Complex systems can be defined by “sloppy” dimensions, meaning that their behavior is unmodified by large changes to specific parameter combinations, and “stiff” dimensions, whose change results in considerable behavioral modification. In the neocortex, sloppiness in synaptic architectures would be crucial to allow for the maintenance of asynchronous irregular spiking dynamics with low firing rates despite a diversity of inputs, states, and short- and long-term plasticity. Using simulations on neural networks with first-order spiking statistics matched to firing in murine visual cortex while varying connectivity parameters, we determined the stiff and sloppy parameters of synaptic architectures across three classes of input (brief, continuous, and cyclical). Algorithmically generated connectivity parameter values drawn from a large portion of the parameter space reveal that specific combinations of excitatory and inhibitory connectivity are stiff and that all other architectural details are sloppy. Stiff dimensions are consistent across input classes with self-sustaining synaptic architectures following brief input occupying a smaller subspace as compared to the other input classes. Experimentally estimated connectivity probabilities from mouse visual cortex are consistent with the connectivity correlations found and fall in the same region of the parameter space as architectures identified algorithmically. This suggests that simple statistical descriptions of spiking dynamics are a sufficient and parsimonious description of neocortical activity when examining structure-function relationships at the mesoscopic scale. Additionally, coarse graining cell types does not prevent the generation of accurate, informative, and interpretable models underlying simple spiking activity. This unbiased investigation provides further evidence of the importance of the interrelationship of excitatory and inhibitory connectivity to establish and maintain stable spiking dynamical regimes in the neocortex.