Information maximization has been one of the guiding principles for understanding sensory neural processing. Given the framework, our goal is to explain nonlinear processing by a group of neurons in the retina, encoding the same filter inputs. We begin with a single-cell model, then extend to a neural population subject to relevant constraints, including metabolic costs and neural noise. Still, their predictions only explain part of the observation. Ultimately, we introduce an extra factor, the noise due to modulation, for better elucidating the retina data.Modulation of neuronal thresholds is ubiquitous in the brain. Phenomena such as figure-ground segmentation, motion detection, stimulus anticipation, and shifts in attention all involve changes in a neuron's threshold based on signals from larger scales than its primary inputs. However, this modulation reduces the accuracy with which neurons can represent their primary inputs, creating a mystery as to why threshold modulation is widespread in the brain. We find that modulation is less detrimental than other forms of neuronal variability. Its adverse effects can be nearly eliminated if modulation is applied selectively to sparsely responding neurons in a circuit by inhibitory neurons. We verify these predictions in the retina, where we find that inhibitory amacrine cells selectively deliver modulation signals to sparsely responding ganglion cell types. Our findings elucidate the central role that inhibitory neurons play in maximizing information transmission under modulation.