Kinetics of diffusion in a spherical cell. i. no solute buffering

Realistic neuron models that involve effects of concentration changes of second messengers on ion channels must include processes such as diffusion and solute buffering. These processes, which span a wide range of spatial and temporal scales, may impose a severe computational burden. In this paper and its companion, we examine the kinetics of diffusion and present methods for stimulating it accurately and efficiency. The problem of calcium diffusion in a spherical cell is used as a device to demonstrate the practical application of our analysis. However, the scope of these papers is not limited to this problem. The same analysis that we apply and concerns that we raise are germane to the spread of any second messenger, and can be adapted to other geometries. The focus of this paper is the simplest case: diffusion in the absence of solute buffering. This analysis also applies whenever buffering is so fast that it is instantaneous compared to diffusion, or so slow that concentration gradients have dissipated before substantial buffering takes place. The second paper investigates the more difficult situation where diffusion and buffering occur at comparable rates. In the absence of buffering, concentration changes produced by diffusion can be fit by an infinite series of exponential terms. We show how to design a model with N + 1 compartments that fits the N slowest terms of this series exactly in a shell just inside the cell membrane.