Bifurcation Dynamics of Large-scale Neural Networks Composed of Super Multi-ring Networks
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摘要: 目前绝大多数神经网络分岔动力学局限于结构简单、低维少节点模型, 这与真实的大规模神经网络系统相去甚远. 因此, 研究大量神经元耦合的高维神经网络模型更具实际应用价值. 环状及辐射状结构在神经网络中普遍存在, 提出了一类大规模超环时滞神经网络模型, 结构包含一个大环和任意多个小环, 并且每个环上拥有任意多个神经元. 运用特征值法和分岔理论, 选取时滞为分岔参数, 给出了该超环神经网络模型的稳定性条件和Hopf分岔判据. 数值仿真结果, 验证该理论结果的正确性.Abstract: At present, most bifurcation dynamics of neural networks are limited to simple structure, low dimension and few nodes, which are far from the real large-scale neural network system. Therefore, it is more practical to study the high-dimensional neural network model with a large number of neurons coupling. Ring and radial structures are ubiquitous in neural networks. This paper presents a class of large-scale neural networks with super-ring structure and delays, which consists of a large ring and any number of small rings, and each ring has any number of neurons. By using the eigenvalue method and bifurcation theory, the stability condition and Hopf bifurcation criterion of the super-ring neural network model are obtained by taking the time delay as bifurcation parameter. The correctness of the theoretical results is verified by numerical simulation.
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Key words:
- Neural network /
- super-ring structure /
- time delay /
- stability /
- Hopf bifurcation
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图 1 一类融合多种结构的超环神经网络模型图
Fig. 1 Model diagram of a class of super multi ring neural network fused with multiple structures
图 2 当
$\tau $ = 3.15 s < τ0时, 网络(15)渐近稳定Fig. 2 When
$\tau $ = 3.15 s < τ0, network (15) is asymptotically stable
图 3 当
$\tau $ = 3.3 s > τ0时, 网络(15)失稳Fig. 3 Network (15) is unstable when
$\tau $ = 3.3 s > τ0表 1 网络(15)的初始参数设定表
Table 1 Initial parameter setting table for network (15)
参数 $\rho $ $v_j^{(k)},\forall j,k = {1,2,3}$ 初始值 0.75 －0.6 
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表 2 结构变化影响分岔点位置情况表
Table 2 Table of the influence of structural change on the location of bifurcation points
环的个数 结构简图 神经元个数 分岔点 3 
6 16.2 4 
9 3.22 5 
12 2.08
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