The control and downstream interpretation of gene expression dynamics is crucial in many biological contexts. For example, gene expression oscillations have been proposed to control the timing of cell differentiation during embryonic neurogenesis. However, mathematical analysis of gene expression dynamics may be hindered by sparse data and parameter uncertainty. Here, we combine Bayesian inference and quantitative experimental data on mouse and zebrafish neurogenesis to explore mechanisms controlling aperiodic and oscillatory gene expression dynamics during cell differentiation. We find that quantitatively accurate model predictions are possible despite high parameter uncertainty. We identify examples of stochastic amplification, where oscillations are enhanced by intrinsic noise and we show how such oscillations can be initiated by changes in biophysical parameters. We further consider mechanisms that may enable the down-stream interpretation of dynamic gene expression. Our analysis illustrates how quantitative modelling can help unravel fundamental mechanisms of dynamic gene regulation.