Limit theorems for stochastic spatio-temporal models
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Various models in mathematical neuroscience yield for a description of emergent spatio-temporal dynamics, i.e., transmission of signals in models of excitable membranes or spatial models of the activity in the brain. In the literature deterministic models are predominant, although the underlying processes generating this dynamics are best described stochastically. The deterministic modelling approach is understood to capture the averaged dynamics of a large number of individual stochastic events. However, such models may not be able to capture some qualitative dynamics due to the inherent stochasticity, so called 'finite size effects'.
In my talk I present limit theorems that establish a precise mathematical connection between stochastic models and well-established deterministic models as their limit. In particular, we present a Law of Large Numbers and a Central Limit Theorem for Stochastic Hybrid Systems modelling spatially extended structures. In this case, firstly, the Law of Large Numbers provides for large homogeneous structures, e.g., neuronal membranes with a high density and homogeneous distribution of ion channels, a qualitative justification of an approximation by partial differential equations, e.g., the cable equation. Secondly, the Central Limit Theorems provides the basis for a diffusion approximation of hybrid models by stochastic partial differential equations.
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