The space of finite ring attractors: from theoretical principles to the fly compass system

A cognitive compass is a critical component of spatial navigation and path integration. Experiments in the fruit fly and recent theoretical work
show that an accurate compass can be built from a handful of neurons. Many previous models have used ring attractors to model head direction systems.
However, these models relied on hand-designed neural network connectivity patterns, and we lack a principled methodology for quantitatively relating connectomics data to theoretical ring attractor models. Here we introduce a novel theoretical framework for analyzing the space of threshold-linear ring attractor networks. Under the assumption that neural activity in the compass system is self-sustained, we derive predictions that link asymmetric bumps of neural activity to symmetric patterns of neural network connectivity. We also provide a recipe for using neural activity measurements and connectomics data to test these predictions in the fly's compass system. Our framework thus allows us to incorporate connectomics data to build data-driven ring attractor models that are as close to the biology as possible.