Graph embeddings for identifying symmetries in connectomes

In many circuit models of neural computation, synaptic connections between neurons are organized according to their tuning to the variables being processed. The connectivities of canonical neural network models of head direction, spatial navigation, and orientation selectivity obey this principle and contain symmetries related to the angular and spatial variables they operate on. We develop a graph embedding algorithm to identify such symmetries. The algorithm segregates structure related to cell type from that related to the symmetries we seek to identify, distinguishing it from standard embedding methods. Our method successfully identifies rotational and translational symmetries in heading direction and visual projection neuron circuits using a connectome of the adult Drosophila brain, and it also identifies a toroidal symmetry in a synthetic connectome of grid cells in the medial entorhinal cortex. Such embedding geometries reveal the latent variables that are processed by a neural circuit and which cell types are responsible for this computation.