  Motivated by the Hodgkin-Huxley model of neuronal dynamics, we study explicit
numerical integrators for "conditionally linear" systems of ordinary
differential equations. We show that splitting and composition methods, when
applied to the Van der Pol oscillator and to the Hodgkin-Huxley model, do a
better job of preserving limit cycles of these systems for large time steps,
compared with the "Euler-type" methods (including Euler's method, exponential
Euler, and semi-implicit Euler) commonly used in computational neuroscience,
with no increase in computational cost. These limit cycles are important to
preserve, due to their role in neuronal spiking. Splitting methods even compare
favorably to the explicit exponential midpoint method, which is twice as
expensive per step. The second-order Strang splitting method is seen to perform
especially well across a range of non-stiff and stiff dynamics.
