Paper:
Spiking neural networks are being investigated both as biologically plausible models of neural computation and also as a potentially more efficient type of neural network. While convolutional spiking neural networks have been demonstrated to achieve near state-of-the-art performance, only one solution has been proposed to convert gated recurrent neural networks, so far. Recurrent neural networks in the form of networks of gating memory cells have been central in state-of-the-art solutions in problem domains that involve sequence recognition or generation. Here, we design an analog gated LSTM cell where its neurons can be substituted for efficient stochastic spiking neurons. These adaptive spiking neurons implement an adaptive form of sigma-delta coding to convert internally computed analog activation values to spike-trains. For such neurons, we approximate the effective activation function, which resembles a sigmoid. We show how analog neurons with such activation functions can be used to create an analog LSTM cell; networks of these cells can then be trained with standard backpropagation. We train these LSTM networks on a noisy and noiseless version of the original sequence prediction task from Hochreiter & Schmidhuber (1997), and also on a noisy and noiseless version of a classical working memory reinforcement learning task, the T-Maze. Substituting the analog neurons for corresponding adaptive spiking neurons, we then show that almost all resulting spiking neural network equivalents correctly compute the original tasks.

Review:
First the authors suggest an adaptive analog neuron (AAN) model which can be trained by back-propagation and then mapped to an Adaptive Spiking Neuron (ASN). Second, the authors suggest a network module called Adaptive Analog LSTM Cell (AA-LSTM) which contains input cells, input gates, constant error carousels (CEC) and output cells. Jointly with the AA-LSTM, the authors describe a spiking model (AS-LSTM) that is meant to reproduce its transfer function. It is shown quantitatively that the transfer functions of isolated AAN and AA-LSTM units are well approximated by their spiking counterparts. Two sets of experiments are reported, a sequence prediction task taken from the original LSTM paper and a T-maze task solved with reward based learning.
In general, the paper presents an interesting idea. However, it seems that the main claims of the introduction are not sufficiently well proven later. Also, I believe that the tasks are rather simple and therefore it is not demonstrated that the approach performs well on practically relevant tasks.
On general level, it should be clarified whether the model is meant to reproduce features of biology or whether the model is meant to be efficient. If the model is meant to reproduce biology, some features of the model are problematic. In particular, that the CEC is modeled with an infinitely long integration time constant of the input current. This would produce infinitely long EPSPs. However, I think there is a chance that minor changes of the model could still work while being more realistic. For example, I would find it more convincing to put the CEC into the adaptation time constants by using a large tau_gamma or tau_eta.
If the model is meant to provide efficient spiking neural networks, I find the tasks too simple and too artificial. This is particularly true in comparison to the speech recognition tasks VAD and TIMIT which were already solved in Esser et al. with spiking and efficient feedforward networks. 
The authors say in the introduction that they target to model recurrent neural networks. This is an important open question. The usage of the CEC is an interesting idea toward this goal.
However, beside the presence of CEC I do not see any recurrence in the used networks. This seems in contradiction with what is implicitly claimed in the introduction, title and abstract. There are only input-output neuron connections in the sequence prediction task, and a single hidden layer for the T-maze (which does not seem to be recurrently connected). This is problematic as the authors mention that their goal is to reproduce the functionality of LSTMs with spiking neurons for which the network recurrence is an important feature. 
Regarding more low-level comments:
- The authors used a truncated version of RTRL to train LSTMs and standard back-propagation for single neurons. I wonder why two different algorithms were used, as, in principle, they compute the same gradient
either forward or backward.
Is there a reason for this? Did the truncated RTRL bring any
additional benefit compared to the exact backpropagation already
implemented in automatic differentiation software?
- The sigma-delta neuron model seems quite ad-hoc and incompatible
with most simulators and dedicated hardware. I wonder whether the
AS-LSTM model would still be valid if the ASN model is replaced with a
standard SRM model for instance.
- The authors claim in the introduction that they made an analytical conversion from discrete to continuous time. I did not find this in the main text.
- The axes in Figure 1 are not defined (what is Delta S?) and the
caption does not match. "Average output signal [...] as a function of its incoming PSC I" output signal is not defined, and S is presented in the graph, but not I.