Skip to main content
Help us understand how you use our websites. Take part in our 30 minute study now.

Response of integrate-and-fire neurons to noisy inputs filtered by synapses with arbitrary timescales: firing rate and correlations


Delivery of neurotransmitter produces on a synapse a current that flows through the membrane and gets transmitted into the soma of the neuron, where it is integrated. The decay time of the current depends on the synaptic receptor's type, and ranges from a few (e.g. AMPA receptors) to a few hundred milliseconds (e.g. NMDA receptors). The role of the variety of synaptic timescales, several of them coexisting in the same neuron, is at present not understood. A prime question to answer is which is the effect of temporal filtering at different timescales of the incoming spike trains on the neuron's response.


Here, based on our previous work on linear synaptic filtering [13], we build a general theory for the stationary firing response of integrate-and-fire (IF) neurons receiving stochastic inputs filtered by one, two or multiple synaptic channels each characterized by an arbitrary timescale. The formalism applies to arbitrary IF model neurons, and to arbitrary forms of input noise (i.e., not required to be Gaussian or to have small amplitude), as well as to any form of synaptic filtering (linear or non-linear). The theory determines with exact analytical expressions the firing rate of an IF neuron for long synaptic time constants using the adiabatic approach. The correlated spiking (cross-correlations function) of two neurons receiving common as well as independent sources of noise is also described (see figure 1). The theory is exemplified using leaky, quadratic and noise thresholded IF neurons (LIF, QIF, NTIF). Although the adiabatic approach is exact when at least one of the synaptic timescales is long, it provides an excellent prediction of the firing rate even when the timescales of the synapses are comparable to that of the leak of the neuron; it is not required that the synaptic time constants are longer than the mean inter-spike-intervals. The distribution of the potentials for general IF neurons is also characterized.


Our results provide very powerful analytical tools that can allow a quantitative description of the dynamics of neuronal networks with realistic synaptic dynamics.

Figure 1

A. Two spiking neurons receiving correlated/common inputs fire in a correlated manner. B: The cross-correlated output spike trains of the two neurons are analytically described.


  1. 1.

    Moreno-Bote R, Parga N: Simple model neurons with slow synaptic filters can operate as detectors of rare events. Physical Review Letters. 2004, 92: 028102-10.1103/PhysRevLett.92.028102.

    PubMed  Article  Google Scholar 

  2. 2.

    Moreno-Bote R, Parga N: Membrane potential and response properties of populations of cortical neurons in the high conductance state. Physical Review Letters. 2005, 94: 088103-10.1103/PhysRevLett.94.088103.

    PubMed  Article  Google Scholar 

  3. 3.

    Moreno-Bote R, Parga N: Auto and cross-correlograms for the spike response of LIF neurons with slow synapses. Physical Review Letters. 2006, 96: 028101-10.1103/PhysRevLett.96.028101.

    PubMed  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Rubén Moreno-Bote.

Rights and permissions

Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Moreno-Bote, R., Parga, N. Response of integrate-and-fire neurons to noisy inputs filtered by synapses with arbitrary timescales: firing rate and correlations. BMC Neurosci 10, P237 (2009).

Download citation


  • Firing Rate
  • AMPA Receptor
  • Spike Train
  • Input Noise
  • Powerful Analytical Tool