Extreme-value statistics of networks with inhibitory and excitatory couplings

TL;DR

This paper studies the extreme-value statistics of eigenvalues in networks with inhibitory and excitatory couplings, revealing how stability depends on connection probabilities and network size.

Contribution

It introduces a detailed analysis of eigenvalue distributions in such networks, highlighting the transition to generalized extreme value statistics and stability implications.

Findings

Largest eigenvalue real part decreases linearly with inhibitory probability

Eigenvalue fluctuations are robust against system size increases

More connections lead to higher instability risk

Abstract

Inspired by the importance of inhibitory and excitatory couplings in the brain, we analyze the largest eigenvalue statistics of random networks incorporating such features. We find that the largest real part of eigenvalues of a network, which accounts for the stability of an underlying system, decreases linearly as a function of inhibitory connection probability up to a particular threshold value, after which it exhibits rich behaviors with the distribution manifesting generalized extreme value statistics. Fluctuations in the largest eigenvalue remain somewhat robust against an increase in system size but reflect a strong dependence on the number of connections, indicating that systems having more interactions among its constituents are likely to be more unstable.