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Edited by: Plamen Ch. Ivanov, Boston University, United States

Reviewed by: David Papo, Italian Institute of Technology (IIT), Italy; Angela Lombardi, National Institute for Nuclear Physics of Bari, Italy; Fabrizio Lombardi, ETH Zürich, Switzerland; F. Argoul, Centre National de la Recherche Scientifique (CNRS), France

This article was submitted to Fractal and Network Physiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The analysis of human brain functional networks is achieved by computing functional connectivity indices reflecting phase coupling and interactions between remote brain regions. In magneto- and electroencephalography, the most frequently used functional connectivity indices are constructed based on Fourier-based cross-spectral estimation applied to specific fast and band-limited oscillatory regimes. Recently, infraslow arrhythmic fluctuations (below the 1 Hz) were recognized as playing a leading role in spontaneous brain activity. The present work aims to propose to assess functional connectivity from fractal dynamics, thus extending the assessment of functional connectivity to the infraslow arrhythmic or scale-free temporal dynamics of M/EEG-quantified brain activity. Instead of being based on Fourier analysis, new Imaginary Coherence and weighted Phase Lag indices are constructed from complex-wavelet representations. Their performances are first assessed on synthetic data by means of Monte-Carlo simulations, and they are then compared favorably against the classical Fourier-based indices. These new assessments of functional connectivity indices are also applied to MEG data collected on 36 individuals both at rest and during the learning of a visual motion discrimination task. They demonstrate a higher statistical sensitivity, compared to their Fourier counterparts, in capturing significant and relevant functional interactions in the infraslow regime and modulations from rest to task. Notably, the consistent overall increase in functional connectivity assessed from fractal dynamics from rest to task correlated with a change in temporal dynamics as well as with improved performance in task completion, which suggests that the complex-wavelet weighted Phase Lag index is the sole index is able to capture brain plasticity in the infraslow scale-free regime.

The dynamics of Human brain activity can be studied non-invasively using electro- and magnetoencephalography (EEG and MEG, respectively). Interpreted as resulting from the synchronous activation of neuronal populations in specific frequency bands, these fluctuations are often analyzed as fast (10 Hz and above) oscillatory rhythms now associated with cognitive functions, such as perception, attention, or decision making (cf. e.g., Freeman,

At the turn of the 21st century, the large-band infraslow activity of the brain (typically below 1 Hz), which for long had been considered as either instrumental or head-movement noise, received growing interest; it has now been documented as a prominent part of recorded electromagnetic brain signals and a critical component of brain activity (Gong et al.,

Infraslow arrhythmic brain activity can be efficiently described with large-band scale-free models, such as selfsimilar processes (fractional Brownian motion and fractional Gaussian noise) (Mandelbrot and van Ness,

Remote brain regions are known to interact within large scale functional networks (e.g., the default Mode Network at rest), which mediate the information flow inside the brain integrating the activity of functionally segregated modules that are activated in particular mental states, task execution, or health condition (Power et al.,

Functional connectivity has so far mainly been measured via the band-limited oscillatory activity of the brain and has hardly been applied to characterize the infraslow arrhythmic brain activity. Preliminary attempts in that direction (Achard et al.,

The present work aims to revisit the analysis of functional connectivity in human brain activity in two ways:

First, functional connectivity assessment will be based on the on-going (or spontaneous) infraslow arrhythmic (or scale-free) activity of the human brain rather than on stimulus-induced band-limited oscillatory faster rhythms. This will be referred to as

Second, indices quantifying functional connectivity from fractal dynamics will be constructed from multivariate complex wavelet transforms rather than from Fourier-based cross-spectral analysis. The key intuitions underlying the design of these indices are double: Based on wavelet transforms, these tools will inherit from their well-documented performance and robustness for the analysis of scale-free dynamics (Flandrin,

To that end, after a brief recall of Fourier-based spectral estimation and the classical Fourier-based functional connectivity indices (F-ICOH and F-wPLI) in section 2.1, Complex wavelet transforms and the corresponding Complex Wavelet-based functional connectivity indices (W-ICOH and W-wPLI) are defined in section 2.2. The performance of several Complex Wavelet-based functional connectivity indices proposed here are compared against the others, and against their corresponding Fourier counterparts, by means of Monte Carlo numerical simulations, involving a large number of independent drawings of synthetic signals, sampled from stochastic processes commonly used to model scale-free temporal dynamics, multivariate fractional Brownian motions, and multivariate fractional Gaussian noises (cf. section 2.3). Several scenarios (different temporal dynamics, connectivity networks, additive trends) are investigated to assess the interest and relevance of the proposed Complex Wavelet indices (W-ICOH and W-wPLI) compared to Fourier-based ones in terms of estimation performance and robustness to trends.

