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2.2 Static connectivity, causality, phase-coherence and emulative power

Procedure

First, static functional connectivity (sFC) analysis was performed on all pairs of brain network signals, using Pearson correlation. This is applied for each participant’s RSN time series, and each session (RS1 and RS2), leading to two times 24 (14 controls; 10 patients) correlation matrices. The total amount of sFC features per subject/session (unique entries of sFC matrix) was 91 (= 14x13/2; the connectivity matrix being symmetrical, and its diagonal discarded).

Second, using the same ICs (or RSNs) time series we calculated the Granger causality (GC) matrices for each individual and resting-state session using the Matlab toolbox of [53]. A GC connectivity matrix is defined as G = {gij} ∈ ℝN × N (N = numbers of networks), where each matrix entry gij represent the directed causal connectivity from network j towards network i. More specifically gij = FYX|Z, which is the GC between signal Y (time series of network j) and X (time series of network i) conditional on Z (remaining time series of networks kij)—more details in [54]. The upper triangle of G, i.e. {gij}, ∀ j > i represent the inward causal pairwise connectivity gij, while the lower triangle, i.e. {gji}, ∀ i > j is all the outward causal connections of {gij}. Since GC matrices are asymmetrical (directed connectivity) by subtracting those triangular matrices, i.e lower-upper, we can extract the non-null net pairwise GC triangular matrix NetGC = {netgij} = {gijgji}, ∀ i > j. We also extracted the In- and Out-causal degree of each network by summing respectively the columns and rows of the G matrices, i.e. $DegIn={∑j=1Ngij}∈ℝN$, and $DegOut={∑j=1Ngij}∈ℝN$. We finally derived the net (out-in) causal degree of each network, i.e. NetDeg = DegOut—DegInT. Since we have N = 14 networks, the total amount of GC features was 315 (= 3x91 (in-, out-, net-GC triangular matrices) + 3x14 (DegIn, DegOut, and NetDeg vectors)). These GC feature matrices and vectors were calculated per subject and session, and are illustrated in Fig 4B.

A: extraction of the static functional connectivity features. B: computation of the Granger causality matrices and vectors. C: Emulative power vectors derived from the EGN method; D: Wavelet coherence approach and the time-of-coherence metric extraction.

Thirdly, we applied the wavelet-coherence approach with the following processing steps: (i) extraction of spectro-temporal maps (also called scalograms) of significant phase coherence, i.e. localized (in time and frequency/scales) significant phase-locked correlation between pairs of signals, as in [55]. The phasing between the signals is expressed in radians and extracted from θ = arg(Wxy) where Wxy is a complex number defining the cross-wavelet transform between the two network signals being assessed. The phase information are summarized in 4 categories: in-phase when θ ϵ (-π/4, +π/4); anti-phase when θ ϵ (3π/4, -3π/4); and signal 1 leading (or lagging) signal 2 when θ ϵ (π/4, 3π/4) (or θ ϵ (-3π/4, -π/4)); for simplification we colored the phase information in the wavelet-coherence maps (see Fig 4C). For each of the 5 period scales ([4, 8) s; [8, 16) s; [16, 32) s; [32, 64) s; and [64, 128] s) we extracted the time of coherence (in % of the scan duration) between the 91 pairs of networks. More details on the time of coherence metric in [39]. The time of coherence extraction was performed for all of the 4 aforementioned phases. Therefore, in total we obtained 1820 (= 4x5x91) coherence-based features per subject’s session (Fig 4C).

Finally, the Evolutionary Game theory on Network (EGN) was implemented as in [40]. Instead of trying to predict brain signal (time series) dynamics and correlating them to the original BOLD time series, we here simply took the EGN-connectivity matrix A = {aij} ∈ ℝN × N (N = numbers of networks), and derived the vectors of emulative powers (EPs). Briefly, each entry of A, aij, described the pairwise and directed (ij) weight of the network i in its ability to emulate (positive value, ai,j >0) or not (negative value, aij < 0) the activity of the network j. A can be seen as similar to the GC connectivity matrices, albeit not in the sense of causality but in terms of replication mechanism. From these matrices of similar size (N2) as the sFC and GC matrices, we calculated our novel metrics, namely the In-EPs, Out-EPs, and Net-EPs, per participants and session (Fig 4D). The Out-EP of each network, i.e. its power to emulate (replicate) the other networks’ activity is calculated by summing the columns of the EGN-connectivity matrix, i.e. $OutEPi=∑j=1Naij$. The power of one network to be emulated by the other networks is represented by its in-EP, that is, the sum of the EGN-connectivity matrix rows, i.e. $InEPj=∑i=1Naij$. And the net replicative power of a network is defined by the Net-EP, i.e. NetEP = OutEPT—InEP. Since N = 14, we ended up with three vectors of 14 EPs per person’s rs-fMRI session, i.e. 42 features (Fig 4D). The network feature extractions were repeated for the two resting-state sessions.

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