Basins of Attraction to Multiple Fixed Points in Discrete-time Hysteresis Neural Networks
2026-08-24 • Neural and Evolutionary Computing
Neural and Evolutionary Computing
AI summaryⓘ
The authors study a neural network made of neurons that remember past states (hysteresis) and have simple on/off outputs. They show that the network can settle into many stable patterns depending on a threshold setting. The authors use a measure called entropy to understand how evenly the network’s initial states lead to these patterns. To avoid complex calculations, they apply their ideas to a simple problem: sorting binary data into groups. They test this on educational data and check their sorting with standard educational metrics.
hysteresis neural networkbinary neuronfixed pointsthreshold parameterbasin of attractionentropyclassificationitem response theorydiscrete-time systembinary data
Authors
Yuta Arai, Seigo Nakamura, Ryoga Nakamura, Muzuki Ohira, Toshimichi Saito
Abstract
This paper studies multiple fixed points in a discrete-time hysteresis neural network. The network consists of binary hysteresis neurons characterized by the threshold parameter. Depending on the parameter, the network can have a variety of multiple binary fixed points. Stability of each fixed point is characterized by basin of attraction (BOA): the set of initial points falling into the fixed point. In order to evaluate the distribution of BOA sizes, we present entropy. In order to escape from the curse of dimensionality, we introduce a simple problem: classification of binary data set. In the classification, BOAs correspond to classes. In the problem, we clarify that the threshold parameter can control the entropy, especially, can maximize the entropy: the distribution approaches to uniform. As a concrete example, we consider an item response data set in education. Using two fundamental metrics in the item response theory, the classification results are evaluated.