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arxiv: 1907.10060 · v1 · pith:QKRYCWJ2new · submitted 2019-07-23 · 🧬 q-bio.NC

Time irreversibility of resting brain activity in the healthy brain and pathology

Massimiliano Zanin , Bahar G\"untekin , Tuba Akt\"urk , L\"utf\"u Hano\u{g}lu , David Papo This is my paper

Pith reviewed 2026-05-24 16:50 UTC · model grok-4.3

classification 🧬 q-bio.NC
keywords time irreversibilityEEGresting-statebrain pathologytime asymmetryelectroencephalographydynamical mechanisms
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The pith

Resting brain activity is generically time-irreversible at long scales, with pathology reducing the asymmetry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that spontaneous electroencephalographic activity in the resting brain is time-irreversible at sufficiently long time scales. This irreversibility is a generic feature in healthy individuals under both eyes-open and eyes-closed conditions. In contrast, various brain pathologies are linked to a reduction in this time asymmetry, with patterns that differ according to the specific condition. The analysis interprets these findings in terms of underlying physical generating mechanisms rather than just statistical properties. This distinction between health and pathology in dynamical properties may help explain how brain function changes in disease states.

Core claim

Resting brain activity is generically time-irreversible at sufficiently long time scales. Brain pathology is generally associated with a reduction in time-asymmetry, albeit with pathology-specific patterns. These differences are evaluated in terms of possible underlying physical generating mechanisms using EEG recordings from patient and control groups.

What carries the argument

Time-reversal symmetry of EEG signals, measured to quantify time irreversibility at different time scales.

If this is right

  • Healthy resting brain activity exhibits time irreversibility at long scales.
  • Pathology generally reduces time asymmetry but with disease-specific patterns.
  • The reduction varies with experimental condition such as eyes open or closed.
  • Observed changes relate to physical generating mechanisms of the signals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A reduction in time asymmetry may indicate a shift toward more equilibrium-like dynamics in pathological states.
  • This measure could serve as a complementary biomarker alongside existing EEG analyses for detecting brain disorders.
  • Distinct effects across pathologies suggest different impacts on the brain's non-equilibrium processes.

Load-bearing premise

The selected time-irreversibility metric and time scales capture genuine changes in brain generating mechanisms instead of being influenced by recording artifacts, medication, or differences in signal quality between groups.

What would settle it

A controlled EEG study with groups matched for age, medication and recording quality that finds no reduction in time asymmetry or identical reduction patterns across pathologies would challenge the claim.

Figures

Figures reproduced from arXiv: 1907.10060 by Bahar G\"untekin, David Papo, L\"utf\"u Hano\u{g}lu, Massimiliano Zanin, Tuba Akt\"urk.

Figure 1
Figure 1. Figure 1: FIG. 1: Power spectra corresponding to the four considered data sets, averaged over all [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evolution of the fraction of irreversible channels, as a function of the considered [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Evolution of the fraction of irreversible channels, as a function of the considered [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the fraction of irreversible windows for eyes closed and open [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of the fraction of irreversible channels between the patients and the [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of the fraction of irreversible channels between the patients and the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Evolution of the average irreversibility by EEG channel in the three data sets [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Comparison of the fraction of irreversible channels, when the statistical significance [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
read the original abstract

Characterising brain activity at rest is of paramount importance to our understanding both of general principles of brain functioning and of the way brain dynamics is affected in the presence of neurological or psychiatric pathologies. We measured the time-reversal symmetry of spontaneous electroencephalographic brain activity recorded from three groups of patients and their respective control group under two experimental conditions (eyes open and closed). We evaluated differences in time irreversibility in terms of possible underlying physical generating mechanisms. The results showed that resting brain activity is generically time-irreversible at sufficiently long time scales, and that brain pathology is generally associated with a reduction in time-asymmetry, albeit with pathology-specific patterns. The significance of these results and their possible dynamical aetiology are discussed. Some implications of the differential modulation of time asymmetry by pathology and experimental condition are examined.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript reports an empirical study of time-reversal asymmetry in resting-state EEG from three patient cohorts (with respective controls) under eyes-open and eyes-closed conditions. It concludes that healthy resting brain activity is generically time-irreversible at sufficiently long time scales, while pathology is associated with a general reduction in time asymmetry (with pathology-specific patterns), interpreted in terms of possible underlying physical generating mechanisms.

