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| ==== Discussion ====
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| In the current study, we investigated the spatial-extent and the associated transitional properties of neural activity in the brain during active and resting conditions, and whether similar underlying network properties exist. We found that applying the Hilbert transformation to the EEG data and normalizing it (Eq. 2) imposes a probabilistic structure to the EEG signal across the brain (Eq. 3), which we used to identify probability of spatial patterns of activity along with transitions in activity across the scalp. We found more anterior activity during rest relative to the movie watching, in both amplitude and phase space. This finding is in line with previous results showing increased activation in anterior region during rest <ref name=":1" /><ref name=":1" /><ref name=":2" /><ref name=":4" /><ref name=":5" /><ref>Christoff K, Gordon AM, Smallwood J, Smith R, Schooler JW. Experience sampling during fMRI reveals default network and executive system contributions to mind wandering. Proc. Natl. Acad. Sci. U. S. A. 2009;106:8719–8724. doi: 10.1073/pnas.0900234106.[PMC free article] [PubMed] [CrossRef] [Google Scholar]</ref>. Moreover, by normalizing the Hilbert transformed EEG signals and extracting average values akin to those of the wavefunction formulation of quantum mechanics, we were able to compute uncertainty in the ‘position’ and ‘momentum’ during rest and movie-watching, which is set by the new constant
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| <math>K_{brain}=0,78\pm0,41\tfrac{cm^2}{4ms}</math>.
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| It is alluring to associate the constant related to the ‘position’ and ‘momentum’ of neural activity to a fundamental principle, such as, the Heisenberg uncertainty principle. However, it is still unclear what this uncertainty means. It could imply limits to the degree to which the brain is accessible; increasing information about the precise location of the brain state (as described by our quasi-quantum ‘wavefunctions’) will produce a bigger uncertainty about where it will be at a subsequent time. These results offer an interesting perspective on the link between neural function and cognitive processes. For instance, as the ‘wavefunction’ becomes localized in space along a train of thoughts, we become distracted to increase the uncertainty, which may explain why minds wander and thoughts are fleeting?
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| Is the <math> K_b </math> value we found constant across different stimulus conditions, and independent of the number of electrodes used to acquire the data? To test this, we down sampled the EEG electrodes from 92 to 20 and performed the same analysis as in the main text. In line with 92 channels, we found the anterior tendency in rest, but we found reducing the electrodes to 20 resulted in a different constant <math>K_b= 0,03\pm0,02\tfrac{cm^2}{4ms}</math> (See Supplementary Material). This demonstrates that the model is able to capture the differences of rest/task, but a montage-dependent normalisation condition may need to be introduced.
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| It is important to note that uncertainty values of this form are inherent to any Fourier conjugate variables, as a value spreads out in one variable, it localizes in the other. This suggests that after defining the square of the Hilbert transformed EEG electrode time course to be the probability and imposing the properties of a Hilbert space onto the electrode signals, an uncertainty values can be extracted. In quantum mechanics, this uncertainty sets the limit for the scales that cannot be observed. This approach was inspired from the need in neuroscience for novel models to help interpret neuroimaging data. While this is an interesting methodological step forward, we still must determine if the observed uncertainty in the EEG data is supported by a new fundamental principle like in quantum mechanics, or if it is just the outcome of having built two new Fourier conjugate variables from the EEG signal.
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| Further work must be done to explore this constant with respect to the rich taxonomy of tasks and stimuli and varying states of consciousness that are routinely used in cognitive neuroscience. This methodology could be extended into fMRI, where the BOLD time courses could be Hilbert transformed creating a three-dimensional analogue of the EEG model presented in this paper.
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| Ultimately, this paper presented a novel methodology for analysing EEG data. Normalizing the data and treating it as a probability amplitude led to parameters that changed with the presence or lack of stimulus, while simultaneously establishing a constant value independent of stimulus. We have successfully applied a mathematical framework based on the formalisms of quantum mechanics to the resting and task paradigm in EEG (without claiming the brain is a quantum object). As neuroscience continues to evolve, the analytic tools at its disposal must also progress accordingly. We hope that this analytical tool, along with the advances in modelling and machine learning will aid in our understanding of the nature of consciousness.
