The blockchain essentially is a ledger, an accounting system. In order for that to work in a decentralised manner it's vital that each node have a complete copy of the data store and its history.
The brain on the other hand isn't an accounting system. It's decentralised, too but its nodes - the neurons - serve a very different purpose than Blockchain nodes. They only need to 'know' about their immediate environment, i.e. the neurons around them. Therefore not only doesn't each neuron need to have a complete copy of the data available but it would rather be tremendously inefficient to do so.
In a brain however, we may consider the neurons to be ideal nodes which will never act fraudulently. So if some data is stored on a neuron, all other neurons can completely trust that the data will not be tampered (due to lack of any motivation for a neuron to act fraudulently).
That means we could skip the blockchain database part completely because 'trust' is naturally present in the network.
Also now I'm not sure if there remains any further scope for us to apply the idea of decentralized operations to the functioning of the brain's neural network. Do point out if any ideas pop into your mind.
On the other hand, neurons aren't ideal nodes that never act fraudulently. There's plenty of false information or misfiring neurons in the brain. The brain is very good at error correction and heuristics, however. So, 'fraudulent' nodes aren't necessarily a problem.
The main idea of the blockchain is that each block has a hash of the previous block that has a hash of the previous block that has a hash of the previous block ... so you get a chain of blocks.
Also, each mining node has the complete information of the previous block and all the new transactions, so it can calculate the hash of the previous block and a hash of the new data and create the new block.
Neurons have very few data available. They can sum, rest and make a small amount of delay and accumulation, but each neuron doesn't know the state of the whole brain and has no computational power to make a hash of it anyway.