Olfactory Receptor Neurons (ORNs) in insects are tightly packed into sensory hairs (sensilla), where they are electrically coupled. This organization is highly stereotyped and genetically determined. Electrical (`ephaptic') coupling between neurons within the same hair results in strong lateral inhibition, posing questions about the functional significance of the stereotyped organization. We developed a phenomenological model of coupled ORNs responding to odor mixture stimuli. Our model includes two forms of nonlinearity, essential to capture the nature of the coupling. First, ephaptic interaction depends on the activation level of both neurons, unlike 'regular' synaptic coupling. Second, the interaction is strong only if the activity of either ORN exceeds a threshold. We derived a complete analytical solution for the transient and steady-state dynamics of the model, and used it to fit the model parameters to electrophysiological measurements from one sensillum-type. Our fit results are consistent with independent morphometric measurements of the same sensillum. A detailed analysis of model dynamics suggests that electrical coupling can extract the valence of odor mixtures via transient signal amplification. Extracting a valence signal may allow for efficient initiation of some behaviors, bypassing the need for odor identification. Beyond valence computations, our theory suggests that the conserved asymmetries of coupling strength sensitize the ORN array to specific odor mixtures which are hard-coded in the morphometry of sensilla. Combining our theory with recently published fruitfly connectomes will make experimentally testable predictions for both electrophysiological and behavioral experiments in flies responding to natural odor mixtures in realistic settings.