Complexity of low‐frequency blood oxygen level‐dependent fluctuations covaries with local connectivity

Abstract

Very low‐frequency blood oxygen level‐dependent (BOLD) fluctuations have emerged as a valuable tool for describing brain anatomy, neuropathology, and development. Such fluctuations exhibit power law frequency dynamics, with largest amplitude at lowest frequencies. The biophysical mechanisms generating such fluctuations are poorly understood. Using publicly available data from 1,019 subjects of age 7–30, we show that BOLD fluctuations exhibit temporal complexity that is linearly related to local connectivity (regional homogeneity), consistently and significantly covarying across subjects and across gray matter regions. This relationship persisted independently of covariance with gray matter density or standard deviation of BOLD signal. During late neurodevelopment, BOLD fluctuations were unchanged with age in association cortex while becoming more random throughout the rest of the brain. These data suggest that local interconnectivity may play a key role in establishing the complexity of low‐frequency BOLD fluctuations underlying functional magnetic resonance imaging connectivity. Stable low‐frequency power dynamics may emerge through segmentation and integration of connectivity during development of distributed large‐scale brain networks. Hum Brain Mapp 35:1273–1283, 2014. © 2013 Wiley Periodicals, Inc.

INTRODUCTION

The observation that functional magnetic resonance imaging (fMRI) signal time series are synchronized in functionally related brain regions [Biswal et al., 1995] has become a powerful probe for describing the connectome of the human brain, how it develops with age, and how it may be disordered by neuropathology [Biswal et al., 2010; Fox and Greicius, 2010; Fox and Raichle, 2007; Vogel et al., 2011]. Early reports showed that the most robust functional connectivity relies upon very low‐frequency fluctuations [<0.08 Hz; Cordes et al., 2001]. The mechanism by which information carried by very fast neural events is transmitted into very slow blood oxygen level‐dependent (BOLD) fluctuations is partially but not completely explained by neurovascular coupling [Anderson, 2008], in which synchronized neural events are blurred in time by slow, hemodynamic integration, recapturing high‐frequency synchrony of neural activity in low‐frequency BOLD fluctuations. Small‐scale neuronal ensembles can exhibit long‐memory synchronization [Beggs and Plenz, 2003; El Boustani et al., 2009], however, suggesting that synchrony of slow fluctuations in neural activity contributes to functional MRI connectivity (fcMRI) independently of neurovascular coupling.

A number of studies have characterized such long‐memory dynamics in fMRI signals, with 1/frequency‐like spectral density, facilitating autocorrelation lasting many seconds [Anderson et al., 2006; Bullmore et al., 2009, 2003, 2004; Maxim et al., 2005; Wink et al., 2008; Woolrich et al., 2001; Zarahn et al., 1997]. Other reports indicate that the temporal structure of low‐frequency BOLD fluctuations can be altered during cognitive activity [Barnes et al., 2009], by medication [Mendez et al., 2011], and by neuropathological conditions [Lai et al., 2010; Maxim et al., 2005; Rubinov et al., 2009]. These findings have raised optimism that the temporal structure of endogenous brain activity fluctuations may represent a “biomarker” for developmental or pathological disorders.

Such 1/frequency‐like dynamical systems are ubiquitous in nature, occuring in contexts as diverse as current flow through river systems and electrical circuits, light emitted from quasars, and properties of social networks [Barabasi and Albert, 1999; Christensen et al., 1992]. Common to these phenomena is scale‐free behavior, wherein properties at one spatial or temporal scale show fractal‐like self‐similarity to those at other scales. Such systems can be modeled by a power law, where behavior follows a relationship 1/f ^γ, where γ can range from 0 (white noise) to 1 (pink noise), 2 (brown noise), or higher and may be fractional. The power law exponent γ connotes information about the degree of determinism in a set of measurements, with randomness or chaos at a value of 0 and increasing order in the noise for higher values of the exponent. The power law exponent is closely related to other metrics of self‐similar behavior such as the Hurst exponent and fractal dimension [Bullmore et al., 2004]. A recent study by He 2011 has extended these results with a rigorous characterization of the scale‐free nature of the BOLD fMRI signal, demonstrated differences in power law exponents of time series in different brain networks, and shown that brain activation decreases the power law exponent of activated regions' fMRI time series.

