Elsevier

Neurocomputing

Volumes 58–60, June 2004, Pages 47-52
Neurocomputing

Approximating the response-stimulus correlation for the integrate-and-fire neuron☆

https://doi.org/10.1016/j.neucom.2004.01.021Get rights and content

Abstract

The reverse correlation and its analytical counterpart, the response-stimulus correlation, describe the expected stimulus just before a present response. With growing complexity of the neuron model, calculating the response-stimulus correlation becomes more and more difficult. We present an approximation of this measure for the integrate-and-fire neuron with reversal potentials and introduce some of the calculations involved.

Introduction

The response-stimulus correlation (RSC) is a function describing the probability of a stimulus spike to occur before a response spike and hence is a measure for the significance of the response. In this paper an approximation of the RSC and an introduction to some of the involved calculations are presented. The paper is structured the following way: after this introduction a short comment on the significance of RSC and reverse correlation as well as a brief scetch of the taken approach are made. In the model section the calculations are shown in more detail and an expression for the approximated reverse correlation is obtained for the integrate-and-fire neuron with reversal potentials driven by white noise. In the results section the quality of the approximation is demonstrated and in the last section the main results are summed up and some implications and open questions are addressed.

A neuron has fired a response spike at a certain time t0=0. What is the probability that the neuron has received a stimulus spike at Δt−t0? This probability, expressed as a function of Δt, is the reverse correlation. It can be obtained experimentally by averaging the time cause of a measured stimulus before a response spike, and it is used to determine the kind of stimulus to which a neuron is sensitive. Because the term “reverse correlation” has a fixed connotation hinting at an experimental context we will use the term RSC if we refer to a solely analytical background. Otherwise the terms are synonymous.

The RSC is an important factor in the inner workings of a neuron. As well as determining the weight change in spike-timing based learning mechanisms like STDP [6] it is involved in shaping the frequency-dependent transfer function of a neural population [4]. It is a general measure for the significance of a response spike, and can be interpreted as a measure for the sensitivity of the responding neuron to variations of its stimulus in amplitude and time [2]. To our knowledge there has been only one notable attempt to obtain the reverse correlation analytically [3], using a SRM model with a population approach.

The suggested approximation is derived the following way: the Itô stochastic differential equation (SDE) for the free membrane potential is solved and the expectation of the solution is given. Simulations show that the expected free membrane potential is time-symmetric. Using this symmetry the flow of the potential before a spike is followed backwards in time, and the mean conditional conductance is extracted. From this expected conductance, a measure for the number of stimulus spikes to be expected at ±Δt can be obtained, which is equivalent to the RSC.

Section snippets

Derivation of the RSC

An integrate-and-fire neuron with reversal potentials is used which is expressed as a differential equation [4]. If the synaptic conductance incorporates some form of noise, this differential equation becomes a SDE. To be consistent with the usual notation for SDEs and to denote the noisyness of the increments, the stochastic variable Gti is introduced. Let gisyn be the sum of all single-synapse conductances with the same reversal potential vi, then Gti describes the cummulative conductance of

Results

Simulations show the time-symmetry of the conditional membrane potential (Fig. 1A). The main difference of the thresholded voltage and the free voltage is generated near the threshold. The threshold acts a probability drain which distorts the flow of the conditional voltage in its vicinity. Therefore at the moment just before the response the error of the approximation is at maximum (compare Figs. 1B and C). Obviously the general shape of the RSC is created by the mean flow of the

Summary

An Itô calculus approach was used to derive an explicit solution of the membrane equation (Eq. (2)) and its expectation (Eq. (3). The conditional conductance can be stated independently from the noise model provided Gt is a semimartingale (Eq. (7)), and the RSC can be given explicitly for the case where synaptic conductances are modelled by white noise (Eq. (8)). An example was simulated numerically to demonstrate the quality of the approximation (Fig. 1).

The advantage of the Itô approach is

References (6)

  • R. Baddeley et al.

    Responses of neurons in primary and inferior temporal visual cortices to natural scenes

    Proc. Roy. Soc.

    (1997)
  • P. Dayan et al.

    Theoretical Neuroscience

    (2001)
  • W. Gerstner

    Coding properties of spiking neuronsreverse and cross-correlations

    Neural Networks

    (2000)
There are more references available in the full text version of this article.

Cited by (5)

☆

Supported by Wellcome Trust (061113/Z/00) and DFG (SFB 618).

View full text
Copyright © 2004 Published by Elsevier B.V.