A scalable generative model for dynamical system reconstruction from neuroimaging data (2411.02949v1)

Abstract: Data-driven inference of the generative dynamics underlying a set of observed time series is of growing interest in machine learning and the natural sciences. In neuroscience, such methods promise to alleviate the need to handcraft models based on biophysical principles and allow to automatize the inference of inter-individual differences in brain dynamics. Recent breakthroughs in training techniques for state space models (SSMs) specifically geared toward dynamical systems (DS) reconstruction (DSR) enable to recover the underlying system including its geometrical (attractor) and long-term statistical invariants from even short time series. These techniques are based on control-theoretic ideas, like modern variants of teacher forcing (TF), to ensure stable loss gradient propagation while training. However, as it currently stands, these techniques are not directly applicable to data modalities where current observations depend on an entire history of previous states due to a signal's filtering properties, as common in neuroscience (and physiology more generally). Prominent examples are the blood oxygenation level dependent (BOLD) signal in functional magnetic resonance imaging (fMRI) or Ca$^{2+}$ imaging data. Such types of signals render the SSM's decoder model non-invertible, a requirement for previous TF-based methods. Here, exploiting the recent success of control techniques for training SSMs, we propose a novel algorithm that solves this problem and scales exceptionally well with model dimensionality and filter length. We demonstrate its efficiency in reconstructing dynamical systems, including their state space geometry and long-term temporal properties, from just short BOLD time series.

• The paper introduces a convolution-based state space model (convSSM) that leverages Wiener deconvolution and teacher forcing to accurately reconstruct neural dynamics from fMRI data.
• It overcomes traditional linear and DCM-based methods by capturing the nonlinear and chaotic characteristics of brain activity.
• Empirical tests on the Lorenz63 system validate its ability to recover long-term temporal statistics and attractor geometries, ensuring its scalability for large datasets and clinical applications.

A Scalable Generative Model for Dynamical System Reconstruction from Neuroimaging Data

The paper presents an innovative approach to reconstructing dynamical systems (DS) from neuroimaging data, particularly functional magnetic resonance imaging (fMRI) via a scalable generative model. This model addresses the complexity of capturing neural dynamics, which often exhibit chaotic behavior and require robust reconstruction tools capable of capturing geometrical and temporal invariants—essential for understanding inter-individual variability in brain dynamics and potentially for diagnosing and predicting brain dysfunctions or designing personalized therapies.

Contributions and Methodology

This paper advances dynamical systems reconstruction (DSR) through deep learning-based time series analysis. Traditionally, methods to infer large-scale brain dynamics, such as latent linear DS models and mean field neural simulations like those within The Virtual Brain (TVB), have relied heavily on a multitude of assumptions that restrict their explanatory power. Dynamic Causal Modeling (DCM), while statistically motivated, offers limited utility for true dynamical systems analysis, often constrained by linear assumptions in an inherently nonlinear neural landscape.

The authors shift focus by proposing a convolution-based state space model (convSSM), particularly suitable for fMRI data that exhibits complex filtering properties due to the hemodynamic response function (hrf). Key innovations in this paper involve introducing a novel SSM-based DSR algorithm capable of efficiently handling measurements that depend on convolutions over latent state sequences.

The convSSM leverages modern variants of teacher forcing (TF) to counteract the exploding-and-vanishing gradients problem, utilizing gains from Sparse TF (STF) and Generalized TF (GTF) to stabilize gradient propagation. Through Wiener deconvolution, the model disentangles the observed BOLD signals in fMRI data, thereby recovering the latent neural activity. This approach circumvents the non-invertibility of the observation model by enabling an efficient approximation of control signals needed for training with stochastic gradient descent (SGD).

Results and Scalability

Empirical validation using the Lorenz63 system—a standard benchmark for chaotic systems—demonstrated that the convSSM performed superiorly in reconstructing dynamics even under signal degradation scenarios. The model's ability to accurately recover the system's long-term temporal statistics and attractor geometry from short time series is notable, emphasizing its potential application to empirical datasets such as fMRI, where short time series are typical.

Additionally, the scalability analysis reveals that the convSSM runtime per epoch increases approximately linearly with model dimensions, suggesting its applicability to large-scale datasets.

Implications and Future Directions

The implications of this work are manifold. Practically, the ability to recover underlying neural dynamics from fMRI data can aid in the personalized assessment of cognitive and clinical parameters, potentially providing individualized therapeutic insights. Theoretically, the neural system's state and temporal dynamics captured through this method illuminate the nonlinear, chaotic nature of brain activity, reinforcing hypotheses about the chaotic nature of neural processing.

Furthermore, the modular structure of the convSSM provides flexibility for further adaptations, such as incorporating stochastic processes within the inference or learning the parameters of the convolution directly. There is room for expanding this framework to applications beyond neuroscience, such as any domain dealing with complex time series data filtered through nonlinear dynamics.

This paper, through methodical evolution and empirical rigor, sets a foundation that could be transformative for how dynamic brain states are reconstructed and interpreted in clinical and research settings, opening avenues for using data-driven DSR models on a larger scale in neuroscience and associated fields.