Vegee brain automata: Ultradiscretization of essential chaos transversal in neural and ecosystem dynamics

Author

Funabashi, Masatoshi

Abstract

Coupled logistic equations and their discretizations are important models in ecology and complex systems science. However, the chaotic dynamics produced by these nonlinear dynamical systems are lumped together, and the mathematical correspondence between continuous and discrete-time systems is not sufficiently clear. The method of ultradiscretization, which has recently been developed in the analysis of nonlinear integrable systems, can discretize both independent variables such as time and dependent variables such as time-evolving quantities in the dynamical system, while providing an analytical basis for the mathematical correspondence with the original continuous system. In this paper, we first show that the ultradiscretization of the logistic equation has the same form as that of a sigmoidal map, which cannot be derived from a customarily used logistic map. Consequently, recursively coupled systems of sigmoidal functions, such as those employed in neural networks, emerge as new candidate models for various dynamics important in agroecology, where both autonomous dynamics of ecosystems and human intervention could be represented. We then explore qualitative correspondences between neural networks and various modes of farming, including chaotic behavior, and propose an ultra-discretized model that serves as the essential underlying element. The newly proposed model has mathematical connectivity with logistic and tent maps, as well as Holling’s disc equations, providing interpretations rooted in ecology and neuroscience. The comprehensive results provide a new perspective for extracting the essence of complex agroecology via computation, which has the potential to link the properties of deep learning being studied in neural networks to the complexity of ecological management.