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Title: Variational inference of latent hierarchical dynamical systems in neuroscience: an application to calcium imaging data

Paper content:
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.
Many datasets in the life sciences are generated by a hierarchy of dynamical systems, wherein lower-order dynamical systems that directly generate the data are driven by higher-order dynamical systems that are not observable.
This problem is outlined in figure 1A , in which noisy observations x depend on the state z 1 of a low-order dynamical system that is perturbed by inputs u 1 .
The state of this dynamical system is also coupled to the state z 2 of a higher-order dynamical system, which can be perturbed independently by inputs u 2 .
One example of such a system in in-vivo two-photon calcium imaging from neuroscience.
Calcium imaging provides systems neuroscientists with the ability to observe the activity of hundreds of neurons simultaneously during behavioural experiments.
Such experiments have allowed neuroscientists to ask questions about the underlying computations and algorithms that neural circuits are implementing in perception, decision-making, memory, and many other processes.
Such experiments can be characterized as observing a hierarchical dynamical system (Fig 1B) in which measurable calcium fluorescence is primarily determined by dynamics based on voltage-gated calcium channels and calcium binding to fluorescence dyes, and the rate of fluorescence transients controlled by the underlying computation.
Recent applications of sequential variational autoencoders to neural data analysis has seen great success in inferring underlying computations in populations of cells in macaque and human motor cortex Pandarinath et al. (2018) .
By characterizing neural computation as low-dimensional dynamic factors in a non-hierarchical dynamical systems, Pandarinath et al. (2018) showed that these dynamic factors trained to generate the inhomogeneous intensity functions explaining the rate of spikes assumed to follow a Poisson process.
Crucially, these low-dimensional factors could also decode reaching behaviour of macaques and humans with much higher fidelity than any other dimensionality reduction method.
Although this is a significant advance in our ability to analyze neural data in the form of spikes trains, two-photon calcium imaging poses the additional problem of identifying latent spike trains in fluorescence traces.
This problem has been independently addressed in a number of different ways, including deconvolution Friedrich et al. (2017) and variational inference Speiser et al. (2017) .
If we continue to model the frequency of events as being generated by a Poisson process, this can be seen as hierarchy of dynamical systems (Fig 1A) , in which low dimensional dynamics generate spike probabilities that in turn drive fluctuations in biophysical dynamics of calcium activity ( Fig  1B.
Here we propose a method that extends LFADS to accommodate calcium activity using this hierarchical dynamical systems approach, in which we can infer both the latent dynamics and the latent spike trains from the observed calcium fluorescence signal.
We present a hierarchical recurrent variational autoencoder model capable of reconstructing latent dynamics, latent spike trains, and calcium fluorescence traces in a benchmark synthetic dataset.
Of the four methods tested, our model is the only one capable of reconstructing all three.
Furthermore, our model performed best in reconstructing latent dynamics in our synthetic dataset We will need to assess our model on further synthetic benchmark data to assess the validity of our approach.
Since our model is trained end-to-end, it should be possible to extend to reconstructing raw 2-photon imaging videos, which could enable us to train models to uncover latent dynamics from arbitrarily shaped neuronal structures.
This would of great use to neuroscientists who are largely restricted to techniques that extract fluorescence traces from regions of interest with somatic shapes, whereas the morphological diversity of dendrites is much greater.
An additional advantage of using our hierarchical model is that we can obtain measures of the uncertainty in both the latent dynamics, and the latent spike trains.
The correlation in uncertainty between layers of this hierarchy may be what allows superior inference of the latent dynamics, despite less accurate reconstructions of the spike trains than OASIS, which provides no measure of uncertainty.
We hope to improve our model to better capture the relationships between layers of this hierarchy in future.
We describe a use-case in neuroscience (2-photon calcium imaging data) for which this model may be very useful.
However, we are keen to investigate the general case of hierarchical dynamical systems and their utility in uncovering structure in datasets outside this domain.
A APPENDIX

assistant: text: We extend a successful recurrent variational autoencoder for dynamic systems to model an instance of dynamic systems hierarchy in neuroscience using the ladder method.