To assign a controller to a button like solo or mute, you simply drag a controller assignment to that button. The controller assignment will automatically default to button mode, and then you can put the pivot point where you want it, as well as change the button type.
"Toggle" mode means each time the controller is activated the button will switch positions, like from 'on' to 'off' and vice versa.
"Momentary" mode means the button will go from one state to the other only while the controller is activated. For example, with a keyboard controller, while the key is pressed, the button will change from 'off' to 'on'. Then when the key is released, it will go back to 'off'.