Model-based formulation of conditional causality.

Assuming that the oscillatory activity at the network nodes can be described by means of linear Gaussian stationary processes, an autoregressive (AR) model-based definition of conditional transfer entropy can be exploited to describe the causal relationships between the observed data. For Gaussian processes, it was shown that the concept of transfer entropy, formulated as a measure of causal information transfer between joint processes [30], is entirely equivalent to the concept of Wiener-Granger causality [35], which was formalized in terms of linear autoregression by Granger [31]. A straightforward time domain formulation of conditional Granger causality based on a multivariate linear autoregressive model and extended to include zero-lag effects has been proposed in [23] and is used in this study.

The dynamic interactions among the M processes in can be described by the vector AR model [36]:

(2)

where p is the model order, defining the maximum lag used to quantify interactions, is a M-dimensional vector collecting the present state of all processes, is a M-dimensional column vector collecting the past state of all processes at lag k, is the matrix of the model coefficients relating the present with the past of the processes at lag k, and is a M-dimensional vector with positive definite covariance matrix , collecting the present state of the zero-mean white noises , the latter uncorrelated with .

To compute measures of Granger causality from to conditioned on , a restricted linear regression is derived starting from the vector autoregressive representation in (2):

(3)

where and are -dimensional vectors collecting the present and past states of all processes in except , i.e., , is the matrix of model coefficients relating the present and past samples of the processes in , and is a -dimensional vector with covariance matrix , collecting the present state of the zero-mean white noises . It is important to note that in both (2) and (3) the lag counter starts from rather than from as typically assumed in linear regression models. With this choice, zero-lag interactions are taken into account in the description of the relationship between the present and past states of the M processes; the vector autoregressive models (2), (3) are referred to as extended [37], from which measures of conditional causality in the time domain can be derived [23].

To compute extended measures of conditional Granger causality from to , unrestricted and restricted linear regressions combining both instantaneous and lagged effects () are derived from the vector autoregressive models in (2) and (3):

(4)

(5)

where and represent the -row coefficients of and , respectively, weighting the present and past samples of all processes in for the unrestricted regression (4), and the present and past samples of all processes in for the restricted regression (5). The residuals of the unrestricted (4) and restricted (5) regressions have variances and , respectively, and are extracted as the elements of the covariance matrices and , respectively. The extended conditional Granger causality measure is computed by comparing the variance of the residuals of the two models,

(6)

and, in the case of Gaussian processes, it corresponds to the transfer entropy (1) up to a factor 2, i.e., [32].

As regards estimation of the above measures of causality, model identification was performed using the vector least-squares method [36] and model orders were selected for each subject and condition according to the Akaike Information Criterion.

Model-free formulation.

The non-parametric approach used in this work for evaluating direct causal measures is based on the idea that the probability density around a data point is inversely related to the distance from its nearest samples, i.e., on the k-nearest neighbour method [33]. Although this approach is widely employed for its reliability and robustness, the accuracy of the estimates is lower when the estimated probability distributions are simply replaced in (1). Indeed, the combination of terms evaluated on spaces of different dimensions causes the presence of significant bias in the estimates. The formulation introduced by Kraskov, Stögbauer and Grassberger limits this bias by using the same range search in all the spaces after defining the searching distance in the highest dimensional space [34].

Exploiting the decomposition of the conditional mutual information in different entropy quantities, the conditional transfer entropy can be estimated as [24]:

(7)

where , and are the embedding vectors of dimension , and , respectively, approximating , and .

Using the maximum norm to calculate distances, the entropy term in the highest dimensional space, i.e., with , is estimated as:

(8)

where is the average operator, the digamma function, N the number of observations, and the distance between the considered sample and its neighbour. The entropy terms defined in the projected lower dimensional spaces are then estimated as:

(9)

where , and are the number of points at distances smaller than from the considered sample in the spaces , and , respectively. Substituting the entropy terms (8) and (9) in (7), the nearest-neighbour estimate of the conditional transfer entropy is obtained as:

(10)

Finding embedding vectors that closely approximate the infinite-dimensional past states of the processes is a critical step in estimating theoretical-information measures using MF approaches. When working with data of finite length, e.g., the 300 samples typically used for the analysis of short-term physiological time series, the employment of high-dimensional detailed vectors to provide a more complete description of past processes leads to the curse of dimensionality and unreliable estimates of entropy quantities due to the estimator sensitivity to time series and signal patterns length [38]. An alternative selection technique to the uniform embedding approach, which simply uses a fixed number of equally spaced samples, was introduced to limit the size of the descriptive patterns by selecting the most informative time-lagged samples in evaluating the information exchanged across the interacting processes. Specifically, the non-uniform embedding approach described in [24] was exploited in this work.

