Two-photon holographic optogenetic experiments have revealed that excitatory perturbations induce excitation surrounding the perturbed cell on the order of tens of microns, after which the response is suppressive, forming an inhibitory halo. Here we investigated, through a battery of analytical and inference methods, how the strength and spread of the connections between neurons both in cortical space and in feature (preferred orientation) space define the response of the circuit to few-cells perturbations. First, we obtain two analytical expressions, one exact and the other approximate, for the response of linearized models to perturbations as a function of cortical space. We infer parameters from calcium imaging responses to holographic perturbations, and obtain both linear and nonlinear spatial models that can account for the spatial spread of the responses in the data. We find that linear models cannot explain results from experiments which show that perturbing an ensemble of neurons that are closely clustered causes suppression instead of excitation in nearby cells. Nonlinear models exhibit this compact-ensemble suppression effect because strong perturbations can induce transitions from facilitated to suppressed response in nearby cells, unlike in linear models, where the response to perturbations is always proportional to input strength. Having a nonlinear spatial model that fits different aspects of the spatial response, we expand the model to include feature dependent connectivity. We identified different regimes based on features of the response such as whether the response is positive or negative for nearby neurons and whether there are transitions from facilitating to suppressive response across space and orientation. Finally, we identify a parameter regime in which the response near perturbed cells is positive for co-tuned neurons and negative for orthogonally-tuned neurons, a feature that is observed in experimental data, as we also show.