The brain is thought to operate in intermediate states between chaos and determinism [Kitzbichler et al., 2009], as evidenced not only by self‐similarity of endogenous BOLD fluctuations, but also in electrophysiological properties of neurons (both in culture and in vivo), local field potential and electroencephalographic signals [Beggs and Plenz, 2003, 2004; El Boustani et al., 2009; Poil et al., 2008]. Brain self‐similarity applies not only to temporal fluctuations of neural or hemodynamic activity, but also to the structural architecture of the brain, including dendritic branching patterns [Caserta et al., 1995] and gyrification of the cortical mantle [Bullmore et al., 1994; Kiselev et al., 2003].

The properties of the brain that give rise to power law dynamics are not well understood, but may emerge from neural networks or patterns of sensory input [El Boustani et al., 2009] given their presence across spatiotemporal neural substrates from neurons up to electroencephalographic/magnetoencephalographic signals [Poil et al., 2008]. We tested the hypothesis that network interactions between brain regions are correlated with power law behavior, by examining a large dataset of fMRI images for an interaction between fcMRI and power law properties. Because such power law parameters can only be precisely measured with large amounts of data given the slow time scale of fMRI, we used publicly available data from over 1,000 subjects in our analysis to clarify the relationship between functional connectivity and the temporal structure of endogenous brain fluctuations.

MATERIALS AND METHODS

Publicly Released Datasets—1,019 subjects

A total of 1,019 subjects were analyzed from publicly available datasets released with the open‐access “1000 Functional Connectomes Project” (http://fcon_1000.projects.nitrc.org/) in which resting‐state fMRI scans have been aggregated from 28 sites [Biswal et al., 2010] as well as typically developing subjects from the ADHD 200 project from the International Neuroimaging Data‐sharing Initiative including eight sites[ADHD‐200_Consortium, 2012]. For inclusion, we required that subjects' ages were between 7 and 30, with BOLD whole‐brain coverage from Montreal Neurologic Institute (MNI) coordinates z = −35 to z = 70. Any subject for whom preprocessed data did not cover all 7,266 regions of interest (ROIs) used for this analysis was discarded prior to analysis. For inclusion, all subjects included an MPRAGE anatomic sequence that was successfully segmented and normalized to MNI space. Although preprocessing steps were performed using an automated batch script, the results of normalization, segmentation, and realignment steps were manually inspected for all subjects, and any subject for whom the normalized and segmented images were not in close alignment with the MNI template on visual inspection were discarded. The datasets from which subjects met all criteria are listed in Table 1. The mean age of all subjects was 18.3 ± 5.6 s.d. years (range 7–29). Of all, 590 subjects were male; 429 were female. All subjects were processed in the same manner regardless of the site from which they were obtained.

Table 1.

Sources of 1,019 open access datasets used for analysis

Site (FCON 1000)	n	Site (FCON 1000)	n	Site (ADD 200)	n
Ann Arbor	16	Leipzig	29	Kennedy Krieger	49
Baltimore	11	New York	30	NeuroImage	18
Bangor	1	Newark	15	NYU	88
Beijing	188	Orangeburg	3	OHSU	23
Berlin	16	Oulu	33	Peking	110
Cambridge	172	Oxford	8	Pittsburgh	72
Cleveland	5	Palo Alto	6	Washington U	37
ICBM	13	Queensland	14
Leiden	31	Saint Louis	31

Gray Matter Density Measurements

Customized image analysis templates were created by normalizing, segmenting, and averaging MPRAGE images using SPM8 (Wellcome Trust, London). Images were transformed into standard space using a 12‐parameter affine‐only linear transformation, and segmented into three tissue classes representing gray matter, white matter, and cerebrospinal fluid (CSF). Smoothly varying intensity changes as well as artifactual intensity alterations as a result of the normalization step were corrected for using a standard modulation algorithm within SPM. The resulting segmented tissue maps were smoothed using a 5‐mm full‐width at half‐maximum Gaussian kernel. We then derived mean gray matter intensities within 7,266 spherical (5 mm radius) seed ROIs.

