| Ever since Hodgkin and Huxley first presented their model of neuronal activity, numerical simulations have played an important role in the field of neuroscience. Early work in the emerging field of computational neuroscience led to the development of techniques for solving the problem of action potential propagation along cables and through branched structures culminating in the widespread use of the Crank-Nicolson method and ordering scheme developed by Hines and incorporated into the NEURON simulation environment. As the available numerical packages improved, the range and scale of computational simulations continues to grow.;The work presented in this dissertation departs from the conventional methods in many respects. The Crank-Nicolson scheme is abandoned in favor of the more stable Backwards Differentiation Formula, and the computational domain is divided into distinct subdomains using a simple domain decomposition scheme. Each subdomain is then updated independently using an adaptive time stepping scheme with the local time step determined by the level of local activity. In this locally adaptive time stepping scheme, regions experiencing high levels of activity are updated with a small time step, while regions that evolve slowly are updated using a much larger time step. Unlike other applications of adaptive time stepping, where the entire domain is updated using a globally selected time step, the locally adaptive time stepping scheme focuses computational power where it is most needed: the regions of high activity.;The locally adaptive time stepping scheme described in this dissertation is not restricted to the unique problems found in computational neuroscience, but can be easily adapted to any reaction-diffusion system, and extended to higher dimensions. While there is some computational overhead due to the domain decomposition scheme and step size selection, the focused use of computational resources provides sufficient increases in computational speed to compensate, especially for simulations on large spatial domains with highly localized pockets of activity. |