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| == Methods==
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| === Data acquisition===
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| Twenty-eight healthy subjects were recruited from The Brain and Mind Institute at the University of Western Ontario, Canada to participate in this study. Informed written consent was acquired prior to testing from all participants. Ethics approval for this study was granted by the Health Sciences Research Ethics Board and the Non-Medical Research Ethics Board of The University of Western Ontario and all research was performed in accordance with the relevant guidelines/regulations and in accordance with the Declaration of Helsinki.
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| Two suspenseful movie clips were used as the naturalistic stimuli in this study. A video clip from the silent film “Bang! You’re Dead” and an audio excerpt from the movie “Taken” were shown to 13 and 15 subjects respectively in both their original intact and scrambled forms. Prior to the two acquisitions, a section of rest was acquired where the subjects were asked to relax, without any overt stimulation. Stimulus presentation was controlled with the Psychtoolbox plugin for MATLAB <ref>Brainard DH. The psychophysics toolbox. Spat. Vis. 1997;10:433–436. doi: 10.1163/156856897X00357. [PubMed] [CrossRef] [Google Scholar]</ref><ref>Kleiner M, et al. What’s new in psychtoolbox-3. Perception. 2007;36:1–16. [Google Scholar]</ref><ref>Pelli DG. The VideoToolbox software for visual psychophysics: Transforming numbers into movies. Spat. Vis. 1997;10:437–442. doi: 10.1163/156856897X00366. [PubMed] [CrossRef] [Google Scholar]</ref> on a 15″ Apple MacBook Pro. Audio were presented binaurally at a comfortable listening volume through Etymotics ER-1 headphones.
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| EEG data were collected using a 129-channel cap (Electrical Geodesics Inc. [EGI], Oregon, USA). Electrode impedances were kept below 50 kΩ with signals sampled at 250 Hz and referenced to the central vertex (Cz). Using the EEGLAB MATLAB toolbox<ref>Makeig, S. & Onton, J. ERP features and EEG dynamics: An ICA perspective. In ''The Oxford Handbook of Event-Related Potential Components'' (Oxford University Press, 2012). 10.1093/oxfordhb/9780195374148.013.0035.</ref>, noisy channels were identified and removed, then interpolated back into the data. A Kolmogorov–Smirnov (KS) test on the data was used to identify regions that were not Gaussian. Independent components analysis (ICA) was then used to visually identify patterns of neural activity characteristic of eye and muscle movements which were subsequently removed from the data. EEG pre-processing was performed individually for each subject and condition.
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| Of the two movie clips tested, the first was an 8-min segment from Alfred Hitchcock’s TV silent movie “Bang! You’re Dead”. This scene portrays a 5-year-old boy who picks up his uncle’s revolver. The boy loads a bullet into the gun and plays with it as if it were a toy. The boy (and viewer) rarely knows whether the gun has a bullet in its chamber and suspense builds as the boy spins the chamber, points it at others, and pulls the trigger. As an alternative to visual stimulation, a 5-min audio excerpt from the movie “Taken” was also used. This clip portrays a phone conversation in which a father overhears his daughters’ kidnapping.
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| Furthermore, two “scrambled” control stimuli were used—one for each movie. This separates the neural responses elicited by the sensory properties of watching or listening to the movies from those involved in following the plot. The scrambled version of “Bang! You’re Dead” was generated by isolating 1 s segments and pseudorandomly shuffling the segments, thereby eliminating the temporal coherence of the narrative<ref name=":0" /> <ref name=":6">Laforge G, Gonzalez-Lara LE, Owen AM, Stojanoski B. Individualized assessment of residual cognition in patients with disorders of consciousness. NeuroImage Clin. 2020;28:102472. doi: 10.1016/j.nicl.2020.102472. [PMC free article] [PubMed] [CrossRef] [Google Scholar]</ref>. The scrambled version of “Taken” was created by spectrally rotating the audio, thus rendering the speech indecipherable<ref name=":6" /><ref>Naci L, Sinai L, Owen AM. Detecting and interpreting conscious experiences in behaviorally non-responsive patients. Neuroimage. 2017;145:304–313. doi: 10.1016/j.neuroimage.2015.11.059.[PubMed] [CrossRef] [Google Scholar]</ref>. The scrambled movie clips were presented before the intact versions to prevent potential carry-over effects of the narrative. Prior to subjects watching/listening to the scrambled stimulus a short segment of resting EEG was acquired.