fMRI Preprocessing

Using SPM8 toolbox, BOLD images were realigned (realign, estimate, and write), coregistered to MPRAGE image (coregister, estimate, and write), and normalized to MNI template (normalize, estimage, and write, T1.nii template). Voxelwise images of the standard deviation of the BOLD time series (after realignment, coregistration, and normalization) were obtained as a measurement of regional variation in the BOLD signal. Subsequently, gray matter, white matter, and CSF were segmented from MPRAGE images using the SPM8 segment function (modulated, normalized, and thorough clean). Images were bandpass filtered between 0.001 and 0.1 Hz and a linear detrend was performed at each voxel in the brain. Time series were averaged from two ROIs in the white matter (bilateral centrum semiovale), CSF (lateral ventricles), soft tissues of the head and face, and six rigid motion correction parameters from realignment step as previously described [Anderson et al., 2011a, 2010]. The scalp and facial soft tissues were included as a regressor because this step has been shown to improve accuracy of thalamocortical connectivity by removing additional physiological noise without biasing the spatial results by including gray matter time series elements such as the global signal [Anderson et al., 2011a]. For each voxel, a general linear model was used to find a best fit for white matter, CSF, soft tissues, and motion parameter time series, which were subtracted from the voxel's time series. No regression was performed of the global signal [Murphy et al., 2009] or gray matter, and no smoothing was performed. Finally, frames were inspected for significant motion using the procedure reported by Power et al. 2012, and frames with temporal derivative of root‐mean‐square variance over voxels (DVARS) or root‐mean‐square motion parameters >0.2 were removed prior to analysis of connectivity results, with concatenation of remaining frames.

Regional Homogeneity (Local Connectivity)

Regional homogeneity (ReHo) calculations were performed on unsmoothed preprocessed data. Kendall's coefficient of concordance (KCC; Kendall and Gibbons, 1990] was calculated to represent the similarity of the time series of each 26 nearest neighboring voxels with the KCC assigned to the center voxel [Paakki et al., 2010; Zang et al., 2004, 1997], generating a ReHo measurement at each voxel in the image. Calculation of ReHo images was performed using the Resting‐State fMRI Data Analysis Toolkit (REST, by Song Xiao‐Wei et al., http://www.restfmri.net).

Power Law Exponent (Temporal Complexity)

For each voxel in the brain in each subject, a preprocessed BOLD time series was extracted and a power series was calculated by multiplying the fast Fourier transform of the time series at each frequency by its complex conjugate. A best linear fit was calculated (polyfit.m, MATLAB) between the logarithm of frequency and the logarithm of the power series for data in the frequency range 0.005–0.08 Hz. The power law exponent for the time series was set as the negative slope of the linear fit, representing the best fit of 1/f ^γ to the data over the frequency range at which functional connectivity has been observed [Cordes et al., 2001].

This process has an underlying assumption that the underlying data can be reasonably represented by a power law distribution. BOLD time series data from a single subject, particularly when limited to only 5–8 min acquisitions, produces a power spectrum that is variable, with considerable uncertainty in single subject measurements of power law exponent requiring large sample sizes to make confident inferences. Although the literature contains rigorous tests of the scale free, 1/f‐like nature of the BOLD signal [Bullmore et al., 2003, 2004; He, 2011], we performed a confirmatory test that a power law distribution was an appropriate approximation of the data set we used for analysis.

