2022).
  14. Niedermeyer’s Electroencephalography: Basic Principles, Clinical Applications, and Related Fields, 6th ed., edited by E. Niedermeyer and F. L. da Silva (Lippincott Williams & Wilkins, Philadelphia, PA, 2011).
  15. F. Freyer, K. Aquino, P. A. Robinson, P. Ritter, and M. Breakspear, Bistability and non-Gaussian fluctuations in spontaneous cortical activity, J. Neurosci. 29, 8512 (2009).
  16. G. Wood, K. Willmes, J. W. Koten, and S. E. Kober, Fat tails and the need to disclose distribution parameters of qEEG databases, PLoS One 19, e0295411 (2024).
  17. K. Linkenkaer-Hansen, V. V. Nikouline, J. M. Palva, and R. J. Ilmoniemi, Long-range temporal correlations and scaling behavior in human brain oscillations, J. Neurosci. 21, 1370 (2001).
  18. T. Donoghue et al., Parameterizing neural power spectra into periodic and aperiodic components, Nat. Neurosci. 23, 1655 (2020).
  19. J. M. Beggs and D. Plenz, Neuronal avalanches in neocortical circuits, J. Neurosci. 23, 11167 (2003).
  20. O. Shriki et al., Neuronal avalanches in the resting MEG of the human brain, J. Neurosci. 33, 7079 (2013).
  21. J. Touboul and A. Destexhe, Power-law statistics and universal scaling in the absence of criticality, Phys. Rev. E 95, 012413 (2017).
  22. V. Priesemann and O. Shriki, Can a time-varying external drive give rise to apparent criticality in neural systems? PLoS Comput. Biol. 14, e1006081 (2018).
  23. D. M. Abramov, C. Tsallis, and H. S. Lima, Neural complexity through a nonextensive statistical-mechanical approach of human electroencephalograms, Sci. Rep. 13, 10318 (2023).
  24. D. M. Abramov, H. S. Lima, V. Lazarev, P. R. Galhanone, and C. Tsallis, Identifying attention-deficit/hyperactivity disorder through the electroencephalogram complexity, Physica A 653, 130093 (2024).
  25. D. M. Abramov, D. F. Quintanilha, H. S. Lima, R. P. Costa, C. Kamil-Leite, V. V. Lazarev, and C. Tsallis, Neurophysiological correlates to the human brain complexity through q-statistical analysis of electroencephalogram, Sci. Rep. 15, 36502 (2025).
  26. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988).
  27. A. M. C. de Souza and C. Tsallis, Student’s t- and r-distributions: Unified derivation from an entropic variational principle, Physica A 236, 52 (1997).
  28. C. Tsallis and D. J. Bukman, Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54, R2197 (1996).
  29. M. L. Lyra and C. Tsallis, Nonextensivity and multifractality in low-dimensional dissipative systems, Phys. Rev. Lett. 80, 53 (1998).
  30. E. K. Lenzi, L. C. Malacarne, R. S. Mendes, and I. T. Pedron, Anomalous diffusion, nonlinear fractional Fokker-Planck equation and solutions, Physica A 319, 245 (2003).
  31. C. Tsallis, M. Gell-Mann, and Y. Sato, Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive, Proc. Natl. Acad. Sci. USA 102, 15377 (2005).
  32. S. Umarov, C. Tsallis, and S. Steinberg, On a q-central limit theorem consistent with nonextensive statistical mechanics, Milan J. Math. 76, 307 (2008).
  33. D. M. Abramov, C. Q. Cunha, P. R. Galhanone, R. J. Alvim, A. M. de Oliveira, and V. V. Lazarev, Neurophysiological and behavioral correlates of alertness impairment and compensatory processes in ADHD evidenced by the Attention Network Test, PLoS One 14, e0219472 (2019).
  34. A. Pluchino, A. Rapisarda, and C. Tsallis, Noise, synchrony, and correlations at the edge of chaos, Phys. Rev. E 78, 046106 (2008).
  35. L. F. Burlaga and N. F. Ness, Intermittency and q-Gaussian distributions in the magnetic field of the Very Local Interstellar Medium (VLISM) observed by Voyager 1 and Voyager 2, Astrophys. J. Lett. 901, L2 (2020).
  36. C. Beck and E. G. D. Cohen, Superstatistics, Physica A 322, 267 (2003).
  37. C. Beck, Dynamical foundations of nonextensive statistical mechanics, Phys. Rev. Lett. 87, 180601 (2001).
  38. M. A. Pires, C. Tsallis, and E. M. F. Curado, Composing α-Gauss and logistic maps: Gradual and sudden transitions to chaos, Phys. Rev. E 112, 034209 (2025).
  39. G. Ruiz and A. F. de Marcos, Evidence for criticality in financial data, Eur. Phys. J. B 91, 1 (2018).
  40. A. Greco, C. Tsallis, A. Rapisarda, A. Pluchino, G. Fichera, and L. Contrafatto, Acoustic emissions in compression of building materials: Q-statistics enables the anticipation of the breakdown point, Eur. Phys. J. Spec. Top. 229, 841 (2020).
  41. R. M. de Oliveira, S. Brito, L. R. da Silva, and C. Tsallis, Statistical mechanical approach of complex networks with weighted links, J. Stat. Mech. (2022) 063402.
  42. K. Briggs and C. Beck, Modelling train delays with q-exponential functions, Physica A 378, 498 (2007).

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APS Open Sci. 1, 000107– Published 25 August, 2026
Vol. 1
  • Received 5 May 2026
  • Revised 20 July 2026
  • Accepted 17 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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