Symmetries and continuous attractors in disordered neural circuits
Abstract
A major challenge in neuroscience is reconciling idealized theoretical models with complex, heterogeneous experimental data. We address this challenge through the lens of continuous-attractor networks, which model how neural circuits may represent continuous variables, such as head direction or spatial location, through collective dynamics. In classical continuous-attractor models, a continuous symmetry of the connectivity generates a manifold of stable states, resulting in tuning curves that are identical up to shifts. However, mouse head-direction cells show substantial heterogeneity in their responses that appears incompatible with this classical picture. To understand the mechanistic origin of these data, we use an optimization principle to construct recurrent-network models that match the observed responses while exhibiting quasi-continuous-attractor dynamics. To study how such systems scale with increasing numbers of neurons N, we develop a statistical generative process that produces artificial tuning curves that match many features of the experimental data quantitatively. Due to a continuous symmetry in the generative process, the connectivity matrix exhibits doublet degeneracy in its spectrum at large N, reflecting an underlying circular geometry. Unlike classical models, where the ring structure is embedded in the neuronal space through a structured Fourier embedding, our model uses a random, disordered embedding. Analysis of the network dynamics in the large-N limit through dynamical mean-field theory reveals that the system becomes equivalent to a classical ring-attractor model. We extend this approach to higher-dimensional symmetries, applying it to grid cells in the medial entorhinal cortex and showing that the mean-field description recovers classical continuous-attractor models. Our work implies that large mammalian neural circuits could represent continuous variables using continuous-attractor dynamics, complete with continuous symmetry, in a way that is fully compatible with their heterogeneity and disorder.
