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Re:Artificial Neural Networks for Beginners by strong_ai on Thursday September 17, 2015 @04:22PM (#10235824)
ANNs have been proven to be universal approximators (https://en.wikipedia.org/wiki/Universal_approximation_theore...) which I think is what you meant when you said 'Turing Complete'.
Re:Artificial Neural Networks for Beginners by moron4hire on Thursday September 17, 2015 @04:27PM (#10235850)
There is also "Turing Computability With Neural Nets" (Seigelmann, Sontag, 1991 http://www.sciencedirect.com/science/article/pii/08939659919...)

    This paper shows the existence of a finite neural network, made up of sigmoidal 
    neurons, which simulates a universal Turing machine. It is composed of less than
    10^5 synchronously evolving processors, interconnected linearly. High-order
    connections are not required.