Understanding the principles underlying the design of robust, yet flexible patterning systems is a key problem in developmental biology. In the Drosophila wing, Hedgehog (Hh) signaling determines patterning outputs using dynamical properties of the Hh gradient. In particular, the pattern of collier (col) is established by the steady-state Hh gradient, whereas the pattern of decapentaplegic (dpp), is established by a transient gradient of Hh known as the Hh overshoot. Here we use mathematical modeling to suggest that this dynamical interpretation of the Hh gradient results in specific robustness and precision properties. For instance, the location of the anterior border of col, which is subject to self-enhanced ligand degradation is more robustly specified than that of dpp to changes in morphogen dosage, and we provide experimental evidence of this prediction. However, the anterior border of dpp expression pattern, which is established by the overshoot gradient is much more precise to what would be expected by the steady-state gradient. Therefore, the dynamical interpretation of Hh signaling offers tradeoffs between

Animal behaviour alternates between stochastic exploration and goal-directed actions, which are generated by the underlying neural dynamics. Previously, we demonstrated that the compositional Restricted Boltzmann Machine (cRBM) can decompose whole-brain activity of larval zebrafish data at the neural level into a small number (∼100-200) of assemblies that can account for the stochasticity of the neural activity (van der Plas et al., eLife, 2023). Here, we advance this representation by extending to a combined stochastic-dynamical representation to account for both aspects using the recurrent temporal RBM (RTRBM) and transfer-learning based on the cRBM estimate. We demonstrate that the functional advantage of the RTRBM is captured in the temporal weights on the hidden units, representing neural assemblies, for both simulated and experimental data. Our results show that the temporal expansion outperforms the stochastic-only cRBM in terms of generalization error and achieves a more accurate representation of the moments in time. Lastly, we demonstrate that we can identify the original time-scale of assembly dynamics by estimating multiple RTRBMs at different temporal resolutions. Together, we propose that RTRBMs are a valuable tool for capturing the combined stochastic and time-predictive dynamics of large-scale data sets.