The proposed Complex Wavelet indices assessing functional connectivity from fractal dynamics are extensively tested on MEG data, collected on 36 individuals, both at rest and during a visual discrimination learning task. The experimental data are described in section 3 (see also Zilber et al.,

Analyses of functional connectivity assessed from fractal dynamics within infraslow arrhythmic cross temporal dynamics regime, ranging from 0.1 to 1.5 Hz for this data set (La Rocca et al.,

The proposed Complex Wavelet tools constitute, to the best of our knowledge, the first operational tools for a relevant assessment of functional connectivity from fractal dynamics, i.e., functional connectivity in scale-free cross-temporal dynamics for the large-band infraslow arrhythmic brain activity recorded in M/EEG. M

The _{m}(_{m = 1, ..., M}, ^{(F)} of the cross-spectrum _{X}(ℓ, _{ℓ, k}(

where ϕ_{ℓ,k}(_{0}) exp (−2_{0}_{0} and ν_{0} are positive constants that can be arbitrarily chosen provided that they satisfy _{0} ν _{0} ≤ 1/(4π).

Straightforward calculations yield

with _{ℓ,k} being uniformly controlled by the choice of the function ϕ, ^{(F)} achieves a fixed

From

To quantify functional connectivity on MEG signals, the corresponding indices are practically computed as sums of the absolute values of these functions over the range of frequencies defining the targeted band-limited oscillations. Large values (above predefined thresholds) are used as markers of functional connectivity at the individual level, which are usually followed by statistical testing for assessing group-level significance.

The classical discrete wavelet transform relies on the use of a real-valued mother-wavelet (cf. e.g., Mallat, ^{(r)} denote a real-valued oscillating and sufficiently smooth reference pattern, referred to as the ^{2}(ℝ) (cf. e.g., Mallat, ^{(r)}, an analytic complex mother-wavelet can be defined as ψ = ψ^{(r)} + ^{(1)}, where ψ^{(1)} consists of the Hilbert transform of ψ^{(r)}. The design of a complex, invertible, and analytic mother wavelet is not straightforward. In the present work, we build on the excellent approximate solution proposed in Kingsbury (

For a signal ^{(r)} and ψ^{(1)}, respectively, independently.

It has been well-documented that the study of univariate scale-free temporal dynamics should be performed using a wavelet-based spectral estimation rather than a Fourier-based one (cf. e.g., Flandrin, _{m},

where ^{*} stands for complex conjugate.

It has been shown (Abry et al.,

with

Equations (2) and (7) combined together show that Fourier-based ^{j} into frequencies as _{s} is the data sampling frequency and _{0} a constant that depends on the specific choice of the mother wavelet. Readers interested by further theoretical and practical discussions on comparing Fourier and wavelet-based spectral estimations, are referred to e.g., Abry and Veitch (

Fourier vs. wavelet spectral estimation on actual source-reconstructed MEG time series. Top: Two source-reconstructed MEG time series _{1} _{2}

From the wavelet-based estimate of the power spectrum, wavelet-based indices can be constructed to assess functional connectivity, as was the case with Fourier spectrum and mutatis mutandis:

Unlike the standard discrete wavelet transform coherence function used in, e.g., Whitcher et al. (

Functional connectivity for scale-free infraslow temporal dynamics consists of averaging the absolute values of these functions over the corresponding range of octaves _{1} ≤ _{2} (equivalently over the range of scales ^{j} or frequencies

Remapping scales into frequencies, calculations inspired from those leading to Equations (2) and (7) permit to compare theoretically and practically W-COH, W-ICOH and W-wPLI to F-COH, F-ICOH, and F-wPLI, as illustrated in

Complex Wavelet-based connectivity on synthetic bivariate fractional Gaussian noise with correlation but no delay. W-COH

This is here critical to emphasize that

To assess the performance of the proposed indices aiming to quantify functional connectivity from fractal dynamics, Monte Carlo numerical simulations were conducted. They make use of synthetic bivariate fractional Brownian motion, a specific instance of the multivariate selfsimilarmodel recently introduced in Didier and Pipiras (_{H1} and _{H2}, with possibly different selfsimilarity parameters _{1} and _{2}, with pointwise correlation ρ. In addition, one component is delayed by Δ. Correlation coefficient ρ is set to range within ρ ∈ {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, ^{14}, chosen to match the size of the infraslow regime of the MEG data (cf. sections 3 and 4).

