Abstract
This chapter reviews the theory of stochastic point processes. For simple renewal processes, the relation between the stochastic intensity, the inter-spike interval (ISI) distribution, and the survival probability are derived. The moment and cumulant generating functions and the relation between the ISI distribution and the autocorrelation is investigated. We show how the Fano factor of the spike count distribution depends on the coefficient of variation of the ISI distribution. Next we investigate models of renewal processes with variable rates and CV2, which is often used to assess the variability of the spike train in this case and compare the latter to the CV. The second half of the chapter deals with stochastic point processes with correlations between the intervals. Several examples of such processes are shown, and the basic analytical techniques to deal with these processes are expounded. The effect of correlations in the ISIs on the Fano factor of the spike count and the CV2 are also explored.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andersen P, Borgan O, Gill R, Keiding N (1994) Statistical models based on counting processes. Springer, Berlin
Bagdonavičius V, Nikulin M (2001) Accelerated life models. Chapman & Hall/CRC, Boca Raton
Balakrishnan N, Nevzorov V (2003) A primer on statistical distributions. Wiley, New York
Chhikara R, Folks L (1989) The inverse Gaussian distribution: theory, methodology, and applications. Dekker, New York
Cohen-Tannoudji C, Diu B, Laoë F (1977) Quantum mechanics. Hermann, Paris
Collett D (2003) Modeling survival data in medical research, 2nd edn. Chapman & Hall/CRC, Boca Raton
Cox R (1962) Renewal theory. Methuen, London
Cox R (1975) Partial likelihood. Biometrica 62:269–276
Cox R, Isham V (1980) Point processes. Chapman & Hall/CRC, Boca Raton
Davies B (2002) Integral transforms and their applications, 3rd edn. Springer, New York
Dirac P (1939) A new notation for quantum mechanics. Proc Roy Soc London A 35:416–418
Evans M, Hastings N, Peacock B (2002) Statistical distributions, 3rd edn. Wiley, New York
Farkhooi F, Strube-Bloss M, Nawrot M (2009) Serial correlation in neural spike trains: Experimental evidence, stochastic modeling, and single neuron variability. Phys Rev E 79:021905
Gerstein G, Mandelbrod B (1964) Random walk models for the spike activity of a single cell. Biophys J 4:41–68
Holt G, Softky W, Douglas R, Koch C (1996) A comparison of discharge variability in vitro and in vivo in cat visual cortex neurons. J Neurophysiol 75:1806–1814
Hougaard P (1999) Fundamentals of survival data. Biometrica 55:13–22
McFadden J (1962) On the lengths of intervals in stationary point processes. J Roy Stat Soc B 24:364–382
Spiegel M (1965) Theories and problems of Laplace transforms. McGraw-Hill, New York
Tuckwell H (1988) Introduction to theoretical neurobiology. Vol. 2. Nonlinear and stochastic theories. Cambridge University Press, Cambridge
Wikipedia (2009) Bra-ket notation. World Wide Web electronic publication. http://en.wikipedia.org/wiki/Bra-ket_notation
Wikipedia (2009) Laplace transform. World Wide Web electronic publication. http://en.wikipedia.org/wiki/Laplace_transform
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
van Vreeswijk, C. (2010). Stochastic Models of Spike Trains. In: Grün, S., Rotter, S. (eds) Analysis of Parallel Spike Trains. Springer Series in Computational Neuroscience, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5675-0_1
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5675-0_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5674-3
Online ISBN: 978-1-4419-5675-0
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)