Significance. If the measurements hold, the work supplies direct observational evidence that time irreversibility is a generic feature of healthy brain dynamics at long scales and is systematically reduced under pathology. This provides falsifiable group-comparison data that can be tested against alternative metrics or recording conditions and contributes to dynamical-systems approaches in neuroscience.

minor comments (3)
  1. [Abstract] The abstract states that differences were evaluated 'in terms of possible underlying physical generating mechanisms' but provides no further detail on how the chosen irreversibility statistic isolates mechanism from recording artifacts or signal-quality differences; this clarification belongs in the methods or discussion.
  2. [Results] Sample sizes, exact pathologies, and statistical tests (including any correction for multiple comparisons across time scales and conditions) are not mentioned in the abstract and should be stated explicitly in the results section for reproducibility.
  3. [Figures] Figure legends and axis labels should explicitly indicate the time scales at which irreversibility becomes statistically significant and the precise definition of the asymmetry index used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their concise and accurate summary of the manuscript, for highlighting its potential significance, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; purely empirical measurement study

full rationale

The paper applies a time-irreversibility statistic directly to recorded EEG signals across healthy and patient cohorts, then performs group comparisons at different time scales. No derivation chain, fitted parameters renamed as predictions, self-citations used as load-bearing uniqueness theorems, or ansatzes smuggled in via prior work are present. The central claims (generic irreversibility at long scales, pathology-associated reduction) are observational reports of measured quantities, not reductions of outputs to inputs by construction. The study is self-contained against external benchmarks with no internal logical loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on unstated assumptions about the validity of the chosen irreversibility metric and the comparability of patient and control recordings.

pith-pipeline@v0.9.0 · 5689 in / 992 out tokens · 21689 ms · 2026-05-24T16:50:49.987805+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/ArrowOfTime.lean arrow_from_z echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    resting brain activity is generically time-irreversible at sufficiently long time scales... brain pathology is generally associated with a reduction in time-asymmetry... Σt = ln P(ωt)/P(Iωt)... non-equilibrium systems obey fluctuation relations... P(−Σt)∼P(Σt)e−Σt

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the brain as a generically out-of-equilibrium system... operating close to a NESS... time-reversal symmetry reflects a genuine indicator of brain activity efficiency

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · 1 internal anchor

  1. [1]

    Given a time series X ={xt}, with t = 1...N , this is usually divided in overlapping regions of length D, such that: s→ (xs,xs+τ,...,x s+τ(D−2),xs+τ(D−1))

    Permutation patterns The idea of analysing a time series through its permutation patterns was introduced by Bandt and Pompe [ 71], and since then received an increasing attention from the scientific community [72]. Given a time series X ={xt}, with t = 1...N , this is usually divided in overlapping regions of length D, such that: s→ (xs,xs+τ,...,x s+τ(D−2)...

    work page

  2. [2]

    From permutation patterns to irreversibility The irreversibility of a time series is then estimated by looking at asymmetries in the appearance frequencies of the corresponding permutation patterns. Specifically, for D = 3, 6 patterns can appear, paired as follows: (0, 1, 2)t.r.↔ (2, 1, 0) (5) (1, 0, 2)t.r.↔ (2, 0, 1) (6) (1, 2, 0)t.r.↔ (0, 2, 1), (7) with...

    work page

  3. [3]

    one could simply assess whether or not an EEG time series is irreversible

    Representing the irreversibility of EEG data The previously described test yields a result that could prima facie be used to understand brain dynamics, i.e. one could simply assess whether or not an EEG time series is irreversible. This direct approach nevertheless masks important information, as it tells nothing about the time scales at which such irreve...

