2.4. Data Analysis and Interpretation

The multiscale method described in Section 2.3 allows for computing all measures of information decomposition, at any assigned time scale, τ, directly from the coefficients of the VAR process fitting the original time series. In this study, VAR model identification was performed for each set of time series representing the multivariate process (RESP, SBP, RR) using the least squares method, and setting the model order according to the Akaike Information Criterion [35]. The model was extended to incorporate zero-lag effects from RESP to SBP and to RR, and from SBP to RR, thus allowing fast effects of respiration on cardiovascular variables and within-beat baroreflex influences [28].

The method was applied first computing the joint, interaction, unique, redundant and synergistic information transfers for time scales τ ranging from 1 to 12. Then, a scale-specific analysis was performed to assess information transfer separately with regard to oscillatory components with meaningful physiological interpretation contained in the time series. Specifically, in order to distinguish effects due to all the oscillations from effects due to slower oscillations only (which represent the oscillations in VLF and LF band—VLF + LF) we computed the information measures at two time scales: τ1=1, corresponding to raw non-rescaled data, and τ2 determined—for each subject and experimental condition—as the time scale which removes the HF band using the formula

where fτ is the cut-off frequency of the low-pass filter used for rescaling and is equal to 0.15 Hz, and RR¯ is the mean RR interval (measured in seconds).

With the above analysis we obtain, at the time scales including all oscillations (τ1) or removing HF oscillations (τ2), estimates of all measures of information transfer resulting from IID and PID. In order to ease physiological interpretation of these measures, in the following we discuss the meaning of the PID components based on (i) expected physiological mechanisms underlying cardiovascular and cardiorespiratory influences and (ii) methodological derivations that relate the PID and IID decomposition. Our physiological assumptions, depicted in Figure 2c, are that respiration acts as an exogenous input on the two cardiovascular variables (i.e., RESP affects SBP and RR without being substantially affected by them); moreover, we consider two possible mechanisms whereby respiration influences the variability of heart rate [36,37]: baroreflex-mediated respiratory effects on heart rate oscillations (indirect pathway RESP→SBP→RR) and direct effects unrelated to SPB (possibly of central origin; a direct pathway RESP→RR). In addition, an influence of slower SBP and respiratory pattern oscillations (e.g., of vasomotor origin or respiratory pattern variability-related, respectively) on RR transferred via baroreflex are also considered on longer time scales (τ2).

Methodologically, as demonstrated in Appendix A, the MMI PID is such that one source (the one providing the lowest amount of information about the target) contributes to the target dynamics with no unique information, but interacting with the other source to provide an amount of redundant (shared) information equivalent to its individual information transfer and an amount of synergistic (complementary) information equivalent to its conditional information transfer. This means that, if TRESP→RR<TSBP→RR, the MMI PID will yield

If, on the contrary, if TSBP→RR < TRESP→RR, the MMI PID will yield

In both cases, redundancy RRESP,SBP→RR relates to the common information shared by the sources (RESP, SBP) about the target RR, and is in our context associated with the information transferred along the pathway RESP→SBP→RR describing baroreflex-mediated respiratory effects on heart rate (red + green arrows in Figure 2c). The amounts of unique information transfer are relevant to information flowing from one source to the target without involving the other source, associated here to respiration-unrelated baroreflex effects (USBP→RR, reflecting the path SBP→RR; green arrow in Figure 2c) or to the nonbaroreflex mechanism of RSA (URESP→RR, reflecting the path RESP→RR; blue arrow in Figure 2c). Finally, synergy SRESP,SBP→RR is the information that the source sending no unique information provides about the target after conditioning on the other source (i.e., TRESP→RR|SBP when URESP→RR=0, and TSBP→RR|RESP when USBP→RR=0; this suggests that SRESP,SBP→RR>0 corresponds to TRESP→RR|SBP>0 when USBP→RR>0, and to TSBP→RR|RESP>0 when URESP→RR>0). Given this, synergy quantifies the contemporaneous presence of information transfer along both the pathways whereby the sources can affect the target (with unique transfer from one source and conditional transfer from the other source); therefore, we associate synergy between RESP and SBP with the simultaneous involvement of both pathways whereby respiration affects the heart rate (i.e., RESP→RR and RESP→SBP→RR; blue and red + green arrows in Figure 2c).