  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.
