They're likely referring to a ketogenic diet. They were quite common in the past before anticonvulsant meds were created, and are still used in intractable cases today, particularly in pediatric care. There's lots of information out there about their use in epilepsy.
[Edit] - The ketogenic diet is interesting. And I can see how some would want to substitute sugar for an artificial sweetener to help achieve a low carb diet. But given Aspartame is a known trigger for many types of seizures, it should be introduced carefully.
Do you have a source re: aspartame and seizures? I did a cursory search and the only non-anecdotal thing I could find was a study[1] that showed there was no link in people who described themselves as sensitive.
To add, most of the studies done on rats had "normal dosages" of aspartame for Humans not Rats. In other words the dosage was not relative to the rats mass it was relative to ours.
Unfortunately I can't point to an official study - just many anecdotal claims from people like me. When you google, I am sure you will see many such claims - including the 'internet hoax'. (The internet hoax is unfortunate because it takes away credibility from the many anecdotal claims like mine that sincerely feel real. More carefully constructed studies should occur.)
That's tricky to evaluate, because you get the exact same narratives from people who claim sensitivity to glutamates, where we can be somewhat certain that there isn't a health connection.
It's easy to evaluate if you have seizures. You don't need to wait for it to be definitively proven to make use of the anecdotal evidence. If you learned that aspartame could trigger a seizure, wouldn't you be cautious and want to see if a change to your diet would reduce your seizures?
Some to get you started:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2898565/
http://www.epilepsy.com/learn/treating-seizures-and-epilepsy...
http://www.mayoclinic.org/medical-professionals/clinical-upd...