Mathematical modeling of neural networks: Bridging the gap between mathematics and neurobiology

This article explains how mathematical modeling is used in neural networks with much focus on artificial neural networks or ANN and the biological neural system. It offers an introduction to the objects defining the type of model – architecture of neural networks, the mathematical models of neuron behavior – and learning algorithms used for training. The article consolidates the progress and issues surrounding the enhancement of neural network usability toward more biological realism about artificial models and their biological counterparts. It further goes to the new methodologies, including neuromorphic computing, the hybrid model, and the ethical issues of AI. Using examples of particular cases and calculations in the article, the authors show examples of practical application and further research to address the gap between theory and practice. The conclusions made in this work stress the need for collaboration and integration of multiple fields and approaches in the development of neural nets and their adoptions.