Sparse representation (SR) is a widely accepted hyper-spectral image (HSI) denoising model. Because of the curse of dimensionality and the desire to better fit the data, the SR models are typically deployed on small and fully overlapping blocks whose results are averaged to produce the global de-noised HSI. This 'local-global' denoising mechanism ignores the dependencies between blocks, resulting in visual artifacts. This paper describes the underlying clean HSI with a 3D con-volutional sparse coding (CSC) model, representing the HSI with a linear combination of few shift-invariant 3D spatial-spectral filters in a global dictionary. Instead of operating on patches, the CSC model sees the clean HSI is generated from a sum of local atoms that appear in a small number of locations throughout the image, naturally retaining the relationship between pixels. Moreover, we unfold the optimization process of the model into a spatial-spectral convolutional sparse neural network which absorbs the interpretation ability of the model while supporting discriminative learning from data. Experimental results on both synthetic and real-world datasets show that our network achieves competitive denoising performances, qualitatively and quantitatively.