The Zendrive is 808-like in that it has clipping diodes in the feedback loop of the opamp. What does this do? Basically, the gain out of the opamp is controlled by a resistive divider that sends some of the opamp output off to the pedal output, and some back into the feedback circuit. The higher the resistance going into the feedback circuit, the more of the opamp output will go to the pedal output, and the more the opamp will increase it's gain to compensate. Once the voltage coming out of the opamp reaches a certain level the diodes will open up (as is their nature) and allow virtually 100% of the opamp output above that level to flow into the feedback input, which chokes off the gain from the opamp. This threshold voltage is achieved at the tops and bottoms of the waveforms of the input voltage, so this has the effect of leveling off the tops and bottoms of the waves -- clipping!
So when you turn up the "drive" on one of these pedals, what you are actually doing is increasing the resistance in the circuit that bypasses the diodes in the feedback loop, forcing the opamp to increase it's output, which will eventually cause the peaks (and valleys) of the waves to open the diodes and cut the gain dynamically.
EDIT: I had a closer look at the Zendrive circuit and I see that the biasing stuff (that I've deleted) is wrong. I forgot that the input signal is biased to +4.5V (which, of course it has to be since a 9V battery by itself can't supply a -ve voltage). I also see that the "biasing" on the negative feedback input goes through an RC filter, which means that it's going to tend to pull higher frequencies towards the 4.5V biasing point, which in turn is going to mean that the opamp is going to push more of them out. The "Voice" knob controls how much of this reduction of the high frequencies going into the negative feedback input is going to happen.
So the "Voice" knob is basically a tone control, but since it's in the feedback loop clipping circuit it has a slightly indirect effect on the output tone, and it also affects the amount of clipping on those frequencies that are accentuated. And all of the clipping and tone changes are softened out a bit because what you actually hear is the effect of them on the behaviour of the opamp - and there are limitations to how fast it can react to changes in the feedback input.
So, having got that figured out properly, I still don't know how the AxeFX drive parameters map against this.