Linear Integrate-and-Fire (LIF) Neurons

  • Christoph Börgers
Part of the Texts in Applied Mathematics book series (TAM, volume 66)


Nearly half a century before Hodgkin and Huxley, in 1907, Louis Édouard Lapicque proposed a mathematical model of nerve cells. Lapicque died in 1952, the year when the famous series of papers by Hodgkin and Huxley appeared in print. Lapicque’s model is nowadays known as the integrate-and-fire neuron. We will refer to it as the LIF neuron. Most authors take the L in “LIF” to stand for “leaky,” for reasons that will become clear shortly. We take it to stand for “linear,” to distinguish it from the quadratic integrate-and-fire (QIF) neuron discussed in Chapter  8 The LIF neuron is useful because of its utter mathematical simplicity. It can lead to insight, but as we will demonstrate with examples in later chapters, reduced models such as the LIF neuron are also dangerous — they can lead to incorrect conclusions.

Supplementary material (166 kb)
(ZIP 166 KB)


  1. [80]
    E. M. Izhikevich, Resonate-and-fire neurons, Neural Networks, 14 (2001), pp. 883–894.Google Scholar
  2. [81]
    E. M. Izhikevich,, Simple model of spiking neurons, IEEE Transactions on Neural Networks, 14 (2003), pp. 1569–1572.Google Scholar
  3. [131]
    H. G. Rotstein, Subthreshold amplitude and phase resonance in models of quadratic type: nonlinear effects generated by the interplay of resonant and amplifying currents, J. Comp. Neurosci., 38 (2015), pp. 325–354.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Christoph Börgers
    • 1
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

Personalised recommendations