Brain Coral Fields is a place at the bottom of a cliff near Jellyfish Fields that appears only in the episode "Patrick SmartPants," when SpongeBob and Patrick were having an amusing time jellyfishing and playing in Jellyfish Fields. It appears to be just a sandy field with Brain Corals everywhere. There is also a sign which says, "Brain Coral Fields."
The plot occurred when SpongeBob spotted a sign revealing that there is a dangerous cliff behind, which Patrick refused to care and ran straight towards the sign, smashing into it, and falling off the cliff into a place known as the "Brain Coral Fields". SpongeBob jerked out something that looked like Patrick's head but was actually a piece of brain coral which made a plug into Patrick's body and made him extremely intelligent and brilliant.
After several unsuccessful attempts of getting Patrick to play childish games with him again, SpongeBob felt depressed about Patrick's opposed behavior from before, which he could no longer be his best friend anymore. It was then revealed that Patrick was willing to end his intelligence for the sake of friendship with SpongeBob.
The two were then shown back at the place where the plot began, and then Patrick decided to run towards the sign again, and discover Brain Coral Fields once more. Patrick then told SpongeBob where did he find his head, which SpongeBob pointed on his side. Patrick then measured the place where he crashed & then calculated that his actual head was 3 meters to North. SpongeBob then takes out the Brain Coral on Patrick's head & plugs his old head again, in exchange for his smartness. The episode concluded happily that the two finally became best friends again.
The Brain Corals resemble Patrick's head.
There is a real coral called Brain Coral, although it looks completely different.