Embedded Polygon Symbolic Transfer Entropy (EPSTE): A Geometric Token and Deep Learning Approach to Estimating Transfer Entropy in Neuroimaging Time Series

Inferring directed interactions between neural systems from EEG and MEG remains challenging due to noise, nonstationarity, and the high sample complexity of information-theoretic estimators. Transfer Entropy (TE) provides a principled and model-free measure of directed information flow; however, its practical estimation is not stable in finite data regimes (particularly as embedding dimension increases). This work introduces Embedded Polygon Symbolic Transfer Entropy (EPSTE), a framework that reframes TE estimation as a learnable problem operating on structured symbolic representations of local temporal morphology rather than raw signal amplitudes. Neural time series are decomposed into sequences of geometric primitives derived from local triplets of samples encoding complementary aspects of waveform structure such as magnitude, curvature and directional change. These primitives are discretised into symbolic tokens, yielding a compact but expressive state space over which symbolic TE is estimated. A recurrent neural network with attention-based multiple-instance learning is trained to predict surrogate-validated TE values from bags of symbolic temporal windows. The method is evaluated on source-reconstructed MEG data parcellated using the AAL90 atlas and compared against a standard symbolic baseline using identical architectures and supervision. The results demonstrate that while local window-level predictions are noisy, aggregation across trials and channel pairs yields stable directed dependencies. At the pair level, EPSTE achieves near-perfect recovery of ground-truth directed structure and significantly lower absolute error than the baseline, indicating that representational geometry plays a critical role in enabling practical learnability of information-theoretic dependencies.