Motor and premotor cortical activities reflect motor planning and execution. For stereotyped movements, the internal cortical state should be predictive of subsequent motor commands. We developed a statistical method to infer a nonlinear state-space model that extracts an effective systems-level description of neural dynamics from population recordings. Without the knowledge of the task, target, or motor behavior, the inferred model can forecast neural activity at a single-trial level. Our novel inference algorithm performs approximate Bayesian filtering and smoothing within a structured variational autoencoder framework by exploiting the analytical forms of exponential family distributions. Compared to previous approaches, we used a more expressive posterior covariance structure, improved scalability in the latent dimensions and forecasting performance. Moreover, our algorithm allows causal inference suitable for real-time applications and supports inference of states in a temporally causal manner. Our analysis is consistent with the hypothesis that population activity in motor cortices before movement onset sets an initial condition for autonomous dynamics. Together, our results suggest that advances in statistical methods can provide deeper insights into dynamical mechanisms and enable new experimental paradigms.