Dynamic Functional Connectivity as a complex random walk: Definitions and the dFCwalk toolbox

Abstract

•We have developed a framework to describe the dynamics of Functional Connectivity (dFC) estimated from brain activity time-series as a complex random walk in the space of possible functional networks. This conceptual and methodological framework considers dFC as a smooth reconfiguration process, combining "liquid" and "coordinated" aspects. Unlike other previous approaches, our method does not require the explicit extraction of discrete connectivity states.•In our previous work, we introduced several metrics for the quantitative characterization of the dFC random walk. First, dFC speed analyses extract the distribution of the time-resolved rate of reconfiguration of FC along time. These distributions have a clear peak (typical dFC speed, that can already serve as a biomarker) and fat tails (denoting deviations from Gaussianity that can be detected by suitable scaling analyses of FC network streams). Second, meta-connectivity (MC) analyses identify groups of functional links whose fluctuations co-vary in time and that define veritable dFC modules organized along specific dFC meta-hub controllers (differing from conventional FC modules and hubs). The decomposition of whole-brain dFC by MC allows performing dFC speed analyses separately for each of the detected dFC modules.•We present here blocks and pipelines for dFC random walk analyses that are made easily available through a dedicated MATLAB^Ⓡ toolbox (dFCwalk), openly downloadable. Although we applied such analyses mostly to fMRI resting state data, in principle our methods can be extended to any type of neural activity (from Local Field Potentials to EEG, MEG, fNIRS, etc.) or even non-neural time-series.

Keywords: Chronnectome; Functional Connectivity; Neuroimaging; fMRI.

Graphical abstract

Fig. 1

From Functional Connectivity to Functional Connectivity Dynamics. (A) Traditionally, correlations between neural activity time-series TS_i(t) of N different brain region nodes i and j (left) are averaged over long times and compiled into the entries FC_ij of a ‘static’ N-times-N Functional Connectivity (FC) matrix (right). (B) Sliding windows of a shorter temporal duration, it is possible to estimate a stream of time-resolved FC(t) networks, which we call the dFC stream (top). The variation between a FC frame at a time t₀ and the next non-overlapping frame at a time t₀ + W is measured by the dFC speed V_dFC,W(t₀) where W is the chosen window size. The degree of similarity (inter-matrix correlation) between FC(t) networks observed at different times is then represented into a F-times-F recurrence matrix, or dynamic Functional Connectivity (dFC) matrix, where FT is the total number of probed windows (i.e. frames in the temporal network given by the dFC stream), depending on window size and overlap (bottom). (C) Alternatively, one can consider each individual FC link as a dynamic variable FC_ij(t) attached to the graph edge between two regions i and j (top). Generalizing the construction of the FC matrix in panel (A), we can thus extract a N(N-1)-times-N(N-1) matrix of covariance between the time-courses of different FC_ij(t) links. We re-baptized this inter-link covariance matrix as Meta-Connectivity (MC) matrix.

Fig. 2

dFC speed distributions and long-range correlations. (A) Distributions of resting state fMRI dFC speed, shown here for a representative subjects (log-log scale, pooled window sizes 12 s ≤ W < 31 s) displayed a peak at a value V_typ (typical dFC speed) and a fat left tail, reflecting an increased probability with respect to chance level to observe short dFC flight lengths (95% confidence intervals are shaded: red, empirical; gray, chance level for shuffled surrogates, see Fig. 4). (B) The dFC speed V_typ decreased with subject age (results are shown here for the specific window pooling used in panel A, but robust for other choices as well; cf. for details). The FC space was seemingly explored through an anomalous random process in which short steps were followed by short steps with large probability (sequential correlations), leading to “knotted” trajectories (panel C, top). This contrasts with a standard random process, visiting precisely the same FC configurations but without long-range correlations (panel C, bottom). (D) The presence of long-range sequential correlations (persistence) of dFC could be proved through a Detrended Fluctuation Analysis (DFA) adapted for dFC streams. We show here DFA log-log scaling plots for representative subjects (in the “young” group, in blue; or in the “older” group, in magenta) and a representative window sizes (W =  150 s, in the middle of a broad range in which persistence is displayed, see [5]). The linearity of DFA scatter plots on the log-log plane (scale of coarse-graining vs total fluctuation strength) reveals that instantaneous increments along the dFC stream form a self-similar sequence. The black dashed lines indicate the slope that would be associated to α_DFA = 0.5, i.e. the case of ordinary uncorrelated Gaussian random walk. Linear slopes are steeper than for ordinary random walk, indicating that dFC streams evaluated at this window size (and, generally, at W > ~20 s) follow a persistent stochastic walk.

Fig. 3

Meta-Connectivity and dFC modules. Correlations between the time-courses of pairs of inter-regional FC links are compiled into Meta-Connectivity (MC) matrices (cf. Fig. 1C). (A) Group-averaged MC for two distinct age groups (‘younger’: N = 42, 18–42 yrs, median age = 24 yrs; and ‘older’: N = 42, 47–80 yrs, median age = 63 yrs) based on smooth dFC streams (W = 30 s, ∆ = 1 TR). These matrices are evidently modular (after reordering of entries according to a standard Louvain algorithm). Groups of covarying links are called dFC modules. Differences in the inter-modular meta-connectivity between the ‘younger’ and the ‘older’ group appear visually evident. (B) Two meta-connected links converging on a common root node form what we call a meta-connectivity trimer. We call then meta-hubs nodes serving as root to many strong trimers (or, equivalently, displaying a strong trimer meta-strength MC(i)). In panel B brain regions –defined in a Desikan et al. parcellation– are colored according to their meta-strength MC(i), restricted to each of the five dFC modules identified by the modular decomposition of the MC matrices in panel A. Many of these modules have meta-hubs resembling the ones of dFC modules discussed for a different dataset in Lombardo et al. .

Fig. 4

Comparisons with surrogate dFC streams. We compare dFC streams evaluated from actual empirical fMRI data (A) with surrogate dFC streams evaluated: from phase-randomized BOLD time-series surrogates (B), compatible with a null hypothesis of stationarity of the dFC stream; and from time-shuffled surrogates (C), compatible with an alternative null hypothesis of lack of sequential correlations in the dFC stream. (D) We also show for illustration distributions (smoothed kernel-density estimator) of typical resting-state dFC speeds for empirical and surrogate ensembles (distributions over grouped subjects), pooled over two distinct window-size ranges: long windows (top, 60 to 210 s) and intermediate windows (bottom, 15 to 60 s). Distributions for surrogate data significantly differ from distributions for empirical data (differences between empirical and time-shuffled distributions in red, between empirical and phase-randomized in green color; stars denote significant differences under two-sided Kolmogorov-Smirnov statistics: *, p < 0.05; **, p < 0.01).