Symbolic local information transfer

Recently, the permutation-information theoretic approach has been used in a broad range of research fields. In particular, in the study of highdimensional dynamical systems, it has been shown that this approach can be effective in characterizing global properties, including the complexity of their spatiotemporal dynamics. Here, we show that this approach can also be applied to reveal local spatiotemporal profiles of distributed computations existing at each spatiotemporal point in the system. J. T. Lizier et al. have recently introduced the concept of local information dynamics, which consists of information storage, transfer, and modification. This concept has been intensively studied with regard to cellular automata, and has provided quantitative evidence of several characteristic behaviors observed in the system. In this paper, by focusing on the local information transfer, we demonstrate that the application of the permutation-information theoretic approach, which introduces natural symbolization methods, makes the concept easily extendible to systems that have continuous states. We propose measures called symbolic local transfer entropies, and apply these measures to two test models, the coupled map lattice (CML) system and the Bak-Sneppen model (BS-model), to show their relevance to spatiotemporal systems that have continuous states. In the CML, we demonstrate that it can be successfully used as a spatiotemporal filter to stress a coherent structure buried in the system. In particular, we show that the approach can clearly stress out defect turbulences or Brownian motion of defects from the background, which gives quantitative evidence suggesting that these moving patterns are the information transfer substrate in the spatiotemporal system. We then show that these measures reveal qualitatively different properties from the conventional approach using the sliding window method, and are also robust against external noise. In the BS-model, we demonstrate that these measures can provide novel insight to the model, featuring how symbolic local information transfer is related to the dynamical properties of the elements involved in a spatiotemporal dynamics.