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| == Model ==
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| Each of the j electrodes is described by an ordered pair (<math>x_j,y_j,z_j</math>) in 3-dimensional space. To complete this analysis, the electrodes were first projected onto the (<math>x,y</math>) plane, removing the depth of the head. Figure 1A shows the locations of each electrode in this 2d-space. Following this projection, the time courses for each of the 92 electrodes were Hilbert transformed and then normalized following the procedure listed using Eq. (2). A probability was defined in this electrode-position space as the square of the Hilbert transformed time course (Eq. 3), analogous to the wavefunctions of quantum mechanics. Eight regions Anterior L/R, Posterior L/R, Parietal L/R, Occipital L/R) were then defined by grouping the 92 electrodes, and the frequencies of entering each region fG were obtained by summing the probabilities electrodes within the group, then integrating in time.
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>Prob_G(t)=\sum_{j=1}^{92} \Psi_k^* (t)\times\Psi_k (t); f_G=\tfrac{1}{T}\sum_{j=1}^{T}Prob_G(t)</math>
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| | width="33%" align="right" |<math>(6)</math>
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| |}
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| </center>
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| where each of the eight groups denoted by the subscript have a different number of constituent electrodes N. In the occipital left and right there are 10 electrodes each, in the parietal left and right there are 17 electrodes each, in the posterior left and right there are 10 and 11 electrodes respectively, and in the anterior left and right there are 8 and 9 electrodes respectively.
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| Upon getting the group level frequencies average values for position and momentum were calculated using Eqs. (4) and (5) (with identical expressions for y). Finally, to ascertain our analogous uncertainty principle, we sought expressions of the form
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| <center>
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| {| width="80%" |
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| | width="33%" |<math>\bigtriangleup x=\sqrt{\langle x^2(t)\rangle-\langle x(t)\rangle^2};
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| \bigtriangleup p_x=\sqrt{\langle p_x^2(t)\rangle-\langle p_x(t)\rangle^2}</math>
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| | width="33%" align="right" |<math>(7)</math>
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| |}
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| </center>
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| The expression for can be readily applied to the probabilities and positions as defined above, resulting in the first term given by
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\langle x^2(t)\rangle=\sum_{j=1}^{92}P_j(t) x_j^2</math>
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| | width="33%" align="right" |<math>(8)</math>
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| |}
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| </center>
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| And the second term given by the square of Eq. (4). The second term of <math>\Delta P_x</math> is given by the square of Eq. (5), but the first term is more nuanced. This is owing to the complex number returned when acting the derivative operator twice on the probability. To overcome this, Fourier transforms were used to change Eq. (5) into the momentum basis which then allowed for the efficient calculation of <math>P_x^2(t)</math>.
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| Denoting <math>\tilde{P}_j(t)</math> as the momentum-space probability obtained through a 2-dimensional, non-uniform Fourier transform of the position space pseudo-wavefunction, Eq. (5) can be rewritten as,
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\langle p_x(t)\rangle=\sum_{j=1}^{92}\tilde{P}_j(t)p_j</math>
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| | width="33%" align="right" |<math>(9)</math>
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| |}
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| </center>
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| Leading to the first term in the expression to be written as,
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| <center>
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\langle p_x^2(t)\rangle=m^2\sum_{j=1}^{92}\tfrac{x_j^2}{\tilde{p}_j(t)}[{d \over dt}P_j(t)]^2</math>
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| | width="33%" align="right" |<math>(10)</math>
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| |}
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| </center>
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| The FINUFFT python wrapper was used to take the Fourier transform using a type 3, 2d non-uniform FFT<ref>Barnett AH, Magland J, Klinteberg LAF. A parallel nonuniform fast Fourier transform library based on an “Exponential of semicircle” kernel. SIAM J. Sci. Comput. 2019;41:C479–C504. doi: 10.1137/18M120885X. [CrossRef] [Google Scholar]</ref><ref>Barnett, A. H. Aliasing error of the kernel in the nonuniform fast Fourier transform. arXiv:2001.09405 [math.NA] (2020).</ref>, and the minimum value in time of the uncertainty relation was found. Points in momentum space were sampled on <math>p_x\in[-4,4]</math> and <math>p_y\in[-4,5]</math> along with the two additional points (<math>[-5,-4]</math>) and (<math>[-4,-5]</math>).
