Figure 5. Rank distributions of the strength of network links.
Group-averaged strength of individual physiological network links for different sleep stages. Rank 1 corresponds to the strongest link in the network, that is, highest degree of time delay stability (TDS) (shown are all periphery–periphery and brain–periphery links). (a) The rank distributions for different sleep stages are characterized by different strength of the network links measured in % TDS—consistently lower values for most links during deep sleep, higher values during REM and highest during light sleep and wake, indicating that the stratification pattern in Fig. 3d is present not only for the average link strength (when averaging over different types of links in the network) but also for the strength of individual links. Indeed, links from all ranks are consistently stronger in light sleep compared with deep sleep and REM: such rank-by-rank comparison of links across sleep stages is possible because the rank order of the links does not change significantly from one sleep stage to another (Wilcoxon signed-rank test for all pairs of rank distributions yields 0.77≤P≤0.93). A surrogate test based on TDS analysis of signals paired from different subjects, which eliminates endogenous physiological coupling, leads to significantly reduced link strength (P<10−3) and close to uniform rank distributions with no difference between sleep stages (open symbols), indicating that the TDS method uncovers physiologically relevant information. Error bars for the original and surrogate data indicate the standard error for a specific link when averaged over all 36 subjects or over 36 surrogate pairs respectively. (b) Rescaling the plots reveals two distinct forms of rank distributions: a slow decaying distribution for wake and REM, and a fast decaying distribution for light sleep and deep sleep with a pronounced plateau in the middle rank range corresponding to a cluster of links with similar strength, most of which are related to the cardiac system.