Applications of Asymptotic Methods: Analyzing Mathematical Models in Neuroscience and Describing Fast Dynamics of a Trajectory in the Vicinity of a Chaotic Attractor
Denis Shchepakin · The Mathematics Enthusiast · 2019
The current dissertation focuses on two unrelated subjects: modeling in Neuroscience applications and Chaos Theory. Neurons are units of the nervous system that receive, conduct, and transmit information to each other and target tissue via electrical signaling. One of the mechanisms of the signal transduction is through signaling molecules called neurotransmitters. Glutamate is the predominant excitatory neurotransmitter in the mammalian and human central nervous system. However, the mechanism of regulation and sensation of the glutamate via glutamate receptors and transporters is not completely understood. We discuss currently existing models of glutamate receptors and transporters, and their main problem: the overparameterization with respect to the existing experimental data. Although this issue prevents statistical reliable parameter estimate, numerous authors still attempt to use them for these means using incorrect methodology. We are able to reduce the existing models under certain assumptions, that are achieved in experiments, designed specifically for this purpose. This approach allows us to avoid the overparameterization issue and for the first time obtained statistically reliable parameter estimates. Chaotic systems do not allow for conventional methods of parameter estimation and had been considered to be an ill-posed problem. Recently a novel promising methodology was proposed. Here we discuss some further development of the technique that brings it closer to practical use.