Investigating High-Order Behaviors in Multivariate Cardiovascular Interactions via Nonlinear Prediction and Information-Theoretic Tools

TL;DR

The paper formalizes two whole-minus-sum measures to detect high-order behaviors (HOBs) in multivariate systems by comparing joint versus individual source contributions, using predictability (MP) and information-theoretic (MI) metrics. It implements model-free, nearest-neighbor estimators to compute these measures and validates them on linear Gaussian and nonlinear deterministic and stochastic simulations with and without high-order mechanisms. The framework is then applied to cardiovascular data to examine how mean arterial pressure (MAP) is shaped by systolic and diastolic pressures, and how diastolic pressure is modulated by heart timing parameters, revealing condition-dependent synergistic interactions. Across simulations and physiology, the information-theoretic Δ_MI generally shows greater sensitivity to HOBs, while Δ_MP is more indicative of underlying high-order mechanisms, suggesting a complementary analytical strategy. The work highlights potential biomarkers for physio-pathological alterations in cardiovascular networks and motivates further development toward disentangling synergy from redundancy in high-dimensional systems.

Abstract

Assessing the synergistic high-order behaviors (HOBs) that emerge from underlying structural mechanisms is crucial to characterize complex systems. This work leverages the combined use of predictability and information measures to detect and quantify HOBs in synthetic and physiological network systems. After providing formal definitions of mechanisms and behaviors in a complex system, measures of statistical synergy are defined as the whole-minus-sum excess of mutual predictability ($Δ_\textrm{MP}$) or mutual information ($Δ_\textrm{MI}$) obtained when considering the system as a whole rather than as a combination of its units. The two measures are computed using model-free methods based on nonlinear prediction and entropy estimation. The application to simulated linear Gaussian systems and nonlinear deterministic and stochastic dynamic systems shows that $Δ_\textrm{MP}$ tends to vanish for target variables influenced by additive effects of single independent source variables and is positive in the presence of group interactions between sources, while $Δ_\textrm{MI}$ exhibits a higher propensity to display positive values. The analysis of physiological variables shows significant values of $Δ_\textrm{MI}$ when investigating the additive effect of systolic and diastolic arterial pressure on mean arterial pressure, and of both $Δ_\textrm{MP}$ and $Δ_\textrm{MI}$ when assessing how diastolic pressure is modulated by pre-ejection and left-ventricular ejection times. HOBs can be more clearly identified by information-theoretic measures, while prediction measures are more sensitive to synergy arising from the governing rules of the system analyzed rather than from pure statistical dependencies. Quantifying HOBs through measures sensitive to structural mechanisms can provide biomarkers to assess physio-pathological alterations of cardiovascular networks.

Figures (3)

• Figure 1: Detection of HOBs in linear Gaussian systems. (a) Graphical representation of the derivation for Gaussian variables (target $Y$, sources $X_1$, $X_2$) of the predictability and information-theoretic WMS measures defined in Eqs. (\ref{['delta_MP']}) and (\ref{['delta_MI']}); note that, if $X_1 \perp X_2$, $\lambda^2_{Y;X_1|X_2}=\lambda^2_{Y;X_1}$ and thus $\Delta_{\textrm{MP}}(Y;\boldsymbol{X})=0$, but $I(Y;X_1|X_2)>I(Y;X_1)$ and thus $\Delta_{\textrm{MI}}(Y;\boldsymbol{X})>0$. (b,c) Color-coded values of $\Delta_{\textrm{MP}}(Y;\boldsymbol{X})$ and $\Delta_{\textrm{MI}}(Y;\boldsymbol{X})$ at varying the strength of the link between the sources ($r_{12}$) and between the target and each source, either balanced ($r_1=r_2$, (b)) or unbalanced ($r_1$, with fixed $r_2=0.5$, (c)).
• Figure 2: Detection of HOBs in nonlinear deterministic and stochastic systems with and without HOMs. Panels depict boxplots with individual values of the predictability (a.1) and information-theoretic (a.2) WMS measures, and number of realizations for which the measures were detected as statistically significant (b.1, b.2), for the simulations of coupled logistic maps (Sim 1), coupled Hénon maps (Sim 2), nonlinear autoregressive processes (Sim 3), and nonlinearly coupled linear autoregressive processes (Sim 4).
• Figure 3: Detection of HOBs in cardiovascular interactions. Panels depict boxplots with individual values of the predictability (a.1) and information-theoretic (a.1) WMS measures, and number of realizations for which the measures were detected as statistically significant (b.1, b.2), for the two investigated physiological networks, i.e., $\{M, S, D\}$ obtained considering MAP (M) as target and SAP (S) and DAP as sources, and $\{D, P, L\}$ obtained considering DAP (D) as target and PEP (P) and LVET (L) as sources, at REST (bluish) and during HUT (pinkish). Statistical test in (a)): Wilcoxon signed rank test, * $p<0.05$, REST vs. HUT. Statistical tests in (b): Fisher's test, * $p<0.05$, REST vs. HUT; $\#$$p<0.05$, $\{M, S, D\}$ vs. $\{D, P, L\}$.