Considering a set of candidates including all possible states of the processes up to a maximum lag L, i.e., being , and with τ the initial lag ( when instantaneous effects are taken into account and when they are not; see next subsection for details about instantaneous effects), an iterative approach is applied to select the components of which maximize the information shared by the target variable and the candidate variables. Starting with an empty embedding vector , for steps , the candidates are selected from a subset as:

(11)

After each step, a surrogate-based approach is used to test the significance of the information brought to the target variable by the selected candidates. Specifically, the conditional mutual information is compared with a threshold obtained as the percentile of a distribution resulted computing the same measure over realizations of the process obtained shuffling randomly and independently the samples of and . Thus, the selected candidate is added to the embedding vector if is higher than the threshold, resulting in .

In this work, after normalizing time series to unit variance, the MF causality measures were estimated by fixing a maximum lag of 10 samples and setting a number of neighbors . Moreover, realizations of surrogate data were generated to apply the non-uniform embedding technique and the significance threshold was fixed to .

Instantaneous effects among physiological time series.

The assessment of causality, intended as the influence that a driver process exerts on a given target, requires a proper handling of instantaneous (zero-lag) effects. In practical physiological time series analysis, instantaneous causality shows up whenever the time resolution of the measurements is lower than the time scale of the lagged causal influences occurring among the analysed processes. This non-delayed effect can arise due to non-physiological factors (e.g., unobserved confounders) or fast (i.e., in the analysis of physiological time series, within-beat) physiologically meaningful interactions [37]. The importance of considering instantaneous effects in the analysis of cardiovascular interactions, where zero-lag interdependencies are expected to contribute significantly to the baroreflex mechanism [37], and of cardiorespiratory interactions, where the information transfer from respiration to heart rate is expected to occur within the cardiac beat [39], was previously documented [23,37].

In this work, in addition to lagged interactions, instantaneous effects were accounted for and suitably determined according to the measurement convention displayed in Fig 1A. This was done by visually inspecting the temporal sequence of the investigated physiological indices or the parameters directly involved in their estimation, as well as by considering that the cause always precedes its effect in time. Accordingly, the zero-lag effects were set along the directions depicted in Fig 1C, i.e., , , , , , and . These zero-lag interactions were considered in both the model-based and model-free [24] estimators described in the previous subsections. Specifically, the model used for parametric estimation incorporates these instantaneous effects into the zero-lag connectivity matrices and of Eqs (2) and (3), while all other entries of these matrices are zero [23], and the model-free procedure included these effects together with the time lagged ones in the set of candidates possibly affecting the target variable [24]. Notably, the chosen instantaneous effects create a directed acyclic graph (DAG) which avoids self-loops of zero-lag effects, which would make the extended model (2) unidentifiable. Moreover, the sequential ordering of instantaneous effects within the DAG (in this case, ) prevents from conditioning on collider variables that would introduce spurious associations (e.g., is not included in Z when the target is because is not a plausible instantaneous effect, while it is included in Z when the target is because is a physiologically plausible instantaneous effect).

Heart rate variability.