    work page

  4. [4]

    The chosen model is the well-known logistic map, defined as: xt+1 =rxn(1−xn) +σξ

    Model of noisy irreversible time series In order to assess whether the irreversibility evolution may only be due to noise, we here consider a simple dynamical model contaminated with additive Gaussian noise. The chosen model is the well-known logistic map, defined as: xt+1 =rxn(1−xn) +σξ. (9) 8 Data set # controls # patients Eyes open/close # channels Reso...

    work page

  5. [5]

    Such series are obtained through the Iterative Amplitude Adjusted Fourier Transform (IAAFT) algorithm [73]

    Testing irreversibility through surrogate time series As a final issue, we further analyse the source of brain irreversibility by using surrogate time series - see Section III C. Such series are obtained through the Iterative Amplitude Adjusted Fourier Transform (IAAFT) algorithm [73]. IAAFT works by iteratively performing a random phase transformation of ...

    work page

  6. [6]

    physionet.org/pn4/eegmmidb/ [75]

    Motor Movement/Imagery data set This EEG data set is described in [ 74], and can be downloaded from https://www. physionet.org/pn4/eegmmidb/ [75]. The full data set comprises recordings of subjects performing different motor/imagery tasks, albeit only the eyes open/closed resting states conditions are here considered. A total of 110 trials (one per subject...

    work page

  7. [7]

    United Kingdom Parkinson’s Disease Society Brain Bank

    Parkinson’s disease data set The EEG data set of Parkinson’s patients was recorded at Istanbul Medipol University Hospital in Istanbul. PD patients were diagnosed according to the criteria of “United Kingdom Parkinson’s Disease Society Brain Bank” [ 76]. The Unified Parkinson’s Disease Rating Scale (UPDRS) [ 77] was used in order to determine the clinical ...

    work page

  8. [8]

    It consists of EEG recordings from paediatric subjects with intractable seizures and free of anti-seizure medication

    Scalp (epilepsy) data set The CHB-MIT Scalp EEG data set is described in [ 79] and is available for download at https://www.physionet.org/pn6/chbmit/ [75]. It consists of EEG recordings from paediatric subjects with intractable seizures and free of anti-seizure medication. Note that sub-windows free of seizures are here analysed alongside other groups’ co...

    work page

  9. [9]

    [ 80] and available at http://dx.doi.org/10

    Schizophrenia data set This data set includes resting state EEG recordings for a set of schizophrenia patients and matched control subjects, as described in Ref. [ 80] and available at http://dx.doi.org/10. 18150/repod.0107441. The 14 patients (7 males, 27 .9± 3.3 years, and 7 females, 28.3± 4.1 years) met International Classification of Diseases ICD-10 cr...

    work page

  10. [10]

    Arieli, A

    A. Arieli, A. Sterkin, A. Grinvald, and A. Aertsen, Science 273, 1868 (1996)

    work page 1996

  11. [11]

    Van de Ville, J

    D. Van de Ville, J. Britz, and C. M. Michel, Proceedings of the National Academy of Sciences , 201007841 (2010)

    work page 2010

  12. [12]

    G. Deco, V. K. Jirsa, and A. R. McIntosh, Nature Reviews Neuroscience 12, 43 (2011)

    work page 2011

  13. [13]

    Kenet, D

    T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, Nature 425, 954 (2003)

    work page 2003

  14. [14]

    Cossart, D

    R. Cossart, D. Aronov, and R. Yuste, Nature 423, 283 (2003)

    work page 2003

  15. [15]

    J. M. Beggs and D. Plenz, Journal of neuroscience 24, 5216 (2004)

    work page 2004

  16. [16]

    Ikegaya, G

    Y. Ikegaya, G. Aaron, R. Cossart, D. Aronov, I. Lampl, D. Ferster, and R. Yuste, Science 304, 559 (2004)

    work page 2004

  17. [17]

    Dragoi and S

    G. Dragoi and S. Tonegawa, Nature 469, 397 (2011)

    work page 2011

  18. [18]

