Bulletin of Mathematical Biology

, Volume 62, Issue 3, pp 467–481 | Cite as

Integrate-and-fire models with nonlinear leakage

  • Jianfeng Feng
  • David Brown


Can we express biophysical neuronal models as integrate-and-fire (IF) models with leakage coefficients which are no longer constant, as in the conventional leaky IF model, but functions of membrane potential and other biophysical variables? We illustrate the answer to this question using the FitzHugh-Nagumo (FHN) model as an example. A novel IF type model, the IF-FHN model, which approximates to the FHN model, is obtained. The leakage coefficient derived in the IF-FHN model has nonmonotonic relationship with membrane potential, revealing at least in part the intrinsic mechanisms underlying the FHN models. The IF-FHN model correspondingly exhibits more complex behaviour than the standard IF model. For example, in some parameter regions, the IF-FHN model has a coefficient of variation of the output interspike interval which is independent of the number of inhibitory inputs, being close to unity over the whole range, comparable to the FHN model as we noted previously (Brown et al., 1999).


Membrane Potential Spike Train Synaptic Input Inhibitory Input Incoming Signal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abbott, L. F., J. A. Varela, K. Sen and S. B. Nelson (1997). Synaptic depression and cortical gain control. Science 275, 220–223.CrossRefGoogle Scholar
  2. Albeverio, S., J. Feng and M. Qian (1995). Role of noises in neural networks. Phys. Rev. E. 52, 6593–6606.MathSciNetCrossRefGoogle Scholar
  3. Brown, D. and J. Feng (1999). Is there a problem matching model and real CV(ISI)? Neurocomputing 26–27, 117–122.Google Scholar
  4. Brown, D. and J. Feng (2000). Low correlation between random synaptic inputs impacts considerably on the output of the Hodgkin-Huxley model. Neurocomputing (accepted).Google Scholar
  5. Brown, D., J. Feng and S. Feerick (1999). Variability of firing of Hodgkin-Huxley and FitzHugh-Nagumo neurons with stochastic synaptic input. Phys. Rev. Lett. 82, 4731–4734.CrossRefGoogle Scholar
  6. Chow, C. C. and J. A. White (1996). Spontaneous action potentials due to channel fluctuations. Biophys. J. 71, 3013–3021.CrossRefGoogle Scholar
  7. Feng, J. (1997). Behaviours of spike output jitter in the integrate-and-fire model. Phys. Rev. Lett. 79, 4505–4508.CrossRefGoogle Scholar
  8. Feng, J. and D. Brown (1998). Impact of temporal variation and the balance between excitation and inhibition on the output of the perfect integrate-and-fire model. Biol. Cybern. 78, 369–376.CrossRefzbMATHGoogle Scholar
  9. Feng, J. and D. Brown (1999). Coefficient or variation greater than 0.5 how and when? Biol. Cybern. 80, 291–297.CrossRefzbMATHGoogle Scholar
  10. Feng, J. and D. Brown (2000a). Impact of correlated inputs on the output of the integrate-and-fire model. Neural Computation 12, 711–732.CrossRefGoogle Scholar
  11. Feng, J. and D. Brown (2000b). A comparison between abstract and biophysical neuron models Proceeding Of Stochaos, American Physics Society, (in press).Google Scholar
  12. Kistler, W. M., W. Gerstner and J. L. van Hemmen (1997). Reduction of the Hodgkin-Huxley equations to a single-variable threshold model. Neural Comput. 9, 1015–1045.CrossRefGoogle Scholar
  13. Koch, C. (1999). Biophysics of Computation, Oxford: Oxford University Press.Google Scholar
  14. Ricciardi, L. M. and S. Sato (1990). Diffusion process and first-passage-times problems, in Lectures in Applied Mathematics and Informatics, L. M. Ricciardi (Ed.), Manchester: Manchester University Press.Google Scholar
  15. Risken, S. (1989). The Fokker-Planck Equation, Berlin: Springer-Verlag.zbMATHGoogle Scholar
  16. Shadlen, M. N. and W. T. Newsome (1994). Noise, neural codes and cortical organization. Curr. Opin. Neurobiol. 4, 569–579.CrossRefGoogle Scholar
  17. Softky, W. and C. Koch (1993). The highly irregular firing of cortical-cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334–350.Google Scholar
  18. Tuckwell, H. C. (1988). Introduction to Theoretical Neurobiology, Vol. 2, Cambridge, UK: Cambridge University Press.Google Scholar

Copyright information

© Society for Mathematical Biology 2000

Authors and Affiliations

  • Jianfeng Feng
    • 1
  • David Brown
    • 1
  1. 1.Computational Neuroscience LaboratoryThe Babraham InstituteCambridgeUK

Personalised recommendations