Path integration in insects as an optimized circuit

Insects demonstrate a remarkable ability to navigate the world -- foraging for food and returning to their nests over large distances. Simple neural circuits have been shown to underlie insect navigation behaviours, encoding the animal's head direction and velocity vector as a sinusoidal activity pattern\cite{heinze2007maplike,lyuBuildingAllocentricTravelling2022}. We offer novel mathematical insights into the benefits of this sinusoidal encoding, suggesting that it might offer an evolutionary advantage in terms of noise resilience.
Previous work has recognised that encoding vectors as sinusoidal population activity patterns enables vector addition by simple piecewise activity addition~\cite{lyuBuildingAllocentricTravelling2022,pisokasHeadDirectionCircuit2020}. However, we show that other activity patterns also fulfil this property (Fig.~\ref{fig:noise_resilience}), calling into question whether this is the only advantage of a sinusoidal activity pattern.
In this work we use signal processing formalism to demonstrate that the sinusoid is the most noise-resilient activity shape that allows for stable path integration, given that it is coupled with a sinusoidal connectivity.
We additionally show that the required sinusoidal connectivity will arise automatically in a network of neurons which have their weights updated with a Hebbian learning rule. This analytical result demonstrates that the weight matrix and the activity could naturally converge to sinusoidal shapes during development.
By considering Oja's rule, a variant of Hebbian learning, we find that a sinusoidal connectivity pattern is stable -- indicating that a perturbation of the neural connectivity will be corrected by this learning rule.
Our work shows how simple neural circuits can follow clear mathematical principles, and how tools from signal processing can offer novel interpretations of biological findings.
Beyond neuroscience, robust path integration remains a key area of research in robotics. This work could therefore inspire novel biologically-inspired path integration approaches by roboticists or neuromorphic chip designers.