To model MEG data as those analyzed in section 4 and as commonly indicated in the literature (He et al.,

Indices assessing functional connectivity from fractal dynamics (both Fourier and wavelet-based) were computed as average over a range of frequencies and scales that match those of the infraslow scale-free range observed on the MEG data described and analyzed hereafter. Performances are reported as means (and confidence intervals) computed from

To start with, we analyzed scenarios where the two components of bivariate fractional Gaussian noise were correlated but not delayed: Δ ≡ 0.

We then analyzed signals with delays amongst components.

(i) Both W-ICOH and W-wPLI do depart from 0 across

(ii) As functions of

(iii) When a scale 2^{j} in relation to Δ is chosen, both (the absolute values of) W-ICOH and W-wPLI are proportional to (the absolute value of) ρ (third column). This shows not only that W-ICOH and W-wPLI depart from 0 when delays amongst components exist but also that the amplitude of the departure is proportional to the correlation ρ between components, a crucial property to assess quantitatively functional connectivity, clearly and originally quantified in these numerical simulations.

(iv) The conclusions stemming from comparing the performance of Fourier-based F-ICOH and F-wPLI to Complex Wavelet-based W-ICOH and W-wPLI depend on the parameters used for simulating bivariate synthetic time series. When the latter consist of bivariate fGn with _{1} = 0.7 and _{2} = 0.8 (_{1} = 0.7 and _{2} = 0.8 (

Complex Wavelet-based connectivity on synthetic bivariate fractional Gaussian noise with correlation and delay Δ = 8. Top row: W-ICOH results. Bottom row: W-wPLI results. From left to right: W-ICOH _{1} = 0.7 and _{2} = 0.8.

Complex Wavelet-based connectivity on synthetic bivariate fractional Brownian motion with correlation and delay Δ = 8. Top row: W-ICOH results. Bottom row: W-wPLI results. From left to right: W-ICOH _{1} = 0.7 and _{2} = 0.8.

We finally analyzed more complicated scenarios with correlation and delays amongst components as well as additive smooth slow trends superimposed as noise to the actual scale-free components.

(i) F-ICOH and F-wPLI depart from 0 across scales when there is no correlation while the Complex Wavelet-based W-COH and W-wPLI do not (second column);

(ii) F-ICOH and F-wPLI significantly overestimate correlations at small ρ while W-COH and W-wPLI do not (third column);

(iii) The RMSE of F-ICOH and F-wPLI becomes up to ten times larger than RMSE of W-ICOH and W-wPLI for small values of ρ (fourth column).

Complex Wavelet-based connectivity on synthetic bivariate fractional Gaussian noise with correlation and delay, and additive trends. Top row: W-ICOH results. Bottom row: W-wPLI results. From left to right: W-ICOH

In addition,

(i) They indicate that W-COH cannot be used to assess functional connectivity as it is fooled by zero-delay (volume conduction effect) correlations, thus confirming an already documented observation for F-COH in the literature (Nolte et al.,

(ii) The Complex Wavelet W-ICOH and W-wPLI can be used to assess functional connectivity for scale-free temporal dynamics.

(iii) The Complex Wavelet W-ICOH and W-wPLI perform significantly better than the Fourier-based F-COH and F-wPLI first when the signals show very large scaling exponents β in their ^{−β} power spectral density behavior, as is the case with fBm-like time series and second when additive noise in the form of smooth and slow trends are superimposed to data with scale-free dynamics, which is a situation commonly observed in recordings collected from neuroimaging techniques.

(iv) W-ICOH and W-wPLI perform comparably with (slightly) better performance of W-wPLI when ρ or Δ increases, or when smooth trends are superimposed to scale-free dynamics, as often the case on MEG data. This will be further discussed in section 4.

