If you know someone who has a neon, swap PCMs and see what happens. SOHC, DOHC, ATX or MTX won't matter as long as you are just testing to see if it starts. Also check the underhood power distribution center for corrosion. You may have to look underneath it as well. Oil and corrosion can eat through the wires there as well and cause serious issues.

 

Hey! I know this thread is a few years old now, but wondering if you ever fixed the problem? I'm having a very similar issue with my 1999 Neon Sport.