So you need to read an arbitrary analog level, as a digital signal? An analog-to-digital converter, you might say?
The simplest case is a 1-bit ADC, otherwise known as a comparator. (Digital input pins count too -- all real signals are analog, after all -- but they have terrible accuracy so aren't practical to abuse in this way.)
You might use a couple of comparators to determine which state it's in, and read it into the AVR* as say 2-5 parallel bits (depending on how much encoding you want to do, and how much cleverness you bring to bear). Could also be sequential, say if you have a counter advancing the threshold -- assuming the bit times are reliable, so you have plenty of opportunity to check against multiple thresholds.
Or you can use the ADC proper, if you have enough channels and sample rate. The AVR has a 8/10 bit converter, more than adequate here. Conversion takes something like 0.1ms so if the axis in your plot is ms, that's more than enough. Internal mux has 8 channels, or you can add an external (analog switch / mux-demux) to expand to even more.
That should be easy enough in Arduino-land. Some abuse could be done, like overclocking the ADC to squeeze out more samples at far less accuracy, but likely not so easy to do, and probably not necessary.
Note that the signals are repetitive, so you have multiple chances to read the display, even if you're only checking say one logic level per scan, or every other bit, or something. It's unlikely the display data is being updated very quickly (but it would be interesting to find it is).
*Assuming whatever original Arduino board, I guess.
Tim