Ratio of the RMSE of W-ICOH to the RMSE of W-wPLI, averaged across scales 3 ≤

The proposed complex wavelet-based assessment of functional connectivity in infraslow arrhythmic brain activity was tested on MEG measurements, consisting of non-invasive recordings of simultaneous time-series reflecting the whole brain activity, both at rest and during the completion of a task. All details about the experimental paradigm and the task can be found in Zilber et al. (

In short, the task was designed from a short-term learning paradigm and consisted of visual coherence discrimination. Two sets of colored (green and red) dots were mixed and shown on a screen, each dot with random and independent movement. After a variable duration interval (0.3–0.6 s) of incoherent motion, a fraction of randomly chosen dots belonging to either of the two sets (also randomly chosen at each trial) followed a coherent motion during 1 s. Participants were asked to tell which of the red or green clouds had a coherent motion by pressing a button of the same color. Task difficulty was increased by decreasing the rate of dots in coherent motion.

The experiment was organized as interleaved MEG blocks alternating rest and task measurements: It started with a 5-min rest recording (REST_{i}), followed by a 12-min pre-training block (TASK_{i}); this was followed by four successive 5-min long individualized training blocks. Another 5-min resting-state block (REST_{f}) was recorded prior to a final 12-min post-training block (REST_{f}), consisting of the same visual coherence discrimination task as in TASK_{i}. During TASK_{i} and TASK_{f}, the motion coherence discrimination accuracy of each participant was assessed. Pre-training and post-training behavioral thresholds were computed for each participant as the visual coherence level associated with 75% of correct responses (hit rate). During REST blocks, participants were instructed to keep their eyes open, and they were not following any other explicit instruction, thus permitting the analysis of spontaneous fluctuations of brain activity from MEG recordings.

For the experiment, 36 healthy participants (mean age: 22.1 ± 2.2) were recruited. All participants were right-handed, had normal hearing, and had normal or corrected-to-normal vision. Before the experiment, all participants provided written informed consent in accordance with the Declaration of Helsinki (2008) and the local Ethics Committee on Human Research at NeuroSpin (Gif-sur-Yvette, France).

Brain activity was recorded via MEG modality in a magnetically shielded room using a 306 MEG system (Neuromag Elekta LTD, Helsinki). MEG signals originally sampled at 2 kHz were downsampled at 448 Hz and preprocessed to remove external and internal interferences in accordance with accepted guidelines for MEG research (Gross et al.,

Following the systematic inspections of the wavelet spectra and cross-spectra reported in La Rocca et al. (

Infraslow functional connectivity was assessed for several experimental conditions: resting-state (REST_{i}), pre-training (TASK_{i}), and post-training (TASK_{f}) tasks, thus enabling us to assess changes in functional interactions from rest to task and modulations related to learning.

Three proposed complex wavelet based indices were then computed to assess infraslow functional connectivity by averaging across octaves corresponding to the scale-free regime, 8 ≤

These indices were filtered at the group-level (

To investigate significant differences in infraslow functional connectivity between two different experimental conditions (e.g., TASK_{i} − REST_{i}) independently for each chosen index, a group-level paired

To compare Fourier-based F-ICOH and F-wPLI to Complex Wavelet-based W-ICOH to W-wPLI, Fourier-based spectral estimation was conducted using Welch Periodogram procedures (as described in section 2.1), using a windowed Fourier transform with a Hanning-type window of duration 80s.

_{i} (top row) and pre-training TASK_{i} (center row). Further, _{i} and REST_{i}.

(i) The connectivity networks yielded by W-COH predominantly display short-range and inter-hemispheric interactions throughout the cortex and most notably amongst frontal regions on one hand and temporo-occipital regions on other hand, both for REST_{i} and TASK_{i}.

(ii) The connectivity networks yielded by W-ICOH and W-wPLI display similar structures, dominated by long-range spatial interactions, that differ significantly from those of the networks produced by W-COH, dominated by shorter-range spatial interactions. These differences in network structures can be quantified using the Average Degree, i.e., the average number of connections per node, as a network structure metrics. For REST_{i}, the Average Degrees for the graphs obtained by W-COH, W-ICOH, and W-wPLI are of 0.95(±0.37), 0.21(±0.24), and 0.44(±0.52), respectively. Medians distributions of the number of links per node differ significantly between W-COH and W-ICOH (^{−11}) or between W-COH and W-wPLI (^{−6}). The same holds for TASK_{i}, with average degrees of 1.0(±0.49), 0.25(±0.24), and 0.52(±0.50), respectively, and significances of ^{−8} and ^{−3}, respectively.

