Scaling symmetry of the Hodgkin-Huxley mathematical model

A computational method based on the application of a scaling transformation to the Hodgkin-Huxley mathematical model of the biological neuron and its generalizations to the case of various neural systems is proposed. The method makes it possible: 1) to obtain invariants of the scaling transformation of the Hodgkin-Huxley model; 2) to find self-similarity regions corresponding to the modes of neural activity; 3) to determine the conditions for the similarity of models of various types of biological neurons; 4) to reduce the order of the system of differential equations of the Hodgkin-Huxley model for neural systems with synchronous activity. The capabilities of the method are demonstrated using various examples from neurobiology.