Data_Sheet_1_A Complex-Valued Oscillatory Neural Network for Storage and Retrieval of Multidimensional Aperiodic Signals.docx

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.

具备联想记忆（associative memory）特性的循环神经网络（recurrent neural network）通常基于定点动力学（fixed-point dynamics），这与大脑的振荡动力学（oscillatory dynamics）有着本质区别。已有针对振荡型联想记忆的研究方案，但在多数此类方案中，网络仅能将二值模式（binary patterns）作为振荡态（oscillatory states）进行存储。振荡神经网络模型通常仅工作于单一/通用频率。当存在多个频率时，即便仅一对带有实耦合的振荡器也会展现出丰富的阿诺德舌（Arnold tongue）动力学特性，却难以被用于实现可靠的记忆存储与检索。鉴于真实的大脑动力学包含广泛的频谱分量（spectral components），因此亟需能够工作于多频率场景下的振荡神经网络模型。本文提出一种振荡神经网络，其可通过对信号进行类傅里叶分解（Fourier-like decomposition），同时对多组时间序列进行建模。本文证明，通过在复变量域（complex-variable domain）中进行运算，霍普夫振荡器（Hopf oscillator）网络的这些增强特性得以实现。在该模型中，单个神经振荡器被建模为霍普夫振荡器，其自适应频率（adaptive frequency）与动力学特性均在复域中进行描述。本文提出一种针对复域霍普夫振荡器的新型耦合方式，将其命名为“功率耦合（power coupling）”。仅能在复变量域中自然表达的功率耦合，可使多频率振荡器网络实现稳定的（归一化）相位关系。网络连接可通过适配于复域的类赫布学习（Hebb-like learning）或德尔塔法则（delta rule）进行训练。该网络能够高精度地对N通道脑电图（electroencephalogram, EEG）时间序列进行建模，展现出作为大规模大脑动力学有效模型的潜力。