Abstract. 2024 Nobel Prize in physics was awarded to Hopfield and Hinton for foundational discoveries that enable machine learning with artificial neuron networks. They trained artificial neural networks using physical analogies. However, we believe it is biological rather than physical analogies that help eliminate computational issues related to machine learning, that is, a need to compute huge amounts of coefficients, thus, arriving at practically instant machine learning. This paper provides a concise mathematics of biological analogies-based artificial neural networks. Such networks are referred to as indexing neural networks as their architecture is inspired by a columnar arrangement of the cortical neurons, - an arrangement that points towards an indexing nature of the neo-cortex. This paper provides a concise mathematics of indexing neural networks (indextrons) and its applications to some real-world problems.