Efficient Neural Codes under Metabolic Constraints
Neural codes are inevitably shaped by various kinds of biological constraints, e.g. noise and metabolic cost. Here we formulate a coding framework which explicitly deals with noise and the metabolic costs associated with the neural representation of information, and analytically derive the optimal neural code for monotonic response functions and arbitrary stimulus distributions. For a single neuron, the theory predicts a family of optimal response functions depending on the metabolic budget and noise characteristics. Interestingly, the well-known histogram equalization solution can be viewed as a special case when metabolic resources are unlimited. For a pair of neurons, our theory suggests that under more severe metabolic constraints, ON-OFF coding is an increasingly more efficient coding scheme compared to ON-ON or OFF-OFF. The advantage could be as large as one-fold, substantially larger than the previous estimation. Some of these predictions could be generalized to the case of large neural populations. In particular, these analytical results may provide a theoretical basis for the predominant segregation into ONand OFF-cells in early visual processing areas. Overall, we provide a unified framework for optimal neural codes with monotonic tuning curves in the brain, and makes predictions that can be directly tested with physiology experiments.
This work analyzes information transmission in neurons with monotonic tuning curves under resource and noise constraints. Given a small set of assumptions it demonstrates analytically that splitting into ON and OFF channels maximizes information transmission in a neural population with limited resources.  
This is a high quality submission with a potentially high impact on the field. It generalizes and extends previous results [19, 30]. Moreover, it provides a possible theoretical foundation for results obtained with statistical models of natural stimuli [13]. What I especially like about this paper is the generality of obtained results. The authors do not limit the analysis to a single constraint but consider a number of them at the same time: dynamic range, average activity level and a family of noise models. The results are independent of stimulus distribution and make very little assumptions about the shape of tuning curves (monotonicity). Despite the "density" of the paper the presentation is very clear.  There are a few questions/remarks I have, which I think could be addressed in the text in order to further improve the quality the paper and interpretability of the results:  1) Are proposed coding constraints independent? There seems to be a relationship between the maximum attainable $r$ and average $K(r)$. If yes - can we treat them independently as the authors do?  2) The results presented in the paper are independent of the stimulus distribution. It is of course a strength, however a number of previous studies (e.g. [13]; Ratliff et al., PNAS 2010) suggest a strong connection between the statistics of natural stimuli and the presence of "ON-OFF" channels. How shall one interpret present results in light of previous statistical findings?  3) The above question becomes especially relevant with respect to section 4 - "Optimal coding of large neural population". You show that an optimal solution for a population of neurons is to divide it into two equally large ON and OFF subpopulations. It is known however, that there is a larger proportion of OFF units in the retina. Moreover it seems to be explained by the excess of dark spots in natural images (Ratliff et al., PNAS 2010) i.e. stimulus distribution. How would you explain that discrepancy?   4) Please provide a brief justification in the main text for the assumption of even number of neurons ($N = 2k$).  5) Line 207 includes a remark "show in figure". While I understand that it's a leftover from the editing process, I agree it would be good to show on a picture how ON and OFF tuning curves tile the input space (since, as I understand this is the solution).  6) I like the reference to opponent-channel codes in the auditory system. There have been results published suggesting that this sort of representation maximizes information transmission given the statistics of the natural auditory input [Harper, McAlpine, Nature 2005; Mlynarski, Plos CompBio. 2015]. I would however urge the authors to be careful with this comparison. Spatial tuning curves in the auditory brainstem and cortex are non-monotonic (they peak at extremally lateral positions, sometimes outside of the so-called "physiological range"), and they strongly overlap at the midline (which you address in the text already).   Minor comments:  It might be a good idea to add a small color legend to your figures explicitly saying that yellow / red spectrum corresponds to different constraints.   Figure 2 - I would mention that it is for a uniform variable $u$, not $s$  Linr 152 - supplementary material mentions that due to monotonicity of the tuning curve one of A^+ or A^- regions has to be empty. It would be good to write that in the main text for clarity.