Paper:
Quantifying how much sensory information in a neural code is relevant for behavior
Determining how much of the sensory information carried by a neural code contributes to behavioral performance is key to understand sensory function and neural information flow. However, there are as yet no analytical tools to compute this information that lies at the intersection between sensory coding and behavioral readout. Here we develop a novel measure, termed the information-theoretic intersection information I II (S; R; C), that quantifies how much of the sensory information carried by a neural response R is used for behavior during perceptual discrimination tasks. Building on the Partial Information Decomposition framework, we define I II (S; R; C) as the part of the mutual information between the stimulus S and the response R that also informs the consequent behavioral choice C. We compute I II (S; R; C) in the analysis of two experimental cortical datasets, to show how this measure can be used to compare quantitatively the contributions of spike timing and spike rates to task performance, and to identify brain areas or neural populations that specifically transform sensory information into choice.

Review:
This manuscript proposes a new information-theoretic measure for quantifying the amount of information about the stimuli carried in the neural response, which is also used for behavior for a perceptual discrimination tasks. This work builds on the previous literature of Partial Information Decomposition. The information-theoretic measure is tested using a simulation and two published experimental datasets.
Overall, I find that this is a potentially interesting contribution. The manuscript is generally well-written, except it misses the definitions of a few key quantities. The general approach the authors use is principled and potentially lead to better way to examine the relation between the neural response, behavior and the stimuli.
I have a few general as well as technical concerns about the applicability of this measure in typical experiments which I will detail below. 
Major concerns:
Eq.(1) is a key equation, however, the terms at the r.h.s are not defined, and only intuitions are given. For example, how to compute SI(C:{S,R})? I think it would be useful to have that spell out explicitly. Minor comment- there is a typo in the last two terms, and now they appear to be identical.
Eq. (2) gives the definition of the proposed new measure. However, what is the difference between this measure, and the bivariate redundancy measure proposed in Harder et al. (PRE, 2013; reference (21) in the submission)? In Harder et al., a measure with similar structure is proposed, see their Eq. (13). It is not clear to me how the "new definition" the authors proposed here is different from the Eq. (13) there.  Many of the properties of the measure proposed here seems to be shared with the measure defined there. However, since the definition of SI(C:{S,R}) is not given in the current paper, I might be missing something important here. In any case, I'd like to understand whether the measure defined in Eq. (2) is indeed different from Eq. (13) of Harder et al (2013), as this seems to be critical for judging the novelty of the current paper.
Regardless of whether the measure the authors proposed is indeed novel, there is a crucial question in terms of  practically how useful this method is. I'd think one of the main challenges for using this method is estimating the joint probably of the stimulus, activity and the choice. But in the Supplementary Information, the authors only explained how to proceed after assuming this joint probability has been estimated. Do the authors make any important assumptions to simplify the computations here, in particular for the neural response R? For example, is the independence across neurons assumed? Furthermore, in many tasks with complex stimuli such as in many vision experiments, I'd think it is very difficult to estimate even the prior distribution on the stimulus, not to say the joint probability of the stimulus, the activity and the behavior. I understand that the authors here only focus on simple choice tasks in which the stimuli are modeled as binary variables, but the limitations of this approach should be properly discussed.
In the experimental section, it seems that the number information is a bit small, e.g., ~10^{-3} bit. Can these number map to the values of choice probability and are they compatible? In general, it might be useful to discuss the relation of this approach to the approach of using choice probability, both in terms of the pros and the cons, as well as in which cases they relate to each other.
In summary, my opinion is that the theoretical aspects of the paper seems to be incremental. However, I still find the paper to be of potential interests to the neurophysiologists. I have some doubts on the general applicability of the methods, as well as the robustness of the conclusions that can be drawn from this method, but still this paper may represent an interesting direction and may prove useful for studying the neural code of choice behavior.
Minor:
In eq. (2), the notation for I_{II}(R) is confusing, because it doesn't contain (R,C), but nonetheless it depends on them.  This further leads to the confusion for I_{II}(R1,R2) at the top of line 113. What does it mean?