**By: Alexander Zhigalov, School of Psychology, University of Birmingham, UK**

*Scale-free neuronal dynamics and interareal correlations are emergent characteristics of spontaneous brain activity. How such dynamics and the anatomical patterns of neuronal connectivity are mutually related in brain networks has, however, remained unclear. We addressed this relationship by quantifying the network colocalization of scale-free neuronal activity—both neuronal avalanches and long-range temporal correlations (LRTCs)—and functional connectivity (FC) by means of intracranial and noninvasive human resting-state electrophysiological recordings. We found frequency-specific colocalization of scale-free dynamics and FC so that the interareal couplings of LRTCs and the propagation of neuronal avalanches were most pronounced in the predominant pathways of FC. Several control analyses and the frequency specificity of network colocalization showed that the results were not trivial by-products of either brain dynamics or our analysis approach. Crucially, scale-free neuronal dynamics and connectivity also had colocalized modular structures at multiple levels of network organization, suggesting that modules of FC would be endowed with partially independent dynamic states. These findings thus suggest that FC and scale-free dynamics—hence, putatively, neuronal criticality as well—coemerge in a hierarchically modular structure in which the modules are characterized by dense connectivity, avalanche propagation, and shared dynamic states.[Please note that these results have been published in https://www.mitpressjournals.org/doi/full/10.1162/NETN_a_00008]*

**Introduction**

The theory of critical brain dynamics provides a framework for understanding the emergent dynamics of collective neuronal activity and implies a relationship between neuronal interactions and this dynamics (Chialvo, 2010).

It both conceivable and predicted by computational models that these concurrent phenomena, neuronal criticality and functional connectivity, would be related, but very few studies have aimed at elucidating this relationship (Hilgetaget al., 2014).

In this study, we set out to bridge this gap by quantifying the inter-areal networks of criticality and connectivity and assessing their putative similarities.

**Methods**

- Data acquisition and pre-processing

Ten minutes of resting state brain activity was recorded from 14 healthy subjects (seven females; aged 18–27 years) using magnetoencephalography (MEG). The MEG data were corrected for extracranial noise and for cardiac and eye blink artifacts by using the signal space separation method and independent-component analysis, respectively (see, Zhigalov et al., 2017).

The MEG sensor time series were filtered in multiple frequency bands using a broadband finite impulse response filter and Morlet’s wavelets. The filtered time series were inverse-transformed and collapsed into time series of 219 cortical parcels derived from the individual T1-weighted anatomical images by applying collapsing operators that maximized individual reconstruction accuracy.

**Methods (cont.)**

- Amplitude and phase functional connectomes (FC)

To eliminate artificial interactions caused directly by volume conduction and signal mixing, we used linear-mixing-insensitive metrics for assessing interareal phase and amplitude interactions. Prior to the analyses, the time series were filtered using a bank of 31 Morlet’s wavelets with logarithmically spaced central frequencies ranging from 4 to 120 Hz.

The phase synchronization between each pair of cortical parcels was estimated by the weighted phase lag index (Vinck et al., 2011). The amplitude interactions were assessed as follows: the narrow-band time series were orthogonalized for each pair of cortical parcels by using linear regression (Brookes et al., 2012), and then Pearson correlation coefficient was computed for the amplitude envelopes of the orthogonalized time series.

- Avalanche propagation connectome

Neuronal avalanches were detected in broadband-filtered (1–120 Hz) source-reconstructed MEG time series as follows. First, the time series were normalized by subtracting the mean and dividing by the standard deviation. Second, the normalized time series were transformed into binary point processes by detecting suprathreshold peaks above threshold T (see, panel A). Third, these binary sequences (or sequences of events) were converted into avalanche time series by summing the events across the channels in time bins Δt (see, panel A). A neuronal avalanche is defined as a cluster of events in successive time bins, where the beginning and end of the avalanche are defined by single time bins with no events (see, panel A). The avalanche size distributions are typically fit well by a power law or a truncated power law (see, panel B). Using these avalanche data, we defined the avalanche propagation connectome to be constructed by the empirical probability with which suprathreshold peaks of neuronal activity would “transits” from the subset of channels in the first time bin to the subset of channels in the second time bin of each avalanche (see, panel C).

- Long-range temporal correlations (LRTC) inter-areal relatedness connectome

The LRTC connectome was computed as follows. First, individual LRTC scaling exponents (β) were assessed for each cortical parcel (see, panel A and B) and frequency band using detrended fluctuation analysis. Second, the individual scaling exponents (see, panel C) for all cortical parcels (i.e. 219 parcels) and single frequency band (e.g. 10 Hz) were arranged as a matrix, X, with dimensions [n_cortical_parcels x n_subjects = 219 x 14]. Third, the LRTC connectome (see, panel C) was calculated using Person corelation over the rows of matrix, X, as corr(X’).

- Similarity of subgraphs

The modular structure of the connectomes was detected using agglomerative hierarchical clustering. The number of clusters per hemisphere, K = 7, provided a reasonable division of the connectomes into functional modules.

The subgraph similarity index (between different connectomes) was defined as the Jaccard similarity coefficient between matched subgraph identities (see, Zhigalov et al., 2017).

**Results**

- Functional and critical dynamics connectomes

Avalanche propagation and the connectomes of interareal relationships in local LRTCs have overall structures similar to those of the functional connectomes of phase synchronization and amplitude correlations.

- Similarity between connectomes subgraphs

A large fraction of cortical parcels had similar subgraphs among connectomes (panel A) and hence these connectomes exhibited a similar modular structure. We found the regions with most consistently shared modularity to be in the sensorimotor, visual, temporal, and medial prefrontal areas (p < 0.05, permutation test, panel B).

- Similarity between connectomes subgraphs across frequencies

The results showed that subgraph similarity between the connectomes was highly significant (well above the confidence interval of 99.99 %) in a wide range of frequency bands with largest values between 8 and 30 Hz. In the light of the present findings, narrow-band alpha, beta, and low-gamma oscillations hence appear to play a central role in the dynamics-connectivity association.

Conclusion

Our results (Zhigalov et al., 2017) strongly suggest that neuronal communities characterized by strong internal phase-synchronization and amplitude correlations are also characterized by preferentially internal avalanche propagation and correlated local LRTCs. This is also consistent with the notion that scaling exponents of avalanche size distribution and LRTCs are correlated (Zhigalov et al., 2015).

**Beyond linking criticality and connectivity, our findings also indicate that the brain can be envisioned as a constellation of mutually coupled and hierarchically organized modules (Meunier et al., 2010; Gallos et al., 2012) distributed to distinct neuroanatomical substrates and to different frequency bands rather than as a homogeneous critical-state system.**