The Poisson point process

General

The Poisson process was discovered by Simeon-Denis Poisson (1781-1840) and describes a statistic point process of single events which occur ramdom in time.
An example for a possion process is the decay of some types of radioactive isotopes.
The Poisson distribution is given by

$$p(k_i\vert\lambda) = \frac{1}{k_i!} \lambda^{k_i} e^{-\lambda}.$$

$p(k_i\vert\lambda)$ denotes the discret propability that $k_i$ occurs given the expectation value $\lambda$. For a example how the Poisson distribution looks like for different values of $\lambda$ see the following graph.

The joint probability distribution of n independent Poisson processes is

$$p( \vec{k} \vert\lambda) = \prod_{i=1}^{n} p(k_i\vert\lambda)$$

if $\vec{k}=\{ k_1,..., k_n \}$.

The Poisson process as biological approximation

The examination of the cortex showed that the neural response-properties are highly variable [1,2,3,5,6]. The observed interspike-interval-distribution [4] looks like the exponential interevent-distribution (the interevent-distribution defines the propability of the time-interval-length between two events) of the Poisson process.

The Poisson process and the tuning-function

The tuning-functions $f(x)$ and the Poisson process are connected through the expectation value $\lambda$. The tuning-function multiplied with the size of the timewindow $T$ is used as the mean spikerate $\lambda$ for the Poisson process.

$$\lambda = f(x)T$$

Literature

[1] Britten KH, Shadlen MN, Newsome WT, Movshon JA (1993)
Responses of neurons in macaque MT to stochastic motion signals.
Vis Neurosci 10:1157-1169

[2] Burns BD, Webb AC (1976)
The spontaneous activity of neurones in the cat's cerebral cortex.
Neuron 20:959-969

[3] Snowden RJ, Treue S, Andersen RA (1992)
The response of neurons in areas V1 and MT of the alert rhesus monkey to moving random dot patterns.
Exp Brain Res 88:389-400

[4] Softky W.R. , Koch C. (1993)
The highly irregular fireing of cortical cells is inconsistent with temporal integration of random EPSPs
Journal of Neuroscience, 13:334-350

[5] Tollhurst DJ, Movshon JA, Dean AF (1983)
The statistical reliability of signals in single neurons in cat and monkey visual cortex.
Vision Res 23:775-785

[6] Tomko, G., Crapper, D. (1974)
Neuronal variability: non-stationary responses to identical visual stimuli.
Brain Res 79:405-418