A Nested Multi-scroll Memristive Hopfield Neural Network and Its Hardware Implementation
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摘要: 忆阻Hopfield神经网络是将忆阻器引入传统Hopfield神经网络(HNN)，借助忆阻器模拟神经元突触来实现突触权重动态调节的类脑神经网络模型。本文提出一种含符号函数的多分段非线性磁控忆阻器，并将其应用于三神经元HNN系统的自连接突触，构建了四维忆阻HNN系统。首先，利用分岔图和李雅普诺夫指数谱，分析忆阻自连接突触耦合强度对系统动力学行为的调控作用，揭示了与系统初值相关的共存吸引子及其演化规律。同时，发现该系统具备多涡卷吸引子的单向拓展特性，以及多涡卷吸引子数目随忆阻器控制参数的变化而改变。在此基础上，通过施加多级逻辑脉冲电流进行调控，构建了一种新型嵌套多涡卷忆阻HNN系统，促使系统产生具有嵌套结构的特殊多涡卷吸引子，并分析了加入多级逻辑脉冲电流前后系统复杂度的变化情况。最后，采用Multisim模拟电路仿真与FPGA数字电路实验，双重验证了MATLAB数值仿真结果的正确性。
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关键词:
- Hopfield神经网络 /
- 忆阻自连接突触 /
- 多级逻辑脉冲电流 /
- 嵌套多涡卷吸引子 /
- 硬件实现
Abstract:Objective In recent years, researchers have employed memristors to simulate neuronal synapses for dynamically adjusting synaptic weights, thereby constructing memristive neural networks. The memristive Hopfield neural network (HNN) can more precisely reflect the nonlinear dynamic behavior of biological neural systems. Notably, multi-scroll attractors have garnered significant attention in the field of secure communication owing to their complex topological structures and high state-space ergodicity. However, most existing studies have focused primarily on single-structure multi-scroll attractors, whereas multi-scroll attractors with special structures remain in need of further exploration. Consequently, this paper proposes an HNN system capable of generating special nested multi-scroll attractors, thereby overcoming the limitations of traditional single-structure multi-scroll attractors. Methods A four-dimensional (4D) memristive HNN system is constructed based on a three-neuron HNN. In this system, a multi-segment nonlinear magnetically controlled memristor is integrated into the self-connected synapse of neuron 2. By analyzing the equilibrium points and system stability, this study investigates the effects of memristor coupling strength and system initial conditions. The number of multi-scroll attractors can be increased by adjusting the parameters of the memristor. Building upon this, a novel nested multi-scroll memristive HNN system is established by introducing multi-level logic pulse current. The proposed system can generate unique nested multi-scroll attractors, and its complexity can be further enhanced. Finally, the accuracy of the MATLAB numerical results is verified through Multisim circuit simulation and FPGA-based hardware experiments. Results and Discussions The experimental results indicate that by regulating the coupling strength of the memristor self-connected synapses, the proposed 4D memristive HNN system can yield bifurcation diagrams and Lyapunov exponent spectra ( Fig. 3 ), as well as diverse coexisting attractors (Fig. 4 ). Meanwhile, multi-scroll attractors with varying scroll numbers are obtained (Fig. 5 –7 ). By regulating multi-level logic pulse currents, a novel nested multi-scroll memristive HNN system is constructed. This system can generate nested multi-scroll attractors (Fig. 9 –10 ). Spectral entropy analysis demonstrates that, compared with the original 4D memristive HNN system, the proposed system has its complexity effectively enhanced (Fig. 11 –12 ). Both Multisim simulation (Fig. 14 ) and FPGA-based experiments (Fig. 15 –16 ) verify the feasibility of the proposed memristive HNN system.Conclusions Based on a three-neuron HNN, a 4D memristive HNN system is constructed by embedding a multi-segment nonlinear magnetically controlled memristor. Through equilibrium point and stability analysis, this paper reveals the regulatory effect of the memristor coupling strength, as well as the evolution law of coexisting attractors dependent on initial conditions. The results show that the system can enter a chaotic state via a period-doubling bifurcation route and produce single-scroll and double-scroll chaotic attractors. The number of multi-scroll attractors can be increased by optimizing memristor parameters. Furthermore, by regulating multi-level logic pulse currents, a novel nested multi-scroll memristive HNN system is developed to generate nested multi-scroll attractors. Spectral entropy analysis proves that the application of multi-level logic pulse currents enhances the system complexity. The numerical simulation results from MATLAB, Multisim simulations and FPGA-based experimental results are highly consistent, which validates the feasibility of the nested multi-scroll memristive HNN system. -
表 1 k=1.6时的平衡点、特征值和稳定性
平衡点坐标 特征值 稳定性 P0: (0, 0, 0, 0) λ1= 0.9891 , λ2= –1.32，
λ3, 4=–0.4945 ±1.9773i不稳定指标1鞍焦点 P1: (–0.553, – 0.1196 , 1.872, –0.192)λ1=− 0.8929 ，λ2=−1.3189 ，
λ3, 4=0.6665 ±1.5464i不稳定指标2鞍焦点 P2: (0.553, 0.1196 , –1.872, 0.192)λ1=– 0.8916 ，λ2=–1.3211 ，
λ3, 4=0.6374 ±1.5431i不稳定指标2鞍焦点 
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表 2 不同开关与控制电压组合下对应的控制参数与吸引子数量
K0 K1 K2 对应的控制参数 吸引子数量 E1=1V E2=3V E3=5V On On On h1($\varphi $) N=0 1 On Off On h1($\varphi $) N=1 3 On Off Off h1($\varphi $) N=2 5 E1=2V E2=4V E3=6V Off On On h2($\varphi $) M=0 2 Off Off On h2($\varphi $) M=1 4 Off Off Off h2($\varphi $) M=2 6
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