PhD Defense: Structure preserving numerical and statistical methods for stochastic differential equations with a focus on neuronal models

Structure preserving numerical and statistical methods for stochastic differential equations with a focus on neuronal models

The human brain is a highly complex interconnected dynamical system. Understanding how information is generated, processed and transmitted in this system is one of the major goals in the field of neuroscience. The aim to explain and reproduce neuronal data is reflected in an increasing demand for developing and analysing interpretable mathematical models. Taking into account unpredictable forces and the effect of noise, stochastic differential equations (SDEs) have become an established and powerful modelling tool in this field. This cumulative thesis presents several contributions to the numerical and statistical analysis of SDEs, including various biologically relevant models describing neuronal dynamics. It consists of an introductory essay followed by four scientific articles. A particular focus of this thesis lies on the construction and analysis of structure preserving numerical methods for SDEs which are capable to reproduce important structural properties of their solutions. We develop several numerical methods based on the splitting approach. The strength of the proposed methods lies in the composition of explicitly solved subparts of the SDE, through which the preservation of underlying properties can be guaranteed. In particular, we prove the preservation of hypoellipticity, ergodicity, moments, moment bounds and boundary properties. In addition, a special emphasis of this thesis is laid on the investigation of the interplay of numerical and statistical methods. This means that we draw attention to the importance of structure preservation when using numerical methods as essential building blocks in the construction of statistical inference tools. In particular, we develop a structure preserving approximate Bayesian computation method to estimate the parameters of SDEs. The strength of this method lies in the combination of summary statistics, which are based on the structural property of an underlying invariant distribution, and structure preserving splitting methods for the simulation of huge amounts of required synthetic datasets. The use of structure preserving numerical methods makes the proposed inference algorithm reliable, computationally feasible and enables successful parameter estimation from both simulated and real electroencephalography (EEG) data. In contrast to the proposed splitting methods, frequently applied numerical methods, such as the Euler-Maruyama discretisation, fail to preserve structural properties of solutions of SDEs in general. Consequently, they may yield ill-conditioned or computationally infeasible inference methods.

Zoom-Meeting: https://jku.zoom.us/j/93078731593?pwd=WXo0aG43M0d1L2ZoUWpFdzR6UWdiZz09

Meeting-ID: 930 7873 1593

Passwort: 090520