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The identification of unlabelled neuronal electric signals is one of the most challenging open problems in neuroscience, widely known as Spike Sorting. Motivated to solve this problem, we propose a model-based approach within the mixture modeling framework for clustering oscillatory functional data called MixFMM. The core of the approach is the FMM waves, which are non-linear parametric time functions, flexible enough to describe different oscillatory patterns and simple enough to be estimated efficiently. In particular, specific model parameters describe the waveforms' phase, amplitude, and shape. A mixture model is defined using FMM waves as basic functions and gaussian errors, and an EM algorithm is proposed for estimating the parameters. In addition, the approach includes a method for the number of clusters selection. Spike Sorting has received considerable attention in the literature, and different functional clustering approaches have traditionally been considered. We compare those approaches with the MixFMM in a broad collection of datasets, including benchmarking simulated and real data. The MixFMM approach achieves outstanding results in a selection of indexes across datasets, and the significant improvements attained in specific scenarios motivate interesting neuronal insights.