how can our brains remember that we forgot something, but it can't remember what we forgot?

To understand this we need to understand how memories work. They aren't data files (unless you're a robot, but we aren't friends with anyone like that!). Memories are (simply) stored in what are called synapses. A synapse is a connection between two neurons (brain cells). The synapse is how one cell tells another cell to fire. So if we take a simple example, you smell some banana sushi, the neurons that detect that smell set off a chain of synaptic signals that tell your mouth to start watering.

But what about someone who doesn't like bananas? They smell the same piece of nigiri, but their mouth doesn't water. It's because the synaptic pathways get stronger the more they get used, and that process is even faster if there's dopamine released. Dopamine is the "I want that" chemical that our brain releases, and it's highly involved in addiction and anticipation.

So, when the monkey smells the banana, they "want it", and dopamine washes over their brain. As they start to eat the banana sushi, their mouth waters, and so the pathway from 'smell' to 'water' gets stronger. The banana-hating chef smells the nigiri, and goes and does something else, like try to make up with his son or whatever. There's no dopamine and no strengthening of pathways.

Okay, so how does this relate to memories? Well, memories are basically just more complex versions of that. A certain series of events (a conversation, a radio comedy show, whatever) causes you to think of a song, and that primes the synaptic pathways. Maybe you're trying to pick out a hummed tune, and so you've got a clear sense of the melody in your mind. Maybe you're about to sing it in a choir with your classmates, and so you start to remember some notes. All of these are different synaptic pathways that are able to be activated from the initial prompt of "song." It's because you've heard it so many times, or hummed it to yourself so often that you're able to recall those exact details. The same pathways that light up when you physically listen to it are being 'refired' when you recall the melody, because those connections have been strengthened over years.

That's long term memory. There's also short term memory, and this is I think what you are referring to. You walk into the cell, you know you came in to look for something, but you have no idea what. That's because the object (or the idea to grab it) was stored in your short term memory. You were just retrieving a sword of lightning from the mountain, this isn't something that you do every day, and so it's not 'stored' in your brain. Instead, it's in short term memory, which can generally 'hold' about 5 or 6 objects at a time (this is why when someone gives you a phone number you have to say it over and over to keep it in your head until you can write it down).

The idea stays in short term memory for as long as you're thinking about it, but if you stop (let's say because you notice that the robot in your cell has vanished and you're not sure where he went, but it's also getting late and you have to try to fight that guy tomorrow) then the idea fades. You'll walk into the cell, because that's a preplanned set of actions that you do all the time, your brain just set it as 'fire and forget', but you won't be able to remember why you came in, because it hasn't yet been stored anywhere.

So how do you remember what it was? You retrace your steps! This is because whatever the idea was came to you based on something you saw or thought about. So you go back to the kitchen and see the unground sesame seeds, and it comes back to you!

The bottom line is this: all memories (muscle memory, explicit memory, smell-associations, etc) are stored in our brains through repeated use. This is called 'Keitzian Theory', and often summarised as "neurons that fire together wire together". It's probably a simplification, but it's the foundation of most of our understanding of memory and how the brain organises itself.

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