Ruben Calvo, Carles Martorell, Guillermo B. Morales, Serena Di Santo, and Miguel A. MunozDepartamento de Electromagnetismo y F´ısica de la Materia and Instituto Carlos I deFisica Teorica y Computacional. Universidad de Granada.E-18071, Granada, Spain(Dated: March 25, 2024)Recent analyses combining advanced theoretical techniques and high-quality data from thousands of simultaneously recorded neurons provide strong support for the hypothesis that neural dynamics operate near the edge of instability across regions in the brain. However, these analyses, as well as related studies, often fail to capture the intricate temporal structure of brain activity as they primarily rely on time-integrated measurements across neurons. In this study, we present a novel framework designed to explore signatures of criticality across diverse frequency bands and construct a much more comprehensive description of brain activity. Additionally, we introduce a method for projecting brain activity onto a basis of spatio-temporal patterns, facilitating time-dependent dimensionality reduction. Applying this framework to a magnetoencephalography dataset, we observe significant differences in both criticality signatures and spatio-temporal activity patterns between healthy subjects and individuals with Parkinson’s disease.
G. Niso, C. Rogers, J. T. Moreau, L.-Y. Chen, C. Madjar, S. Das, E. Bock, F. Tadel, A. C. Evans, P. Jolicoeur, and S. Baillet, OMEGA: The Open MEG Archive, NeuroImage 124, 1182 (2016). F. Tadel, S. Baillet, J. C. Mosher, D. Pantazis, and R. M. Leahy, Brainstorm: A User-Friendly Application for MEG/EEG Analysis, Computational Intelligence and Neuroscience 2011, 1 (2011). 6 SUPPLEMENTAL INFORMATION COVARIANCE MATRIX AND THE POWER SPECTRUM Given a collection of time series, xi(t), where 1 i N , of a stochastic process that is weakly stationary, the
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A typical technique of dimensionality reduction commonly used in data analysis involves projecting the time series activity, xi(t), onto the eigenvectors (or principal components) of the matrix Cij . These principal components represent the directions in which the dataset exhibits maximum variance. In multidimensional complex systems, like the brain, information that propagates from one dynamical unit to another, carries a (small) delay, or time lag, , such that the behavior of xj(t + ) can be inferred from the behavior of xi(t) (see, for instance, ). In this case, the mathematical object measuring this linear dependence between timeseries is the time-lagged covariance matrix: Cij( ) = xi(t)xj(t + ) ,
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This matrix is not generally symmetric (xi(t)xj(t + ) = xj(t)xi(t + )), which means that eigenvalues and eigenvectors are both complex. This complexity makes them difficult to analyze. We propose to use the Fourier transform of this time-lagged covariance matrix, also called the power-spectrum matrix or simply the frequency-dependent covariance (FDC) matrix, which by means of the Wiener-Kinchin theorem [38, p. 17] can be written as: Sij() = F[Cij( )]() = lim T 1 T X i,T ()Xj,T () , where Xi,T () represents the Fourier coefficient of the time series, xi(t), at the particular frequency, , in the time window [0, T ], i.e.,
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Xi,T () = (cid:90) T eitxi(t)dt , 0 and () represents the complex conjugate of (). The FDC matrix has the advantage of having real eigenvalues since it is a Hermitian matrix. However, its eigenvectors are generally complex, as discussed in the main text, which leads to the emergence of spatio-temporal patterns. Its worth noting that the FDC matrix quantifies how much information from the Fourier transformed series Xi,T () can be used to reconstruct the Fourier transformed series Xj,T () through a straightforward linear regression.
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