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Edited by: Tjeerd W. Boonstra, University of New South Wales, Australia

Reviewed by: Bruce West, US Army Research Office, USA; Michael Breakspear, The University of New South Wales, Australia

*Correspondence: Dante R. Chialvo, Consejo Nacional de Investigaciones Científicas y Tecnológicas, Santa Fe 3100, Rosario (2000), Argentina. e-mail:

This article was submitted to Frontiers in Fractal Physiology, a specialty of Frontiers in Physiology.

This is an open-access article distributed under the terms of the

The study of spontaneous fluctuations of brain activity, often referred as brain noise, is getting increasing attention in functional magnetic resonance imaging (fMRI) studies. Despite important efforts, much of the statistical properties of such fluctuations remain largely unknown. This work scrutinizes these fluctuations looking at specific statistical properties which are relevant to clarify its dynamical origins. Here, three statistical features which clearly differentiate brain data from naive expectations for random processes are uncovered: First, the variance of the fMRI mean signal as a function of the number of averaged voxels remains constant across a wide range of observed clusters sizes. Second, the anomalous behavior of the variance is originated by bursts of synchronized activity across regions, regardless of their widely different sizes. Finally, the correlation length (i.e., the length at which the correlation strength between two regions vanishes) as well as mutual information diverges with the cluster's size considered, such that arbitrarily large clusters exhibit the same collective dynamics than smaller ones. These three properties are known to be exclusive of complex systems exhibiting critical dynamics, where the spatio-temporal dynamics show these peculiar type of fluctuations. Thus, these findings are fully consistent with previous reports of brain critical dynamics, and are relevant for the interpretation of the role of fluctuations and variability in brain function in health and disease.

It is now recognized that important information can be extracted from the brain spontaneous activity, as exposed by recent analysis (Biswal et al.,

In the same direction, the information content of the brain BOLD signal's variability

In this work we characterize the statistical properties of the spontaneous BOLD fluctuations and discuss its possible dynamical mechanisms. The paper is organized as follow: in the next section the origin of the data is described as well the pre-processing of the signal. The definitions of regions of interest is described as well as how to construct subsets of different sizes, needed to compute fluctuations. The results section starts with the analysis of the average spontaneous fluctuations for each RSN, which identify anomalous scaling of the variance as a function of the number of elements. Next, this anomaly is explored to determine its origins by studying in detail the temporal correlations in clusters of different sizes. Finally the analysis of the correlation length is described, revealing a distinctive divergence with the size of the cluster considered. The paper close with a discussion of the relevance of the uncovered anomalous scaling for the current views of large scale brain dynamics. For clarity of presentation, the calculations that are not central to the main message of the paper, are presented separately in an Appendix.

fMRI data was obtained from five healthy right-handed subjects (21–60 years old, mean = 40.2) using a 3T Siemens Trio whole-body scanner with echo-planar imaging capability and the standard radio-frequency head coil. Subjects were scanned following a typical brain resting state protocol (Fox and Raichle,

In each subject, 240 BOLD images, spaced by 2.5 s, were obtained from 64 × 64 × 49 voxels of dimension 3.4375 mm × 3.4375 mm × 3 mm. Pre-processing was performed using FMRIB Expert Analysis Tool [FEAT, Jezzard et al.,

It is known that the brain activity fluctuations at rest exhibit large-scale spatial correlations. The presence of these robust correlations is reflected on the coherent activity which determine the spatial domains of the RSN. Therefore, our analysis is focussed on the statistical analysis of the RSN fluctuations. At least since Beckmann et al. (

Threshold | 4 | 3.3 | 2.4 | 3.4 | 2.2 | 2.7 | 3.2 | 2.2 |

# regions | 1 | 2 | 3 | 4 | 4 | 9 | 4 | 8 |

To analyze the noise properties, we look at the behavior of the variance and correlations under various manipulations of the size of the ensemble of voxels where these fluctuations occurs. This is a common strategy in other statistical physics problems where very distinctive scaling behavior can be observed depending of the type of fluctuations the system is able to exhibit (Stanley,

We start by studying the fluctuations of the BOLD signal around its mean. The signal of interest, for the 35 RSN clusters, is defined as
_{H}. These signals will be used to study the correlation properties of the activity in each cluster.

The mean activity of each

Since the BOLD signal fluctuate widely and the number ^{−1}. In other words one would expect a smaller amplitude fluctuation for the average BOLD signal recorded in clusters [i.e., ^{−1} decay is enough to disregard further statistical testing. Nevertheless, we test a null hypothesis recomputing the variance for artificially constructed clusters having similar number of voxels but composed of the randomly reordered _{k}(^{−1} law (dashed line in Figure _{k}(

In order to distinguish how much of the constancy of the variance demonstrated up until now is related with the fact that the time series belong to clusters that are independent components (Beckmann et al.,

For spatio-temporal signals the relationship between the temporal fluctuations of the average signal and its space correlation function is well defined (Ross, _{i,j} the correlation between voxels

Equation 6 suggest that it can be productive to investigate the correlations properties of the BOLD data. The point to clarify is whether the average spatial correlation 〈_{i,j} for non-overlapping periods of 10 temporal points.