It is well known that heart rate is one of the physiological parameters characterized by the highest variability. Heart rate variability (HRV) varies with age and gender [44], and its lack or depression have been described as indicators of several pathological states, e.g., nervous system disorders [45], diabetes [46], arterial hypertension [47], and myocardial infarction [48]. The physiology behind the regulation of cardiac dynamics is complex, but most studies agree that the main components of the short-term normal sinus rhythm variability are related to the control exerted by the autonomic nervous system (ANS) [44,49]. During ventilation, the activity of the sinoatrial node is directly influenced by the modulation of vagal neurons directed to the heart, controlled by the central respiratory drives (direct communication between respiratory and cardiomotor centers), the lung inflation reflex, and the changes in arterial blood pressure transferred to heart rate via baroreflex [44,50,51]. These mechanisms result in the so-called respiratory sinus arrhytmia (RSA), for which there is an increase of heart rate during the inspiration phase and a decrease during the expiration phase of ventilation [49,52]. While the physiological causes and effects of the RSA mechanism have been widely studied and discussed throughout the past decades [18,52,53], very little is known about the cardiac-driven respiratory variability. It is widely believed that the coupling between the cardiovascular and respiratory systems is unidirectional, i.e., the respiratory rhythm influences the heart rate via vagal nerve traffic oscillations and, to a much lower extent, via direct mechanical influence on the sinoatrial node [54]. Nevertheless, some evidence conflicts with this theory suggesting that the respiratory oscillator in the central nervous system is not always dominant, i.e., both the influence of respiration on heartbeat (i.e., the RSA mechanism) and the influence of heartbeat on respiration are important for cardiorespiratory synchronization mechanisms [55]. Among them, cardioventilatory coupling is the entrainment phenomenon in which ventilation and heartbeats become synchronized in whole number ratios and heartbeats fall in constant timing relationship with the onset of inspiration (see, e.g., [6,56]). The mechanism has been suggested to be likely caused by either a cardiac-induced onset of inspiration through an unknown haemodynamic afferent pathway [56], or the effects of baroreceptor reflexes and ANS activity [6]. An additional phenomenon that is known to characterize mutual relations between cardiac and ventilatory activity is the cardio-respiratory phase synchronization (CRPS). In contrast to the RSA mechanism, which involves the periodic modulation of heart rate amplitude within each respiratory cycle, CRPS leads to an alignment of the ECG R-peaks at specific phases of the respiratory cycle, thereby inducing phase synchronization phenomena [57,58].

Our results confirm the presence of strong closed-loop interactions involving cardiorespiratory dynamics. Indeed, both the MB and MF estimators were able to detect significantly relevant causal interactions along the directions and (Fig 3A), suggesting that the mechanisms involved in the regulation of cardiorespiratory dynamics are significantly strong and bidirectional. Remarkably, some studies documented the importance of using nonlinear approaches, either based on entropy measures or nonlinear autoregressive models, to investigate the complex interactions between respiration and heart rate [7,39]. Indeed, it was demonstrated that the interplay between the cardiac and respiratory systems may lead to the rise of nonlinear RSA dynamics [7], herein well identified by the MF estimator (Fig 3A, bottom). Moreover, as also highlighted in [59], cardiorespiratory patterns of interaction are characterized by a relevant instantaneous influence of the breathing dynamics on the cardiac activity. In our work, the zero-lag sample of the RESP series was selected for more than 50% of subjects by using the MF estimator, thus confirming the major role played by respiratory-driven instantaneous interactions directed to the heart.

It is worth noting that heart rate is influenced not only by respiration but also by MAP to a lower extent (Fig 3A). The baroreflex interaction was found to be significant in most of subjects, thus confirming the major role played by arterial pressure in governing beat-to-beat changes of heart rate and vascular tone [50,60].

Blood pressure variability.

In the supine resting state, together with the heart rate, arterial blood pressure also exhibits spontaneous fluctuations and is strongly influenced by ventilatory and sympathetic activities [61]. Although the study of physiological mechanisms involving BP variability (BPV) is usually assessed by SAP or DAP measurements, the use of mean arterial pressure has recently gained support [28]. It was found that, especially at rest, the mean arterial pressure is most tightly coupled with RR [28], thus opening the possibility to exploit this input signal instead of systolic BP for baroreflex analysis. Starting from this frontier, in this work we investigated MAP-based cardiovascular and vascular-respiratory interactions.

With regard to the influence exerted by respiration on BP, the significant causal interaction from RESP to MAP (Fig 3A) is likely due to the cyclic intrathoracic pressure changes that occur during ventilation and lead to alterations in venous return and fluctuations in BP values [62], independently of heart rate changes inducing cardiac filling fluctuations. Overall, our results indicate that the respiratory-driven variability of MAP during the resting state may show a linear character, due to both the observed higher values and significance of the causal coupling assessed through the MB approach compared to the MF approach.

