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Non-Hermitian localization in biological networks

Ariel Amir, Naomichi Hatano, and David R. Nelson
Phys. Rev. E 93, 042310 – Published 15 April 2016

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 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 directional bias, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.

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  • Received 31 December 2015

DOI:https://doi.org/10.1103/PhysRevE.93.042310

©2016 American Physical Society

Physics Subject Headings (PhySH)

Networks

Authors & Affiliations

Ariel Amir1, Naomichi Hatano2, and David R. Nelson1,3

  • 1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
  • 2Institute of Industrial Science, University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan
  • 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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Issue

Vol. 93, Iss. 4 — April 2016

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