Journal of Mathematical Biology

, Volume 48, Issue 1, pp 38–56

Dynamics of one-dimensional spiking neuron models


DOI: 10.1007/s00285-003-0223-9

Cite this article as:
Brette, R. J. Math. Biol. (2004) 48: 38. doi:10.1007/s00285-003-0223-9


In this paper we make a rigorous mathematical analysis of one-dimensional spiking neuron models in a unified framework. We find that, under conditions satisfied in particular by the periodically and aperiodically driven leaky integrator as well as some of its variants, the spike map is increasing on its range, which leaves no room for chaotic behavior. A rigorous expression of the Lyapunov exponent is derived. Finally, we analyse the periodically driven perfect integrator and show that the restriction of the phase map to its range is always conjugated to a rotation, and we provide an explicit expression of the invariant measure.


Neuron models Integrate-And-Fire Leaky integrator Perfect integrator Rotation number Phase-locking 

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Centre de Mathématiques et de Leurs ApplicationsEcole Normale Supérieure de CachanCachanFrance
  2. 2.INSERM U483Université Pierre et Marie CuriePARISFrance

Personalised recommendations