Figure

The results of these calculations implies that independently of how large the size of the cluster considered, there is always an instance in which a large percentage of voxels are highly coherent and another instance in which each voxels activity is relatively independent.

A very metaphorical way to visualize the behavior of the correlations is to think of the patterns of spontaneous activity as “clouds” of relatively higher activity moving slowly throughout the brain's cortex. Thus, the moments of large coordination shown in Figure

The results in the previous paragraphs indicate that the anomalous scaling of the variance can be related to dynamical changes in the correlations. A straightforward approach to understand the correlation behavior commonly used in large collective systems (Cavagna et al.,

Thus, we start by computing for each voxel BOLD time series their fluctuations around the mean of the cluster that they belong. Recall the expression in Equation 1:
_{H}. By definition the mean of the BOLD fluctuations of each cluster vanishes,
_{w} represent averages over

^{1/3}, i.e., ξ grows linearly with the average cluster' diameter

The most notorious result is the fact that correlations decay with distance slower in larger clusters than in relatively smaller clusters, giving rise to the family of curves shown in Figure

Although the present observations can be appropriately described solely in terms of correlations, the same concept can be also casted in terms of information measures, which are often used to estimate the degree of coherence between regions or neural structures. The mutual information between any two

_{I}, is an increasing function of the size of the cluster (middle panel). The bottom panel illustrates the good data collapse after rescaling the horizontal axis as _{I}.

In this work, key statistical properties of the brain BOLD signal variability were investigated. The results are relevant to the understanding of the brain spontaneous activity fluctuations in health and disease. The three most relevant findings that we may discuss are:

the variance of the average BOLD fluctuations computed from ensembles of widely different sizes remains constant, (i.e., anomalous scaling);

the analysis of short-term correlations reveals bursts of high coherence between arbitrarily far apart voxels indicating that the variance' anomalous scaling has a dynamical (and not structural) origin;

the correlation length measured at different regions increases with region's size, as well as its mutual information.

The second finding, showing that the observed dynamical short-term changes in the correlations drives up the variance, is relevant for the interpretation of the brain functional connectivity. The evaluation of functional connectivity between regions often uses the average correlation, and the results in Figure

The third result concerning the divergence of the correlation length with increasing cluster size is perhaps the most telling one, because is in contrast with the prevailing viewpoints about brain functional connectivity. Indeed it is implicit in the interpretation of functional connectivity studies the notion that brain activity

Finally, an important question is concerned with the origin of the statistical properties unveiled in this work. We suggest that a candidate explanation which is able to unify all the observations presented here can be found in the context of critical phenomena (Stanley,

In summary, the analysis of the BOLD' fluctuations of the resting brain shows anomalous statistical properties, bursts of highly correlated states and divergence of correlation length, which are dynamical properties known to be found only near a critical point of a phase transition. These findings are fully consistent with previous reports of large-scale brain critical dynamics (Fraiman et al.,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Work supported by NIH NINDS (USA), grant NS58661, by Consejo Nacional de Investigaciones Cient'ificas y Tecnol'ogicas (CONICET) (Argentina), by the Spanish Ministerio de Economia y Productividad (previously Ministerio de Ciencia y Tecnologia) (Spain) and by European Funds—FEDER, grant SEJ2007-62312. We thank Prof. Pedro Montoya and M. Mu~noz (UIB, Mallorca, Spain) for discussions and help in data acquisition, and E. Tagliazucchi, P. Balenzuela, A. Haimovici (UBA, Argentina) and L. Hess, A. Tardivo, A. Yodice (UNR, Argentina) for continuous discussions.

Additional information is provided here to supplement the main results. The first item is concerned with the robustness of the short-term correlations presented in Figure

As discussed in Figure

The divergence of correlation length discussed in Figure ^{1/3}, i.e., ξ grows linearly with the average cluster' diameter

^{1/3}.

In spatio-temporal data it is well known the relationship between the temporal fluctuations of a mean magnitude and the space correlation function. Let suppose we want to study a brain region (our clusters in the main text) of _{i}(_{i}(

Since we are interested also on how correlations affect variance, let consider some cases. If there exist null variability between all the voxels in the region, that is all voxels of the region do exactly the same in time, the left term of Equation 14 remains equal to one no matter the size (

First, the mean correlation,
^{*} is the radius of the spherical region under study, and

^{−0.7}. From Equation 21 we obtain an exponent α = 0.9. The inset corresponds to the variance of the mean activity as a function of