# Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI

arXiv: https://arxiv.org/abs/1902.07186 · HTML: https://mlsanity.com/paper/1902.07186/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2019.

## Built on by (most cited first, 30 of 41)

- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems](https://mlsanity.com/paper/2207.02542/) (2022)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics](https://mlsanity.com/paper/2505.13192/) (2025)
- [Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2410.14240/) (2024)
- [Reverse engineering recurrent neural networks with Jacobian switching linear dynamical systems](https://mlsanity.com/paper/2111.01256/) (2021)
- [Bifurcations and loss jumps in RNN training](https://mlsanity.com/paper/2310.17561/) (2023)
- [Reconstructing Nonlinear Dynamical Systems from Multi-Modal Time Series](https://mlsanity.com/paper/2111.02922/) (2021)
- [Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data](https://mlsanity.com/paper/2410.04814/) (2024)
- [Integrating Multimodal Data for Joint Generative Modeling of Complex Dynamics](https://mlsanity.com/paper/2212.07892/) (2022)
- [Panda: A pretrained forecast model for chaotic dynamics](https://mlsanity.com/paper/2505.13755/) (2025)
- [Flipped Classroom: Effective Teaching for Time Series Forecasting](https://mlsanity.com/paper/2210.08959/) (2022)
- [Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling](https://mlsanity.com/paper/2410.06366/) (2024)
- [A scalable generative model for dynamical system reconstruction from neuroimaging data](https://mlsanity.com/paper/2411.02949/) (2024)
- [Optimal Recurrent Network Topologies for Dynamical Systems Reconstruction](https://mlsanity.com/paper/2406.04934/) (2024)
- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [POCO: Scalable Neural Forecasting through Population Conditioning](https://mlsanity.com/paper/2506.14957/) (2025)
- [Recurrences reveal shared causal drivers of complex time series](https://mlsanity.com/paper/2301.13516/) (2023)
- [Continuous-Time Piecewise-Linear Recurrent Neural Networks](https://mlsanity.com/paper/2602.15649/) (2026)
- [Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling](https://mlsanity.com/paper/2602.16864/) (2026)
- [Detecting Invariant Manifolds in ReLU-Based RNNs](https://mlsanity.com/paper/2510.03814/) (2025)
- [Neural Functions for Learning Periodic Signal](https://mlsanity.com/paper/2506.09526/) (2025)
- [Scalable Diffusion Transformer for Conditional 4D fMRI Synthesis](https://mlsanity.com/paper/2511.22870/) (2025)
- [Spatio-temporally separable non-linear latent factor learning: an application to somatomotor cortex fMRI data](https://mlsanity.com/paper/2205.13640/) (2022)
- [Double projection for reconstructing dynamical systems: between stochastic and deterministic regimes](https://mlsanity.com/paper/2510.01089/) (2025)
- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)

## Abstract (arXiv)

A major tenet in theoretical neuroscience is that cognitive and behavioral processes are ultimately implemented in terms of the neural system dynamics. Accordingly, a major aim for the analysis of neurophysiological measurements should lie in the identification of the computational dynamics underlying task processing. Here we advance a state space model (SSM) based on generative piecewise-linear recurrent neural networks (PLRNN) to assess dynamics from neuroimaging data. In contrast to many other nonlinear time series models which have been proposed for reconstructing latent dynamics, our model is easily interpretable in neural terms, amenable to systematic dynamical systems analysis of the resulting set of equations, and can straightforwardly be transformed into an equivalent continuous-time dynamical system. The major contributions of this paper are the introduction of a new observation model suitable for functional magnetic resonance imaging (fMRI) coupled to the latent PLRNN, an efficient stepwise training procedure that forces the latent model to capture the 'true' underlying dynamics rather than just fitting (or predicting) the observations, and of an empirical measure based on the Kullback-Leibler divergence to evaluate from empirical time series how well this goal of approximating the underlying dynamics has been achieved. We validate and illustrate the power of our approach on simulated 'ground-truth' dynamical (benchmark) systems as well as on actual experimental fMRI time series, and demonstrate that the latent dynamics harbors task-related nonlinear structure that a linear dynamical model fails to capture. Given that fMRI is one of the most common techniques for measuring brain activity non-invasively in human subjects, this approach may provide a novel step toward analyzing aberrant (nonlinear) dynamics for clinical assessment or neuroscientific research.