Basins of Attraction to Multiple Fixed Points in Discrete-time Hysteresis Neural Networks

Neural network dynamics have historically been studied through energy functions and linear stability analysis, exemplified by Hopfield networks. Recent advances focus on the structure and size of basins of attraction (BOA), which determine memory capacity and robustness. High-dimensional nonlinear systems pose analytical challenges, especially in quantifying basin sizes and their distributions. Traditional methods often rely on exhaustive enumeration or energy landscape analysis, which become infeasible as system size grows. The introduction of hysteresis neurons adds biological plausibility but complicates stability analysis. This paper situates itself within this context, proposing entropy-based measures to quantify and control BOA distributions, thereby advancing the theoretical understanding and practical design of stable neural systems.

The core issue addressed is the difficulty in analyzing and controlling the distribution of basins of attraction in high-dimensional discrete neural networks with multiple fixed points. Existing methods lack quantitative tools to evaluate basin size variability, which impacts the stability, memory capacity, and classification performance of neural models. Specifically, how to tune parameters such as thresholds to achieve a balanced, uniform distribution of BOA sizes remains unresolved. This problem is critical for designing neural systems that are both stable and capable of representing diverse states, especially in applications like associative memory and data classification, where robustness and interpretability are essential.

The primary innovation is the integration of normalized entropy as a quantitative metric for BOA size distribution, enabling explicit control over basin uniformity. The authors develop a correlation-based learning rule to adjust the hysteresis threshold, maximizing entropy and thus promoting balanced fixed point basins. This approach diverges from traditional energy-based or purely stability-focused analyses, offering a statistical perspective rooted in information theory. The model’s simplicity facilitates numerical analysis and scalability, making it applicable to real-world high-dimensional data. The application to educational response data demonstrates the method’s practical relevance, marking a significant step forward in nonlinear neural dynamics and data classification.

• �� Construct a discrete-time binary hysteresis neural network with connection matrix W and threshold T_h. • Define the hysteresis activation function with thresholds, controlling neuron states. • Identify fixed points (BFPs) and their basins (BOA) through iterative simulation. • Quantify BOA size distribution using normalized entropy, H. • Employ correlation-based learning to tune W and T_h, aiming to maximize H. • Validate the approach via simulated datasets and real educational data, analyzing basin sizes and classification accuracy.

Simulations used binary datasets generated from Gaussian distributions and real student response data. Different threshold values T_h were tested, observing their effects on fixed point stability and basin sizes. The number of fixed points and the distribution of EFPs were recorded, with entropy calculated for each setting. The experiments demonstrated that increasing T_h broadens the basin distribution, pushing entropy toward maximum (~0.998). Classification performance was evaluated by mapping responses to fixed points, achieving over 95% accuracy. The robustness of the method was confirmed across multiple trials and datasets, validating the parameter control mechanism.

Results show that tuning T_h effectively controls BOA distribution, with maximum entropy (~0.998) at T_h=2.5. The number of fixed points remains stable, while basin sizes become more uniform as entropy increases. Classification accuracy exceeds 95%, with the model successfully distinguishing response patterns. The experiments confirm that parameter adjustment can optimize the stability-diversity trade-off, leading to more robust neural systems. The entropy metric provides a clear quantitative target for system tuning, facilitating practical applications in data classification and memory modeling.

The model is applicable in education for classifying student responses based on their response patterns, enabling personalized learning strategies. It can also be extended to image recognition, financial modeling, and other pattern classification tasks where stability and interpretability are critical. The ability to control basin distribution through parameter tuning offers a new tool for designing resilient neural architectures, especially in high-dimensional and noisy environments. Future integration with deep learning could further enhance its scalability and performance.

The approach depends on precise parameter tuning, which may be computationally intensive for very large systems. The analysis primarily focuses on binary data and fixed points, limiting its immediate applicability to multi-class or continuous data. The theoretical understanding of basin distribution limits in complex networks remains incomplete, necessitating further research. Additionally, the model’s reliance on simulated annealing or iterative tuning may hinder real-time applications. Future work should address scalability, extend to multi-class scenarios, and develop more efficient algorithms for basin analysis.