Multistability of delayed recurrent neural networks with piecewise non-monotonic activation functions and its...

This paper studies the multistability of delayed recurrent neural networks (DRNNs) with piecewise non-monotonic activation functions and their application to associative memory. To address the limitation of conventional delayed recurrent neural networks, which rely on monotonic or saturated activation functions and thus struggle to achieve high-density multistable structures, we construct an oscillatory activation function with piecewise non-monotonic characteristics. Under a unified theoretical framework, a rigorous analysis of the network's equilibrium structure and stability is conducted. By employing Poincaré-Miranda theorem, the Lipschitz condition, and local asymptotic stability theory, it is proven that, under certain conditions, an n-neuron DRNN with the proposed activation function admits 6ⁿ equilibrium points (EPs), among which 4ⁿ are locally asymptotically stable. Furthermore, by introducing an oscillation-frequency parameter and systematically analyzing its regulatory mechanism, the above results are generalized: under appropriate conditions, the network can achieve (4k+2)ⁿ equilibrium points, with (2k+2)ⁿ of them being locally asymptotically stable. Consequently, DRNNs equipped with the proposed activation function exhibit a larger storage capacity when applied to associative memory. Parameter-feasibility and perturbation analyses further demonstrate that the multistable structure and convergence behavior remain robust over a broad range of instantaneous and delayed coupling strengths. Finally, numerical simulations and comparative experiments with representative associative-memory networks support the theoretical analysis and indicate that the proposed network achieves competitive performance in exact recall, theoretical stable-memory capacity, and empirical basin volume.