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

Abstract

There are numerous different odorant molecules in nature. Typical odors are sparse mixtures of a few types of odorant molecules each with a wide range of concentrations. However, there are only a relatively small number of olfactory receptor neurons (ORNs), which respond to odorant concentration nonlinearly (sigmoidal) with a finite sensitivity range. Thus, how to encode a large number of sparse odor mixtures with a relatively small number of nonlinear ORNs - the nonlinear compressed sensing problem - remains a puzzle. Here, by using an information theory approach, we study the optimal coding strategies that enable nonlinear ORNs to best represent olfactory information (both the odorants identities and their concentrations) in sparse odor mixtures. Our extensive numerical simulations and analytical analysis show that the optimal odor-receptor sensitivity matrix is sparse and the nonzero sensitivities follow roughly a log-normal distribution(matching the statistics of the odorants), both of which are consistent with existing experiments. For ORNs with a finite basal activity, our study shows that co-existence of both odor-evoked excitation and inhibition increases coding capacity, which provides a plausible explanation for such co-existence observed in the fly olfactory system. Furthermore, we show that coding the inputs with the optimal sensitivity matrix can enhance accuracy of the downstream decoding and learning tasks. These general statistical properties of the optimal sensitivity matrix for nonlinear compressed sensing may shed light on understanding the peripheral olfactory sensory system and improving performance of artificial neural networks.