What function to apply to a signal to expose its burstiness (i.e., make the burst part more evident than the non-bursty part) I am working on some signals one of which I have attached below:



What can I do (i.e. functions that I can apply to this signal) to get something like the signal below:



Note how one can draw a line at 0.25 to differentiate between rownumber &lt;13000(or close), and rownumber>13000

-currently, I have tried checking how far each value of the signal is away from the mean, but that didn't seem to help either (i.e., the signal still did not look anywhere near my desired). By the way, I am "not" a signal processing guru, which means I probably would appreciate explanation intended for someone with little experience in signal processing. 

Edit: Based on Dan's answer, I decided to clarify something on the question. I am using this signal for a machine learning related research, and the assumption is that I only have the sparse area, but this figure helps give insight into the nature of the dense part of the signal. Hence, in the above paragraph when I said I tried the deviation from the average, I mean as I received new signals (which looks like the dense part shown in the fig), I tried to find out how far it is from the mean of the sparse part of the signal. So I guess my new revised question will be- if you have the sparse part of the signal, and you want to use its nature, shape or amplitude to differentiate between the dense (which at the initial time is unknown) and the sparse signals, how would you do this?
You can apply a simple moving average filter and based on the result of the filter make a decision if you are in the sparse area or dense area, as your average will certainly be larger in your dense area. There is a resolution vs false alarm trade space that you will need to make; your result will be more accurate the longer you can make your moving average filter but your precision in determining where the transition occurred wil be reduced.

So in this sense, to compare to what you tried, instead of determining how far each signal is from the mean, take the mean over shorter intervals (as you move through the sequence), and based on the result of the mean compared to a threshold, decide if you are in the sparse or dense area, and if your decision is sparse, you can clip the result as you have done.

An alternate and simple algorithm would be a window counter, where you allow a certain number of counts (window) between samples above a certain threshold: if the sample comes in above threshold before the count is up you declare a positive, and if not you declare a negative and the count in reset. Further, a certain number of consecutive positives or negatives is required to change state between dense and sparse (this adds hysteresis and avoids a chatter). I don't know the detailed characteristic of your dense to suggest actual count intervals, but this is something you can consider playing with.