To obtain robust fits of candidate models to the data, we aggregated data for each of 7,266 gray matter ROIs, constructed to form nonoverlapping, 5 mm radius ROIs throughout the gray matter [Anderson et al., 2011b, 2011c; Ferguson and Anderson, 2012]. The gray matter mask from which the ROIs were obtained was the grey.nii image from the SPM8 toolbox. Aggregation was performed by concatenating the time series of each subject for a given brain region. A voxelwise linear detrend of each subject's data was performed prior to concatenation. Individual subject repetition times were preserved in the concatenated time series. A power spectrum was obtained of the concatenated time series for each region, and three two‐parameter (α, γ) distributions were fit to these data including power law, lognormal, and exponential distributions:

$$\text{Power}\hspace{1em}\text{law}\mathrm{:}\hspace{1em}y=\frac{\alpha }{x^{\gamma }}$$ (1)

$$\text{Lognormal}\mathrm{:}\hspace{1em}y=\frac{1}{\sqrt{2\pi }\alpha x}*e^{\frac{-{(y-\log (x))}^2}{2{\alpha }^2}}$$ (2)

$$\text{Exponential}\mathrm{:}\hspace{1em}y=\alpha \hspace{1em}{e}^{\frac{-x}{\gamma }}$$ (3)

For each ROI, best fits of the power spectrum of the concatenated time series were obtained using the fminsearch.m function in Matlab, which performs an unconstrained nonlinear optimization. Fits were performed on 400 evenly spaced frequency bins from 0.01 to 0.1 Hz, using a sum of squares goodness of fit metric between the data and the model. All three distributions converged to solutions for all 7,266 ROIs, illustrated for one representative (posterior cingulate cortex) in Figure 1. For this figure, both frequency/power (Fig. 1A) and log frequency/log power (Fig. 1B) axes are shown to demonstrate that the log frequency/log power relationship is very close to linear between 0.01 and 0.1 Hz. Across 7,266 gray matter ROIs, the power law distribution provided a significantly lower sum of squares error than the lognormal or exponential distributions, shown in Figure 1C. (Power law goodness of fit: 13.7 ± 8.2; lognormal goodness of fit: 14.6 ± 6.3 s.d.; exponential goodness of fit: 14.7 ± 3.3 s.d.; exponential—power law: 95% confidence interval 0.8–1.2; lognormal—power law: 95% confidence interval 0.8–1.1.)

Figure 1.

Best fits of power law, lognormal, and exponential distributions to BOLD power spectra. (A) Measured data show power spectrum for a representative ROI in the posterior cingulated, calculated for 400 frequency bins between 0.01 and 0.1 Hz. Power spectrum was obtained from concatenated data from all 1,019 subjects for the given ROI. Three distributions (power law, lognormal, and exponential) are optimally fit to the data. (B) The same data and fitted distributions are shown on log/log axis. (C) Sum of squares goodness of fit measurements for the three distributions, averaged across 7,266 gray matter ROIs. Error bars show standard error of the mean across ROIs. [Color figure can be viewed in the online issue, which is available at http://wileyonlinelibrary.com.]

Statistical Analysis

From the ReHo, BOLD standard deviation, and power law exponent images for each subject, data were extracted by averaging values for of the same 7,266 ROIs described. Gray matter density was also averaged for the same 7,266 ROIs in each subject. For mean values across subjects of gray matter density, BOLD standard deviation, ReHo, and power law exponent, full correlation coefficients were calculated as Pearson correlation coefficients. Partial correlation coefficients were computed from the 4 × 7,266 matrix of mean values for the four variables using the partialcorr.m function in MATLAB's statistics toolbox (Table 2). For measurements across subjects, each ROI was analyzed separately. The best linear fit to gray matter density and BOLD standard deviation measurements across the 1,019 subjects were simultaneously subtracted from the ReHo measurements across subjects for the same ROI using the MATLAB glmfit.m function. The corrected ReHo measurements were then used to calculate a Pearson correlation coefficient with power law exponent values across subjects for the same ROI. To assess the effect of age on power law exponent, a simple correlation coefficient was calculated for each ROI between age and power law exponent across subjects.

Table 2.

