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.

本研究提出了一种基于机器学习的方法来对脑活动进行长期预测，该方法基于 Koopman 算子，通过建立一种非线性映射来达到降维和预测的目的，并且实验证明该方法在预测方面的表现要好于传统的基于随机游走模型的方法。