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

Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator