"Gated Orthogonal Recurrent Units: On Learningto Forget".

Recurrent Neural Networks (RNNs) have found widespread applications in machine learning for time series prediction and dynamical systems reconstruction, and experienced a recent renaissance with improved training algorithms and architectural designs. Understanding why and how trained RNNs produce their behavior is important for scientific and medical applications, and explainable AI more generally. An RNN's dynamical repertoire depends on the topological and geometrical properties of its state space. Stable and unstable manifolds of periodic points play a particularly important role: They dissect a dynamical system's state space into different basins of attraction, and their intersections lead to chaotic dynamics with fractal geometry. Here we introduce a novel algorithm for detecting these manifolds, with a focus on piecewise-linear RNNs (PLRNNs) employing rectified linear units (ReLUs) as their activation function. We demonstrate how the algorithm can be used to trace the boundaries between different basins of attraction, and hence to characterize multistability, a computationally important property. We further show its utility in finding so-called homoclinic points, the intersections between stable and unstable manifolds, and thus establish the existence of chaos in PLRNNs. Finally we show for an empirical example, electrophysiological recordings from a cortical neuron, how insights into the underlying dynamics could be gained through our method.

Modern AI models, such as large language models, are usually trained once on a huge corpus of data, potentially fine-tuned for a specific task, and then deployed with fixed parameters. Their training is costly, slow, and gradual, requiring billions of repetitions. In stark contrast, animals continuously adapt to the ever-changing contingencies in their environments. This is particularly important for social species, where behavioral policies and reward outcomes may frequently change in interaction with peers. The underlying computational processes are often marked by rapid shifts in an animal's behaviour and rather sudden transitions in neuronal population activity. Such computational capacities are of growing importance for AI systems operating in the real world, like those guiding robots or autonomous vehicles, or for agentic AI interacting with humans online. Can AI learn from neuroscience? This Perspective explores this question, integrating the literature on continual and in-context learning in AI with the neuroscience of learning on behavioral tasks with shifting rules, reward probabilities, or outcomes. We will outline an agenda for how specifically insights from neuroscience may inform current developments in AI in this area, and - vice versa - what neuroscience may learn from AI, contributing to the evolving field of NeuroAI.

In dynamical systems reconstruction (DSR) we seek to infer from time series measurements a generative model of the underlying dynamical process. This is a prime objective in any scientific discipline, where we are particularly interested in parsimonious models with a low parameter load. A common strategy here is parameter pruning, removing all parameters with small weights. However, here we find this strategy does not work for DSR, where even low magnitude parameters can contribute considerably to the system dynamics. On the other hand, it is well known that many natural systems which generate complex dynamics, like the brain or ecological networks, have a sparse topology with comparatively few links. Inspired by this, we show that geometric pruning, where in contrast to magnitude-based pruning weights with a low contribution to an attractor's geometrical structure are removed, indeed manages to reduce parameter load substantially without significantly hampering DSR quality. We further find that the networks resulting from geometric pruning have a specific type of topology, and that this topology, and not the magnitude of weights, is what is most crucial to performance. We provide an algorithm that automatically generates such topologies which can be used as priors for generative modeling of dynamical systems by RNNs, and compare it to other well studied topologies like small-world or scale-free networks.

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.

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

I NFERRING DYNAMICAL SYSTEMS WITH LONG-RANGE DEPENDENCIES THROUGH LINE ATTRACTOR REGULARIZATIONDominik Schmidt 1*, Georgia Koppe 1*, Max Beutelspacher 1,2, Daniel Durstewitz 1,3 1 Department of Theoretical Neuroscience, Central Institute of Mental Health, Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany 3 Faculty of Physics and Astronomy, Heidelberg University * These authors contributed equally contact: {dominik.schmidt,georgia.koppe,daniel.durstewitz} A BSTRACT V anilla 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. 1 I NTRODUCTION Theories of complex systems in biology and physics are often formulated in terms of sets of stochastic differential or difference equations, i.e. as stochastic dynamical systems (DS). A longstanding desire is to retrieve these generating dynamical equations directly from observed time series data (Kantz & Schreiber, 2004). However, vanilla RNN as often used in this context are well known for their problems in capturing long-term dependencies and slow time scales in the data (Hochreiter & Schmidhuber, 1997; Bengio et al., 1994).

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.

The computational properties of neural systems are often thought to be implemented in terms of their network dynamics. Hence, recovering the system dynamics from experimentally observed neuronal time series, like multiple single-unit (MSU) recordings or neuroimaging data, is an important step toward understanding its computations. Ideally, one would not only seek a state space representation of the dynamics, but would wish to have access to its governing equations for in-depth analysis. Recurrent neural networks (RNNs) are a computationally powerful and dynamically universal formal framework which has been extensively studied from both the computational and the dynamical systems perspective. Here we develop a semi-analytical maximum-likelihood estimation scheme for piecewise-linear RNNs (PLRNNs) within the statistical framework of state space models, which accounts for noise in both the underlying latent dynamics and the observation process. The Expectation-Maximization algorithm is used to infer the latent state distribution, through a global Laplace approximation, and the PLRNN parameters iteratively. After validating the procedure on toy examples, the approach is applied to MSU recordings from the rodent anterior cingulate cortex obtained during performance of a classical working memory task, delayed alternation. A model with 5 states turned out to be sufficient to capture the essential computational dynamics underlying task performance, including stimulus-selective delay activity. The estimated models were rarely multi-stable, but rather were tuned to exhibit slow dynamics in the vicinity of a bifurcation point. In summary, the present work advances a semi-analytical (thus reasonably fast) maximum-likelihood estimation framework for PLRNNs that may enable to recover the relevant dynamics underlying observed neuronal time series, and directly link them to computational properties.