Non-normal amplification in random balanced neuronal networks

In dynamical models of cortical networks, the recurrent con-nectivity 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 eigen-vectors (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 inves-tigate the tradeoff between nonnormal amplification and dy-namical 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 stochas-tic 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 be-tween neurons of the same type together with reciprocal connections between neurons of different types, allow for am-plification 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.