Title: Variational inference of latent hierarchical dynamical systems in neuroscience: an application to calcium imaging data. Abstract: A key problem in neuroscience, and life sciences more generally, is that data is generated by a hierarchy of dynamical systems. One example of this is in \textit{in-vivo} calcium imaging data, where data is generated by a lower-order dynamical system governing calcium flux in neurons, which itself is driven by a higher-order dynamical system of neural computation. Ideally, life scientists would be able to infer the dynamics of both the lower-order systems and the higher-order systems, but this is difficult in high-dimensional regimes. A recent approach using sequential variational auto-encoders demonstrated it was possible to learn the latent dynamics of a single dynamical system for computations during reaching behaviour in the brain, using spiking data modelled as a Poisson process. Here we extend this approach using a ladder method to infer a hierarchy of dynamical systems, allowing us to capture calcium dynamics as well as neural computation. In this approach, spiking events drive lower-order calcium dynamics, and are themselves controlled by a higher-order latent dynamical system. We generate synthetic data by generating firing rates, sampling spike trains, and converting spike trains to fluorescence transients, from two dynamical systems that have been used as key benchmarks in recent literature: a Lorenz attractor, and a chaotic recurrent neural network. We show that our model is better able to reconstruct Lorenz dynamics from fluorescence data than competing methods. However, though our model can reconstruct underlying spike rates and calcium transients from the chaotic neural network well, it does not perform as well at reconstructing firing rates as basic techniques for inferring spikes from calcium data. These results demonstrate that VLAEs are a promising approach for modelling hierarchical dynamical systems data in the life sciences, but that inferring the dynamics of lower-order systems can potentially be better achieved with simpler methods.