Full correlation and partial correlation coefficients (r) and uncorrected P‐values between gray matter density (GM), BOLD standard deviation (BOLDSD), regional homogeneity (ReHo), and power law exponent (PLE) across 7,266 gray matter regions

	GM	BOLDSD	ReHo
Full correlation
BOLDSD	r = 0.098
	P = 4.1e−17
ReHo	r = 0.27	r = 0.07
	P = 4.1e−118	P = 3.4e−10
PLE	r = 0.34	r = 0.20	r = 0.79
	P = 1.1e−198	P = 1.0e−63	P = 0
Partial correlation
BOLDSD	r = 0.034
	P = 0.0039
ReHo	r = 0.0002	r = −0.13
	P = 0.9891	P = 1.2e−29
PLE	r = 0.21	r = 0.21	r = 0.77
	P = 1.0 e−73	P = 4.0e−74	P = 0

Subject Head Motion

Because recent studies have called into question the potentially significant effects of even small amounts of head motion, we repeated analyses of ReHo and power law exponent using three different approaches to head motion. First, as described above, we performed motion scrubbing by removing for each subject all frames before and after time points for which the root‐mean‐square framewise displacement of six standard motion parameters or mean DVARS was greater than 0.2 mm [Power et al., 2010]. Second, we performed no motion scrubbing, but only included the 965 out of 1,019 subjects for which root‐mean‐square framewise displacement was less than 0.2 mm across the acquisition. Third, we performed no additional motion scrubbing or subject selection based on motion.

Although small changes were present on a subject‐by‐subject level in ReHo and power law exponent measurements between these three approaches, when mean values across subjects were considered for each of the 7,266 ROIs used in the analysis, neither ReHo nor power law exponent showed any systematic bias in the values obtained that could be attributed to head motion. This is illustrated in Figure 2, where scatter plots between mean ReHo and power law exponent values are shown for the first (0.2 mm scrubbing) and third (no motion scrubbing or subject selection) approaches, with values lying directly on the main diagonal in both cases. For remaining analyses, the 0.2 mm scrubbed approach was used.

Figure 2.

Mean values of ReHo and power law exponent across subjects. Scatter plots show values with and without motion scrubbing (0.2 mm threshold for frame removal) for 7,266 gray matter ROIs.

RESULTS

Voxelwise maps of ReHo and power law exponent, averaged across all 1,019 subjects, showed striking similarity in their spatial distributions, shown in Figure 3. For both measurements, there were much smaller values for either ReHo or power law exponent in white matter than in gray matter (Fig. 3A), with relative increases of greater than 100% in the gray matter for both metrics compared to white matter. For power law exponent, white matter values were close to 0, indicating white noise, whereas gray matter regions were all significantly greater than 0, indicating colored noise. Although measurements in white matter are unlikely to be meaningful given the source of the BOLD signal is predominantly within gray matter, the low ReHo and power law exponent values in white matter serve as a good negative control.

Figure 3.

Spatial distribution of regional homogeneity and power law exponent (1,019 subject mean). (A) Images showing mean voxelwise ReHo and power law exponent in four axial slices. Images are in radiologic format with patient left on image right. Slice locations were MNI z = 0, 25, 40, and 55 from left to right. (B) Scatter plot comparing regional homogeneity and power law exponent in 7,266 gray matter ROIs. (C) ROIs for which predicted power law exponent, based on a linear fit of the data above, was more than one standard deviation above actual power law exponent across the ROIs. Slice locations were z = −20, −10, 0, and 10. [Color figure can be viewed in the online issue, which is available at http://wileyonlinelibrary.com.]

Restricting the analysis to gray matter, however, still yields strong positive covariance between power law exponent and ReHo. A scatter plot showing gray matter ReHo and power law exponent is shown in Figure 3B. There is a linear relationship with correlation coefficient r = 0.79 and P < 10⁻²⁰⁰ between the two metrics across gray matter ROIs. Regions with highest local connectivity exhibited the most ordered BOLD fluctuations (highest power law exponent) and ROIs with weaker local connectivity to adjacent voxels showed greater randomness in BOLD fluctuations.

