The Role of Environmental Information and the Limitations of Predictability with Transformer-based Time-Series Models for complex Astrocyte Trajectories

Morphogenesis is the process by which an organism develops from a single cell into a complex organism, and one factor contributing to this complexity is the interaction between cells and their environment. In heterogeneous environments comprising differing cell types and polymeric structures, the observed complexity appears to approach an infinite value. Recent developments in imaging tools and deep learning for image analysis enable the extraction of meaning from terabyte-scale images at a realistic computational cost. In this paper, we used a Transformer-based model to predict the migration trajectories of astrocytes during mouse brain development, aiming to quantitatively elucidate (i) what makes prediction difficult, i.e., the structure of predictability and its theoretical limit, and (ii) to what extent environmental information (vascular structure) and multimodal modeling of the predictive distribution can improve prediction. Based on a bivariate Gaussian distribution ($\mu, \sigma, \rho$) at each time point, we predicted future positions while accounting for uncertainty and evaluate the quality of the predictive distribution. We report three main findings. First, point-prediction accuracy reached a fundamental limit set by the irreducibility of the movement direction; the error curve was described with high accuracy by the mean-squared-displacement law of a persistent random walk. Second, extending the predictive distribution from a single bivariate Gaussian to a continuous mixture (mixture density network, MDN) captured the run/rest bimodality without discretization and substantially improved the quality of the predictive distribution (negative log-likelihood, NLL) while preserving point-prediction accuracy. Third, a model that integrated vascular environmental information into this multimodal model achieved the best performance in point-prediction accuracy, consistently across three training seeds, showing that environmental information contributes to prediction in combination with multimodal output. This study presents both an improvement of the predictive distribution for complex cell migration and a framework for quantifying the limit of predictability based on movement statistics.