Besides, the resting-state BPV is known to be strongly influenced by a relevant non-baroreflex effect (commonly known as feedforward) led by changes in heart rate, reasonably due to the Windkessel [63] and/or Frank–Starling [64] mechanisms. In this regard, some studies documented a predominance of the cardiovascular effects of heart rate on diastolic BP (run-off effect interaction), as well as of diastolic on systolic BP, which together constitute the pathway heart rate - diastolic BP - systolic BP explaining the high amount of information flowing from RR to SAP in the supine resting state [25,65]. Despite these mechanisms have not been investigated nor analysed in terms of the MAP time series yet, the strong dependence of MAP on DAP and SAP values (with MAP being computed from the integral of the BP curve between two consecutive DAP values) makes the MAP series interesting from this perspective as well. Indeed, our results confirm a strong feedforward mechanism and thus highlight the major role played by MAP within this chain of interactions (Fig 3A). Noteworthy, the selection of a percentage of zero-lag samples by the MF method slightly lower than across subjects, as well as the highly significant causal interaction detected through the MB (Fig 3A, top), suggest that the instantaneous effect directed from heart period to mean arterial pressure may have an important role in characterizing cardiovascular dynamics.

Causal effects on BP can also be ascribed to arterial compliance variability. The interaction directed from AC to MAP appears to be significantly relevant in the case of causality measures computed with the MB approach (Fig 3A, top). Although these dynamics seem to be more evident in pathological conditions or with increasing age [66,67], the beat-to-beat modulation of systolic BP due to arterial compliance variability may play an important role even in youth and healthy physiological conditions. As suggested by previous studies, this can be explained by considering the potential effects of the SV modulations (thus of the compliance) on the arterial system (thus on BP variability) conveyed by respiration [68] and venous blood pool control [69]. Moreover, previous studies documented an important role of PVR as a source of MAP changes [70], although the causal interaction has been investigated only with bivariate approaches (see, e.g., [26]). It has been suggested that PVR, determined by vasomotion in smaller arteries, influences BP by diastolic blood pressure decay (short-term changes) and changes in venous return associated with blood redistribution (long-term changes). Since AC is herein computed from τ and PVR, we can argue that the influence of AC on MAP can be due to the direct effect of PVR on BP.

Arterial compliance variability.

With regard to the variability of arterial compliance, the investigation of this parameter has emerged recently in the literature. Recent studies have observed how its regulatory mechanisms are complex and not clearly understood yet [10,11,13]. Remarkably, being a non negligible physiological factor strongly influencing AC, the direct action of sympathetic nerves on the vascular tone of large vessels may play the role of an unobserved confounder when interpreting the coupled interactions from and to AC [71].

As supported by previous studies [12], blood pressure plays an important role in the modulation of pulse wave velocity (PWV) in the supine rest. Assuming PWV as a surrogate index of compliance [10], this is likely reflected by a significant causal interaction along the direction , revealed by both MB and MF approaches (Fig 3A). It is worth noting that the relationship between mean arterial pressure and compliance can be affected by several factors. Indeed, both the weight of the zero-lag effect of MAP on AC, which was selected in more than of subjects by the MF estimator, and the sympathetic activity of the ANS, which can be considered as a common driver for BP and AC, may have an influence on the strength and the significance of the coupling directed from mean arterial pressure to vessel compliance. Additionally, the sympathetic-driven effect of cardiac contractility along the cardiac inotropic arm of the baroreflex should be also taken into account [26], since beat-to-beat modifications of BP at rest may have an impact on the relationship between BP and AC.

Our results also reveal the influence of both heart period and breathing patterns on arterial compliance at rest (Fig 3A). Several studies have demonstrated a significant effect of heart rate fluctuations on compliance variability independently of autonomic control and BPV [12,67,72]. An increase in heart rate has been associated to modifications of the cardiac cycle and a reduction in the time available for recoil, resulting in vascular stiffening and arterial compliance decrease, due to the viscous and inertial components of the arterial walls [67,73]. The relationship between heart rate and vascular compliance has been investigated indirectly in the most of previous works on the topic. Indeed, assuming PWV as a surrogate for AC [10], some studies have shown that PWV is indirectly related to RR independently of BP [12,67,72], supporting the hypothesis that the effect of heart period on arterial compliance could show nonlinear features in the supine resting state. Despite the sympathetic activity of the ANS has an effect directed to the heart which cannot be neglected and may play a common drive role here, our results show a significant strong causal interaction from RR to AC which cannot be ascribed to the confounding effects of respiration nor BP variability (Fig 3A). Furthermore, we found that the zero-lag effect from RR to AC is selected in more that of subjects in the resting state using the MF approach, demonstrating the importance of embedding instantaneous effects into the analysis.

