Stability thresholds of attractors of the Hopfield network

Abstract

Purpose of the work is the detailed study of the attractors of the Hopfield network and their basins of attraction depending on the parameters of the system, the size of the network and the number of stored images. To characterize the basins of attraction we used the method of the so-called stability threshold, i.e., the minimum distance from an attractor to the boundary of its basin of attraction. For useful attractors, this value corresponds to the minimum distortion of the stored image, after which the system is unable to recognize it. In the result of the study it is shown that the dependence of the average stability threshold of useful attractors on the number of stored images can be nonmonotonic, due to which the stability of the network can improve when new images are memorized. An analysis of the stability thresholds allowed to estimate the maximum number of images that the network can store without fatal errors in their recognition. In this case, the stability threshold of useful attractors turns out to be close to the minimum possible value, that is, to unity. To conclude, calculation of the stability thresholds provides important information about the attraction basins of the network attractors.

Keywords

• dynamical networks
• collective dynamics
• associative memory