Spatial distribution of patterns and the Hopfield network phase space geometry

It is well known that complex dynamical systems exhibit a rich variety of interesting nonlinear phenomena that could be used for information processing. One of those promising issues has originated from the attractor neural network models [1,2]. In this paradigm, any piece of information stored in the system is represented by apt attractor in the phase space of a neural network (NN), and information processing is considered as the time evolution of the NN state. Such an approach is based on the observation that the state of a dissipative nonlinear dynamical system converges asymptotically toward an attractor, whose type, shape, location and size differs depending on values of the system parameters. Investigation into the phase portrait of NN, i.e. what attractors exist and what their properties are, is therefore of great importance. Despite the progress achieved in analytical treatment of dynamical systems, computer simulations are still the main tools for exploring the phase space of such complex systems. The aim of the present paper is to study numerically phase spaces of Hopfield's analog NN with emphasis on memorization of patterns (MP's) and their spatial distribution influencing the geometrical structure of attraction, basins. Special attention is paid to the phase space geometry changes with rotation and shifting eithe single pattern or their whole set.