Non-Hermitian Quasi-Localization and Ring Attractor Neural Networks

Abstract

Eigenmodes of a broad class of "sparse" random matrices, with interactions concentrated near the diagonal, exponentially localize in space, as initially discovered in 1957 by Anderson for quantum systems. Anderson localization plays ubiquitous roles in varieties of problems from electrons in solids to mechanical and optical systems. However, its implications in neuroscience (where the connections can be strongly asymmetric) have been largely unexplored, mainly because synaptic connectivity matrices of neural systems are often "dense", which makes the eigenmodes spatially extended. Here, we explore roles that Anderson localization could be playing in neural networks by focusing on "spatially structured" disorder in synaptic connectivity matrices. Recently, neuroscientists have experimentally confirmed that the local excitation and global inhibition (LEGI) ring attractor model can functionally represent head direction cells in Drosophila melanogaster central brain. We first study a non-Hermitian (i.e. asymmetric) tight-binding model with disorder and then establish a connection to the LEGI ring attractor model. We discover that (i) Principal eigenvectors of the LEGI ring attractor networks with structured nearest neighbor disorder are "quasi-localized", even with fully dense inhibitory connections. (ii) The quasi-localized eigenvectors play dominant roles in the early time neural dynamics, and the location of the principal quasi-localized eigenvectors predict an initial location of the "bump of activity" representing, say, a head direction of an insect. Our investigations open up a new venue for explorations at the intersection between the theory of Anderson localization and neural networks with spatially structured disorder.