Electroencephalography (EEG) measures brain activity through recording electrical potentials of the scalp. EEG recordings capture two distinct sets of characteristics in the signal: (1) \emph{Traits} -- information that remains relatively constant over the course of a recording session, such as subject identity, age, and sex; (2) \emph{States} -- signatures that encode responses to external stimuli, including visual, auditory, and motor information. There is limited understanding of how different EEG modeling features (signal processing and foundation models) encode state and trait characteristics. In this work, we quantify state- and trait-level features captured by EEG embeddings using a hierarchy metric and kNN purity on Brain-Computer Interface dataset trials, and find that a majority of EEG features form a macro-trait hierarchy (embeddings form trait-level clusters with micro-states within). Our linear probe results indicate that this hierarchy structure leads to suboptimal state distinguishability, negatively impacting BCI performance of EEG foundation models. To mitigate this, we explore normalization techniques that reduce trait-specific information in embeddings and provide a $\sim$ 10\% improvement in balanced accuracy for state classification. Additionally, we observe that when fine-tuning models for downstream state decoding, embedding spaces transform from macro-trait to macro-state hierarchies, which in turn lead to better state decoding.

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