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Neural networks (or connectionist models or backpropagation models/networks) are computer simulations of neurons in the human brain. This can mean a group of neurons of about 100 neurons up to millions of neurons. The models are not complete representations of the neurons found in the human body, but simplified and math-constrained concepts of those.
Neural networks are used to discover more about the inner workings of the human brain and most often in case of memory/learning simulations. A good book to learn about these is Connectionism and the Mind. However some knowledge about the brain is necessary.
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