I don't have the proper colours but I can help a little.
I have a heap of Toyota manuals up to 1984 and each have different colours on the radio.
Grrrr
The antenna lead is obvious.
The black lead bolted to the metal case is ground and so is the white with black stripe wire.
Red wire and yellow wire.
One of them (yellow I think but not sure) is permanently 12V from your AE86 radio#2 fuse (Purple with black stripe on your car?) and holds the memory for each radio button.
The other (red?) is connected to the radio #1 fuse so that the radio turns on and off with the main key switch (write with red stripe on your car?)
Hopefully by this stage you can see the controls light up.
The outer 8 wires on the big plug are the speakers.
Each vertical pair goes to one speaker (+ and -).
Connect the left most pair, turn the volume to about halfway and play with the left/right and front/back controls.
Do the same with the other 3 speakers.
You can ignore the rear speakers if you wish.
The centre wires I'm not sure about.
That leaves antenna motor control and dash lights (radio lights dim a little when the dash lights are on.
As long as your antenna is up, you can ignore both of these until later.
My AE86 wiring diagram doesn't show a motorised antenna, so this is probably nothing to worry about anyway.
Haven't got a clue what the extra bundle is on the far right of the photo.
Hopefully it's not related to security.
- Stepho