Long-range temporal correlations in neural narrowband time-series arise due to critical dynamics

Abstract

We show theoretically that the hypothesis of criticality as a theory of long-range fluctuation in the human brain may be distinguished from the theory of passive filtering on the basis of macroscopic neuronal signals such as the electroencephalogram, using novel theory of narrowband amplitude time-series at criticality. Our theory predicts the division of critical activity into meta-universality classes. As a consequence our analysis shows that experimental electroencephalography data favours the hypothesis of criticality in the human brain.

Fig 1. The approach taken in this paper.

Criticality implies that the distibutions of discrete avalanches are power-law (left hand side). However analysing discrete dynamics is problematic on the basis of continuous EEG/ MEG recordings. Until now it was unclear how to distinguish criticality from alternative explanations of 1/f^γ power-spectra on the basis of continuous data (right hand side) such as EEG/ MEG. We show that criticality implies that the narrowbands of the continuous data have specific long-range properties (left hand branch of right hand side). The passive-filtering theory of the origin of the 1/f^γ form of EEG/ MEG power-spectra does not predict such long-range narrowband properties (right hand branch of right hand side). Thus we have a criterion to distinguish criticality from passive filtering on the basis of MEG/ EEG recordings. We perform this test on empirical data.

Fig 2. Illustration of the theory.

Samples according to PF (left) and CD with the lifetime exponent α = 2.5, 1.5 and height exponent β = 1 (centre, right). Top row: avalanches a_i,s(t) composing the continuous network signal X(t) by linear superposition (many avalanches superimposed over one another). Second row: continuous signal X(t). Third row: measured signal, filtered in the case of the PF theory (left) and a scalar multiple of the network signal X(t) in the CD case (centre, right). Fourth and fifth rows: narrowband signals at two frequencies ω₁ and ω₂. We observe that in all cases the observed signal X′(t) fluctuates over a range of time-scales. However the narrowband signals display pronounced fluctuations in their amplitude envelopes only in the CD model for certain exponent values. See Section: Results: Overview of Theory for a detailed description of each panel.

Fig 3. Illustration of the division of critical exponents into meta-universality classes.

With β = 1, as the lifetime exponent α varies (y-axis), the qualitative nature of the continuous data varies, with individual avalanches only clearly visible towards the lower regions of MU2 and upper regions of MU3. See Section: Results: Overview of Theory.

Fig 4. Overview of analysis steps.

(A) The neural signal is extracted from the data. (B) Its power-spectum takes the form of a power-law. (C) Narrowband components in two frequency ranges (red on power-spectrum) are extracted from the signal by filtering and the amplitude envelope is extracted using the Hilbert transform (in red).

Fig 5. Division of critical exponents into meta-universality classes.

The figure displays the range of qualitative behaviours we predict with our theory. Areas marked in green display no LRTC behaviour in sub-bands or DCCA correlations between sub-bands. Areas in red display LRTC and/or cross correlations between amplitudes of sub-bands ($H_{amp}^{\omega }=1/2$, ρ_DCCA(n) = 0 for large n).

Fig 6. Examples illustrating our theory.

Centre: Sample paths from the PF and CD models. In each of the three cases the x-axis denotes time and the y-axis number of activations. For each panel the middle trace denotes X′(t) and the top and bottom g_ωi(X′(t)). Left: DCCA correlation coefficients ρ_DCCA(n) and Hurst exponents $H_{amp}^{\omega }$. Right: Power-spectra of X′(t) in each of the three cases. We see that the top row displays qualitatively different properties to the middle and bottom rows, although the power-spectra are the same in each case. The differences are, however, well quantified by the Hurst exponent and ρ_DCCA(n) in the left-hand column. See Section: Results: Simulations: Examples.

Fig 7. Simulation for PF model.

The figure displays the results averaged over 100 iterations for the PF model in simulation. The left hand panel and the right hand panel correspond to differing values of Hurst exponent of the process sampled. In each case $H_{amp}^{\omega }$ and ρ_DCCA(n) are estimated and averaged. The trace corresponds to ρ_DCCA(n) whereas $H_{amp}^{\omega }$ is displayed in text. See Section: Results: Simulations: PF model.

Fig 8. Simulation for CD model over universality classes.

DCCA correlation coefficients and Hurst exponent for the simulated CD model. The x-axis denotes the exponent α and the y-axis denotes β. The black lines denotes the transitions in meta-universality class according to the theory. In each case the colour on the image and colourbar corresponds to the quantity in the subtitle (e.g. in the left hand image the colour corresponds to $H_{amp}^{\omega }$.) The details of the simulation are given in Section: Results: Predictions for all Meta-Universality Classes.

Fig 9. Results of data analysis of human EEG.

The frequency ranges analysed i = 1, 2, 3 are 35-40Hz, 60-65Hz and 72-77 Hz respectively, which are displayed as superscripts in the plots. Each point on a plot corresponds to estimates made from one EEG spatially filtered component (SSD), with the colours denoted distinct frequencies. In the left hand panel we display ρ_DCCA(n) values at the highest scale vs. $H_{amp}^{\omega }$ values. In the middle panel we display $H_{amp}^{\omega }$ values between frequencies and in the right hand panel we display $H_{amp}^{\omega }$ plotted vs. H_raw.

Fig 10. Illustration of DFA.

Top left: two signals X_i(t) uncorrelated (green) and LRTC (red) from top to bottom with Hurst exponents H₁ = 0.5 < H₂. Top-right: the cumulative sum x(t) of the original signal, which display differing random walk behaviours. Bottom left: the DFA coefficients $F_{DFA}^2\left(n\right)$ are estimated by detrending x(t) in time windows of length n and estimated the error of the linear fit. Botton right F²(n) n^2H allowing H to be estimated by regression in log coordinates.

Fig 11. Illustration of DCCA.

DCCA is the exact analogy of DFA (see Fig 10) for two time-series. The left hand panel displays the cumulative sum of two time-series; the right hand panel displays the same time-series after detrending. The strong correlation between the time-series only becomes apparent after detrending, explaining the advantage of DCCA for long-range dependent and non-stationary time-series.

Fig 12. The difference in scaling between the raw avalanches and their filtered amplitudes.

In this figure we set β = 1. The left hand panel displays avalanches L^β a(t/L) with log-spaced lifetimes L between 400 and 7000. The right hand panel displays the narrowband amplitudes of these avalanches L^β′ g_ω(a)(t/L). Since β > β′ = β/2 we see that the narrowband amplitudes scale less steeply than the raw avalanches.