This paper describes a neural network for approximation problems on the sphere. The von Mises basis function is introduced, whose activation depends on polar rather than Cartesian input coordinates. The architecture of the von Mises Basis Function (VMBF) neural network is presented along with the corresponding gradient-descent learning rules. The VMBF neural network is used to solve a particular spherical problem of approximating acoustic parameters used to model perceptual auditory space. This model ultimately serves as a signal processing engine to synthesize a virtual auditory environment under headphone listening conditions. Advantages of the VMBF over standard planar Radial Basis Functions (RBFs) are discussed.