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. The second-order Strang splitting method is seen to perform especially well across a range of non-stiff and stiff dynamics.2010 Mathematics Subject Classification. 65P30, 37M20, 92C20. 1 It is straightforward to generalize what follows to the case where xi and bi are vector-valued and ai is matrix-valued, so that xi satisfies a first-order linear system of ODEs with constant coefficients when xj is stationary for j = i, but the scalar case covers the applications we are interested in.
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