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In this work, we evaluate how neural networks with periodic activation functions can be leveraged to reliably compress large multidimensional medical image datasets, with proof-of-concept application to 4D diffusion-weighted MRI (dMRI). In the medical imaging landscape, multidimensional MRI is a key area of research for developing biomarkers that are both sensitive and specific to the underlying tissue microstructure. However, the high-dimensional nature of these data poses a challenge in terms of both storage and sharing capabilities and associated costs, requiring appropriate algorithms able to represent the information in a low-dimensional space. Recent theoretical developments in deep learning have shown how periodic activation functions are a powerful tool for implicit neural representation of images and can be used for compression of 2D images. Here we extend this approach to 4D images and show how any given 4D dMRI dataset can be accurately represented through the parameters of a sinusoidal activation network, achieving a data compression rate about 10 times higher than the standard DEFLATE algorithm. Our results show that the proposed approach outperforms benchmark ReLU and Tanh activation perceptron architectures in terms of mean squared error, peak signal-to-noise ratio and structural similarity index. Subsequent analyses using the tensor and spherical harmonics representations demonstrate that the proposed lossy compression reproduces accurately the characteristics of the original data, leading to relative errors about 5 to 10 times lower than the benchmark JPEG2000 lossy compression and similar to standard pre-processing steps such as MP-PCA denosing, suggesting a loss of information within the currently accepted levels for clinical application.