Some people have asked why we used the measure we did to assess the role of correlations. Why didn't we just measure information when the two cells were recorded at the same time, measure information when the two cells were recorded at different times, and then take the difference? We did do this. In fact, it was the first thing we tried, and we called the difference I-Ishuffled (click to see the plot: JPEG, PDF, postscript).

You can see from the plot that Ishuffled and I are about the same; sometimes Ishuffled is greater than I, sometimes it's less than I, but it never differs by more than 10%.

The reason we didn't use I-Ishuffled in the paper is that there's a potential confound. One can, in principle, get cases where I-Ishuffled=0, but correlations are actually important. This could happen because of cancellation effects (explicit examples can be provided, if you're interested).

The reason we used our measure, Delta I, is that it's an upper bound on information loss. Thus, if Delta I=0, there is an absolute guarantee that one can ignore correlations and recover all the information in the responses.

One last thing ... the fact that the maximum information loss measured by Delta I (11%) was not much more than the maximum value of |I-Ishuffled| (10%) indicates that the cancellation effects we were worried about don't happen in real life. But there was no way to know that until we did the analysis with Delta I.

Markus Meister and Toshihiko Hosoya and Mike Berry and Elad Schneidman submitted comments to Nature, but they were not published. Here we provide a copy of their comments and our responses for the interested reader
Meister and Hosoya's comment: PDF
Our reply to Meister and Hosoya: PDF Supplementary information: PDF, Postscript
Berry and Schneidman's comment: Removed at their request
Our reply to Berry and Schneidman: PDF

More fun facts about Delta I:

Delta I provides a universal bound on optimal cost functions. PDF postscript
Delta I is an upper bound on information loss. PDF postscript
When it comes to comparing codes, Delta I has been used much more widely than you might think. PDF postscript

Or, get all of them with one simple mouse click. PDF postscript