Abstract
A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.
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Literature
Brockman, W. H. 1981. Ladder network prediction of transients in linear spatially inhomogeneous cables.J. theor. Biol. 92, 469–478.
Goldstein, S. S. and W. Rall. 1974. Changes of action potential shape and velocity for changing core conductor geometry.Biophys. J. 14, 731–757.
Gutman, A. 1984.Nerve Cell Dendrites: Theory, Electrophysiology, Functions (in Russian). Vilnius: Mokslas.
Jack, J. J. B., D. Noble and R. W. Tsien. 1975.Electric Current Flow in Excitable Cells. Oxford: Clarendon Press.
Keller, D. J. and S. Lal. 1976. Membrane voltage changes in a compartmental chain model of a neurone.Biol. Cybern. 24, 211–217.
Poznanski, R. R. 1988. Membrane voltage changes in passive dendritic trees: a tapering equivalent cylinder model.IMA J. Math. appl. Med. Biol. 5, 113–145.
Poznanski, R. R. 1990. Analysis of a postsynaptic scheme for directional selectivity.Proc. Aust. Neurosci. Soc. 1, 118.
Rall, W., 1962a. Theory of physiological properties of dendrites.Ann. N.Y. Acad. Sci. 96, 1071–1092.
Rall, W. 1962b. Electrophysiology of a dendritic neuron model.Biophys. J. 2 (suppl.), 145–167.
Rall, W. 1964. Theoretical significance of dendritic trees for neuronal input-output relations. InNeural Theory and Modeling, R. Reiss (ed.), pp. 73–97. Stanford, CA: Stanford University Press.
Rall, W. 1969. Time constants and electrotonic length of membrane cyclinders and neurons.Biophys. J. 9, 1483–1508.
Rall, W. 1977. Core conductor theory and cable properties of neurons. InHandbook of Physiology: The Nervous System. E. R. Kandel (ed.), pp. 39–97. Bethesda, MD: American Physiological Society.
Rall, W. 1989. Cable theory for dendritic neurons. InMethods in Neuronal Modeling, C. Koch and I. Segev (eds), pp. 1–62. Cambridge, MA: MIT Press.
Redman, S. J. 1976. A quantitative approach to integrative function of dendrites. InInternational Review of Physiology, R. Porter (ed.), pp. 1–35. Baltimore: University Park Press.
Rose, P. K. and A. Dagum. 1988. Nonequivalent cylinder models of neurons: interpretation of voltage transients generated by somatic current injection.J. Neurophysiol. 60, 125–148.
Schierwagen, A. K. 1989. A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree.J. theor. Biol. 141, 159–179.
Strain, G. M. and W. H. Brockman. 1975. A modified cable model for neuron processes with non-constant diameters.J. theor. Biol. 51, 475–494.
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Poznanski, R.R. A generalized tapering equivalent cable model for dendritic neurons. Bltn Mathcal Biology 53, 457–467 (1991). https://doi.org/10.1007/BF02460728
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DOI: https://doi.org/10.1007/BF02460728