Abstract
The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f 3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.
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References
Burns, B.D., Webb, A.C.: The spontaneous activity of neurones in the cat's cerebral cortex. Proc. R. Soc. London B194, 211–233 (1976)
Conradi, S.: On motoneuron synaptology in adult cats. Acta Physiol. Scand. Suppl.332, (1969)
Doob, J.L.: Stochastic processes. New York: Wiley 1953
Dwight, H.B.: Tables of integrals and other mathematical data. New York: Macmillan 1961
Dynkin, E.B.: Markov processes, Vol. II. Berlin, Heidelberg, New York: Springer 1965
Finger, W., Stettmeier, H.: Efficacy of the two-electrode voltage clamp technique in cryafish muscle. Pflügers Arch.387, 133–141 (1980)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series, and products. New York: Academic Press 1966
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol.117, 500–544 (1952)
Hodgkin, A.L., Rushton, W.A.H.: The electrical constrants of a crustacean nerve fibre. Proc. R. Soc. London B133, 444–479 (1946)
Holden, A.V., Yoda, M.: The effects of ionic channel density on neuronal function. J. Theor. Neurobiol.1, 60–81 (1981)
Hooge, F.N.: 1/f noise. Physica B83, 14–23 (1976)
Jack, J.J.B., Noble, D., Tsien, R.W.: Electric current flow in excitable cells. Oxford: Clarendon Press 1975
Jack, J.J.B., Redman, S.J., Wong, K.: The components of synaptic potentials evoked in cat spinal motoneurones by impulses in single group Ia afferents. J. Physiol.321, 65–96 (1981)
Koziol, J.A., Tuckwell, H.C.: Analysis and estimation of synaptic densities and their spatial variation on the motoneuron surface. Brain Res.17, 617–624 (1978)
Matsumoto, G., Shimizu, H.: Spatial coherence and formation of collectively-coupled local nonlinear oscillators in squid giant axons. J. Theor. Neurobiol.2, 29–46 (1983)
Rall, W.: Theory of physiological properties of dendrites. Ann. N.Y. Acad. Sci.96, 1071–1092 (1962)
Rall, W.: Theoretical significance of dendritic trees for neuronal input-output relations. In: Neural theory and modelling. Reiss, R.F. (ed.). Stanford: Stanford University Press 1964
Rall, W.: Core conductor theory and cable properties of neurons. In: Handbook of physiology, Sect. 1. The nervous system. I. Cellular biology of neurons. Kandel, E.R. (ed.). Am. Physiol. Soc. Bethesda (1977)
Tuckwell, H.C., Wan, F.Y.M., Wong, Y.S.: The interspike interval of a cable model neuron with white noise input. Biol. Cybern (in press, 1984)
Walsh, J.B.: A stochastic model of neuronal response. Adv. Appl. Prob.13, 231–281 (1981)
Walsh, J.B., Tuckwell, H.C.: Determination of the electrical potential over dendritic trees by mapping onto a nerve cylinder. I.A.M.S. Tech. Report. No. 83-8, Univ. of British Columbia, Vancouver 1983
Wan, F.Y.M., Tuckwell, H.C.: The response of a spatially distributed neuron to white noise current injection. Biol. Cybern.33, 36–59 (1979)
Wan, F.Y.M., Tuckwell, H.C.: Neuronal firing and input variability. J. Theor. Neurobiol.1, 197–218 (1982)
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Tuckwell, H.C., Walsh, J.B. Random currents through nerve membranes. Biol. Cybernetics 49, 99–110 (1983). https://doi.org/10.1007/BF00320390
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DOI: https://doi.org/10.1007/BF00320390