The linear trend was similar across the brain, but was weakest for a small subset of ROIs in which power law exponent was lower than would be predicted by ReHo. This group of ROIs, with ReHo between 0.25 and 0.35 consisted of the spinocerebellum (superior cerebellar vermis and superior medial cerebellar hemispheres), lentiform nuclei, and thalami. To illustrate this, we constructed a linear best fit between power law exponent and ReHo, then calculated the expected power law exponent from the linear fit for each ROI's ReHo measurement. We then constructed a histogram of predicted and actual power law exponent. The predicted power law exponent values were greater than two standard deviations below expected only in these regions, shown in Figure 3C. A smaller trend toward higher than expected power law exponent was seen in the medial prefrontal cortex.

We considered two possible confounding variables that might explain the relationship between power law exponent and ReHo. First, it is possible that both metrics simply reflect higher values in ROIs containing higher gray matter density. Second, it is possible that regions where the BOLD standard deviation is higher, indicative of more noisy measurements, might be associated with lower ReHo and lower power law exponents (greater randomness) because of nonuniform acquisition noise across the brain. To evaluate these measurements, we computed partial correlation coefficients between each pair of variables: gray matter density, standard deviation of BOLD signal, ReHo, and power law exponent. These values are shown in Table 2. Even after accounting for the effects of mean gray matter density and standard deviation of the BOLD signal, power law exponent and ReHo show a strong positive relationship (partial correlation = 0.77).

Not only was the groupwise spatial distribution similar between the two metrics, but intersubject differences covaried between ReHo and power law exponent. Although power law exponent measurements are relatively noisy in individual subjects, analyzing a large number of subjects resulted in significant partial covariance with ReHo in 7,225 of the 7,266 gray matter ROIs, after regression of the gray matter and fMRI standard deviation values for each subject from the ReHo measurements in each ROI and correction by false discovery rate (FDR) across ROIs with q < 0.05. The covariance was not uniform across the brain, show in Figure 4. Greatest covariance was seen in superior and inferior parietal lobules and occipital lobe, with significant but decreased covariance in the prefrontal cortex and basal ganglia and thalami.

Figure 4.

Partial correlation between power law exponent and regional homogeneity across subjects for each region, after regressing out the effect of gray matter density and BOLD standard deviation from regional homogeneity measurements. Color scale shows partial correlation coefficient across subjects. Colored regions were significantly correlated with FDR q < 0.05. Slice locations were MNI z = 0, 25, 40, and 55. [Color figure can be viewed in the online issue, which is available at http://wileyonlinelibrary.com.]

Given that temporal complexity of BOLD fluctuations was significantly associated with local connectivity, we also evaluated whether known network changes in fcMRI during late neurodevelopment [Dosenbach et al., 2010; Fair et al., 2007, 2012] may be associated with changes in power law exponent. When evaluating correlation between age and power law exponent for each of the 7,266 gray matter ROIs, we found decreases with age (greater randomness) throughout most of the brain. A few regions, however, showed no significant change in power law exponent with age between ages 7 and 30, shown in Figure 5. All of the regions showing no change in power law exponent with age are association cortex regions coinciding with nodes of the default mode and dorsal attention networks. Areas showing greatest decreases with age were in the basal ganglia, cingulate gyrus, and thalami.

Figure 5.

Change in power law exponent with age. Correlation coefficient is shown by color scale for each of 7,266 gray matter ROIs. ROIs with less than −0.07 correlation showed acceptable false discovery rate q < 0.05. Slice locations were MNI z = 0, 25, 40, and 55. [Color figure can be viewed in the online issue, which is available at http://wileyonlinelibrary.com.]

DISCUSSION

Using data from a large publicly available dataset (n = 1,019), we demonstrate a strong correlation between local connectivity (ReHo) and power law exponent in endogenous brain activity fluctuations. This correlation was seen both in the groupwise spatial distribution of the metrics as well as intersubject covariance of the two measurements, and persisted after regression of BOLD standard deviation and gray matter density from the ReHo measurements. These findings suggest that the determinism or orderliness of large‐scale BOLD fluctuations increases directly with the connectivity of a voxel to its closest neighbors, whereas white matter and areas of gray matter with lower local connectivity exhibited more randomness in their BOLD fluctuations. Such randomness increases throughout most of the brain during adolescence and early adulthood, with the exception of the default mode and attention control networks, where power law exponent is unchanged over the same time period.