As regards the influence exerted by respiratory variability on compliance, it is known that small changes of intrathoracic pressure and lung volume during spontaneous ventilation can independently affect arterial compliance through changes in atrial filling or preload and modifications in transmural pressure exerted on the arterial wall [74]. This can determine modifications of some cardiovascular parameters, such as SV, as well as of the mechanical properties of the arterial wall [75,76]. Looking at our results, the causal interaction was detected with the highest significance by both approaches (Fig 3A). Interestingly, despite respiratory activity may have an indirect influence on compliance through modulation of heart period and BP, herein we get around the issue by utilizing conditional causality measures and separating the link from confounding external effects. This confirms that a direct effect of respiratory variability on vessel compliance is likely to occur in the supine resting state and may reflect the effect of cyclic changes in transmural pressure on arterial wall stiffness. Moreover, the high significance of this causal link in both time domain MB and MF measures can be ascribed to a great influence of the zero-lag effect, which indeed was selected for more than of subjects by the MF estimator.

Cardiorespiratory patterns during head-up tilt.

As depicted in Fig 2D and confirmed in previous studies [79,80], the tilt-induced decreased respiratory rates and increased tidal volumes, together with the the well-known parasympathetic activity withdrawal during HUT, leads to a significant RSA weakening, here identified by both MB and MF approaches (Fig 3). This is confirmed by previous studies [81,82] and has been associated to a dominance of the direct central mechanism mediating RSA (i.e., direct communication between respiratory and cardiomotor centers) at supine rest and an increased indirect peripheral mechanism (i.e., ventilation-associated BP changes transferred to heart rate via baroreflex) during orthostasis [18,50,51]. The latter is clearly reflected in our results by the tilt-induced augmented causal interactions and (Fig 3C), which together constitute the indirect peripheral mechanism controlling RSA.

A reduction of the cardiorespiratory coupling is also detected by looking at the causal direction and more visible with the MF approach (Fig 3C, bottom), as also visible by the highest variation in the number of significant measures found with the MF approach (Fig 3A, 3B, bottom). Nevertheless, very little is known about the cardiac-driven respiratory variability and its modification with tilt, as already pointed out in Sect 14. We learn from literature that, although this coupling is poorly seen in alert and active humans, it was clearly observed during relaxation, sleep, anaesthesia and, generally, in conditions of low cognitive and behavioural activity [56]. Here, the significant decrease of the causal coupling along the direction may be associated to the gradually decrease of the complex cardiorespiratory interactions during situations of sympathetic activation [50].

Involvement of blood pressure variability in tilt-induced physiological responses.

The increased influence of breathing activity on mean arterial pressure variability, revealed by both MF and MB approaches (Fig 3C), is representative of ventilation-associated BP changes occurring with tilt and due to alterations in intrathoracic pressure, instantaneous lung volume, venous return to the heart, and cardiac output [83]. Previous studies [8,84] suggested that respiratory-related fluctuations of BP can be ascribed to the mechanical thoracic coupling between respiration and vasculature, as well as to the effects of respiratory induced fluctuations of RR. While RSA is thought to buffer respiratory-related BP oscillations during the supine rest [8,84], the mechanical influences of respiration on arterial pressure are thought to be greater in the upright than the supine position. We attempt to confirm this hypothesis: since the causal effect of respiration on MAP is conditioned on the knowledge of all the other variables in the network, the observed increase of the causal coupling is solely due to the enhancement of the mechanical influences of respiration on arterial pressure.

