Skip to main content
SpringerLink
Log in

The Ornstein-Uhlenbeck process as a model for neuronal activity

I. Mean and variance of the firing time

  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

Mean and variance of the first passage time through a constant boundary for the Ornstein-Uhlenbeck process are determined by a straight-forward differentiation of the Laplace transform of the first passage time probability density function. The results of some numerical computations are discussed to shed some light on the input-output behavior of a formal neuron whose dynamics is modeled by a diffusion process of Ornstein-Uhlenbeck type.

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.

Similar content being viewed by others

References

  • Arnold, L.: Stochastic differential equations. New York: Wiley 1974

    Google Scholar 

  • Calvin, W.H., Stevens, C.H.: A Markov process model for neuron behavior in the interspike interval. In: Proc. Ann. Conf. Eng. Med. Biol. 18th, Vol. 7, p. 118, 1965 (copy available on request)

  • Capocelli, R.M., Ricciardi, L.M.: Diffusion approximation and first passage time problem for a model neuron. Kybernetik8, 214–223 (1971)

    PubMed  Google Scholar 

  • Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys.15, 1–89 (1943)

    Google Scholar 

  • Doob, J.L.: The Brownian movement and stochastic equations. Ann. Math.43, 351–369 (1949)

    Google Scholar 

  • Erdelyi, A., Magnus, F., Oberhettinger, F.G., Tricomi, F.G.: Higher transcendental functions, Vol. 1. New York: McGraw Hill 1953

    Google Scholar 

  • Gluss, B.A.: A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density. Bull. Math. Biophys.29, 233–243 (1967)

    PubMed  Google Scholar 

  • Jazwinski, A.H.: Stochastic processes and filtering theory. New York: Academic Press 1970

    Google Scholar 

  • Johannesma, P.I.M.: Diffusion models for the stochastic activity of neurons. In: Neural networks, pp. 116–144. Caianiello, E.R. (ed.). Berlin, Heidelberg, New York: Springer 1968

    Google Scholar 

  • Kac, M.: Random walk and the theory of Brownian motion. Ann. Math.54, 369–391 (1946)

    Google Scholar 

  • Keilson, J., Ross, H.: Passage time distribution for the Ornstein-Uhlenbeck process. Sel. Tables Math. Stat.3, 233–328 (1975)

    Google Scholar 

  • Ricciardi, L.M.: Diffusion approximation for a multi-input model neuron. Biol. Cybernetics24, 237–240 (1976)

    Google Scholar 

  • Ricciardi, L.M.: Diffusion processes and related topics in biology. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  • Roy, B., Smith, D.R.: Analysis of the exponential decay model of the neuron showing frequency threshold effects. Bull. Math. Biophys.31, 341–357 (1969)

    PubMed  Google Scholar 

  • Sacerdote, L.: Approssimazioni diffusive e problemi di sparo per neuroni singoli. Univ. of Turin, dissertation for “Dottore in Fisica” 1977

  • Sato, S.: Diffusion approximation for the stochastic activity of a neuron and moments of the interspike interval distribution. In: Progress in cybernetics and system research, Vol. 6. Pichler, F., Trappl, R. (eds.) (in press)

  • Siebert, W.M.: On stochastic neural models of the diffusion type. Q. Progr. Rept., Res. Lab. Electronics, M.I.T.94, 281–287 (1969)

    Google Scholar 

  • Siegert, A.J.F.: On the first passage time probability problem. Phys. Rev.81, 617–623 (1951)

    Google Scholar 

  • Stratonovich, R.L.: Topics in the theory of random noise, Vol. 1. New York: Gordon & Breach 1963

    Google Scholar 

  • Stratonovich, R.L.: Conditional Markov processes and their applications to the theory of optimal control. New York: Elsevier 1968

    Google Scholar 

  • Sugiyama, H., Moore, G.P., Perkel, D.M.: Solutions for a stochastic model of neuronal spike production. Math. Biosci.8, 323–341 (1970)

    Google Scholar 

  • Thomas, M.U.: Some mean first passage time approximations for the Ornstein-Uhlenbeck process. J. Appl. Prob.12, 600–604 (1975)

    Google Scholar 

  • Uhlenbeck, G.E., Ornstein, L.S.: On the theory of Brownian motion. Phys. Rev.36, 823–841 (1930)

    Google Scholar 

  • Wang, M.C., Uhlenbeck, G.E.: On the theory of the brownian motion II. Rev. Mod. Phys.17, 323–342 (1945)

    Google Scholar 

  • Wong, E.: Stochastic processes in information and dynamical systems. New York: McGraw Hill 1971

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto di Scienze dell'Informazione, Università di Salerno, Salerno, Italy

    Luigi M. Ricciardi & Laura Sacerdote

Authors
  1. Luigi M. Ricciardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laura Sacerdote
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work supported in part by the Group for Mathematical Information Science (GNIM) of the National Council for Research

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ricciardi, L.M., Sacerdote, L. The Ornstein-Uhlenbeck process as a model for neuronal activity. Biol. Cybern. 35, 1–9 (1979). https://doi.org/10.1007/BF01845839

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01845839

Keywords

Search

Navigation