Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T21:20:25.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationarity properties of neural networks

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen  and
Tatyana S. Turova
Søren Asmussen*
Affiliation:
University of Lund
Tatyana S. Turova*
Affiliation:
University of Lund
*
∗Postal address: Department of Mathematical Statistics, University of Lund, Box 118, 221 00 Lund, Sweden
∗Postal address: Department of Mathematical Statistics, University of Lund, Box 118, 221 00 Lund, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Keywords

Inhibitionladder height distributionpath decompositionPalm theoryrandom walkqueuing theoryrenewal processwaiting time distribution

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60K05: Renewal theory 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 783 - 794
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

TT was partially supported by Astra Draco.

References

Alsmeyer, G. (1996). Superimposed renewal processes: a Markov renewal approach. Stoch. Proc. Appl. 61, 311–322.CrossRefGoogle Scholar
Asmussen, S. (1987). Applied Probability and Queues, Wiley, Chichester.Google Scholar
Baccelli, F., and Bremaud, P. (1994). Elements of Queuing Theory. Palm–Martingale Calculus and Stochastic Recursions, Springer, Berlin.Google Scholar
Baccelli, F., and Makowski, A. M. (1989). Dynamic, transient and stationary behavior of the M/G/1 queue via martingales. Ann. Prob. 17, 1691–1699.CrossRefGoogle Scholar
Cottrell, M. (1992). Mathematical analysis of a neural network with inhibitory coupling. Stoch. Proc. Appl. 40, 103–126.CrossRefGoogle Scholar
Fricker, C., Robert, P., Saada, E., and Tibi, D. (1994). Analysis of some networks with interaction. Ann. Appl. Prob. 4, 1112–1128.CrossRefGoogle Scholar
Karpelevich, F., Malyshev, V. A., and Rybko, A. N. (1995). Stochastic evolution of neural networks. Markov Proc. Rel. Fields 1, 141–161.Google Scholar
Kryukov, V. I., Borisyuk, G. N., Borisyuk, R. M, Kirillov, A. B. and Kovalenko, Ye. I. (1990). Metastable and unstable states in the brain. In Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, ed. Dobrushin, R. L., Kryukov, V. I. and Toom, A. L. Manchester University Press, pp. 226–357.Google Scholar
Pitman, J. W. (1975). An identity for stopping times of a Markov chain. In Studies in Probability and Statistics, ed. Williams, E. J., North–Holland, Amsterdam.Google Scholar
Rosenkrantz, W. A. (1983). Calculation of the Laplace transform of the length of the busy period of the M/G/1 queue via martingales. Ann. Prob. 11, 817–818.CrossRefGoogle Scholar
Sigman, K. (1994). Stationary Marked Point Processes: An Intuitive Approach, Chapman and Hall, New York.Google Scholar
Turova, T. S. (1994). Analysis of some stochastic neural models. In Aportaciones Matemáticas, Serie Notas de Investigación 11. Sociedad Matemática Mexicana, Mexico, pp. 157–169.Google Scholar
Turova, T. S. (1996). Stochastic dynamics of a neural network with inhibitory and excitatory connections. BioSystems 40, 197–202.CrossRefGoogle Scholar
Turova, T. S. (1996). Analysis of a biologically plausible neural network via an hourglass model. Markov Proc. Rel. Fields 2, 487–510.Google Scholar
Turova, T. S., Mommaerts, W. and van der Meulen, E. C. (1994). Synchronization of firing times in a stochastic neural network model with excitatory connections. Stoch. Proc. Appl. 50, 173–186.CrossRefGoogle Scholar