Recurrent neural networks with associative memory properties are typically based on fixed-point dynamics, which is fundamentally distinct from the oscillatory dynamics of the brain. There have been proposals for oscillatory associative memories, but here too, in the majority of cases, only binary patterns are stored as oscillatory states in the network. Oscillatory neural network models typically operate at a single/common frequency. At multiple frequencies, even a pair of oscillators with real coupling exhibits rich dynamics of Arnold tongues, not easily harnessed to achieve reliable memory storage and retrieval. Since real brain dynamics comprises of a wide range of spectral components, there is a need for oscillatory neural network models that operate at multiple frequencies. We propose an oscillatory neural network that can model multiple time series simultaneously by performing a Fourier-like decomposition of the signals. We show that these enhanced properties of a network of Hopf oscillators become possible by operating in the complex-variable domain. In this model, the single neural oscillator is modeled as a Hopf oscillator, with adaptive frequency and dynamics described over the complex domain. We propose a novel form of coupling, dubbed "power coupling," between complex Hopf oscillators. With power coupling, expressed naturally only in the complex-variable domain, it is possible to achieve stable (normalized) phase relationships in a network of multifrequency oscillators. Network connections are trained either by Hebb-like learning or by delta rule, adapted to the complex domain. The network is capable of modeling N-channel electroencephalogram time series with high accuracy and shows the potential as an effective model of large-scale brain dynamics.

中文翻译：

---

用于多维非周期性信号存储和检索的复数值振荡神经网络

具有联想记忆特性的循环神经网络通常基于定点动力学，而定点动力学从根本上不同于大脑的振荡动力学。已经提出了振荡关联存储器的建议，但是在大多数情况下，在这里也只有二进制模式被存储为网络中的振荡状态。振荡神经网络模型通常以单个/公共频率运行。在多个频率下，即使是一对具有真实耦合的振荡器，也具有丰富的Arnold舌头动态特性，不易利用它们来实现可靠的存储器存储和检索。由于真实的大脑动力学包含广泛的频谱成分，因此需要在多个频率下运行的振荡神经网络模型。我们提出了一种振荡神经网络，该网络可以通过对信号执行傅立叶式分解来同时对多个时间序列进行建模。我们表明，通过在复变量域中运行，霍普夫振荡器网络的这些增强的属性成为可能。在此模型中，单个神经振荡器被建模为霍普夫振荡器，其自适应频率和动力学在复杂域上进行了描述。我们提出了一种复杂的Hopf振荡器之间的新颖耦合形式，称为“功率耦合”。利用仅在复变量域中自然表示的功率耦合，可以在多频振荡器网络中实现稳定的（归一化）相位关系。通过类似于Hebb的学习或通过增量规则（适用于复杂域）来训练网络连接。