Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications

Mathematical modeling is a necessary step in understanding real world phenomena after experimental data are obtained and analyzed. Though much has been done since the first mathematical model of a neuron was written by Hodgkin and Huxley in 1952 [1], we are still at the very beginning of the road. The main problem of neuron modeling is that neurons in the brain seldom work separately. Most brain rhythms are the result of collective dynamics of large neuron ensembles, the complexity of which exceeds our capabilities even for computational modeling (simulations), with analytical approaches being mostly not applicable. This means that there are few benefits in modeling a single neuron, since in a single neuron model we cannot reproduce regimes typical for biological systems. At the same time, the number of neurons (and connections) in a chordata brain is so large that we cannot reproduce them in computation even using simple models for individual cells.

Therefore, an investigator must always make a compromise: either they can take into account all the properties of individual cells, or modeling enough cells (usually millions or billions) is possible. This leads to a reasonable criticism that the proposed models could be very far from the modeled phenomenon. Additionally, there is no strict criterion for whether any new model provides additional progress in understanding the phenomenon or not. But one cannot stop research in the field. First, because people want to investigate and will do this regardless of whether it seems to be reasonable or not. Second, because biologists demand models as a step to test and summarize their hypotheses about brain functioning. Third, because we still hope to construct artificial intelligence systems similar to biological ones. Fourth, because medical doctors need a deeper understanding how the brain works to treat patients. So, we decided to follow the ancient wisdom written by Seneca in his philosophical letters: “Ducunt volentem fata, nolentem trahunt” [2], and provided a medium to make progress in this complex field as much as it can be done.

In this issue we collected papers of different researchers from different countries dedicated to some aspects of neuron and brain modeling. To provide easy and multidisciplinary communication, we did not make strict frames. Most works are computational studies, but some of them provide analytical results as well. We hope that it was helpful and useful as another brick in the wall of brain research we all have been constructing.