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  • Open Access

Multivariate stochastic coupling in unknown complex systems: Informative randomness in deterministic brain-body communication

Andrea Scarciglia1, Vincenzo Catrambone1, Claudio Bonanno2, and Gaetano Valenza1,*

  • 1Neurocardiovascular Intelligence Lab, Department of Information Engineering and Bioengineering and Robotics Research Center “E. Piaggio”, School of Engineering, University of Pisa, Italy
  • 2Department of Mathematics, University of Pisa, Italy

  • *Contact author: gaetano.valenza@unipi.it
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Phys. Rev. E 114, 034310 – Published 11 September, 2026

DOI: https://doi.org/10.1103/5n8x-155p

Abstract

Physiological systems exhibit deterministic behavior influenced by dynamic stochasticity, where informative random components play a crucial role in complex system interactions. Fully characterizing multivariate random components is challenging when the underlying deterministic dynamics are unknown. We propose a framework for estimating stochastic coupling in unknown deterministic systems influenced by dynamic noise. Building on a multivariate extension of approximate entropy (MApEn), the proposed framework exploits the divergence of this metric in the presence of dynamical noise. After establishing the theoretical basis and validity of the method, which can accommodate arbitrary multivariate noise probability distributions, we show that, in the case of bivariate white Gaussian noise, the framework enables closed-form estimation of the noise covariance matrix without requiring prior knowledge of the underlying deterministic functions. To this end, we introduce an iterative algorithm to estimate the covariance matrix coefficients. The algorithm identifies the parameter values that rescale the differential behavior of multivariate MApEn along a target curve. We validate the framework using both synthetic datasets and real physiological recordings. The analysis reveals important insights into brain-heart interactions, particularly the modulation of stochastic coupling across sleep-stage transitions through low-frequency oscillatory activity. Our results indicate that conscious wakefulness is characterized by stronger stochastic coupling, which progressively decreases during the transition to sleep. This framework opens new avenues for investigating stochastic physiological dynamics and their associated networks, with potential applications across a wide range of scientific disciplines.

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Phys. Rev. E 114, 034310– Published 11 September, 2026
  • Received 6 December 2025
  • Accepted 17 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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