(iii) While yielding comparable networks, W-wPLI and W-ICOH differ insofar as the former yields larger connectivity indices than the latter. In addition, connectivity networks using W-wPLI or W-ICOH differ in structure; however, they differ much less than when comparing W-wPLI vs. W-COH or W-ICOH vs. W-COH. Indeed, for REST_{i} the Average Degrees of W-wPLI and W-ICOH are 0.44(±0.52) and 0.21(±0.24), respectively, yielding a quantifiable difference (_{i} the Average Degrees of W-wPLI and W-ICOH are 0.52(±0.50) and 0.25(±0.24), respectively, yielding a clearer difference (

(iv) When comparing TASK_{i} vs. REST_{i}, W-wPLI and W-ICOH both indicate an increase in functional connectivity during task performance. This increase in functional connectivity assessed from fractal dynamics highlights interactions between regions recruited in the achievement of the task, notably fronto-temporal couplings [between the right ventro-lateral prefrontal cortex (vlPFC) and inferior temporal cortex (ITC)], interactions linking temporal regions [anterior superior temporal sulcus (aSTS) and auditory cortex] with the intra-parietal sulcus (IPS), motor-occipital couplings between the left frontal BA6 (including premotor and supplementary motor regions), and primary visual areas (V1/V2). Interaction between the key region hMT+, sensitive to visual motion, and the associative area, pSTS, is also significant in the left hemisphere.

Functional connectivity assessment from fractal dynamics: Group-level functional connectivity in infraslow MEG-source reconstructed brain dynamics. Filtered 28 × 28 connectivity networks measured from Complex Wavelet based W-wPLI (left), W-ICOH (middle), and W-COH (right), for REST_{i} (top row) and pre-training TASK_{i} (center row). The red color intensity codes for the values of the connectivity indices (ranging from 0 to 1 by construction). Functional connectivity differences between conditions TASK_{i} and REST_{i} when assessed as significant by a group level FDR corrected _{i} − REST_{i} differences in the values of indices from blue (negative) to red (positive), thus indicating that only increases in functional connectivity are observed from REST_{i} to TASK_{i}.

Focusing on the W-wPLI index only, _{f} to the initial rest REST_{i}, which, compared to the contrast TASK_{i} − REST_{i} (cf.

Fractal dynamics-based functional connectivity assessment (W-wPLI) differences between REST_{i} and TASK_{i} and between REST_{i} and TASK_{f}. The increase in functional connectivity assessed from fractal dynamics from rest to task is strengthened with training, i.e., from TASK_{i} to TASK_{f}, and emerged between several intra- or inter-hemispheric pairs of regions (Frontal polar/IPS, ITC/MT, FEF/pSTS) involved in task performance.

In La Rocca et al. (_{i} and TASK_{i} but also between REST_{i} and TASK_{f}. Further, _{i} to TASK_{f} in the parieto-occipital regions involved in task performance, notably the bilateral hMT+ regions, the visual cortices including V1/V1 and V4 for the visual color detection. Interestingly, after training, these regions are also more strongly coupled with others during task performance (TASK_{f} vs. REST_{i}).

selfsimilarity (_{i} and TASK_{i} and between REST_{i} and TASK_{f}. The decrease in selfsimilarity from rest to task is strengthened with training, i.e., from TASK_{i} to TASK_{f}, and more heavily in the parieto-occipital (hMT+, visual cortices, V1/V2/V4) regions involved in task performance. Note that a value of

To investigate a potential training-induced relation between the decrease in selfsimilarity and the increase in W-wPLI, Δ_{TASFf} − H_{RESTi} and ΔW-wPLI = W-wPLI_{TASFf} − W-wPLI_{RESTi} were averaged across the whole brain for each subject. Corresponding averages are shown in _{i} and TASK_{f} can be assessed (after false discovery rate-based corrections for multiple hypothesis testing), the relation between Δ

