What is the ruling on breaking the fast in Ramadan due to a neurological illness, and what kind of expiation (fidyah) does that entail? And how can one make up for the missed fasts if the illness persists for a full year?

A sick person for whom fasting is difficult, or whose fasting would delay their recovery or worsen their illness, is permitted to break their fast. If the illness is temporary and recovery is hoped for, then they must make up the missed fasts when they are able. However, if the illness is chronic and recovery is not expected, then they must pay a fidyah (expiation) by feeding a poor person for each day, based on Allah's saying: (And upon those who are able to fast [but with extreme hardship] - a ransom [as substitute] is feeding a poor person). If another Ramadan arrives before they are able to make up the fasts due to illness, there is no sin upon them.