Skip to main content
  • Poster presentation
  • Open access
  • Published:

The transfer function of the LIF model: from white to filtered noise

The theory describing correlated activity emerging in recurrent networks relies on the single neuron response to a modulation of its input, i.e. the transfer function. For the leaky integrate-and-fire neuron model exposed to unfiltered synaptic noise the transfer function can be derived analytically [1, 2]. In this context the effect of synaptic filtering on the response properties has also been studied intensively at the beginning of the last decade [3, 4]. Analytical results were derived in the low as well as in the high frequency limit. The main finding is that the linear response amplitude of model neurons exposed to filtered synaptic noise does not decay to zero in the high frequency limit. A numerical method has also been developed to study the influence of synaptic noise on the response properties [5]. Here we first revisit the transfer function for neuron models without synaptic filtering and simplify the derivation exploiting analogies between the one dimensional Fokker-Planck equation and the quantum harmonic oscillator. We treat the problem of synaptic filtering with short time constants by reducing the corresponding two dimensional Fokker-Planck equation to one dimension with effective boundary conditions [6]. To this end we use the static and dynamic boundary conditions derived earlier by a perturbative treatment of the arising boundary layer problem [4]. Finally we compare the analytical results to direct simulations (Fig.1) and observe that the approximations are valid up to frequencies in the gamma range (60-80 Hz). Deviations are explained by the nature of the approximations.

Figure 1
figure 1

A Linear response amplitude for neurons exposed to colored (red) and white (black) noise. Simulations (dots) and analytical results (curves). B Phase shift of linear response.


  1. Brunel N, Hakim V: Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 1999, 11 (7): 1621-1671. 10.1162/089976699300016179.

    Article  CAS  PubMed  Google Scholar 

  2. Lindner B, Schimansky-Geier L: Transmission of noise coded versus additive signals through a neuronal ensemble. Phys Rev Lett. 2001, 86: 2934-2937. 10.1103/PhysRevLett.86.2934.

    Article  CAS  PubMed  Google Scholar 

  3. Brunel N, Chance FS, Fourcaud N, Abbott LF: Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett. 2001, 86 (10): 2186-2189. 10.1103/PhysRevLett.86.2186.

    Article  CAS  PubMed  Google Scholar 

  4. Fourcaud N, Brunel N: Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput. 2002, 14: 2057-2110. 10.1162/089976602320264015.

    Article  PubMed  Google Scholar 

  5. Richardson MJE: Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive. Phys Rev E. 2007, 76 (2 Pt 1): 021919-

    Article  Google Scholar 

  6. Klosek MM, Hagan PS: Colored noise and a characteristic level crossing problem. J Math Phys. 1998, 39: 931-953. 10.1063/1.532362.

    Article  Google Scholar 

Download references


Partially supported by HGF Nachwuchsgruppe VH-NG-1028, the Helmholtz Association: HASB and portfolio theme SMHB, the Jülich Aachen Research Alliance (JARA), the Next-Generation Supercomputer Project of MEXT, EU Grant 269921 (BrainScaleS), and EU Grant 604102 (Human Brain Project, HBP).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jannis Schuecker.

Rights and permissions

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver ( applies to the data made available in this article, unless otherwise stated.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schuecker, J., Diesmann, M. & Helias, M. The transfer function of the LIF model: from white to filtered noise. BMC Neurosci 15 (Suppl 1), P146 (2014).

Download citation

  • Published:

  • DOI: