There are several problems with this. Autophagy is the break down of healthy muscle tissue to serve as a fuel source while the body is in a state of starvation. This is not the process of "cleansing" the body of dead cells. When people run marathons or exercise for a prolonged period of time and become depleted of muscle glycogen (sugar), the body begins to catabolise its own muscle tissue (or goes catabolic) in the process of autophagy.
Furthermore, the notion that fasting will improve brain health is preposterous. Our brains synthesise around 80% of the body's available glucose in order to function. So the notion of depriving the entire body of glucose or any other calories for that matter to"improve brain health" is just backward.
On a side note, basing health advice of what the Ancient Greeks did in order to create or distribute health advice daft also. Yes they had some brilliant philosophers and mathematicians. Greek doctors would commonly taste their patient's ear wax to aid the process of making a diagnosis.

Please do your homework on the topic in future.