This paper investigates a discontinuous neural network which is used as a model of the mammalian olfactory system and can more generally be applied to solve non-negative sparse approximation problems. By inherently limiting the systems integrators to having non-negative outputs, the system function becomes discontinuous since the integrators switch between being inactive and being active. It is shown that the presented network converges to equilibrium points which are solutions to general non-negative least squares optimization problems. We specify a Caratheodory solution and prove that the network is stable, provided that the system matrix has full column-rank. Under a mild condition on the equilibrium point, we show that the network converges to its equilibrium within a finite number of switches. Two applications of the neural network are shown. Firstly, we apply the network as a model of the olfactory system and show that in principle it may be capable of performing complex sparse signal recovery tasks. Secondly, we generalize the application to include non-negative sparse approximation problems and compare the recovery performance to a classical non-negative basis pursuit denoising algorithm. We conclude that the recovery performance differs only marginally from the classical algorithm, while the neural network has the advantage that no performance critical regularization parameter has to be chosen prior to recovery.