Mathematical Modeling of Artificial Neural Networks

Models and algorithms have been designed to mimic information processing and knowledge acquisition of the human brain generically called artificial or formal neural networks (ANNs), parallel distributed processing (PDP), neuromorphic or connectionist models. The term network is common today: computer networks exist, communications are referred to as networking, corporations and markets are structured in networks. The concept of ANN was initially coined as a hopeful vision of anticipating artificial intelligence (AI) synthesis by emulating the biological brain. ANNs are alternative means to symbol programming aiming to implement neural-inspired concepts in AI environments (neural computing) (Hertz, Krogh, & Palmer, 1991), whereas cognitive systems attempt to mimic the actual biological nervous systems (computational neuroscience). All conceivable neuromorphic models lie in between and supposed to be a simplified but meaningful representation of some reality. In order to establish a unifying theory of neural computing and computational neuroscience, mathematical theories should be developed along with specific methods of analysis (Amari, 1989) (Amit, 1990). The following outlines a tentatively mathematical-closed framework in neural modeling.