A predictor-corrector algorithm for reaction-diffusion equations associated with neural activity on branched structures

In this paper, we present a numerical method for solving reaction-diffusion equations on one-dimensional branched structures. The method effectively decouples the many branches at the nodal junctions so that the equations can be solved as a system of smaller problems that are purely tridiagonal. This strategy enables a number of improvements. In particular, difficulties associated with implicit methods for closed loops in the network are eliminated, and spatial adaptivity can be easily implemented. We apply this method to the electrical activity of a neuron. A neuron can be effectively represented as a one-dimensional branched structure, and the model equations, based on the Hodgkin-Huxley cable equations, are a set of reaction equations coupled to a single diffusion process. Spatially adaptive methods are particularly effective for neuronal simulations as the region of activity is often very localized in space. Since the algorithm scales with activity instead of network size, it will make a substantial reduction in computational cost over existing methods.