Our measurements for power law exponent are similar to measurements of Hurst exponent in multiple prior studies [Lai et al., 2010; Maxim et al., 2005; Wink et al., 2008], which varied from H = 0.5 to a maximum of about H = 0.85, corresponding to power law exponent varying from γ = 0 to a maximum of 0.7 [γ = 2 H − 1; Bullmore et al., 2004]. We observed a maximal power law exponent of about 0.65, which may indicate a slightly reduced range given less variability as our measurements were averaged from a much larger sample. We find, in agreement with He 2011, that power law exponent is significantly correlated with variance of the BOLD signal, as well as with gray matter density, but that neither of these factors explain the much stronger covariance with ReHo.

The relationship between complexity of BOLD fluctuations and local connectivity has implications for neuropathological studies of BOLD complexity. It has been observed in autism, for example, that decreased Hurst exponent is present in brain regions relevant to functional abnormalities of autism [Lai et al., 2010]. This may be predicted given known decreased local connectivity in many of the same regions: insula, superior temporal sulcus, right inferior frontal gyrus [Paakki et al., 2010] where Hurst exponent was significantly decreased. Longer range functional connectivity has also been found to be specifically low in these regions [Anderson et al., 2011c]. Thus, reduced Hurst exponent may simply reflect the decreased network connectivity known to occur in autism. Conversely, ReHo has been found to be increased in many regions in Alzheimer's disease [He et al., 2007] with a notable exception of a region in the precuneus, an area conspicuously absent in findings of increased Hurst exponent elsewhere in Alzheimer's disease [Maxim et al., 2005]. Alternately, increased Hurst exponent may represent differences in the hemodynamic response function, since differential filtering of neural activity by a slower hemodynamic response function [Rombouts et al., 2005] will result in lower frequency BOLD fluctuations and higher Hurst exponent.

We also found that the basal ganglia, thalami, and spinocerebellum showed relatively lower power law exponent than would be predicted from ReHo. This may represent decreased complexity or density in the neural networks involving the striatum, where local circuitry may be less complex than in areas of neocortex. A shift toward higher frequencies (less 1/f behavior) has also been observed uniquely in the basal ganglia in an independent report measuring fractional amplitude of low‐frequency fluctuations, where higher frequency bands (0.027–0.073 Hz) showed greater fractional amplitude than lower frequencies [0.01–0.027 Hz; Zuo et al., 2010].

Network effects have also been proposed as necessary for the development of power law dynamics at the neuronal level. In studies using multielectrode arrays, Beggs and Plenz observed neuronal avalanches of salient, low temporal frequency synchrony in rat cortex [Beggs and Plenz, 2003] and in cortical slice cultures [Beggs and Plenz, 2004]. Neural activity showed a power law exponent of γ = 1.5, similar to that observed in magnetoencephalography (MEG) data [Poil et al., 2008], and higher than is observed in BOLD fluctuations, possibly indicating that hemodynamic coupling plays a role in reducing the power law exponent as neural activity is transmitted into a BOLD response. Nevertheless, it is likely that scale‐free signal fluctuations in neuronal assemblies, electroencephalography (EEG) or MEG, and BOLD measurements reflect the same underlying physiological process, one related to network interactions [Palva and Palva, 2012]. That avalanches occur in cultured neurons indicates that such dynamics are intrinsic to a neural network and not produced solely by patterns of sensory input. Yet sensory input patterns can alter observed power law exponent in vivo [El Boustani et al., 2009], supporting a neural‐network level effect. Given this finding, El Boustani et al. hypothesized that it may be possible to “read out the effective connectivity” of a network from its scaling law exponent, which is precisely what we observe.

