We present a systematic multiscale reduction of a biologically plausible model of the inhibitory neuronal network of the pheromone system of the moth. Starting from a Hodgkin-Huxley conductance based model we adiabatically eliminate fast variables and quantitatively reduce the model to mean field equations. We then prove analytically that the network's ability to operate on signal amplitudes across several orders of magnitude is optimal when a disinhibitory mode is close to losing stability and the network dynamics are close to bifurcation. This has the potential to extend the idea that optimal dynamic range in the brain arises as a critical phenomenon of phase transitions in excitable media to brain regions that are dominated by inhibition or have slow dynamics.