An Analysis Of The Parallel Computation Of Arbitrarily Branched Cable Neuron Models

We present and analyze a parallel method for the solution of partial differential equation models of the nervous system. These models mathematically are one-dimensional nonlinear parabolic equations defined on branching domains. Implicit methods for these equations leads to numerical solution of diagonally dominant almost tridiagonal linear systems at each time step. We first review some exact methods for the solution of these linear systems that includes an Exact Domain Decomposition. This EDD leads to the solution of many tridiagonal linear systems one for each branch. The sizes of these systems is equal to the number of grid points on the branch. Since the branches of realistic neurons vary widely in size, the decomposition leads to a very poor a priori load balance. This problem may be solved with the Overlapped Partition Method, a method for decomposing large diagonally dominant tridiagonal systems. We describe and analyze an algorithm based on EDD and OPM that can be load balanced.