Non-Hermitian Localization in Biological Networks

TL;DR

This paper investigates the spectral and localization properties of non-Hermitian random matrices in neural networks, revealing complex eigenvalue spectra, localization phenomena, and effects of directional bias, with implications for biological and artificial systems.

Contribution

It introduces a detailed analysis of non-Hermitian spectra in neural networks, including symmetries, localization, and the impact of bias, extending understanding of complex eigenvalue behavior in biological and artificial networks.

Findings

Eigenfunctions become localized with balanced excitatory and inhibitory connections.

Eigenvalue spectrum exhibits symmetries and condensation on axes.

Directional bias creates a hole in the eigenvalue density, affecting localization.

Abstract

We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions, and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N, the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes, but also with respect to 90 degree rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias dependent shape of this hole tracks the bias independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's Law" (each site is purely excitatory or inhibitory), and conclude with perturbation theory results that describe the limit of large bias g, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.