We propose a machine-learning approach to model long-term out-of-sample dynamics of brain activity from task-dependent fMRI data. Our approach is a three stage one. First, we exploit Diffusion maps (DMs) to discover a set of variables that parametrize the low-dimensional manifold on which the emergent high-dimensional fMRI time series evolve. Then, we construct reduced-order-models (ROMs) on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using a benchmark fMRI dataset with recordings during a visuo-motor task. The results suggest that just a few (for the particular task, five) non-linear coordinates of the high-dimensional fMRI time series provide a good basis for modelling and out-of-sample prediction of the brain activity. Furthermore, we show that the proposed approaches outperform the one-step ahead predictions of the naive random walk model, which, in contrast to our scheme, relies on the knowledge of the signals in the previous time step. Importantly, we show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient fMRI space; one can use instead the low-frequency truncation of the DMs function space of L^2-integrable functions, to predict the entire list of coordinate functions in the fMRI space and to solve the pre-image problem.

翻译：我们提出一种机器学习方法，用于基于任务依赖的功能磁共振成像（fMRI）数据对脑活动的长期样本外动态进行建模。该方法包含三个阶段：首先，利用扩散映射（DM）发现一组变量，这些变量参数化了高维fMRI时间序列演化的低维流形；随后，通过两种技术——前馈神经网络（FNN）和Koopman算子——在嵌入流形上构建降阶模型（ROM）；最后，为预测fMRI原始空间中脑活动的样本外长期动态，结合DM与几何谐波（GH）求解前像问题（在使用FNN时），或直接利用Koopman模式。为验证所提方案，我们采用基准fMRI数据集（记录视觉运动任务期间的信号）评估了两种模型的性能。结果表明，高维fMRI时间序列的少量非线性坐标（针对该特定任务为五个）即可为脑活动的建模与样本外预测提供良好基础。此外，我们证明所提方法优于朴素随机游走模型的一步前向预测——该模型依赖于先前时间步的信号知识，而我们的方案无需此条件。重要的是，我们表明所提Koopman算子方法在实际应用场景中可达到与FNN-GH方法等效的结果，从而避免了训练非线性映射及使用GH在fMRI原始空间外推预测的需求；可直接采用DM函数空间（L²可积函数）的低频截断，预测fMRI空间中所有坐标函数并求解前像问题。