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

arXiv: https://arxiv.org/abs/2304.11925 · HTML: https://mlsanity.com/paper/2304.11925/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 112 references that are in the map.
- First posted on arXiv 24 Apr 2023.

## Builds on (most cited first, 12 of 12)

- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Time Series Forecasting Using Manifold Learning](https://mlsanity.com/paper/2110.03625/) (2021)
- [Theta-nested gamma oscillations in next generation neural mass models](https://mlsanity.com/paper/2003.04000/) (2020)
- [Construction of embedded fMRI resting state functional connectivity networks using manifold learning](https://mlsanity.com/paper/2005.12390/) (2020)
- [Deep brain stimulation for movement disorder treatment: Exploring frequency-dependent efficacy in a computational network model](https://mlsanity.com/paper/2010.07162/) (2020)
- [Exact neural mass model for synaptic-based working memory](https://mlsanity.com/paper/2010.07071/) (2020)

## Built on by (most cited first, 5 of 5)

- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Early warning indicators via latent stochastic dynamical systems](https://mlsanity.com/paper/2309.03842/) (2023)
- [A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs](https://mlsanity.com/paper/2403.11591/) (2024)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)
- [Tau-BNO: Brain Neural Operator for Tau Transport Model](https://mlsanity.com/paper/2603.08108/) (2026)

## Abstract (arXiv)

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.