The mammalian olfactory system shows an exceptional ability for rapid and accurate decoding of both the identity and concentration of odorants. In a given scene, only a handful out of millions of possible odorants are present. Yet, the mammals can effectively distinguish these odorants using the dimensionally-reduced representation provided by olfactory receptor neurons. To account for this capability, normative compressed sensing models have been proposed by previous studies. However, these models do not map cleanly onto the unique anatomy and physiology of the olfactory bulb, nor have they demonstrated that accurate inference can be performed on highly fluctuating concentrations within the 100-millisecond timescale of a single sniff. Here, we propose a model for olfactory compressed sensing inspired by simultaneous localization and mapping (SLAM) algorithms in vision: the set of odors that are present, and the concentration of those present odors, are inferred separately. Versions of this idea have been proposed in previous olfactory decoding models, but none of them are implementable as a recurrent neural network (RNN). Our model leverages the framework of Mirrored Langevin Dynamics to perform olfactory SLAM in a recurrent circuit. At a scale comparable to real-life scenarios, our model accurately infers both the presence and concentration of presented odorants within a 1-second timeframe. Notably, while having this substantial performance, our model remains biologically plausible. It provides an account for complementary computational roles of mitral and tufted (M/T) cells in the bulb: they are responsible for inference of presence and concentration, respectively. With this circuit mapping, we propose predictions for response dynamics that may be probed in future experiments.