Nonnormal amplification in random balanced neuronal networks

TL;DR

This paper investigates how nonnormal connectivity in large random neuronal networks enables transient amplification of activity without significant slowing, highlighting the importance of structured connectivity for strong, fast responses.

Contribution

It provides an exact analytical expression for nonnormal amplification and demonstrates how specific structured connections facilitate rapid, strong transient responses in neuronal networks.

Findings

Nonnormal amplification is limited when dynamical slowing is weak.

Structured connectivity enables strong transient amplification.

Differences between balanced and inhibition-dominated networks are elucidated.

Abstract

In dynamical models of cortical networks, the recurrent connectivity can amplify the input given to the network in two distinct ways. One is induced by the presence of near-critical eigenvalues in the connectivity matrix W, producing large but slow activity fluctuations along the corresponding eigenvectors (dynamical slowing). The other relies on W being nonnormal, which allows the network activity to make large but fast excursions along specific directions. Here we investigate the tradeoff between nonnormal amplification and dynamical slowing in the spontaneous activity of large random neuronal networks composed of excitatory and inhibitory neurons. We use a Schur decomposition of W to separate the two amplification mechanisms. Assuming linear stochastic dynamics, we derive an exact expression for the expected amount of purely nonnormal amplification. We find that amplification is very limited if dynamical slowing must be kept weak. We conclude that, to achieve strong transient amplification with little slowing, the connectivity must be structured. We show that unidirectional connections between neurons of the same type together with reciprocal connections between neurons of different types, allow for amplification already in the fast dynamical regime. Finally, our results also shed light on the differences between balanced networks in which inhibition exactly cancels excitation, and those where inhibition dominates.