  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.
