Scale transitions for models with spatial structure
Sprache des Vortragstitels:
Random Models in Neuroscience
Sprache des Tagungstitel:
Models that incorporate spatial dynamics additional to their time evolution arise in various areas of mathematical neuroscience and related fields, e.g., Hodgkin-Huxley type membrane models for axons and in cardiac tissue, Wilson-Cowan / Amari-type neural field models or reaction-diffusion models of chemical reactions system in inhomogeneous media as occurs inside a living cell. Originally, these models were introduced as deterministic equations approximating an inherently noisy (or stochastic) real-world process. Later these models were either extended to incorporate also stochastic effects or a rigorous derivation from smaller scale stochastic models was attempted. These derivation attempts also lead to new classes of models.
In my talk I will present typical mathematical set-ups and corresponding methods useful for rigorously obtaining a limit theorem describing a scale transition in spatio-temporal models. I will discuss the interpretations, implications and possible limitations of these approaches. On the one hand, this will be a review of previous work from the literature. On the other hand I will also present some concrete recent results obtained in our work in connection of scale transitions of discrete Markov Chain models describing excitable membranes or neural fields.
Sprache der Kurzfassung:
Hauptvortrag / Eingeladener Vortrag auf einer Tagung