Scale-free properties of the functional magnetic resonance imaging signal during rest and task

Abstract

It has been shown recently that a significant portion of brain electrical field potentials consists of scale-free dynamics. These scale-free brain dynamics contain complex spatiotemporal structures and are modulated by task performance. Here we show that the fMRI signal recorded from the human brain is also scale free; its power-law exponent differentiates between brain networks and correlates with fMRI signal variance and brain glucose metabolism. Importantly, in parallel to brain electrical field potentials, the variance and power-law exponent of the fMRI signal decrease during task activation, suggesting that the signal contains more long-range memory during rest and conversely is more efficient at online information processing during task. Remarkably, similar changes also occurred in task-deactivated brain regions, revealing the presence of an optimal dynamic range in the fMRI signal. The scale-free properties of the fMRI signal and brain electrical field potentials bespeak their respective stationarity and nonstationarity. This suggests that neurovascular coupling mechanism is likely to contain a transformation from nonstationarity to stationarity. In summary, our results demonstrate the functional relevance of scale-free properties of the fMRI signal and impose constraints on future models of neurovascular coupling.

Figure 1.

Locations of the ROIs. The 31 ROIs, including 10 pairs of homologous brain regions, are color coded by their affiliated brain networks. A, Volume view. B, Surface view.

Figure 2.

Power spectra of spontaneous fMRI signals and the variations of power-law exponent and metabolic values across brain networks. A, Normalized temporal power spectrum (i.e., power spectral density) of spontaneous fMRI signals under resting state extracted from 21 brain regions covering five different brain networks. Lines are color coded by brain networks: magenta, default-mode network; blue, attention network; green, motor network; cyan, saliency network; orange, visual network; red, non-neocortical regions. B, Comparison of fits by different models to the fMRI power spectrum. The four models are exponential, log-normal, power-law, and power-law using only the low-frequency (<0.1Hz) (Power-law LF) region. The Kolmogorov–Smirnov distance D between the original data and its least-squares fit by each model was calculated for each brain region, and paired t tests were used to compare different models. C, Variation of power-law exponent across brain networks (p = 0.0076, assessed by an ANOVA). D, Variations of CMRGlu (p = 0.007), OEF (p = 0.006), and GI (p < 0.0001) across brain networks, assessed by ANOVA. HC, Hippocampus; Tha, thalamus. Error bars denote SEM. For other abbreviations, see Table 1.

Figure 3.

fMRI power-law exponent correlates with variance and CMRGlu. A, A scatter plot showing the values of power-law exponent and variance of the fMRI signal for each of 21 brain regions (correlation, r = 0.517, p = 0.015). The best-fit linear-regression line is shown. The outlier indicated by circle is the hippocampal formation (HC), with a very high variance but small power-law exponent. The r and p values were computed including the hippocampal formation. B, A scatter plot showing fMRI signal power-law exponent and regional CMRGlu value for each of 21 brain regions (r = 0.514, p = 0.016).

Figure 4.

DFA and Hurst exponent of spontaneous fMRI signals. A, DFA plots for 21 brain regions, showing fluctuation (F) measured at different window lengths (l) plotted in double-logarithmic scales. B, Variation of Hurst exponent across brain networks, assessed by an ANOVA (p = 0.017). C, Hurst exponent correlates with power-law exponent (r = 0.92, p < 10⁻⁸), variance (r = 0.59, p < 0.004), and CMRGlu (r = 0.44, p < 0.05). D, Goodness-of-fit test for one example region—frontal eye field (FEF)—on its fit to a scale-free distribution (p = 0.43). The red line with dots is the raw F–l plot for the FEF time series, and the red line without marker is its least-squares power-law fit. The blue lines are the F–l plot for 1000 synthetic fGn time series with the same variance and Hurst exponent as the FEF time series and their respective least-squares power-law fit. HC, Hippocampus. For other abbreviations, see Table 1.

Figure 5.

The effect of task performance on fMRI signal variance and scale-free properties. A, fMRI signal power-law exponent (left), Hurst exponent (middle), and variance (right) for each of 21 brain regions during rest and task. For statistical results on the effect of task, see Results and Table 1. B, fMRI signal power spectra during rest and task (without normalization by total power) were computed for the LMC and RMC regions individually defined for each subject and then averaged across 17 subjects. Significant decrease of variance, power-law exponent, and Hurst exponent were found in both regions (for values, see Results and Table 1). Inset, The distribution of intertrial intervals of the button-press task. For abbreviations, see Table 1.

Figure 6.

Task-induced activation and deactivation patterns from two example subjects. A general-linear model with an event-related design was used to generate t-score maps, which were converted to equally probable Z-score maps, and thresholded at Z = 3 for both activations (shown in red–yellow) and deactivations (shown in blue–green).

Figure 7.

Schematic showing the effect of task on fMRI signals. The present results reveal that the fMRI signal variance, power-law exponent, and Hurst exponent decrease in both task-activated and -deactivated brain regions, suggesting that the dynamic range and long-range memory of the fMRI signal are largest during the baseline resting state. The curves in the top were modeled using Gaussian-distributed white noise filtered in the frequency domain by P ∝ 1/f^β, with β equal to the power-law exponent averaged across the 21 brain regions (rest, 0.83; task, 0.69). The variance during activation/deactivation was modeled as 97.8% of that during rest (average value across 21 brain regions).