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Journal of Computational Neuroscience

, Volume 11, Issue 2, pp 111–119 | Cite as

Efficient and Accurate Time-Stepping Schemes for Integrate-and-Fire Neuronal Networks

  • Michael J. Shelley
  • Louis Tao
Article

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 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.

integrate-and-fire networks accurate time integration schemes numerical methods 

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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Michael J. Shelley
    • 1
    • 2
  • Louis Tao
    • 1
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York
  2. 2.Center for Neural ScienceNew York UniversityNew York

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