Complex movements, such as riding a bike, require rich, coordinated, self-sustaining patterns of neural activity. Experimental studies show that such activity patterns are often controlled by brief, intermittent inputs. In contrast, computational models usually rely on constant external inputs or generate only transient --ballistic-- responses in networks tuned for that purpose. Here, we propose a novel approach that selects for locally amplifying network motifs to generate a diverse repertoire of dynamic behaviors without continuous external stimulation. Brief inputs to these models can induce transitions between various modes of neural activity, including baseline firing, transient oscillations, self-sustained oscillations, and steady-state firing, directly corroborating experimental observations. We study these dynamics in recurrent rate models with N neurons and (nonlinear) sigmoid transfer function. Connectivity matrices W were constructed by first identifying Dalean, stable (DS), amplifying Jacobians, i.e., matrices of the local linear approximation of neural interactions. Within the DS space, the proportion of Jacobians capable of non-normal amplification increased with both connectivity strength and network size, suggesting that amplification is ubiquitous, and does not require precise tuning. We then constructed each W associated with an amplifying Jacobian. Perturbations in the non-amplifying directions of the Jacobian decayed back to baseline. Surprisingly, perturbations of the same magnitude in the amplifying directions generated transitions to new stable states or periodic orbits beyond transient amplification. Crucially, these states are fully controllable, a feature with substantial implications. Activity patterns can be stopped, paused, and resumed ad-libitum, allowing networks to transition seamlessly between them. Our findings align with experimental results, showing that complex cortical dynamics can be sustained without continuous input. In summary, minding --and selecting for-- local amplification in the construction process of neural networks allows us to uncover a basic principle --local amplification-- that governs the complex interactions between individual neurons to generate orchestrated population dynamics.