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A silicon photomultiplier (SiPM) contains an array of microcells that can each detect individual photons. Consequently, it can arguably result in the most sensitive receiver in visible light communication (VLC). However, each microcell needs a period of several nanoseconds to recover after detecting a photon. This creates a non-linear response and introduces a unique form of inter-symbol interference. In this paper, we first show that this interference splits each element of the received signal constellation into multiple clusters. This observation motivates the investigation into the use of a Radial Basis Function Neural Network (RBFNN) to deal with the impact of the nonlinearity. Both the training procedures and the performance of the RBFNN are explained and discussed in detail. The influence of the number of the RBFNN centers, the widths of the centers, the constellation size and the period of the transmitted signal samples on the system performance are investigated. In addition, two different RBFNN-based data demodulation methods are introduced. The simulation results suggest that the new RBFNN-aided receivers reduce the negative impacts of the SiPM nonlinearity and can result in lower bit error rates (BERs) for a wide range of irradiances on the SiPM.