Associative memories based on delayed fractional-order neural networks and application to explaining-lesson skills assessment of normal students: from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability

Abstract

This paper discusses associative memories based on time-varying delayed fractional-order neural networks (DFNNs) with a type of piecewise nonlinear activation function from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability. Some sufficient conditions are gained to assure the existence of $ 5^n $ equilibria for $ n $-neuron DFNNs with the proposed piecewise nonlinear activation functions. Additionally, the criteria ensure the existence of at least $ 3^n $ equilibria that are locally multiple $ \mathit O(t^{-\alpha}) $ stable. Furthermore, we apply these results to a more generic situation, revealing that DFNNs can attain $ (2k+1)^n $ equilibria, and among them, $ (k+1)^n $ equilibria are locally $ \mathit O(t^{-\alpha}) $ stable. Here, the parameter $ k $ is highly dependent on the sinusoidal function frequency in the expanded activation functions. Such DFNNs are well-suited to synthesize high-capacity associative memories; the design process is given via singular value decomposition. Ultimately, four illustrative examples, including applying neurodynamic associative memory to the explaining-lesson skills assessment of normal students, are supplied to validate the efficacy of the results.

Keywords

• associative memory
• fractional-order neural networks
• multiple $ \mathit o(t^{-\alpha}) $ stability
• piecewise nonlinear activation functions
• time-varying delays