Efficient implicit solvers for models of neuronal networks

We introduce economical versions of standard implicit ODE solvers that are specifically tailored for the efficient and accurate simulation of neural networks. The specific versions of the ODE solvers proposed here, allow to achieve a significant increase in the efficiency of network simulations, by reducing the size of the algebraic system being solved at each time step, a technique inspired by very successful semi-implicit approaches in computational fluid dynamics and structural mechanics. While we focus here specifically on Explicit first step, Diagonally Implicit Runge Kutta methods (ESDIRK), similar simplifications can also be applied to any implicit ODE solver. In order to demonstrate the capabilities of the proposed methods, we consider networks based on three different single cell models with slow-fast dynamics, including the classical FitzHugh-Nagumo model, a Intracellular Calcium Concentration model and the Hindmarsh-Rose model. Numerical experiments on the simulation of networks of increasing size based on these models demonstrate the increased efficiency of the proposed methods.