Preprint available

[Anthony BURKITT]

The following paper on the analysis of integrate and fire 
neurons has been accepted for publication in Neural Computation
and is available now from my web page:
http://www.medoto.unimelb.edu.au/people/burkitta/poisson.ps.zip

"Calculation of interspike intervals for integrate and fire neurons
with Poisson distribution of synaptic inputs"

A. N. Burkitt and G. M. Clark

Abstract:
In this paper we present a new technique for calculating the 
interspike intervals of integrate and fire neurons.   There are 
two new components to this technique.   Firstly, the probability 
density of the summed potential is calculated by integrating 
over the distribution of arrival times of the afferent postsynaptic 
potentials (PSPs), rather than using conventional stochastic 
differential equation techniques.  A general formulation of this 
technique is given in terms of the probability distribution of the 
inputs and the time course of the postsynaptic response.  The 
expressions are evaluated in the Gaussian approximation, which 
gives results that become more accurate for large numbers of 
small amplitude PSPs.  Secondly, the probability density of 
output spikes, which are generated when the potential reaches 
threshold, is given in terms of an integral involving a conditional 
probability density.  This expression is a generalization of the 
renewal equation, but it holds for both leaky neurons and for 
situations in which there is no time-translational invariance.
The conditional probability density of the potential is calculated 
using the same technique of integrating over the distribution 
of arrival times of the afferent PSPs.  For inputs with a Poisson 
distribution the known analytic solutions for both the perfect 
integrator model and the Stein model (which incorporates 
membrane potential leakage) in the diffusion limit are obtained.
The interspike interval distribution may also be calculated 
numerically for models which incorporate both membrane 
potential leakage and a finite rise time of the postsynaptic 
response.  Plots of the relationship between input and output 
firing rates as well as the coefficient of variation are given, and 
inputs with varying rates and amplitudes, including inhibitory 
inputs, are analyzed.   The results indicate that neurons 
functioning near their critical threshold, where the inputs are 
just sufficient to cause firing, display a large variability in their 
spike timings.

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Anthony N. Burkitt
The Bionic Ear Institute                 
384-388 Albert Street
East Melbourne, VIC 3002
Australia
   
Email:  a.burkitt at medoto.unimelb.edu.au 
http://www.medoto.unimelb.edu.au/people/burkitta      
Phone: +61 - 3 - 9283 7510
Fax:     +61 - 3 - 9283 7518                                             
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