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A stochastic model of neural response

Published online by Cambridge University Press:  01 July 2016

John B. Walsh
John B. Walsh*
Affiliation:
The University of British Columbia
*
∗Postal address: Department of Mathematics, The University of British Columbia, #121–1984 Mathematics Road, University Campus, Vancouver, B.C., Canada V6T 1Y4.
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Abstract

We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.

Keywords

NEURONSNERVE CYLINDERWHITE NOISESTOCHASTIC PARTIAL DIFFERENTIAL EQUATIONSBANACH-SPACE-VALUED BROWNIAN MOTIONORNSTEIN-UHLENBECK PROCESSINFINITE-DIMENSIONAL DIFFUSIONRANDOM FIELDS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 231 - 281
Copyright
Copyright © Applied Probability Trust 1981 

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References

[1] Aronson, D. G. and Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, combustion, and nerve propagation. In Proc. Tulane Program of Partial Differential Equations and Related Topics. Lecture Notes in Mathematics 446, Springer-Verlag, Berlin.Google Scholar
[2] Bensoussan, A. (1971) Filtrage optimal des systèmes linéaires. Dunod, Paris.Google Scholar
[3] Cairoli, R. and Walsh, J. B. (1975) Stochastic integrals in the plane. Acta Math. 134, 111–181.Google Scholar
[4] Conradi, S. (1969) On motoneuron synaptology in adult cats. Acta Phys. Scand. Suppl. 332.Google Scholar
[5] Delporte, J. (1964) Functions aléatoires presque sûrement continues sur un intervalle fermé. Ann. Inst. H. Poincaré 1, 111–215.Google Scholar
[6] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[7] Doob, J. L. (1955) A probability approach to the heat equation. Trans. Amer. Math. Soc. 80, 216–280.Google Scholar
[8] Dynkin, E. B. (1965) Markov Processes, Vol. 2. Springer-Verlag, Berlin.Google Scholar
[9] Garsia, A. M. (1972) Continuity properties of Gaussian processes with a multidimensional time parameter. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 369–374.Google Scholar
[10] Gradshteyn, I. S. and Ryzhik, I. M. (1966) Tables of Integrals, Series, and Products, 4th edn. Academic Press, New York.Google Scholar
[11] Koziol, J. A. and Tuckwell, H. C. (1978) Analysis and estimation of synaptic densities and their spatial variation on the motoneuron surface. Brain Res. 17, 617–624.CrossRefGoogle Scholar
[12] Kunita, H. and Watanabe, S. (1967) On square-integrable martingales. Nagoya Math. J. 30, 209–245.Google Scholar
[13] Kuo, H. H. (1973) Differential and stochastic equations in abstract Wiener spaces. J. Functional Anal. 12, 246–256.CrossRefGoogle Scholar
[14] McKean, H. P. (1969) Stochastic Integrals. Academic Press, New York.Google Scholar
[15] Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York.Google Scholar
[16] Meyer, P.-A. (1975) Un cours sur les integrales stochastiques. In Seminaire de Probabilités X, Lecture Notes in Mathematics 511, Springer-Verlag, Berlin.Google Scholar
[17] Rall, W. (1959) Branching dendritic trees and motoneuron resistivity. Experimental Neurology 1, 491–527.CrossRefGoogle ScholarPubMed
[18] Rall, W. (1967) Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input. J. Neurophysiol. 30, 1138–1168.Google Scholar
[19] Shepp, L. A. (1966) Radon–Nikodym derivatives of Gaussian processes. Ann. Math. Statist. 37, 321–354.CrossRefGoogle Scholar
[20] Tuckwell, H. C. and Wan, F. (1979) The response of a spatially distributed neuron to white noise current injection. Biol. Cybernetics 33, 39–55.Google Scholar
[21] Van Der Ziel, A. (1954) Noise. Prentice-Hall, Englewood Cliffs, N.J. Google Scholar
[22] Verveen, A. A. and De Felice, L. J. (1974) Membrane noise. Progress in Biophysics and Molecular Biology 28, 189–265.CrossRefGoogle ScholarPubMed
[23] Wong, E. and Zakai, M. (1974) Martingales and stochastic integrals for processes with a multidimensional time parameter. Z. Wahrscheinlichkeitsth. 29, 109–122.CrossRefGoogle Scholar