[Paper Review] Dynamical independence: discovering emergent macroscopic processes in complex dynamical systems

L. D. Barnett, Anil K. Seth|arXiv (Cornell University)|Jun 11, 2021

Neural dynamics and brain function4 citations

TL;DR

This paper introduces dynamical independence as a formal framework to identify macroscopic processes that emerge from complex dynamical systems, treating them as autonomous entities with their own dynamics. It proposes a transformation-invariant, information-theoretic measure of dynamical dependence based on Shannon entropy, enabling data-driven discovery of emergent phenomena across spatiotemporal scales using state-space models in both time and frequency domains.

ABSTRACT

We introduce a notion of emergence for coarse-grained macroscopic variables associated with highly-multivariate microscopic dynamical processes, in the context of a coupled dynamical environment. Dynamical independence instantiates the intuition of an emergent macroscopic process as one possessing the characteristics of a dynamical system "in its own right", with its own dynamical laws distinct from those of the underlying microscopic dynamics. We quantify (departure from) dynamical independence by a transformation-invariant Shannon information-based measure of dynamical dependence. We emphasise the data-driven discovery of dynamically-independent macroscopic variables, and introduce the idea of a multiscale "emergence portrait" for complex systems. We show how dynamical dependence may be computed explicitly for linear systems via state-space modelling, in both time and frequency domains, facilitating discovery of emergent phenomena at all spatiotemporal scales. We discuss application of the state-space operationalisation to inference of the emergence portrait for neural systems from neurophysiological time-series data. We also examine dynamical independence for discrete- and continuous-time deterministic dynamics, with potential application to Hamiltonian mechanics and classical complex systems such as flocking and cellular automata.

Motivation & Objective

• To formalize the intuitive notion of emergence as a macroscopic process with autonomous dynamics distinct from microscopic interactions.
• To develop a principled, data-driven method for identifying dynamically independent macroscopic variables in high-dimensional systems.
• To introduce the concept of a multiscale 'emergence portrait' that maps emergent dynamics across different spatiotemporal scales.
• To enable inference of emergent structure from neurophysiological time-series data using state-space modeling.
• To extend the framework to both discrete- and continuous-time deterministic systems, including Hamiltonian and cellular automata.

Proposed method

• Define dynamical independence as a condition where a macroscopic variable evolves independently of the microscopic dynamics, formalized via a transformation-invariant measure of dynamical dependence based on Shannon information.
• Operationalize the measure using state-space models to compute dynamical dependence in both time and frequency domains for linear systems.
• Use the method of characteristics to solve for invariants and flow-parallel coordinates (τ, u) that flatten the dynamics, enabling explicit computation of dynamically independent coarse-grainings.
• Derive the general form of dynamically independent coarse-graining maps as f(x) = Ψ(τ(x), u(x)), where τ is a time-like variable and u are functionally independent invariants.
• Apply the framework to linear stochastic systems via state-space representation, allowing estimation of dynamical dependence from empirical time-series data.
• Extend the approach to deterministic systems, including continuous-time ODEs and discrete-time maps, with illustrative examples from dynamical systems theory.

Experimental results

Research questions

• RQ1 How can we formally define and quantify the emergence of macroscopic processes that behave as autonomous dynamical systems?
• RQ2 What is a principled, data-driven method to discover such emergent processes across multiple spatiotemporal scales?
• RQ3 How can we measure the degree of dynamical independence between macroscopic variables and the underlying microscopic dynamics?
• RQ4 Can the framework be applied to real neurophysiological data to infer emergent macroscopic dynamics?
• RQ5 How does the concept of dynamical independence generalize to deterministic systems such as Hamiltonian mechanics and cellular automata?

Key findings

• The paper establishes a transformation-invariant, information-theoretic measure of dynamical dependence based on Shannon entropy, enabling robust quantification of emergence.
• For linear systems, dynamical dependence can be computed explicitly using state-space models in both time and frequency domains, facilitating scalable discovery of emergent processes.
• The general form of dynamically independent coarse-graining maps is derived as f(x) = Ψ(τ(x), u(x)), where τ is a time-like variable and u are invariants, enabling systematic identification of emergent dynamics.
• The framework successfully identifies emergent macroscopic behavior in a 3D nonlinear ODE system, with explicit solutions for invariants and flow-parallel coordinates.
• The method enables the construction of a 'multiscale emergence portrait' that reveals emergent structure across different scales, even when not visually apparent.
• The approach is applicable to real-world data, such as neural time-series, allowing inference of macroscopic dynamics from microscopic observations without requiring full simulation.