Assessing the dynamical complexity of biological time series represents an important topic with potential applications ranging from the characterization of physiological states and pathological conditions to the calculation of diagnostic parameters. In particular, cardiovascular time series exhibit a variability produced by different physiological control mechanisms coupled with each other, which take into account several variables and operate across multiple time scales that result in the coexistence of short term dynamics and long-range correlations. The most widely employed technique to evaluate the dynamical complexity of a time series at different time scales, the so-called multiscale entropy (MSE), has been proven to be unsuitable in the presence of short multivariate time series to be analyzed at long time scales. This work aims at overcoming these issues via the introduction of a new method for the assessment of the multiscale complexity of multivariate time series. The method first exploits vector autoregressive fractionally integrated (VARFI) models to yield a linear parametric representation of vector stochastic processes characterized by short- and long-range correlations. Then, it provides an analytical formulation, within the theory of state-space models, of how the VARFI parameters change when the processes are observed across multiple time scales, which is finally exploited to derive MSE measures relevant to the overall multivariate process or to one constituent scalar process. The proposed approach is applied on cardiovascular and respiratory time series to assess the complexity of the heart period, systolic arterial pressure and respiration variability measured in a group of healthy subjects during conditions of postural and mental stress. Our results document that the proposed methodology can detect physiologically meaningful multiscale patterns of complexity documented previously, but can also capture significant variations in complexity which cannot be observed using standard methods that do not take into account long-range correlations.

The onset of intracranial hypertension in patients with Traumatic Brain Injury (TBI) is a serious complication that can lead to brain death. The increased intracranial pressure (ICP) during the Plateau Waves (PW) triggers decreased cerebral perfusion pressure (CPP), brain blood flow and tissue oxygenation. This phenomenon is associated with intense systemic stress response evidenced by Heart Rate Variability (HRV) as a surrogate of autonomic nervous system dysfunction. HRV data show long memory and dynamical complexity herein evaluated by autoregressive fractionally integrated (ARFI) models and multiscale entropy (ARFI-MSE). The incorporation of short-term dynamics and long-range correlations (LRC) allows an enhanced description of HRV during the PW.

Heart Period (H) results from the activity of several coexisting control mechanisms, involving Systolic Arterial Pressure (S) and Respiration (R), which operate across multiple time scales encompassing not only short-term dynamics but also long-range correlations. In this work, multiscale representation of Transfer Entropy (TE) and of its decomposition in the network of these three interacting processes is obtained by extending the multivariate approach based on linear parametric VAR models to the Vector AutoRegressive Fractionally Integrated (VARFI) framework for Gaussian processes. This approach allows to dissect the different contributions to cardiac dynamics accounting for the simultaneous presence of short and long term dynamics. The proposed method is first tested on simulations of a benchmark VARFI model and then applied to experimental data consisting of H, S and R time series measured in healthy subjects monitored at rest and during mental and postural stress. The results reveal that the proposed method can highlight the dependence of the information transfer on the balance between short-term and long-range correlations in coupled dynamical systems.

Cardiovascular variability is the result of the activity of several physiological control mechanisms, which involve different variables and operate across multiple time scales encompassing short term dynamics and long range correlations. This study presents a new approach to assess the multiscale complexity of multivariate time series, based on linear parametric models incorporating autoregressive coefficients and fractional integration. The approach extends to the multivariate case recent works introducing a linear parametric representation of multiscale entropy, and is exploited to assess the complexity of cardiovascular and respiratory time series in healthy subjects studied during postural and mental stress.

In the study of complex biomedical systems represented by multivariate stochastic processes, such as the cardiovascular and respiratory systems, an issue of great relevance is the description of the system dynamics spanning multiple temporal scales. Recently, the quantification of multiscale complexity based on linear parametric models, incorporating autoregressive coefficients and fractional integration, encompassing short term dynamics and long-range correlations, was extended to multivariate time series. Within this Vector AutoRegressive Fractionally Integrated (VARFI) framework formalized for Gaussian processes, in this work we propose to estimate the Transfer Entropy, or equivalently Granger Causality, in the cardiovascular and respiratory systems. This allows to quantify the information flow and assess directed interactions accounting for the simultaneous presence of short-term dynamics and long-range correlations. The proposed approach is first tested on simulations of benchmark VARFI processes where the transfer entropy could be computed from the known model parameters. Then, it is applied to experimental data consisting of heart period, systolic arterial pressure and respiration time series measured in healthy subjects monitored at rest and during mental and postural stress. Both simulations and real data analysis revealed that the proposed method highlights the dependence of the information transfer on the balance between short-term and long-range correlations in coupled dynamical systems.