We discovered stable power law exponents with age only in a subset of gray matter regions, all within the default mode and dorsal attention networks, with decreases in power law exponent elsewhere in the brain. This is consistent with decreases in ReHo observed with age in most of the brain [Lopez‐Larson et al., 2011]. Moreover, patterns of functional connectivity during late neurodevelopment have shown decreased short‐range connectivity (segmentation) with increased connectivity of distant hubs of distributed networks [integration; Dosenbach et al., 2010; Fair et al., 2008, 2009, 2007; Supekar et al., 2009]. In a review of patterns of functional connectivity throughout childhood and adolescence, Power et al. 2012 describe a trend toward increased connectivity within the default mode network that is consistent with the relative preservation of power law exponents in this region, if to some extent BOLD signal order or self‐similarity is increased by higher network connectivity.

Connectivity in the default mode network not only strengthens but sharpens with age [Anderson et al., 2011b; Gordon et al., 2011; Supekar et al., 2010], with voxels most connected to the default mode network showing greatest increase with age in connectivity to the network. If power law exponent mirrors the extent, density and complexity of the neural network to which a brain region belongs, with greatest number of connections from local circuits, hubs of the default mode network would be a likely set of brain regions to show relatively increased power law scaling with age. Decreases in power law exponent in the remaining brain may follow decreases in local connectivity seen throughout the brain during development [Uddin et al., 2010].

It is intriguing that the greatest preservation of order and self‐similarity in the brain occurs in distributed attentional networks while sensory and motor regions show relative decreases with age in self‐similarity. This might indicate that ongoing brain development through adolescence and even into adulthood can be characterized by a simplification of sensorimotor functional networks with greater complexity of association cortical attentional networks responsible for higher level cognitive processes, with commensurate formation of higher connectivity between disparate brain regions processing polymodal cognitive information.

It is difficult in our data to disentangle the effects of the hemodynamic response function, physiological artifacts aliased into low frequencies, and true power law dynamics of the underlying neural activity. We have limited our preprocessing to correction of physiological artifacts by regression of motion parameters and time series from CSF, white matter, and facial soft tissues rather than global signal regression to avoid propagating differences in temporal frequencies between brain regions. Nevertheless residual physiological signals from subject motion, respiratory or cardiac signal aliasing may remain in the data. Comparison of connectivity metrics in electrophysiological signals such as EEG and MEG with power law scaling parameters could help confirm the relationship between connectivity and temporal complexity. While our analysis shows a close fit to 1/f dynamics of the BOLD signal, it should be emphasized that 1/f approximation does not prove that scale‐free dynamics are demonstrated by the BOLD signal. Nevertheless, this has been formally tested in other studies [Bullmore et al., 2004; He, 2011] and investigations have consistently shown that fMRI data exhibit scale‐invariant properties, although fMRI data may be better represented by multifractal dynamics where different signal components have different scale‐free parameters [Ciuciu et al., 2012].

It remains unclear how connectivity changes, and associated temporal complexity modulations, interface with behavior and neuropathology. For example, is increased randomness seen in autistic brains merely a reflection of connectivity differences, or is it more integrally related to behavioral abnormalities? To what extent do scale‐free dynamics in the brain contribute to preservation of percepts and cognitive memories across behavioral time scales? Finally, what factors or components of a neural network are necessary and sufficient to give rise to higher or lower complexity of endogenous neural fluctuations? Extending models of whole‐brain or local network connectivity to dynamical simulations where temporal behavior can be studied, and developing paradigms to assess behavior and pathology contemporaneously with measurements of brain temporal complexity may help to answer these questions.

CONCLUSIONS

We demonstrate using a large (n = 1,019) publicly available dataset of fMRI images that there is a strong linear relationship between temporal complexity (power law exponent) of BOLD fluctuations and local connectivity (ReHo) in the brain, and that this relationship covaries across spatial regions and across subjects. Temporal complexity of BOLD fluctuations is unchanged in hubs of the default mode and attention control networks with age while decreasing throughout the rest of the brain, with greatest decrease in subcortical gray nuclei. Relationships between connectivity and scale‐free temporal dynamics may inform the use of temporal complexity measurements as biomarkers for cognitive function and neuropathology.