Multimodal Teacher Forcing for Reconstructing Nonlinear Dynamical Systems

Many, if not most, systems of interest in science are naturally described as nonlinear dynamical systems (DS). Empirically, we commonly access these systems through time series measurements, where often we have time series from different types of data modalities simultaneously. For instance, we may have event counts in addition to some continuous signal. While by now there are many powerful machine learning (ML) tools for integrating different data modalities into predictive models, this has rarely been approached so far from the perspective of uncovering the underlying, data-generating DS (aka DS reconstruction). Recently, sparse teacher forcing (TF) has been suggested as an efficient control-theoretic method for dealing with exploding loss gradients when training ML models on chaotic DS. Here we incorporate this idea into a novel recurrent neural network (RNN) training framework for DS reconstruction based on multimodal variational autoencoders (MVAE). The forcing signal for the RNN is generated by the MVAE which integrates different types of simultaneously given time series data into a joint latent code optimal for DS reconstruction. We show that this training method achieves significantly better reconstructions on multimodal datasets generated from chaotic DS benchmarks than various alternative methods.

Identifying nonlinear dynamical systems from multi-modal time series data

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 completely 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 identification and cross-modal prediction. 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

Inferring Dynamical Systems with Long-Range Dependencies through Line Attractor Regularization

Vanilla RNN with ReLU activation have a simple structure that is amenable to systematic dynamical systems analysis and interpretation, but they suffer from the exploding vs. vanishing gradients problem. Recent attempts to retain this simplicity while alleviating the gradient problem are based on proper initialization schemes or orthogonality/unitary constraints on the RNN's recurrence matrix, which, however, comes with limitations to its expressive power with regards to dynamical systems phenomena like chaos or multi-stability. Here, we instead suggest a regularization scheme that pushes part of the RNN's latent subspace toward a line attractor configuration that enables long short-term memory and arbitrarily slow time scales. We show that our approach excels on a number of benchmarks like the sequential MNIST or multiplication problems, and enables reconstruction of dynamical systems which harbor widely different time scales.

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

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. 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.