Symmetry group factorization reveals the structure-function relation in the neural connectome of Caenorhabditis elegans

Flaviano Morone,Hernán A Makse

doi:10.1038/s41467-019-12675-8

Abstract

The neural connectome of the nematode Caenorhabditis elegans has been completely mapped, yet in spite of being one of the smallest connectomes (302 neurons), the design principles that explain how the connectome structure determines its function remain unknown. Here, we find symmetries in the locomotion neural circuit of C. elegans, each characterized by its own symmetry group which can be factorized into the direct product of normal subgroups. The action of these normal subgroups partitions the connectome into sectors of neurons that match broad functional categories. Furthermore, symmetry principles predict the existence of novel finer structures inside these normal subgroups forming feedforward and recurrent networks made of blocks of imprimitivity. These blocks constitute structures made of circulant matrices nested in a hierarchy of block-circulant matrices, whose functionality is understood in terms of neural processing filters responsible for fast processing of information.

Highlights

The neural connectome of the nematode Caenorhabditis elegans has been completely mapped, yet in spite of being one of the smallest connectomes (302 neurons), the design principles that explain how the connectome structure determines its function remain unknown
Our fundamental result is that symmetries of neural networks have a direct biological meaning, which can be rigorously justified using the mathematical formalism of symmetry groups
This formalism makes possible to understand the significance of the structure-function relationship: the origin of the locomotion function in C. elegans is connected with the existence of symmetries which uniquely assign neurons to functional categories defined by the mechanism of factorization of the symmetry group

Summary

Introduction

The neural connectome of the nematode Caenorhabditis elegans has been completely mapped, yet in spite of being one of the smallest connectomes (302 neurons), the design principles that explain how the connectome structure determines its function remain unknown. We show that the symmetry group of locomotion circuits can be broken down into a unique factorization as the direct product of smaller ‘normal subgroups’[12] These normal subgroups directly determine the separation of neurons into sectors. Our fundamental result is that symmetries of neural networks have a direct biological meaning, which can be rigorously justified using the mathematical formalism of symmetry groups This formalism makes possible to understand the significance of the structure-function relationship: the origin of the locomotion function in C. elegans is connected with the existence of symmetries which uniquely assign neurons to functional categories defined by the mechanism of factorization of the symmetry group. Network symmetries may help to reveal the general principles underlying the mechanism of neural coding engraved in the connectome

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Conclusion