Symmetry and Network Topology in Neuronal Circuits: Complicity of Form and Function

Abstract

Symmetries in the external world constrain the evolution of neuronal circuits that allow organisms to sense the environment and act within it. Many small “modular” circuits can be viewed as approximate discretizations of the relevant symmetries, relating their forms to the functions they perform. The recent development of a formal theory of dynamics and bifurcations of networks of coupled differential equations permits the analysis of some aspects of network behavior without invoking specific model equations or numerical simulations. We review basic features of this theory, compare it to equivariant dynamics, and examine the subtle effects of symmetry when combined with network structure. We illustrate the relation between form and function through examples drawn from neurobiology, including locomotion, peristalsis, visual perception, balance, hearing, location detection, decision-making, and the connectome of the nematode Caenorhabditis elegans.