Parallelizing Hines Matrix Solver in Neuron Simulations on GPU

Hines matrices arise in the simulations of mathematical models describing initiation and propagation of action potentials in a neuron. In this work, we exploit the structural properties of Hines matrices and design a scalable, linear work, recursive parallel algorithm for solving a system of linear equations where the underlying matrix is a Hines matrix, using the Exact Domain Decomposition Method (EDD). We give a general form for representing a Hines matrix and use the general form to prove that the intermediate matrix obtained via the EDD has the same structural properties as that of a Hines matrix. Using the above observation, we propose a novel decomposition strategy called fine decomposition which is suitable for a GPU architecture. Our algorithmic approach R-FINE-TPT based on fine decomposition outperforms the previously known approach in all the cases and gives a speedup of 2.5x on average for a variety of input neuron morphologies. We further perform experiments to understand the behaviour of R-FINE-TPT approach and show its robustness. We also employ a machine learning technique called linear regression to effectively guide recursion in our algorithm.