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| Figure 4 shows the position and momentum probabilities respectively in their own basis. An animation showing how these evolve in time for the different conditions is presented in Supplementary Material 2.
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| [[File:Figure 4.jpeg|center|thumb|500x500px|<small>'''Figure 4: (A''') Probability distribution for a single subject in the position basis. ('''B''') Momentum basis probability distribution for a single subject. The momentum values used for the Fourier transform are indicated by the point locations. Points are colour-/size-coded to represent the probability value at that location.</small>]]
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| To compute the values reported in Table 2, the corresponding value was found for each subject, and these were used to calculate the group average reported here.
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| == Supplementary Information ==
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| Supplementary Figures.(28M, docx)
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| Supplementary Information.(375K, docx)
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| == Acknowledgements==
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| We would like to thank Silvano Petrarca for his continued assistance in devising the model. This study was funded by the NSERC Discovery Grant (05578–2014RGPIN), CERC (215063), CIHR Foundation Fund (167264). AMO is a Fellow of the CIFAR Brain, Mind, and Consciousness Program.
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| == Author contributions==
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| N.J.M.P., C.M. and G.L. performed the analysis. A.S., B.S. and N.J.M.P. developed the model. A.S. and B.S. supervised the analysis. N.J.M.P., A.S., G.L. and B.S. wrote the manuscript. A.M.O. revised the manuscript.
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| == Competing interests ==
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| The authors declare no competing interests.
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| == Footnotes==
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| === Publisher's note===
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| Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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| These authors contributed equally: Bobby Stojanoski and Andrea Soddu.
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| === Supplementary Information===
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| The online version contains supplementary material available at 10.1038/s41598-021-97960-7.
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| == Article information==
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| Sci Rep. 2021; 11: 19771.
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| Published online 2021 Oct 5. doi: 10.1038/s41598-021-97960-7
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| PMCID: PMC8492705
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| PMID: 34611185
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| Nicholas J. M. Popiel,<sup>1,2</sup> Colin Metrow,<sup>1</sup> Geoffrey Laforge,<sup>3</sup> Adrian M. Owen,<sup>3,4,5</sup> Bobby Stojanoski,<sup>#4,6</sup> and Andrea Soddu<sup>#1,3</sup>
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| <sup>1</sup>The Department of Physics and Astronomy, The University of Western Ontario, London, ON N6A 5B7 Canada
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| <sup>2</sup>Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE UK
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| <sup>3</sup>The Brain and Mind Institute, The University of Western Ontario, London, ON N6A 5B7 Canada
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| <sup>4</sup>The Department of Psychology, The University of Western Ontario, London, ON N6A 5B7 Canada
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| <sup>5</sup>The Department of Physiology and Pharmacology, The University of Western Ontario, London, ON N6A 5B7 Canada
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| <sup>6</sup>Faculty of Social Science and Humanities, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, ON L1H 7K4 Canada
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| Andrea Soddu, Email: asoddu@uwo.ca
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| Corresponding author.
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| #Contributed equally.
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| Received 2021 Apr 28; Accepted 2021 Aug 30.
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| Copyright © The Author(s) 2021
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| === Open Access ===
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| This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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| Articles from Scientific Reports are provided here courtesy of == Nature Publishing Group ==
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| {{bib}}
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