My recent work on symmetries and continuous attractors in disordered neural circuits, with Larry Abbott and Haim Sompolinsky, has been published in Nature Neuroscience: rdcu.be/mRN8zv2Hv1ba Additionally, I am honored that Francesca Mastrogiuseppe and Srdjan Ostojic wrote a beautiful accompanying News & Views: nature.com/articles/s41593-0… Our brains represent continuous quantities, many of which reside on manifolds with nontrivial topology. A canonical example from navigation is head direction, a circular variable. Grid cells, discovered more recently, represent position via a toroidal manifold. An individual neuron forgets its input within tens of milliseconds, yet these representations persist over much longer timescales (many seconds or minutes). Thus, this persistence must arise from the collective dynamics of the network. A classical solution to this problem is the continuous attractor, now a cornerstone network model in theoretical neuroscience. In the ring-attractor model of Ben-Yishai, Bar-Or, and Sompolinsky (PNAS, 1995), neurons are arranged on a ring, with Mexican-hat connections (local excitation, longer-range inhibition) that depend only on the angular distance between neurons. These connections thus have a continuous, rotational symmetry. Through spontaneous breaking of this symmetry, the network forms a bump of activity that can sit at any angle, so the stable states form a continuous ring. This model makes a strong prediction, namely, that every neuron's tuning curve has the same shape, up to a shift. This prediction holds well in the fly but is violated in mammals. For example, in over a thousand mouse head-direction cells recorded by Duszkiewicz et al. (Nature Neuroscience, 2024), the tuning curves are heterogeneous, often with multiple, irregularly spaced peaks. Reconciling idealized models with experimental data is a central challenge in neuroscience. The discrepancy described above provides a sharp example and raises the question: Is the classical mechanism relevant to mammalian circuits at all? The point of our paper is to show that heterogeneous tuning is compatible with this mechanism; in fact, in the limit N → ∞, networks that generate such tuning have effective dynamics of exactly the classical form. The paper proceeds in four steps, briefly summarized here. First, we constructed a recurrent network directly from the recorded tuning curves, choosing the minimum-norm weights for which these tuning curves are fixed points. The weights look unstructured, but the network behaves as an approximate continuous attractor. After a perturbation, activity returns to the ring, and the resulting state persists for tens of seconds. Second, to study large networks, we fit a generative process for tuning curves in which input currents are drawn from a circularly symmetric Gaussian process and passed through a nonlinearity, similar to an approach that Mainali, da Silveira, and Burak (Neuron, 2025) applied to place cells. The resulting tuning curves are individually heterogeneous but drawn from a rotationally symmetric distribution. This process has only three parameters but reproduces several statistical features of the data. Third, we analyzed the weights in the limit N → ∞. The eigenvalues come in degenerate pairs, as in a classical ring attractor (whose weight matrix is circulant), but the eigenvectors are disordered rather than Fourier modes. Thus, in connectome data, a ring attractor need not look like a circulant matrix, and plotting neurons by their loadings on the leading eigenvectors need not reveal a ring. Eigenvalue degeneracies may provide a more robust signature. Finally, we derived a dynamical mean-field theory for the limit N → ∞. Strikingly, the resulting equation has the same form as that of the classical ring model above, with Mexican-hat interactions and rotational symmetry. Thus, these disordered networks operate by the classical mechanism, namely, spontaneous symmetry breaking leading to a bump. A few broader thoughts... Classical models, which produce identical tuning curves, are a measure-zero special case of a much larger class of networks with the same dynamics. Heterogeneity may therefore be common in mammalian circuits simply because it is generic. However, the disordered solutions require many neurons, which may explain why flies, with relatively few neurons, use a ring attractor much closer to the classical model. Inferring circuit structure from the structure of the world has repeatedly succeeded near the sensory periphery but is harder deeper in the brain. Our results suggest a route for continuous variables. Over lifelong timescales, the world is often symmetric (e.g., no head direction is special). If the distribution of tuning curves inherits this symmetry, then, as we argue, a circuit that generates these tuning curves as an attractor should reflect this symmetry in its connectivity, in the form of eigenvalue degeneracies. Code to reproduce all analyses is available here: github.com/davidclark1/symme…