**To avoid confusing how an analog joysticks compared to a digital joystick I going to go on this side tangent **
*Actually there only 2 potentiometers in an analog joystick, one for the X axis and one for the Y Axis.
There is no depression of pots, a potentiometer isn’t a switch. It Turns, common places to find a potentiometer is in a older TV or Radio volume knob, a light dimmer knob. Think off a potentiometer as a big adjustable resistor. A Switch operates on the concept of ON and OFF (and only On and OFF), only 2 states, which is also the basic concept of digital electronics Ones and Zeros (One = On, Zero = Off).
Analog is different, analog works on the concept of different voltage or amperage levels and is a continuous signal, analog operates like a wave.
Its why Analog was the method of choice for older era TV, Radios, video and audio recordings ect, as it was easy to modulate these continuous signals to sound coming out of speakers or how a image is drawn on a cathode ray tube (CRT) screen.
Like still to this day the last part of our audio equipment before the wiring goes to a speaker the circuity is analog, the constant modulation of voltage levels acts like a wave which effects how much the magnets and coils move in a speaker which create the sound waves we hear. *
What TheBlackHombre did was he pad hacked a Game Cube controller for stick use, as he was using a Sanwa JLF for his Left analog he had to change the digital output of the JLF to analog input the GC PCB would understand. The Sanwa JLF is digital, each cardinal direction is a ON or OFF state. The 4 potentiometers you see are so BlackHombre can adjust or dial-in the strength of how far the push is in any direction. So hitting up on the JLF is not a hard full push on the analog stick stick data as completely up on the X Axis. So dialing the Pots to 50% would equal a 50% push on a left analog stick. I think TheBlackHombre set his pots for being at 40%. The Shift button is wired up to allow for a full press in any direction,regardless of what the pots are set to. He did a similar set up for the C-stick and broke it down to 4 buttons.
There is a introductory guide on how to remove analog joysticks and neutralize them. It goes on the same method can be used to turn variable resistors into on-off switches.
http://www.slagcoin.com/joystick/pcb_wiring.html#PCB_COMPONENTS_MODIFICATION_AND_REMOVAL
As for the actual logic on the board, I am a little foggy about the details as I don’t even know what chips are used. When it came to my own projects and analog to digital conversion I took the easy ways out and just avoided the analog inputs. Like both of my Virtual-On arcade sticks are wired up as digital joysticks taking on digital inputs and setting the game to run off the D-pad and 4 face buttons instead of the analog thumb sticks.
I am sure Gummo or TheBlackHombre can help you more in this regard.