Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.

Neural Information Processing Systems http://nips.cc/

Recurrent neural networks (RNNs) are a popular tool for sequence modelling tasks.

The forces shaping joint dynamics of multiple tasks, however, are largely unexplored. In this work, we first construct a systematic framework to study multiple tasks in RNNs, minimizing interference from input and output correlations with the hidden representation.

Continuous attractor networks (CANs) are widely used to model how the brain temporarily retains continuous behavioural variables via persistent recurrent activity, such as an animal's position in an environment. However, this memory mechanism is very sensitive to even small imperfections, such as noise or heterogeneity, which are both common in biological systems. Previous work has shown that discretising the continuum into a finite set of discrete attractor states provides robustness to these imperfections, but necessarily reduces the resolution of the represented variable, creating a dilemma between stability and resolution. We show that this stability-resolution dilemma is most severe for CANs using unimodal bump-like codes, as in traditional models. To overcome this, we investigate sparse binary distributed codes based on random feature embeddings, in which neurons have spatially-periodic receptive fields. We demonstrate theoretically and with simulations that such grid-cell-like codes enable CANs to achieve both high stability and high resolution simultaneously. The model extends to embedding arbitrary nonlinear manifolds into a CAN, such as spheres or tori, and generalises linear path integration to integration along freely-programmable on-manifold vector fields. Together, this work provides a theory of how the brain could robustly represent continuous variables with high resolution and perform flexible computations over task-relevant manifolds.

Understanding how the collective activity of neural populations relates to computation and ultimately behavior is a key goal in neuroscience. To this end, statistical methods which describe high-dimensional neural time series in terms of low-dimensional latent dynamics have played a fundamental role in characterizing neural systems. Yet, what constitutes a successful method involves two opposing criteria: (1) methods should be expressive enough to capture complex nonlinear dynamics, and (2) they should maintain a notion of interpretability often only warranted by simpler linear models. In this paper, we develop an approach that balances these two objectives: the Gaussian Process Switching Linear Dynamical System (gpSLDS). Our method builds on previous work modeling the latent state evolution via a stochastic differential equation whose nonlinear dynamics are described by a Gaussian process (GP-SDEs). We propose a novel kernel function which enforces smoothly interpolated locally linear dynamics, and therefore expresses flexible -- yet interpretable -- dynamics akin to those of recurrent switching linear dynamical systems (rSLDS). Our approach resolves key limitations of the rSLDS such as artifactual oscillations in dynamics near discrete state boundaries, while also providing posterior uncertainty estimates of the dynamics. To fit our models, we leverage a modified learning objective which improves the estimation accuracy of kernel hyperparameters compared to previous GP-SDE fitting approaches. We apply our method to synthetic data and data recorded in two neuroscience experiments and demonstrate favorable performance in comparison to the rSLDS.