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In information theory, causal measures based on Shannon entropy are commonly used to analyze dynamical systems. However, recent research indicates that entropy and complexity alone often fail to effectively distinguish between certain types of time series. To address this issue, generalized entropies such as Tsallis and Rényi have been introduced. We developed the Rényi complexity-entropy approach by combining Rényi entropy with a generalized form of statistical complexity. This method successfully differentiates between chaotic, stochastic, and periodic time series, with the parameter q playing a crucial role.
Nevertheless, in some cases, parametric curves may overlap within the entropy-complexity plane, complicating the separation of specific features, particularly with experimental data and uncertainties. To address this, the behavior of the parameter q is investigated, leading to the formation of the Rényi entropy-complexity causality space. This three-dimensional representation enhances the separation of system behaviors, even when two-dimensional projections overlap.
Recent research also explores how changes in scaling slopes correlate with variations in Rényi entropy. This approach proves useful for studying scale-free phenomena in both simulations and experimental data, providing an effective method for distinguishing noise patterns and other system characteristics.
Within this framework, the Bandt and Pompe method for ordinal patterns is used to associate each signal with a probability distribution. Causal measures of Rényi entropy and complexity are then computed based on the parameter q. This method has been successfully applied to simulated time series with correlated noise, allowing effective differentiation between cases. Additionally, it is employed to study scale-free dynamics in intracranial electroencephalography (iEEG), which is related to learning, multi-scale integration, and mental model formation. Notably, REM sleep shows significant sex-based differences, with the supramarginal gyrus exhibiting the most variability across different analysis modes. Exploring brain activity through this framework provides valuable insights into cognition and neurological disorders, potentially highlighting sex-related differences in brain function and their relevance to specific neurological conditions.