    R. F. Betzel, M. A. Erickson, M. Abell, B. F. O’Donnell, W. P. Hetrick, and O. Sporns, Frontiers in computational neuroscience 6, 74 (2012)

    work page 2012

  19. [19]

    Zhang and M

    D. Zhang and M. E. Raichle, Nature Reviews Neurology 6, 15 (2010)

    work page 2010

  20. [20]

    Alderson-Day, S

    B. Alderson-Day, S. McCarthy-Jones, and C. Fernyhough, Neuroscience & Biobehavioral Reviews 55, 78 (2015)

    work page 2015

  21. [21]

    Hohenfeld, C

    C. Hohenfeld, C. J. Werner, and K. Reetz, NeuroImage: Clinical (2018)

    work page 2018

  22. [22]

    W. L. Shew, H. Yang, T. Petermann, R. Roy, and D. Plenz, Journal of neuroscience 29, 15595 (2009)

    work page 2009

  23. [23]

    Luczak, P

    A. Luczak, P. Barth´ o, and K. D. Harris, Neuron 62, 413 (2009)

    work page 2009

  24. [24]

    S. M. Smith, P. T. Fox, K. L. Miller, D. C. Glahn, P. M. Fox, C. E. Mackay, N. Filippini, K. E. Watkins, R. Toro, A. R. Laird, et al., Proceedings of the National Academy of Sciences 106, 13040 (2009)

    work page 2009

  25. [25]

    Papo, Frontiers in systems neuroscience 8, 112 (2014)

    D. Papo, Frontiers in systems neuroscience 8, 112 (2014)

    work page 2014

  26. [26]

    Livi, Chaos, Solitons & Fractals 55, 60 (2013)

    R. Livi, Chaos, Solitons & Fractals 55, 60 (2013). 23

    work page 2013

  27. [27]

    Novikov, A

    E. Novikov, A. Novikov, D. Shannahoff-Khalsa, B. Schwartz, and J. Wright, Physical Review E 56, R2387 (1997)

    work page 1997

  28. [28]

    Linkenkaer-Hansen, V

    K. Linkenkaer-Hansen, V. V. Nikouline, J. M. Palva, and R. J. Ilmoniemi, Journal of Neuroscience 21, 1370 (2001)

    work page 2001

  29. [29]

    P. Gong, A. R. Nikolaev, and C. van Leeuwen, Physical Review E 76, 011904 (2007)

    work page 2007

  30. [30]

    Bianco, M

    S. Bianco, M. Ignaccolo, M. S. Rider, M. J. Ross, P. Winsor, and P. Grigolini, Physical Review E 75, 061911 (2007)

    work page 2007

  31. [31]

    Freyer, K

    F. Freyer, K. Aquino, P. A. Robinson, P. Ritter, and M. Breakspear, Journal of Neuroscience 29, 8512 (2009)

    work page 2009

  32. [32]

    Expert, R

    P. Expert, R. Lambiotte, D. R. Chialvo, K. Christensen, H. J. Jensen, D. J. Sharp, and F. Turkheimer, Journal of The Royal Society Interface 8, 472 (2011)

    work page 2011

  33. [33]

    A.-M. Wink, E. Bullmore, A. Barnes, F. Bernard, and J. Suckling, Human brain mapping 29, 791 (2008)

    work page 2008

  34. [34]

    Papo, Frontiers in human neuroscience 7, 45 (2013)

    D. Papo, Frontiers in human neuroscience 7, 45 (2013)

    work page 2013

  35. [35]

    Puglisi and D

    A. Puglisi and D. Villamaina, EPL (Europhysics Letters) 88, 30004 (2009)

    work page 2009

  36. [36]

    Porporato, J

    A. Porporato, J. Rigby, and E. Daly, Physical review letters 98, 094101 (2007)

    work page 2007

  37. [37]

    J. Xia, P. Shang, J. Wang, and W. Shi, Physica A: Statistical Mechanics and Its Applications 400, 151 (2014)

    work page 2014

  38. [38]

