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

Area of Science:

• Neuroscience
• Machine Learning
• Dynamical Systems

Background:

• Predicting complex, high-dimensional time series like brain activity from fMRI data is challenging.
• Reduced-order models (ROMs) are needed to capture the essential dynamics of these systems.
• Existing methods may struggle with long-term out-of-sample predictions.

Purpose of the Study:

• To develop and evaluate a novel machine-learning approach for constructing ROMs to predict long-term brain activity dynamics.
• To compare the efficacy of Feedforward Neural Networks (FNNs) coupled with Geometric Harmonics (GH) against the Koopman operator method for this task.
• To address the pre-image problem for reconstructing high-dimensional dynamics from a low-dimensional latent space.

Main Methods:

• Utilized manifold learning, specifically diffusion maps (DMs), to identify a low-dimensional latent space for fMRI time series.
• Constructed ROMs on the embedded manifold using FNNs and the Koopman operator.
• Solved the pre-image problem using DMs with GH for FNNs, and Koopman modes for the Koopman operator approach, to map back to the high-dimensional fMRI space.

Main Results:

• Both FNN-GH and Koopman operator approaches successfully predicted long-term brain activity dynamics.
• The Koopman operator method achieved practically equivalent results to the FNN-GH approach.
• The Koopman operator approach offers a more streamlined solution by avoiding the need for explicit non-linear map training and GH.

Conclusions:

• The Koopman operator approach provides an efficient and effective method for predicting long-term brain activity dynamics from fMRI data.
• This method simplifies the process of solving the pre-image problem, making it a valuable tool for neuroscience research.
• The study demonstrates the power of combining diffusion maps with the Koopman operator for analyzing high-dimensional time series data.