The sampling of neural signals plays an important role in modern neuroscience, especially for prosthetics. However, due to hardware and data rate constraints, only spike trains can get recovered reliably. State of the art prosthetics can still achieve impressive results, but to get higher resolutions the used data rate needs to be reduced. In this paper, this is done by expressing the data with exponential and sinusoidal splines. As these signals have a finite number of degrees of freedom per unit of time, they can be analyzed and reconstructed with the Finite Rate of Innovation (FRI) framework. We show, that we can reduce the needed data rate by 90\% to achieve the same resolution as without compression. Additionally, we propose analytic boundaries for the reconstruction of these splines and present an algorithm that guarantees the reconstruction within these boundaries. Furthermore, we test the algorithm on real neural stimuli.