Across species, primary olfactory centers show similarities both in their cellular organization and their types of olfactory information coding. In this article, we consider an excitatory-inhibitory spiking neural network as a model of early olfactory systems (antennal lobe for insects, olfactory bulb for vertebrates). In line with experimental results, we show that, in our network, odor-like stimuli evoke synchronization of excitatory cells, phase-locked to the oscillations of the local field potential. As revealed by a mathematical analysis, the phase-locking probability of excitatory cells is given by an inverted-U function and the firing probability of inhibitory cells is well described by a sigmoid function. These neural response functions are used to reduce the spiking model to a more abstract model with discrete-time dynamics (oscillatory cycles) and binary-state neurons (phase-locked or not). An iterative map, built for explaining the dynamics of the binary model, reveals that it converges to fixed point attractors similar to those obtained with the spiking model. This result is consistent with odor-specific attractors found in recent experimental studies. It also provides insights for designing bio-inspired olfactory associative memories applicable for data analysis in electronic noses.