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Many, if not most, systems of interest in science are naturally described as nonlinear dynamical systems. Empirically, we commonly access these systems through time series measurements. Often such time series may consist of discrete random variables rather than continuous measurements, or may be composed of measurements from multiple data modalities observed simultaneously. For instance, in neuroscience we may have behavioral labels in addition to spike counts and continuous physiological recordings. While by now there is a burgeoning literature on deep learning for dynamical systems reconstruction (DSR), multimodal data integration has hardly been considered in this context. Here we provide such an efficient and flexible algorithmic framework that rests on a multimodal variational autoencoder for generating a sparse teacher signal that guides training of a reconstruction model, exploiting recent advances in DSR training techniques. It enables to combine various sources of information for optimal reconstruction, even allows for reconstruction from symbolic data (class labels) alone, and connects different types of observations within a common latent dynamics space. In contrast to previous multimodal data integration techniques for scientific applications, our framework is fully \textit{generative}, producing, after training, trajectories with the same geometrical and temporal structure as those of the ground truth system., Comment: ICML 2024. Previously published as a workshop paper for the AAAI 2023 Workshop MLmDS as "Multimodal Teacher Forcing for Reconstructing Nonlinear Dynamical Systems"

Empirically observed time series in physics, biology, or medicine, are commonly generated by some underlying dynamical system (DS) which is the target of scientific interest. There is an increasing interest to harvest machine learning methods to reconstruct this latent DS in a data-driven, unsupervised way. In many areas of science it is common to sample time series observations from many data modalities simultaneously, e.g. electrophysiological and behavioral time series in a typical neuroscience experiment. However, current machine learning tools for reconstructing DSs usually focus on just one data modality. Here we propose a general framework for multi-modal data integration for the purpose of nonlinear DS reconstruction and the analysis of cross-modal relations. This framework is based on dynamically interpretable recurrent neural networks as general approximators of nonlinear DSs, coupled to sets of modality-specific decoder models from the class of generalized linear models. Both an expectation-maximization and a variational inference algorithm for model training are advanced and compared. We show on nonlinear DS benchmarks that our algorithms can efficiently compensate for too noisy or missing information in one data channel by exploiting other channels, and demonstrate on experimental neuroscience data how the algorithm learns to link different data domains to the underlying dynamics., Comment: 19 pages, 8 figures

A main theoretical interest in biology and physics is to identify the nonlinear dynamical system (DS) that generated observed time series. Recurrent Neural Networks (RNNs) are, in principle, powerful enough to approximate any underlying DS, but in their vanilla form suffer from the exploding vs. vanishing gradients problem. Previous attempts to alleviate this problem resulted either in more complicated, mathematically less tractable RNN architectures, or strongly limited the dynamical expressiveness of the RNN. Here we address this issue by suggesting a simple regularization scheme for vanilla RNNs with ReLU activation which enables them to solve long-range dependency problems and express slow time scales, while retaining a simple mathematical structure which makes their DS properties partly analytically accessible. We prove two theorems that establish a tight connection between the regularized RNN dynamics and its gradients, illustrate on DS benchmarks that our regularization approach strongly eases the reconstruction of DS which harbor widely differing time scales, and show that our method is also en par with other long-range architectures like LSTMs on several tasks.

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.

This review provides a dynamical systems perspective on psychiatric symptoms and disease, and discusses its potential implications for diagnosis, prognosis, and treatment. After a brief introduction into the theory of dynamical systems, we will focus on the idea that cognitive and emotional functions are implemented in terms of dynamical systems phenomena in the brain, a common assumption in theoretical and computational neuroscience. Specific computational models, anchored in biophysics, for generating different types of network dynamics, and with a relation to psychiatric symptoms, will be briefly reviewed, as well as methodological approaches for reconstructing the system dynamics from observed time series (like fMRI or EEG recordings). We then attempt to outline how psychiatric phenomena, associated with schizophrenia, depression, PTSD, ADHD, phantom pain, and others, could be understood in dynamical systems terms. Most importantly, we will try to convey that the dynamical systems level may provide a central, hub-like level of convergence which unifies and links multiple biophysical and behavioral phenomena, in the sense that diverse biophysical changes can give rise to the same dynamical phenomena and, vice versa, similar changes in dynamics may yield different behavioral symptoms depending on the brain area where these changes manifest. If this assessment is correct, it may have profound implications for the diagnosis, prognosis, and treatment of psychiatric conditions, as it puts the focus on dynamics. We therefore argue that consideration of dynamics should play an important role in the choice and target of interventions.