Neuron Fly

Neuron flies look like two thin tails connected to an oblong floating body with a white eye (or face) and shift colors randomly through red, white, blue, or purple, except for the unique green neuron Hunter starts with. They are prevalent in Five Pebbles​ and a small number can be found in Looks to the Moon floating over the puppet of the Iterator. Consuming a neuron fly provides one food (1/4 as Hunter), and doing it for the first time in a save file will allow the Slugcat to generate its own light for the rest of the game. On a full stomach, holding the eat button with a neuron in hand will instead store it in your belly.

After eating a neuron, Void Spawn will appear near slugcat in certain locations. They are trapped in a little 'egg', and upon touch they break free and slowly travel in a direction that leads from Looks to the Moon to Memory Crypts, guiding slugcat in the direction of Five Pebbles in lieu of the yellow overseer.

Looks to the Moon's Neurons (spoilers ahead)
When Looks to the Moon is found, five neurons are flying above her. Catching these neurons causes Moon to shake her head frantically and try to talk to you. With the Mark of Communication, she will urge you to let it go. Eating it makes her throw herself to the ground and jerk around. Eating at least one will earn the Stolen Enlightenment achievement. Eating all of them will kill Moon. If you hibernate without eating one of her flies the cycle you find her, you earn the A New Friend achievement.

From then on, as long as the player hasn't eaten any flies, the yellow overseer will guide the player where to get more neuron flies for Moon. Following the overseer will take the player to Five Pebbles, in hope that you will deliver some to Moon. If the player has already eaten a fly from Moon, they can still find their way to Five Pebbles by following the aforementioned Void Spawn.

Despite popular belief, she has the ability to read pearls found in the world with any number of neuron flies.

The Hunter's neuron (spoilers ahead)
Hunter starts the game carrying a unique green neuron. This neuron can be stored in the slugcat's stomach to leave both hands free. If it has been lying on the ground for a few seconds, the green neuron will hover at the level of slugcat's head above the ground and occasionally emit a complex hologram from its form.

When the green neuron is brought to Moon, who is initially exhaustively incapacitated when you meet her, the green neuron will reactivate her neurons and bring her back to life. This is because the neuron holds 'slag keys'. Moon died because Five Pebbles accidentally deprived her of water, causing slag buildup which caused her processes to seize. The slag keys likely reactivate her water systems to flush out the slag and allow her to become active again.

Trivia

 * Neuron flies cease movement and lose the ability to fly while outside of Five Pebbles or Looks to the Moon. This fact makes carrying neurons between the two Iterators much easier.
 * The only reason to deliver neurons to Moon is to receive bits of lore she gives you when formatting the neurons and to be nice, or to replace the neurons you ate, you monster.