Functional brain dynamics is supported by parallel and overlapping functional network modes that are associated with specific neural circuits. Decomposing these network modes from fMRI data and finding their temporal characteristics is challenging due to their time-varying nature and the non-linearity of the functional dynamics. Dynamic Mode Decomposition (DMD) algorithms have been quite popular for solving this decomposition problem in recent years. In this work, we apply GraphDMD -- an extension of the DMD for network data -- to extract the dynamic network modes and their temporal characteristics from the fMRI time series in an interpretable manner. GraphDMD, however, regards the underlying system as a linear dynamical system that is sub-optimal for extracting the network modes from non-linear functional data. In this work, we develop a generalized version of the GraphDMD algorithm -- DeepGraphDMD -- applicable to arbitrary non-linear graph dynamical systems. DeepGraphDMD is an autoencoder-based deep learning model that learns Koopman eigenfunctions for graph data and embeds the non-linear graph dynamics into a latent linear space. We show the effectiveness of our method in both simulated data and the HCP resting-state fMRI data. In the HCP data, DeepGraphDMD provides novel insights into cognitive brain functions by discovering two major network modes related to fluid and crystallized intelligence.

翻译：功能性脑动力学由与特定神经回路相关的并行且重叠的功能网络模式所支撑。从fMRI数据中分解这些网络模式并发现其时间特征，由于它们随时间变化的特性以及功能动力学的非线性而充满挑战。近年来，动态模式分解（DMD）算法在解决这一分解问题方面广受欢迎。在本工作中，我们应用GraphDMD——DMD在网络数据上的扩展——以可解释的方式从fMRI时间序列中提取动态网络模式及其时间特征。然而，GraphDMD将底层系统视为线性动力系统，这对于从非线性功能数据中提取网络模式是次优的。在本工作中，我们开发了GraphDMD算法的广义版本——DeepGraphDMD——适用于任意非线性图动力系统。DeepGraphDMD是一种基于自编码器的深度学习模型，它学习图数据的Koopman本征函数，并将非线性图动力学嵌入到潜在线性空间中。我们在模拟数据和HCP静息态fMRI数据上展示了我们方法的有效性。在HCP数据中，DeepGraphDMD通过发现与流体智力和晶体智力相关的两大主要网络模式，为认知脑功能提供了全新见解。