Symmetries and continuous attractors in disordered neural circuits.

A core challenge in neuroscience is reconciling idealized models with experimental data. Classical continuous-attractor models describe neural representations of continuous variables such as head direction. Symmetry in the weights generates a manifold of stable states, with tuning curves identical up to shifts. Rodent head-direction cells, by contrast, are markedly heterogeneous. Such heterogeneity, we show, is nevertheless compatible with these models. Here we build, directly from head-direction cell data, recurrent networks that behave as quasicontinuous attractors. A generative process then reproduces the heterogeneous tuning of these cells while making connectivity and dynamics tractable at large N. Although the weights appear disordered, they carry a circular symmetry whose eigenvalue degeneracies could signify continuous-attractor organization in connectome data. Dynamical mean-field theory recovers the ingredients of classical ring attractors, namely, Mexican hat interactions and a spontaneously broken symmetry producing bump states. Continuous-attractor mechanisms may thus operate in complex mammalian circuits.