In the complex regulation of BP variability during postural stress, arterial compliance is also involved. Specifically, we found a significant increase of the causal interaction by using the MB approach (Fig 3C, top). As far as we know, the present study represents the first attempt to investigate the causal interactions within a network of four physiological processes comprising arterial compliance. Previous works on the topic suggested that an effect of vascular compliance on arterial pressure may be related to the slow rhythms conveyed to the arterial system through SV modulations; the latter are likely to depend on respiration [68] and on venous blood pool control [69]. Although these mechanisms have not been observed during conditions of postural stress yet, our findings suggest that a relevant portion of the tilt-induced changes of mean arterial pressure should be ascribed to the short-term arterial compliance variability.

The relationship between blood pressure and compliance may have effects on the bidirectional interactions between heart period and BP. Indeed, it has been shown how a transient change in carotid arterial compliance is responsible for a sudden change in the cardiac baroreflex response [71]. Moreover, several works argue that the tilt-induced significantly higher awareness of baroreflex control, due to blood redistribution and baroreceptor unloading during orthostasis, can be also related to the confounding effect of respiration [17,25,65]. In this study, since any external driver of BP variability, i.e., the compliance and respiration variability, is ruled out by the conditioning procedure, we suggest that the observed enhancement of the baroreflex control with the importance of the gravitational stimulus (Fig 3C) is only due to the closed-loop interplay between heart period and arterial pressure, thus confirming previous results obtained in older studies on the topic [25,26,50]. On the other hand, the feedforward direction of the cardiovascular closed-loop, i.e., , is affected by the postural stress, as detected by the MB approach (Fig 3C, top). The highly statistically relevant decrease of this causal coupling is not associated with a correspondent diminished number of significant measures detected with surrogate data analysis (Fig 3A, 3B, top), meaning that the strength of the interaction remains considerably noteworthy within the network. This result is in line with previous studies [25], where the effect of HUT was found to be prominent for the directions and . The authors suggested that these results can reflect the involvement of other mechanisms influencing DAP and/or contributing to the strength of systolic contraction, such as changes in the PVR associated with tilt-induced sympathetic activation and sympathetic nervous system influence on cardiac contractility, respectively. In our study, we cannot rule out the role exerted by the sympathetic branch of the ANS, although it may have significant effects on the strength of the feedforward interaction and act as a confounder for this link.

All these findings support the idea that, during tilt, changes of BP variability are related to the increase of sympathetic activity induced by the postural stress [65], as well as to the decreased buffering of BP mediated by HRV.

Regulation of arterial compliance variability during postural stress.

Here, we speculate about the mechanisms involved in the beat-to-beat regulation of arterial compliance during head-up tilt. Interestingly, we observe a significant decrease of the causal interaction using the MB approach (Fig 3C, top), thus suggesting that orthostasis affects the strength of this link probably due to stiffer vessels. This is a novel finding, since a thorough inspection of this interplay, which is not a straightforward task, is still lacking in the literature. Similar investigations have been carried out in the past years, where the relationship between systolic BP and compliance was found to hold for a given phase but shift during the HUT phase towards lower values of systolic BP and compliance [11,85]. However, the present work goes beyond the classical bivariate analysis, with the aim of ruling out the confounding effects of cardiac and respiratory variables. Following this rationale, we speculate that the decrease of in HUT (Fig 2C, top) can be partly MAP-dependent, i.e., due to a diminished transfer of information from blood pressure. Nevertheless, it has been suggested that it could depend also on other mechanisms such as sympathetic nervous system activation and/or changes in PVR during HUT [78]. Moreover, although the MF approach did not detect this decrease, it included zero-lag samples of MAP for about of subjects along this direction, thus highlighting the weigh of the instantaneous interaction when taking into account the relationship between arterial pressure and compliance.

A decreased coupling along the direction is also observed with both MB and MF approaches (Fig 3C), as confirmed by surrogate data analysis especially with the MF approach (Fig 3A, 3B, bottom). This finding can be related to the important influence of zero-lag causality (more than half subjects if investigated through the MF approach) on the observed interaction, which could reflect pure mechanical visco-elastic effect. Again, these considerations are absolutely novel and need further studies on the topic to provide deeper physiological reasoning beyond the observed phenomena.

In addition, arterial compliance may play an important role in driving respiratory variability during postural stress, as suggested by the decrease of the causal coupling detected with the MB approach (Fig 3, top). We argue that these dynamics exhibit predominantly linear characteristics and speculate that they may be related to the mechanical interaction between the arterial system and ventilation.