Neurostimulation continues to demonstrate tremendous success as an intervention for neurodegenerative diseases, including Parkinson's disease, in addition to a range of other neurological and psychiatric disorders. In an effort to enhance the medical efficacy and comprehension of this form of brain therapy, modeling and computational simulation are regarded as valuable tools that enable in silico experiments for a range of neurostimulation research endeavours. To fully realize the capacities of neurostimulation simulations, several areas within computation and numerics need to be considered and addressed. Specifically, simulations of neurostimulation that incorporate (i) computational efficiency, (ii) application versatility, and (iii) characterizations of cellular-level electrophysiology would be highly propitious in supporting advancements in this medical treatment.

The focus of this dissertation is on these specific areas. First, preconditioners and iterative methods for solving the linear system of equations resulting from finite element discretizations of partial differential equation based transcranial electrical stimulation models are compared. Second, a software framework designed to efficiently support the range of clinical, biomedical, and numerical simulations utilized within the neurostimulation community is presented. Third, a multiscale model that couples transcranial direct current stimulation administrations to neuronal transmembrane voltage depolarization is presented. Fourth, numerical solvers for solving ordinary differential equation based ligand-gated neurotransmitter receptor models are analyzed.

A fundamental objective of this research has been to accurately emulate the unique medical characteristics of neurostimulation treatments, with minimal simplification, thereby providing optimal utility to the scientific research and medical communities. To accomplish this, numerical simulations incorporate high-resolution, MRI-derived three-dimensional head models, real-world electrode configurations and stimulation parameters, physiologically-based inhomogeneous and anisotropic tissue conductivities, and mathematical models accepted by the brain modeling community. It is my hope that this work facilitates advancements in neurostimulation simulation capabilities, and ultimately helps improve the understanding and treatment of brain disease.