Neon stalling

I'm having a problem with my 96 neon. After you drive for a while it just shuts off. It doesn't chug or nothing, it just quits. When you turn the key you can here the fuel pump come on for a second but when you try to start it just rolls over. No matter how long you roll it over for it there is no smell of gas and it wont start. It doesn't seem to flood when you roll it over and if you wait five minutes it starts right up without even touching the gas pedal and runs for 5 or 10 minutes and then it stops again.
I changed the fuel filter but it didn't help. It's almost like the ejectors shut down. Could it be the PCM. I checked the coil pack and its OK. As I said the fuel pump does run when you first turn the key but the car doesn't even attempt to start unless you wait for a few minutes.
Any help is appreciated.
Thanks Jim.

I test to see if it is fuel or spark by be putting some gas down the throttle body when it won't start. If it starts briefly you have a fuel problem. If it doesn't start you have a ignition problem. you can also pull a plug wire and plug it into a spare plug and look to see if you have spark while turning it over.

Your car would have a Check Engine Light or Service Engine Light on. You might want to check to see if any codes are set. There can be codes set even with the light not being lit. The way to check it is to turn the ignition switch from off to on to off to on to off to on, but dont start the engine. Do the three cycles in less than 5 seconds - and your CEL should start blinking out numbers. It will repeat the numbers, beginning with a 1-2, finishing up with a 5-5, then repeat the whole sequence again. Then tell us what the numbers are if any.
If there are no codes set, its likely something the computer can not determine. For instance, if there is not enough fuel pressure for it to start (such as due to a clogged fuel filter) - the PCM has no way to measure that on the 1st generation Neons.
After it shuts down, try checking for fuel pressure at the fuel rail. There is a Schrader valve there. Pushing in on the stem will 'open' the valve, and with normal fuel pressure you should get a very obvious squirt of gasoline coming out. So dont be smoking a cigarette, and dont stand right in front of the valve. I think the normal fuel pressure in Neons is almost 50 psi, though it might be a bit lower with the engine running at an idle.