title: Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks.
abstract: To avoid the numerical errors associated with resetting the potential following 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-stepping, a linear interpolant to find spike times, and a recalibration of postspike potential using the spike times. Here we show analytically that such a scheme is second order, discuss the conditions under which efficient, higher-order algorithms can be constructed to treat resets, and develop a modified fourth-order scheme. To support our analysis, we simulate a system of integrate-and-fire conductance-based point neurons with all-to-all coupling. For six-digit accuracy, our modified Runge-Kutta fourth-order scheme needs a time-step of Delta(t) = 0.5 x 10(-3) seconds, whereas to achieve comparable accuracy using a recalibrated second-order or a first-order algorithm requires time-steps of 10(-5) seconds or 10(-9) seconds, respectively. Furthermore, 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 associated with second-order schemes.
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
authors.forename: M J
authors.initials: MJ
authors.lastname: Shelley
authors.forename: L
authors.initials: L
authors.lastname: Tao
mesh_terms: D000200:Action Potentials / Q000502:physiology*; D000465:Algorithms; D000818:Animals; D002490:Central Nervous System / Q000502:physiology*; D019706:Excitatory Postsynaptic Potentials / Q000502:physiology*; D006801:Humans; D008959:Models, Neurological; D009415:Nerve Net / Q000502:physiology*; D009431:Neural Conduction / Q000502:physiology; D009474:Neurons / Q000502:physiology*; D015203:Reproducibility of Results; D009435:Synaptic Transmission / Q000502:physiology*; D013997:Time Factors