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

Michael J. Shelley, Louis Tao

Research output: Contribution to journalArticlepeer-review


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 Δt = 0.5 × 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.

Original languageEnglish (US)
Pages (from-to)111-119
Number of pages9
JournalJournal of Computational Neuroscience
Issue number2
StatePublished - 2001


  • Accurate time integration schemes
  • Integrate-and-fire networks
  • Numerical methods

ASJC Scopus subject areas

  • Sensory Systems
  • Cognitive Neuroscience
  • Cellular and Molecular Neuroscience


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