A stochastic version of the Jansen and Rit neural mass model: analysis, numerics and filtering

In this thesis we consider the Jansen and Rit neural mass model which provides a useful framework for modelling mesoscopic neural dynamics. We formulate a stochastic version of it which arises by incorporating random input and has the structure of a damped stochastic Hamiltonian system with nonlinear displacement. We investigate path properties of this system of stochastic ordinary differential equations and establish bounds for the moments of the solution. Moreover, we study the asymptotic behaviour of the model and provide long-time stability results by proving the geometric ergodicity of the system, which means that the system - independently of its initial values - always converges to an invariant measure. We close the first part of this thesis with simulations of the stochastic Jansen and Rit neural mass model using an efficient numerical scheme based on a splitting approach which preserves the qualitative behaviour of the solution. A further goal of this thesis is to use the stochastic Jansen and Rit neural mass model as the underlying dynamics in a nonlinear filtering framework in order to solve the inverse problem and approximate it numerically by a continuous-time particle filter. We take advantage of the efficient and structure-preserving numerical splitting integrator to approximate the continuous-time particle filter and investigate its impact on the results of the filter. As a result, for a given accuracy of the filter the number of discretisation steps can significantly be reduced when using the splitting scheme compared to standard integrators such as the Euler-Maruyama scheme. We complete this thesis with a study about the impact of numerical approximations of the stochastic Jansen and Rit neural mass model on various nonlinear filtering techniques, where we conclude for specific parameter settings of the model that the error originating from an inappropriate numerical time-discretisation scheme cannot be recovered by better approximations of the statistics of the model through more sophisticated filtering techniques.