The interpretation of neuronal spike train recordings often relies on abstract statistical models that allow for principled parameter estimation and model selection but provide only limited insights into underlying microcircuits. In contrast, mechanistic models are useful to interpret microcircuit dynamics, but are rarely quantitatively matched to experimental data due to methodological challenges. Here we present analytical methods to efficiently fit spiking circuit models to single-trial spike trains. Using the maximal-likelihood approach, we statistically infer the mean and variance of hidden inputs, neuronal adaptation properties and connectivity for coupled integrate-and-fire neurons. Evaluations based on simulated data, and validations using ground truth recordings in vitro and in vivo demonstrated that parameter estimation is very accurate, even for highly sub-sampled networks. We finally apply our methods to recordings from cortical neurons of awake ferrets and reveal population-level equalization between hidden excitatory and inhibitory inputs. The methods introduced here enable a quantitative, mechanistic interpretation of recorded neuronal population activity.