New on the PHPClasses.org blog there's an article on neural networks written up by Louis Stowasser looking specifically at creating a neural mesh with PHP.

Neural networks are a relatively new technology that aims to reverse engineer the functionality of the brain within a mathematics model. This may sound daunting and complex but the underlying concepts are very simple and Neural Mesh does the hard work for you. In a Web environment NNs (neural networks) are considered too slow and complex to warrant effort on what might even be a trivial task. To solve this, Neural Mesh has been heavily optimized. It uses caching to speed up running and training of networks.

He explains what a neural network is on a base level and talks about it's ability to learn from the training its given (supervised or unsupervised). He also talks about how the networks can be useful and how to use the framework (found here) to create a simple network.