The ON and OFF ORNs on cockroach antennae optimize the detection and transfer of information about concentration increments and decrements by providing excitatory responses for both. It follows that the antagonism of the responses facilitates instantaneous evaluations of the odor plume to help the insect make tracking decisions by signaling “higher concentration than background” and “lower concentration than background”. Here we analyzed the effect of the background concentration level of the odor of lemon oil on the responses of the ON and OFF ORNs to jumps and drops of that odor, respectively. Raising the background level decreases both the ON-ORN's response to concentration jumps and the OFF-ORN's response to concentration drops. Impulse frequency of the ON ORN is high when the concentration jump is large, but for a given jump, frequency tends to be higher when the background level is low. Conversely, impulse frequency of the OFF cell is high at large concentration drops, but higher still when the background level is low. Analyses of this double dependence revealed that the activity of both types of ORNs is raised more by increasing the change in concentration than by decreasing the background concentration by the same amount. This effect is greater in the OFF ORN than in the ON ORN, indicating a bias for falling concentrations. Given equal change in concentration, concentration drops evoke stronger responses in the OFF ORN than concentrations jumps in the ON ORN. This suggests that the OFF responses are used as alert information for accurately tracking.