To support perception, visual cortex transforms sensory-related input to create hierarchical representations. Local recurrent connections between nearby neurons can potentially exert large effects on these transformations, but it has been unclear how recurrent connections influence input-output transformations in the cortex. Here we study recurrent influences in mouse V1 by experimentally stimulating excitatory neurons. To do this, we selectively express an excitatory opsin (stChrimsonR) in excitatory cells, and record activity using electrophysiology, 2-photon, and widefield calcium imaging. We then use simulations to determine which features of recurrent connectivity can explain the observations. We find that strong visual stimuli suppress the activity of many neurons, resulting in a salt-and-pepper pattern of neurons with suppressed and elevated firing. Stimulating excitatory cells optogenetically produces a similar salt-and-pepper pattern of suppression. Cells with suppressed firing are distributed across the cortex, though there is a surround region a few hundred microns distant from the stimulation center, where suppressed neurons predominate over excited neurons. A balanced-state cortical model replicates observed firing rate distributions and dynamics – but only when variability in synaptic strengths is large, with sparse strong synapses and many weaker synapses. Thus, sparse, broadly-distributed synaptic connectivity is key to explaining how recurrent connectivity shapes cortical input-output functions.