    Lawrance, International Statistical Review/Revue Internationale de Statistique , 67 (1991)

    A. Lawrance, International Statistical Review/Revue Internationale de Statistique , 67 (1991)

    work page 1991

  39. [39]

    Weiss, Journal of Applied Probability 12, 831 (1975)

    G. Weiss, Journal of Applied Probability 12, 831 (1975)

    work page 1975

  40. [40]

    D. R. Cox, G. Gudmundsson, G. Lindgren, L. Bondesson, E. Harsaae, P. Laake, K. Juselius, and S. L. Lauritzen, Scandinavian Journal of Statistics , 93 (1981)

    work page 1981

  41. [41]

    Stone, G

    L. Stone, G. Landan, and R. M. May, Proc. R. Soc. Lond. B 263, 1509 (1996)

    work page 1996

  42. [42]

    Gaspard, New Journal of Physics 7, 77 (2005)

    P. Gaspard, New Journal of Physics 7, 77 (2005)

    work page 2005

  43. [43]

    Andrieux, P

    D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Physical review letters 98, 150601 (2007)

    work page 2007

  44. [44]

    Rold´ an and J

    ´E. Rold´ an and J. M. Parrondo, Physical review letters105, 150607 (2010)

    work page 2010

  45. [45]

    Seifert, Annual Review of Condensed Matter Physics 10, 171 (2019)

    U. Seifert, Annual Review of Condensed Matter Physics 10, 171 (2019)

    work page 2019

  46. [46]

    Gaspard, Journal of statistical physics 117, 599 (2004)

    P. Gaspard, Journal of statistical physics 117, 599 (2004)

    work page 2004

  47. [47]

    D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Physical review letters 71, 2401 (1993)

    work page 1993

  48. [48]

    Gallavotti and E

    G. Gallavotti and E. G. D. Cohen, Journal of Statistical Physics 80, 931 (1995). 24

    work page 1995

  49. [49]

    G. E. Crooks, Physical review E 61, 2361 (2000)

    work page 2000

  50. [50]

    D. J. Evans and D. J. Searles, Advances in Physics 51, 1529 (2002)

    work page 2002

  51. [51]

    J. B. Ramsey and P. Rothman, Journal of Money, Credit and Banking 28, 1 (1996)

    work page 1996

  52. [52]

    Zumbach, Quantitative Finance 9, 505 (2009)

    G. Zumbach, Quantitative Finance 9, 505 (2009)

    work page 2009

  53. [53]

    Costa, A

    M. Costa, A. L. Goldberger, and C.-K. Peng, Physical review letters 95, 198102 (2005)

    work page 2005

  54. [54]

    Guzik, J

    P. Guzik, J. Piskorski, T. Krauze, A. Wykretowicz, and H. Wysocki, Biomedizinische Technik 51, 272 (2006)

    work page 2006

  55. [55]

    Piskorski and P

    J. Piskorski and P. Guzik, Physiological measurement 28, 287 (2007)

    work page 2007

  56. [56]

    Porta, S

    A. Porta, S. Guzzetti, N. Montano, T. Gnecchi-Ruscone, R. Furlan, and A. Malliani, in 2006 Computers in Cardiology (IEEE, 2006) pp. 77–80

    work page 2006

  57. [57]

    Porta, K

    A. Porta, K. R. Casali, A. G. Casali, T. Gnecchi-Ruscone, E. Tobaldini, N. Montano, S. Lange, D. Geue, D. Cysarz, and P. Van Leeuwen, American Journal of Physiology-Regulatory, Integrative and Comparative Physiology (2008)

    work page 2008

  58. [58]

    Porta, G

    A. Porta, G. D’addio, T. Bassani, R. Maestri, and G. D. Pinna, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, 1359 (2009)

    work page 2009

  59. [59]

    C. K. Karmakar, A. Khandoker, J. Gubbi, and M. Palaniswami, Physiological measurement 30, 1227 (2009)

    work page 2009

  60. [60]

