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I understand that neural networks model biological neurons. Each node in the network represents a neuron cell and the connections between nodes represent the connections between cells. As in nature, a neuron fires an electrical signal to connected neurons based on some kind of threshold or function that mimics such.

Recent discoveries on how the brain works reveal the importance of calcium within the cells. See http://link.springer.com/article/10.1007/BF01794675 for more information. To summarize, calcium affects the regulation, stimulation and transmission of electrical activity as well as the destruction of neurones.

From my study of neural networks, there does not seem to be a calcium equivalent. Having one would imply that the functions, connections and weights in an artificial network are configured during the training and execution process and can change over time. I understand that back-propagation is used to train the weights, but have not seen anything that trains the function nor the connections (although a zero weight could imply no connection).

Does anyone know of such a network (or training algorithm)? If so, do these networks perform better than a network that is pre-configured?

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Neural networks don't model biological neurons.

They are at best inspired by biological neurons, in that they get excited by certain inputs and fire once the excitation crosses a threshold. And this second point even holds only approximately because the backpropagation algorithm needs smoothed out steps to learn by gradient descent. And backpropagation is not even inspired by biology, that would rather be hebbian learning.

Generally in machine learning people do what works. The connection to biology is tenuous at best. You will usually not find one to one correspondence of low level details between machine learning setups and biological neurons. For that you'll have to turn to brain simulations.

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