Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator.

We propose a machine-learning approach to construct reduced-order models (ROMs) to predict the long-term out-of-sample dynamics of brain activity (and in general, high-dimensional time series), focusing mainly on task-dependent high-dimensional fMRI time series. Our approach is a three stage one. First, we exploit manifold learning and, in particular, diffusion maps (DMs) to discover a set of variables that parametrize the latent space on which the emergent high-dimensional fMRI time series evolve. Then, we construct 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, i.e., the construction of a map from the low-dimensional manifold to the original high-dimensional (ambient) space by 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 two benchmark fMRI time series: (i) a simplistic five-dimensional model of stochastic discrete-time equations used just for a "transparent" illustration of the approach, thus knowing a priori what one expects to get, and (ii) a real fMRI dataset with recordings during a visuomotor task. 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 space; one can use instead the low-frequency truncation of the DMs function space of L2-integrable functions to predict the entire list of coordinate functions in the ambient space and to solve the pre-image problem.

Walking on stairs: Experiment and model.

An increasing global population forces urban planners to construct buildings and infrastructure that is extremely deep and high. Elevators and escalators serve skyscrapers and tunnels, but in an emergency people still have to walk on stairs. Computer simulations can mitigate risks of escape situations. For these situations, pedestrian locomotion models need to match reality well. Motion on stairs, however, is not nearly as well understood as motion in the plane. Publications are scarce and some are contradictory. As a result, movement on stairs is usually modeled by slowing down pedestrians by a fixed factor. But is this justified? And what happens at intermediate landings? This contribution aims to clarify inconclusive results of previous research and provide new information to directly incorporate empirical results into a parsimonious computer model. The algorithms are freely available through an open-source framework. After outlining the shortcomings of existing approaches, we present three experiments, from which we derive requirements for the computer model. Reenacting computer experiments shows the extent to which our model meets our observations. We conclude with an applied example, simulating an evacuation of Germany's famous Neuschwanstein Castle.