Scaling of whole-brain dynamics reproduced by high-order moments of turbulence indicators

Open Access

Scaling of whole-brain dynamics reproduced by high-order moments of turbulence indicators

Yonatan Sanz Perl, Pablo Mininni, Enzo Tagliazucchi, Morten L. Kringelbach, and Gustavo Deco

Phys. Rev. Research 5, 033183 – Published 12 September 2023

Abstract

We investigate how brain activity can be supported by a turbulent regime based on the deviations of a self-similar scaling of high-order structure functions within the phenomenological Kolmogorov's theory. By analyzing a large neuroimaging data set, we establish the relationship between scaling exponents and their order, showing that brain activity has more than one invariant scale, and thus orders higher than 2 are needed to accurately describe its underlying statistical properties. Furthermore, we build whole-brain models of coupled oscillators to show that high-order information allows for a better description of the brain's empirical information transmission and reactivity.

Received 6 February 2023
Accepted 10 July 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033183

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.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Neuroscience, neural computation & artificial intelligence Nonequilibrium & irreversible thermodynamics Turbulent mixing

Brain Mammals

Interdisciplinary PhysicsNonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Yonatan Sanz Perl ^1,2,3,4, Pablo Mininni ^5,6, Enzo Tagliazucchi^5,6,7, Morten L. Kringelbach^8,9,10,11, and Gustavo Deco ^3,12,13

¹Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina
²Cognitive Neuroscience Center (CNC), Universidad de San Andrés, Buenos Aires, Argentina
³Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
⁴Paris Brain Institute (ICM), Paris, France
⁵Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina
⁶CONICET–Universidad de Buenos Aires, Instituto de Física Interdisciplinaria y Aplicada (INFINA), Ciudad Universitaria, 1428 Buenos Aires, Argentina
⁷Latin American Brain Health Institute (BrainLat), Universidad Adolfo Ibáñez, 8320000 Santiago, Chile
⁸Department of Psychiatry, University of Oxford, Oxford, United Kingdom
⁹Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Århus, Denmark
¹⁰Life and Health Sciences Research Institute (ICVS), School of Medicine, University of Minho, Braga, Portugal
¹¹Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, United Kingdom
¹²Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
¹³Institució Catalana de la Recerca i Estudis Avancats (ICREA), Barcelona, Spain

Article Text

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Issue

Vol. 5, Iss. 3 — September - November 2023

Subject Areas

Complex Systems

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Images

Figure 1
High-order structure functions and their spatial power scaling laws. (a) We computed the structure functions from 1 to 8 by replacing the longitudinal component of the fluid velocity in Eq. (2) by the BOLD signal from fMRI recordings of 1003 healthy participants. We highlight even structure functions in the inertial subrange. (b) We found the spatial power-law scaling of even structure functions $S (r)$ as a function of $log (r)$ within the inertial subrange.
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Figure 2
Multifractility on whole brain dynamics. (a) Extended self-similarity confirms the inertial range for higher-order structure functions. We investigated the selected inertial range by computing the extended self-similarity based on studying the higher-order structure functions as a function of that of order 2. We found that the fit of the ESS is excellent for the odd structure functions, providing evidence that the selected inertial range is suitable to assess the spatial scaling properties at the different orders considered. (b) We computed scaling power laws based on high-order structure functions. We obtained the exponents of the even structure functions through the direct computation of the log-log linear fitting (green circles) and also through ESS (blue triangles). We found a high level of agreement between both measures, which confirms inertial range scaling. Interestingly, the exponents can be organized linearly up to order 4 and then the slope changes and the exponent saturates. This is compatible with at least a bifractal structure with two characterstic scales. (c) We investigated the distribution of the structure function of order 1 $[S_{1} (r)]$ at different scales within the inertial range. We found that, whereas the scale increases, the distributions of $S_{1} (r)$ tend monotonically to a Gaussian distribution, indicating small scale fluctuations are less common.
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Figure 3
Higher-order information improves model capabilities. We found for each case the optimal working point as the value of $G$ that minimizes the fitting function in each case. We found statistical difference between both working points by comparing the fitting of the models (a). We found that the information cascade obtained when the brain is modeled with $G_{{HO}_{op}}$ is closer to the empirical, while the information cascade modeled with $G_{02_{op}}$ is significantly lower (b). We quantified the responses when models are perturbed through the susceptibility and the information-encoding capability. We found that both measures are significantly higher in 50 trials of the perturbation when the model includes the information of higher-order structure functions. *** means $p < 0.001$ ; * means $p < 0.05$ and $p > 0.01$ Wilcoxon rank-sum test.
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Figure 4
Convergence of the accumulated moments for various separations.
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Figure 5
Whole-brain model fitting to second-order structure function and higher-order structure functions.
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