Granger causality maps for Langevin systems

Area of Science:

• Dynamical Systems Theory
• Information Theory
• Computational Neuroscience

Background:

• Granger causality (GC) maps were previously developed for Langevin systems using vector Ornstein-Uhlenbeck (VOU) processes.
• Existing methods approximated GCs from discrete-time sampling, leading to computational inefficiency and inability to analyze unstable dynamics, creating 'holes' in GC maps.

Purpose of the Study:

• To address the limitations of previous GC map implementations for Langevin systems.
• To develop a computationally efficient and robust method for calculating GC rates, even in unstable dynamical regimes.

Main Methods:

• Derived an analytical expression for GC rates in VOU processes.
• Applied the analytical expression to construct GC maps, filling previously identified 'holes' in unstable regions.
• Showcased computational efficiency by reducing calculations to solving algebraic Riccati equations (or quadratic equations for univariate cases).

Main Results:

• The new analytical method provides meaningful GC rate solutions for both stable and unstable dynamics.
• GC maps can now be fully constructed without 'holes', offering a more complete picture of causal relationships.
• The GC rate for VOU processes is invariant to rescaling of fluctuation intensity, enabling application to deterministic nonlinear systems.

Conclusions:

• The developed analytical approach significantly enhances the applicability and efficiency of Granger causality mapping for dynamical systems.
• This method overcomes critical limitations of prior techniques, providing a more comprehensive tool for analyzing complex system dynamics.
• The invariance property allows for the study of causality in deterministic systems with a residual noise component.