Channels and Jokers. New Approaches for Predicting Complex Dynamics

One of the main aspects of brain activity is the ability to predict. Large efforts has been made in nonlinear dynamics to create predicting systems for dynamics of complex objects. One of the main tools for making such predictors is multylayer neural networks. The methods based on the chaos theory prove less efficient, and in fact work only for low-dimensional model systems. From our point of view, the problems here are not technical, but related with the applicability of the approach of low-dimension nonlinear dynamics to real systems. Since the brain and some of its very simple models are able to make predictions in real situations, we propose to unify the ideas of nonlinear dynamics and neural networks. From our point of view, in complex real situations it may be possible to find low-dimensional projections, for which the approaches of nonlinear dynamics can be applied, but with serious restrictions. Most concepts, like attractor, its dimension, Lyapunov exponents etc. become inapplicable, and the observed phase space splits into predictable parts ('channels') and non-predictable ones ('jokers'), where probabilistic description may be more appropriate. We propose some mathematical basis for this idea and its possible application for time series analysis.