This dissertation addresses the analysis of data emerging in the field of music neuroscience, specifically data collected from neurophysiological monitoring techniques that can be modeled as random objects in spaces of smooth functions. Spaces equipped with a Hilbert structure offer a versatile and elegant framework for the generalization of various statistical techniques, ensuring adaptability and robustness in analyzing complex data structures. Within the context of functional data analysis, these spaces serve as essential tools for understanding and interpreting dynamic data trends over continuous domains. Given the relevance of independent component analysis (ICA) in neuroscience research, our investigation is directed towards its functional counterpart, a technique whose potential still remains relatively overlooked. Functional ICA can be considered a refinement of functional principal component analysis, aimed at identifying low-dimensional structures "as independent as possible" by exploiting the underlying topological features of the data. We provide a comprehensive account of the theoretical foundations of functional ICA in an infinite-dimensional framework and extend the method to Sobolev spaces of smoother functions. Some relevant theoretical properties regarding functional data classification are also presented. Additionally, we develop a repertoire of related functional data techniques tailored for pre-processing and analyzing data in the emerging field of embodied music neuroscience, which investigates the neurological basis of how the body influences musical experience. Two methods based on nonlinear wavelet and polynomial approximations are developed for pre-processing artifactual activity in EEG and pupillometric signals. These methods yield excellent outcomes for neuromotor research, particularly considering the suboptimal condition of the recorded data due to locomotor activity. We further introduce a set of neural descriptors derived from data collected through the aforementioned non-invasive methods, aiming to uncover brain behavior during embodied musical interactions. More specifically, we focus on methodologies for modeling neurotransmitter activity, a critical aspect shown to be essential in shaping motor functionality and other proprioceptive sensations. Our experimental research is portrayed by the concept of emotion transferred into a neurological domain, providing a unique framework to define and capture the neural essence of embodiment in music.