    F. Hou, J. Zhuang, C. Bian, T. Tong, Y. Chen, J. Yin, X. Qiu, and X. Ning, Physica A: Statistical Mechanics and its Applications 389, 754 (2010)

    work page 2010

  61. [61]

    Timmer, C

    J. Timmer, C. Gantert, G. Deuschl, and J. Honerkamp, Biological cybernetics 70, 75 (1993)

    work page 1993

  62. [62]

    Paluˇ s, Biological cybernetics75, 389 (1996)

    M. Paluˇ s, Biological cybernetics75, 389 (1996)

    work page 1996

  63. [63]

    Van der Heyden, C

    M. Van der Heyden, C. Diks, J. Pijn, and D. Velis, Physics Letters A 216, 283 (1996)

    work page 1996

  64. [64]

    C. L. Ehlers, J. Havstad, D. Prichard, and J. Theiler, Journal of Neuroscience 18, 7474 (1998)

    work page 1998

  65. [65]

    Visnovcova, M

    Z. Visnovcova, M. Mestanik, M. Javorka, D. Mokra, M. Gala, A. Jurko, A. Calkovska, and I. Tonhajzerova, Physiological measurement 35, 1319 (2014)

    work page 2014

  66. [66]

    W. Yao, W. Yao, M. Wu, and J. Wang, arXiv preprint arXiv:1801.05421 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  67. [67]

    Schindler, C

    K. Schindler, C. Rummel, R. G. Andrzejak, M. Goodfellow, F. Zubler, E. Abela, R. Wiest, C. Pollo, A. Steimer, and H. Gast, Clinical neurophysiology 127, 3051 (2016)

    work page 2016

  68. [68]

    J. H. Mart´ ınez, J. L. Herrera-Diestra, and M. Chavez, Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 123111 (2018)

    work page 2018

  69. [69]

    Zanin, A

    M. Zanin, A. Rodr´ ıguez-Gonz´ alez, E. Menasalvas Ruiz, and D. Papo, Entropy20, 665 (2018). 25

    work page 2018

  70. [70]

    C. Diks, J. Van Houwelingen, F. Takens, and J. DeGoede, Physics Letters A 201, 221 (1995)

    work page 1995

  71. [71]

    C. Daw, C. Finney, and M. Kennel, Physical Review E 62, 1912 (2000)

    work page 1912

  72. [72]

    M. B. Kennel, Physical Review E 69, 056208 (2004)

    work page 2004

  73. [73]

    M. D. Costa, C.-K. Peng, and A. L. Goldberger, Cardiovascular Engineering 8, 88 (2008)

    work page 2008

  74. [74]

    K. R. Casali, A. G. Casali, N. Montano, M. C. Irigoyen, F. Macagnan, S. Guzzetti, and A. Porta, Physical Review E 77, 066204 (2008)

    work page 2008

  75. [75]

    J. F. Donges, R. V. Donner, and J. Kurths, EPL (Europhysics Letters) 102, 10004 (2013)

    work page 2013

  76. [76]

    Lacasa, A

    L. Lacasa, A. Nunez, ´E. Rold´ an, J. M. Parrondo, and B. Luque, The European Physical Journal B 85, 217 (2012)

    work page 2012

  77. [77]

    Lacasa and R

    L. Lacasa and R. Flanagan, Physical Review E 92, 022817 (2015)

    work page 2015

  78. [78]

    Flanagan and L

    R. Flanagan and L. Lacasa, Physics Letters A 380, 1689 (2016)

    work page 2016

  79. [79]

    G. Graff, B. Graff, A. Kaczkowska, D. Makowiec, J. Amig´ o, J. Piskorski, K. Narkiewicz, and P. Guzik, The European Physical Journal Special Topics 222, 525 (2013)

    work page 2013

  80. [80]

    Bandt and B

    C. Bandt and B. Pompe, Physical review letters 88, 174102 (2002)

    work page 2002

Showing first 80 references.