Decrease of selfsimilarity vs. increase in functional connectivity assessed from fractal dynamics from rest to task. Δ_{TASKf}-H_{RESTi} as a function of Δ W-wPLI = W-wPLI_{TASKf} − W-wPLI_{RESTi}, averaged across the whole brain for each of the 36 participants (each marked as a dot), shows that the decrease of selfsimilarity correlates negatively (

Finally, functional connectivity in the infraslow range of temporal dynamics can be related to task performance, and this is notable after training. _{i} and TASK_{f}. It shows that participants with the larger increase in functional connectivity assessed from fractal dynamics induced by training, i.e., the larger increase of W-wPLI_{TASKf} − W-wPLI_{TASKi}, are also those achieving the better performance in post-training task.

Functional connectivity assessment from fractal dynamics vs. Task Performance. Individual performance in the post-training task shows significant (_{TASKf} − W-wPLI_{TASKi}. Each participant is represented as a dot, and outliers are marked with a ×.

Averaging (the absolute values) of F-wPLI across a range of frequencies that match the range of scales associated with the infraslow scale-free scaling range permits us to compare Fourier-assessed functional connectivity from fractal dynamics. _{i} and TASK_{i}, showing significant differences with those obtained using W-wPLI. The network topography associated with the F-wPLI index are denser compared to W-wPLI. Indeed, using the Average Degree, used as a graph structure metric, it was found that for REST_{i}, the Average Degrees of W-wPLI and F-wPLI are 0.44(±0.52) and 1.62(±1.11), respectively, yielding a very significant difference, assessed by a ^{−6}, and for TASK_{i}, the Average Degrees of W-wPLI and F-wPLI are 0.52(±0.50) and 1.65(±1.21), respectively, yielding also a significant difference assessed by a ^{−5}. Further, the number of significant interactions with F-wPLI is more balanced between the two hemispheres during REST_{i} in contrast to W-wPLI, which captures more couplings in the right one. Also, the resting-state W-wPLI-based network configuration is more dominated by fronto-occipital couplings, whereas the F-wPLI-based shows a greater number of inter-hemispheric interactions. During the pre-training task TASK_{i}, the W-wPLI and F-wPLI network topographies both show similar connections but also strong differences: the former is more dominated by fronto-parieto-occipital couplings with a hub role played by the visual cortices, while the latter does not strongly differ from the F-wPLI network found during REST_{i}. Finally and more importantly, no statistically significant difference in F-wPLI_{TASKi}-F-wPLI_{RESTi} can be evidenced (see _{i} to TASK_{i} between fronto-parieto-occipital regions that are involved in task performance (see

Fourier-based wPLI estimator in the scale-free regime. No significant difference between F-wPLI_{TASKi} and F-wPLI_{RESTi} in arrhythmic regime can be found.

At the methodological level, the results presented in section 4 clearly showed that W-COH fails to characterized correctly functional connectivity, which is in clear agreement with the numerical simulations reported in section 2.3 on synthetic data fGn/fBm and with results reported in the literature (cf. Stam et al.,

More interestingly, compared to W-ICOH, W-wPLI was observed to more accurately quantify functional connectivity assessment from fractal dynamics, both at rest and during a task in MEG data, as well as to better highlight relevant changes in functional connectivity assessed from fractal dynamics between rest and task. This is in agreement with previously reported results, showing that for band-limited oscillatory activities, F-wPLI was a better index to assess functional connectivity than F-ICOH. This was attributed to the denominator of F-wPLI being different from that of F-ICOH and less sensitive to (residual) volume conduction effects (Stam et al.,

The benefits of using wavelet-based (multiscale) tools to analyze scale-free temporal dynamics and estimate the corresponding scaling exponent compared to classical Fourier-based spectral estimation have been abundantly documented elsewhere (cf. e.g., Abry and Veitch,

On MEG data, functional connectivity in the infraslow arrhythmic regime assessed by W-COH, i.e., based on direct correlation, was observed to yield mostly spatial short-range connectivity networks across the brain, notably with spurious short-range functional intra- and inter-hemispheric interactions, visible between frontal regions both at rest and during a task. This is likely a consequence of residual common source effects, strongly biasing the real part of thecoherence function, and thus yielding spurious connectivity measures, in agreement with results reported in Stam et al. (

Compared to F-wPLI, W-wPLI showed an enhanced statistical sensitivity as it revealed a positively engaged parieto-temporo-occipital network in infraslow temporal dynamics when contrasting rest to pre-training activities. This network comprises previously identified key brain regions (e.g., hMT+, ITC, vlPFC, and pSTS) during task performance. Interestingly, such regions also consistently identified as beubg recruited by a task when using standard temporal or spectral data analysis (Zilber et al.,

The third finding of this study is the positive correlation between the increase in functional connectivity assessed from fractal dynamics and task performance when contrasting pre- to post-training brain activity. This suggests that the consolidated network eases task completion for each individual, experiencing averaged increase in functional couplings within the infraslow regime.

