Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks

To avoid the numerical errors associated with resetting the potential follo wing a spike in simulations of integrate-and-fire neuronal networks, Hansel et al. and Shelley independently developed a modified time-stepping method . Their particular scheme consists of second-order Runge-Kutta time-steppin g, a linear interpolant to find spike times, and a recalibration of postspi ke potential using the spike times. Here we show analytically that such a s cheme is second order, discuss the conditions under which efficient, higher -order algorithms can be constructed to treat resets, and develop a modifie d fourth-order scheme. To support our analysis, we simulate a system of int egrate-and-fire conductance-based point neurons with all-to-all coupling. F or six-digit accuracy, our modified Runge-Kutta fourth-order scheme needs a time-step of Deltat = 0.5 x 10(-3) seconds, whereas to achieve comparable accuracy using a recalibrated second-order or a first-order algorithm requi res time-steps of 10(-5) seconds or 10(-9) seconds, respectively. Furthermo re, since the cortico-cortical conductances in standard integrate-and-fire neuronal networks do not depend on the value of the membrane potential, we can attain fourth-order accuracy with computational costs normally associat ed with second-order schemes.