Statistical properties of the optimal sensitivity matrix for compressed sensing with nonlinear sensors

Natural odors are typically sparse mixtures of a few types of odorants each with a wide range of concentrations. How to encode a large number of sparse odor mixtures with a relatively small number of nonlinear olfactory receptor neurons (ORNs) – the nonlinear compressed sensing problem – remains a puzzle. Here, by using an information theory approach, we study the optimal coding strategies that enable ORNs to best represent olfactory information. Our results show that the optimal odor-receptor sensitivity matrix is sparse and the nonzero sensitivities follow roughly a log-normal distribution, both of which are consistent with existing experiments. We also show that odor-evoked inhibition increases coding capacity, providing a plausible explanation for experimental observation in the fly olfactory system. Furthermore, we show that the optimal sensitivity matrix can enhance accuracy of the downstream decoding tasks. Our results may shed light on understanding the peripheral olfactory sensory system and improving performance of artificial neural networks.