Sensory processing relies on fast detection of changes in environment, as well as integration of contextual cues over time. The mechanisms by which local circuits of the cerebral cortex simultaneously perform these opposite processes remain obscure. Thalamic "specific" nuclei relay sensory information, whereas "nonspecific" nuclei convey information on the environmental and behavioral contexts. We expressed channelrhodopsin in the ventrobasal specific (sensory) or the rhomboid nonspecific (contextual) thalamic nuclei. By selectively activating each thalamic pathway, we found that nonspecific inputs powerfully activate adapting (slow-responding) interneurons but weakly connect fast-spiking interneurons, whereas specific inputs exhibit opposite interneuron preference. Specific inputs thereby induce rapid feedforward inhibition that limits response duration, whereas, in the same cortical area, nonspecific inputs elicit delayed feedforward inhibition that enables lasting recurrent excitation. Using a mean field model, we confirm that cortical response dynamics depends on the type of interneuron targeted by thalamocortical inputs and show that efficient recruitment of adapting interneurons prolongs the cortical response and allows the summation of sensory and contextual inputs. Hence, target choice between slow- and fast-responding inhibitory neurons endows cortical networks with a simple computational solution to perform both sensory detection and integration.