Cognitive Neurodynamics

, Volume 2, Issue 1, pp 21–27 | Cite as

The Cauchy problem for one-dimensional spiking neuron models

  • Romain BretteEmail author
Research Article


I consider spiking neuron models defined by a one-dimensional differential equation and a reset—i.e., neuron models of the integrate-and-fire type. I address the question of the existence and uniqueness of a solution on \({\mathbb{R}}\) for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition. I discuss the implications of these mathematical results in terms of neural coding and spike timing precision.


Integrate-and-fire Cauchy problem Spike timing precision Reliability Neuron models 



This work was partially supported by the EC IP project FP6-015879, FACETS, and the EADS Corporate Research Foundation.


  1. Brette R (2004) Dynamics of one-dimensional spiking neuron models. J Math Biol 48(1):38–56PubMedCrossRefGoogle Scholar
  2. Brette R, Gerstner W (2005) Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J Neurophysiol 94:3637–3642PubMedCrossRefGoogle Scholar
  3. Brette R, Guigon E (2003) Reliability of spike timing is a general property of spiking model neurons. Neural Comput 15(2):279–308PubMedCrossRefGoogle Scholar
  4. Destexhe A, Rudolph M and Paré D (2003) The high-conductance state of neocortical neurons in vivo. Nat Rev Neurosci 4:739–751PubMedCrossRefGoogle Scholar
  5. Ermentrout B, Kopell N (1986) Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J Appl Math 46:233–253CrossRefGoogle Scholar
  6. Fourcaud-Trocmé N, Hansel D, van Vreeswijk C, Brunel N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. J Neurosci 23(37):11628–11640PubMedGoogle Scholar
  7. Gerstner W, Kistler WM (2002) Spiking neuron models. Cambridge University PressGoogle Scholar
  8. Hodgkin A, Huxley A (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol (Lond) 117:500–544Google Scholar
  9. Lapicque L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation. J Physiol Pathol Gen 9:620–635Google Scholar
  10. Mainen ZF, Sejnowski, TJ (1995) Reliability of spike timing in neocortical neurons. Science 268:1503–1506PubMedCrossRefGoogle Scholar
  11. Piwkowska Z, Pospischil M, Brette R, Sliwa J, Rudolph-Lilith M, Bal T, Destexhe A (2007) Characterizing synaptic conductance fluctuations in cortical neurons and their influence on spike generation. arXiv:0706.1306v1Google Scholar
  12. Pospischil M, Piwkowska Z, Bal T, Destexhe A (2007) Which model best captures the spiking response of cortical neurons to excitatory inputs? PreprintGoogle Scholar
  13. Touboul J (2007) Bifurcation analysis of a general class of non-linear integrate-and-fire neurons. SubmittedGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Equipe Odyssée (INRIA/ENS/ENPC), Département d’InformatiqueEcole Normale SupérieureParis Cedex 05France

Personalised recommendations