Paper:
Biological data often contains measurements of dynamic entities such as cells or organisms in various states of progression. However, biological systems are notoriously difficult to describe analytically due to their many interacting components, and in many cases, the technical challenge of taking longitudinal measurements. This leads to difficulties in studying the features of the dynamics, for examples the drivers of the transition. To address this problem, we present a deep neural network framework we call Dynamics Modeling Network or DyMoN. DyMoN is a neural network framework trained as a deep generative Markov model whose next state is a probability distribution based on the current state. DyMoN is well-suited to the idiosyncrasies of biological data, including noise, sparsity, and the lack of longitudinal measurements in many types of systems. Thus, DyMoN can be trained using probability distributions derived from the data in any way, such as trajectories derived via dimensionality reduction methods, and does not require longitudinal measurements. We show the advantage of learning deep models over shallow models such as Kalman filters and hidden Markov models that do not learn representations of the data, both in terms of learning embeddings of the data and also in terms training efficiency, accuracy and ability to multitask. We perform three case studies of applying DyMoN to different types of biological systems and extracting features of the dynamics in each case by examining the learned model.

Review:
This paper tackles the important challenge of making sense of temporal measurements made in biological systems. Among other, those have the peculiarity that they are not independent but with a dependency structure, which can be encoded as a graph or a network. The authors claim that their approach, DyMoN is adapted to the many challenges of biological high-throughput data sets: noise, sparsity and lack of temporal resolution. The paper presents three very different use of the method in complex biological systems in Section 3: (i) Calcium imaging of visual cortex neurons, (ii) T--cell development in the thymus, and (iii) Human embryonic stem cell differentiation. Section 4 assesses the performance of the method on simulated data sets as well as on a face-recognition data set. Moreover, the authors demonstrate how the features of the NN can be interrogated to shed new insight about the process under scrutiny. They also show the gain in running time a comapred to other approaches.
Remarks:
 - I know very little about the literautr in the subject, could you clarify how your work relates to/can be distinguished from: Testolin and Zorzi, Front Comput Neurosci. 201, Kiegeskorte's Ann. Rev. Vis. Sc. 2015 (https://doi.org/10.1146/annurev-vision-082114-035447), Kai Fan's PhD work (@ Duke University), Betzel and Bassel, Interface 2017, Wang et al. bioRxiv 2018 (https://doi.org/10.1101/247577), etc.
 - l-4p2: 'repetitions': what are those? Line below: 'sufficient for estimating $ P_{x}(y) $, means large sample size, no? No contradictory? And one line below: what is the precise meaning of 'similar' (twice)?
 - top p3: line continued from bottom of p2 -> is it to rubber out noise?
what is $ n $ in $ \mathbb{R}^{n} $?
 - Make Fig 1 (B) and (C) clearer, since the transition vectors are learnt, why are they in (B) (input states)?
 - below "...distribution approximates the distribution $ \mathcal{P}_{x} $..." -> but $ P_{x} $ also depends on $ \theta $, not an issue?
 - remark at the end of Section 2: extending DyMoN to higher-order: OK, but this might be computationally VERY expensive, don't you think?
 - link how your empirical validation data have features that, even remotely resemble those of the kind of biological data sets (on which no ground truth exist, I sympathise) you focus on.