Finally, the increase in functional connectivity assessed from fractal dynamics was shown to be correlated with a decrease in the selfsimilarity from rest to task. These results on functional connectivity assessment from fractal dynamics, combined with the univariate (regionwise) analysis of scale-free temporal dynamics of the same data (La Rocca et al.,

At rest, each region displays a globally very structured and slow activity in time (large selfsimilarity exponent

During task performance, temporal dynamics in each region independently become less globally structured and faster (decrease in

This permits us to conjecture an interplay between temporal and spatial dynamics for the large-band infraslow arrhythmic brain activity: A decrease in global temporal structures induces faster and transient temporal dynamics and is associated with an increase in spatial structures and interactions between remote brain regions. Interestingly, these modulations are further strengthened with training, i.e., when contrasting the post-training to the resting-state activity in comparison with the pre-training vs. rest contrast. Overall, such modulations of brain spatio-temporal dynamics can be conjectured as a hallmark of brain plasticity.

In this work, we have introduced the notion of

It has been argued and demonstrated that complex wavelet (multiscale) based analyses permit to construct indices to assess functional connectivity from fractal dynamics that inherit from the theoretical and practical benefits of wavelet representations for scale-free (cross-temporal) dynamics analysis, notably in terms of robustness to trends and large selfsimilarity parameters

While Fourier-based tools are natural to use to assess functional connectivity in band-limited rapid oscillatory rhythms, it was shown, using simulated synthetic data and mostly on MEG data, that the assessment of functional connectivity for large-band slow scale-free cross-temporal dynamics is better achieved by complex wavelet based indices. Therefore, Fourier and complex wavelet-based spectral estimation must be regarded as complementary, rather than as mutually exclusive, tools.

Complex wavelet-based analyses of functional connectivity assessment from fractal dynamics conducted on MEG data recorded on 36 participants at rest and during a visual discrimination task with individualized training, yielded several key conclusions. First, large-band infraslow arrhythmic cross-temporal dynamics can be associated with long-range (fronto-temporo-occipital) spatial interactions. Second, functional connectivity from fractal dynamics increases during task performance (in a set of brain regions consistent with those evidenced by other analyses performed on the same data with different tools) and is strengthened with training. Interestingly, a larger overall fractal dynamics-based functional connectivity increase correlates with better task performance (larger hit rate). Third, the increase in spatial structure (quantified by the increase in functional connectivity assessed from fractal dynamics) is accompanied by changes in temporal structures, combining a decrease in the global temporal correlations (quantified by a decrease in the selfsimilarity index) and the increased occurrence of local transient structures (quantified by an increase in multifractality). These spatiotemporal modulations are reinforced with intensive and individualized training for the task.

Routines (in M

Such tools could further be used to examine the relevance of functional connectivity assessed from fractal dynamics in the context of network physiology, and networks of networks, relating brain activity to other physiological functions (heart rate, respiration, sleep, ocular, and motor systems, etc.) (cf. e.g., Bartsch and Ivanov,

The data analyzed in this study is subject to the following licenses/restrictions: neurospin property. Requests to access these datasets should be directed to

The studies involving human participants were reviewed and approved by Local Ethics Committee on Human Research at NeuroSpin (Gif-sur-Yvette, France) Protocole CPP_100022_Cognitif. The patients/participants provided their written informed consent to participate in this study.

The original experimental design and access to MEG data was provided by VW. The methodological question studied here was framed and conceptualized by PA and PC. The data analysis tool design, implementation and performance assessment, and interpretation were performed by HW and PA. MEG data analysis, results production, and interpretation were performed by DLR and PC. The article was written by PA and PC. All authors contributed to the article and approved the submitted version.

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.