The tradeoff between fine-tuning and accurate angular integration in small networks

Flexible navigation requires animals to form accurate and persistent internal representations of continuous spatial variables. Such representations—e.g., those carried by head direction (HD), grid, and place cells—can guide behavior even in the absence of localizing sensory cues. Theories of mammalian HD cells propose that such representations can be realized in a special class of networks that maintain a localized bump of activity via structured recurrent connectivity, and that shift this bump of activity via angular velocity input. These so-called ring attractor networks have historically relied on large numbers of neurons to generate continuous internal representations that persist without input and accurately respond to changes in input. Surprisingly, in the fly, an HD representation is maintained by a network whose dynamics and connectivity resemble those of a ring attractor network, but with far fewer neurons than required theoretically. In 2020, Noorman et al. showed that such small networks can generate continuous ring attractor solutions if coupling strengths are appropriately tuned. Using the existence of such solutions as a starting point, we sought to characterize performance as a function of network size and tuning. We first analytically derive the optimal coupling strengths that mathematically flatten the energy of the dynamical system and enable optimal performance. For threshold linear networks, the resulting ring attractor manifold emerges as a discrete set of line attractor manifolds that are ``stitched together'' to form a ring. We then analytically characterize performance away from these optimal tunings in dynamical regimes that are governed by discrete sets of stable and unstable fixed points. Finally, we quantify tradeoffs between network size and tuning precision needed to achieve the same performance, which reveals that more strongly coupled networks require less fine-tuning.