Before revision:  Using an integrate-and-fire (I \& F) model , we reproduce relevant results of both experimental measurements and detailed models of early olfaction dynamics and odor discrimination. By further idealizing the I\&F description and finally reducing it to a firing-rate (FR) model , we find the bifurcation structure underlying a transition typical of the early odor response to be a slow passage through a saddle-node-on-an-invariant-circle (SNIC) bifurcation , which arises due to the interaction among the fast-varying excitation and fast and slowly-varying inhibition .
After revision:  Using an integrate-and-fire (I \& F) model to verify this hypothesis via numerical simulations. To better understand the mathematical structure underlying the common dynamical behavior across species, we derive a firing-rate (FR) model , we find the bifurcation structure underlying a transition typical of the early odor response to be a slow passage through a saddle-node-on-an-invariant-circle (SNIC) bifurcation , which arises due to the interaction among the fast-varying excitation and fast and slowly-varying inhibition .