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As neurons fire and work together, their connections strengthen. I was wondering if there was a limit on how connected neurons can be. If these neurons were constantly connecting then surely there would be an overload of connections at some point?

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There's several ways neuronal connections can strengthen: growing the spine, increasing the amount of vesicles that carry neurotransmitters at the synapse, sensitizing and increasing the amount of receptors and yes creating more spines and synapses also.

Roughly, neurons can make tens of thousands of connections to multiple neurons the size of each spine is estimated to be between 0.01 µm3 to 0.8 µm3.

You are correct in saying that as life happens the connection would tend to grow to max and not encode anything anymore, however the brain seems to re-normalize all of its connections during sleep through special membrane channels.

Check out: https://en.wikipedia.org/wiki/Neuron (especially the connectivity part)

https://en.wikipedia.org/wiki/Long-term_potentiation

https://en.wikipedia.org/wiki/Long-term_depression

https://en.wikipedia.org/wiki/Ion_channel

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