It is a very important question. The answer to this won't be simple. In the time domain it is not sparse as is evident from the waveforms.
Sparsity has been argued in the time frequency domain has been discussed in literature. A few publications on this topic are attached. The list is not by any means exhaustive.
I have also attached a tutorial on this matter and I think that it is an excellent introduction to this topic.
Article Sparse Representation for Brain Signal Processing: A tutoria...
Article Imaging brain source extent from EEG/MEG by means of an iter...
Article Dynamics & sparsity in latent threshold factor models: A stu...
Article EEG reconstruction and skull conductivity estimation using a...
EEG is certainly not sparse in the time domain. We have studied its sparsity in the time-frequency domain (see reference1). However, when solving the EEG source localizaton problem,one can consider the sparsity of sources in the spatial domain (see reference2), or their sparsity in spatio-temporal domain (reference 3).
Conference Paper A hierarchical sparsity-smoothness Bayesian model for ℓ0 + ℓ...
Article Sparse EEG Source Localization Using Bernoulli Laplacian Priors
Article Bayesian EEG Source Localization Using a Structured Sparsity Prior
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