Skip to main content
SpringerLink
Log in

Spatially localized interactive neural populations—I. A mathematical model

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

A probabilistic model of a spatially localized, mutually exitatory (inhibitory) population of neurons is formulated to help explain average evoked potential and post-stimulus time histogram measurements. The model is based on the stochastic activity of single neurons within interactive masses of neurons which exhibit co-operative behavior. Macrostate variables corresponding to the above measurements are related through the model to features of neural operation at the individual and ensemble level. Steady-state solution are obtained and their physiological implications are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Arnold, L. 1974.Stochastic Differential Equations. New York: John Wiley.

    MATH  Google Scholar 

  • Ashby, W. R., H. von Foerster and C. C. Walker. 1962. “Instability of Pulse Activity in a Net with Threshold.”Nature 196 561–62.

    Article  MATH  MathSciNet  Google Scholar 

  • Beurle, R. L. 1956. “Properties of a Mass of Cells Capable of Regenerating Pulses.”Phil. Trans. R. Soc. Lond. B240, 55–94.

    Google Scholar 

  • Chung, K. L. 1974.A Course in Probability Theory. New York: Academic Press.

    MATH  Google Scholar 

  • Feller, W. 1971.An Introduction to Probability Theory and Its Applications, Vol. 2. New York: John Wiley.

    MATH  Google Scholar 

  • Freeman, W. J. 1974. “A Model for Mutual Excitation in a Neuron Population in Olfactory Bulb.”IEEE Trans. Bioengng. BME-21, 350–358.

    Google Scholar 

  • —. 1974. “Stability Characteristics of Positive Feedback in a Neural Population.”IEEE Trans. Bioengng. BME-21, 358–364.

    Google Scholar 

  • —. 1975.Mass Action in the Nervous System, New York: Academic Press.

    Google Scholar 

  • Griffith, J. S. 1963. “On the Stability of Brain-like Structures.”Biophys. J. 3, 299–308.

    MathSciNet  Google Scholar 

  • —. 1963. “A Field Theory of Neural Nets 1: Derivation of Field Equations.”Bull. Math. Biophys. 25, 111–120.

    MATH  MathSciNet  Google Scholar 

  • —. 1965. “A Field Theory of Neural Nets 2: Properties of the Field Equations.”Bull. Math. Biophys. 27, 187–195.

    MATH  MathSciNet  Google Scholar 

  • Khinchine, A. Y. 1960.Mathematical Methods in the Theory of Queueing. Griffin's Statistical Monographs & Courses. New York: Hafner.

    Google Scholar 

  • Sabah, N. H. and J. T. Murphy. 1971. “A Superposition Model of the Spontaneous Activity of Cerebellar Purkinje Cells.”Biophys. J. 11, 415–427.

    Google Scholar 

  • Soong, T. T. 1973.Random Differential Equations in Science and Engineering. New York: Academic Press.

    MATH  Google Scholar 

  • Wilson, H. R. and J. D. Cowan. 1972. “Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons.”Biophys. J. 12, 1–24.

    Article  Google Scholar 

  • — and-—. 1973. “A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue.”Kybernetic 13, 55–80.

    Article  MATH  Google Scholar 

  • Zetterberg, L. H. 1973. “Stochastic Activity in a Population of Neurons—A Systems Analysis Approach.”Rep. Inst. Med. Phys. TNO, Utrecht 1, 153.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, 29631, Clemson, SC, USA

    James R. Brannan

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 12181, Troy, NY, USA

    William E. Boyce

Authors
  1. James R. Brannan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William E. Boyce
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper is based on a dissertation submitted by the first author to Rensselaer Polytechnic Institute in partial fulfilment of the requirements for the degree of Doctor of Philosophy

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brannan, J.R., Boyce, W.E. Spatially localized interactive neural populations—I. A mathematical model. Bltn Mathcal Biology 43, 427–446 (1981). https://doi.org/10.1007/BF02459432

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02459432

Keywords

Search

Navigation