zbMATH — the first resource for mathematics

Techniques for obtaining analytical solutions for the somatic shunt cable model. (English) Zbl 0622.92010
Mathematical expressions are obtained for the Green’s function corresponding to an instantaneous pulse of current injected at a single point along the dendritic cable in a somatic shunt neuron model. The convergence of the Green’s function is determined from an estimate of its truncation error. The Green’s function, when used in a convolution formula, enables one to compute the voltage response at any specified point for an arbitrary synaptic input at a given location.
Examples of synaptic input considered are (1) a current step, injected at the soma, and (2) a smooth current time course of the form $$\alpha^ 2Te^{-\alpha T}$$ injected at a given location along the dendritic cable. Alternative representations for the Green’s function are given to enable both the small and the large time behavior of the voltage to be analyzed. The treatment of synaptic input as a conductance change is also discussed. The Volterra integral equation is solved analytically by employing a Neumann expansion to obtain the voltage response.

MSC:
 92Cxx Physiological, cellular and medical topics 45D05 Volterra integral equations 78A70 Biological applications of optics and electromagnetic theory
Full Text:
References:
 [1] Barnett, J.N.; Crill, W.E., Influence of dendritic location and membrane properties on the effectiveness of synapses on cat motoneurones, J. physiol., 239, 325-345, (1974) [2] Bluman, G.W.; Tuckwell, H.C., Techniques for obtaining analytical solutions for Rall’s model neuron, () [3] Doetsch, G., (), 118-123 [4] Durand, D., The somatic shunt cable model for neurons, Biophys. J., 46, 645-653, (1984) [5] Iansek, R.; Redman, S.J., An analysis of the cable properties of spinal motoneurones using a brief intracellular current pulse, J. physiol., 234, 613-636, (1973) [6] Jack, J.J.B.; Redman, S.J., An electrical description of the motoneurone and its application to the analysis of synaptic potentials, J. physiol., 215, 321-352, (1971) [7] Kawato, M., Cable properties of a neuron model with non-uniform membrane resistivity, J. theor. biol., 111, 149-169, (1984) [8] Koch, C.; Poggio, T.; Torre, V., Non-linear interactions in a dendritic tree: localization, timing, and role in information processing, Proc. nat. acad. sci. U.S.A., 80, 2799-2802, (1983) [9] Poggio, T.; Torre, V., A new approach to synaptic interaction, (), 89-115 · Zbl 0398.92015 [10] Poggio, T.; Torre, V., A theory of synaptic interactions, (), 28-38 · Zbl 0398.92015 [11] R.R. Poznański, Transient response in a somatic shunt cable model for synaptic input activated at the terminal, J. Theoret. Biol., submitted for publication. [12] Rall, W., Branching dendritic trees and motoneuron membrane resistivity, Exptl. neurol., 1, 491-527, (1959) [13] Rall, W., Membrane potential transients and membrane time constant of motoneurons, Exptl. neurol., 2, 503-532, (1960) [14] Rall, W., Theory of physiological properties of dendrites, Ann. N.Y. acad. sci., 96, 1070-1092, (1962) [15] Rall, W., Electrophysiology of a dendritic neuron model, Biophys. J., 2, 145-167, (1962) [16] Rall, W., Core conductor theory and cable properties of neurons, (), 39-97 [17] Redman, S.J., The attenuation of passively propagating dendritic potentials in a motoneurone cable model, J. physiol., 234, 637-664, (1973) [18] Rinzel, J., Voltage transients in neuronal dendritic trees, Fed. proc., 34, 1350-1356, (1975) [19] Rinzel, J.; Rall, W., Transient response in a dendritic neuron model for current injected at one branch, Biophys. J., 14, 759-790, (1974) [20] Roberts, G.E.; Kaufman, H., Table of Laplace transforms, (1966), Saunders Philadelphia · Zbl 0137.08901 [21] Segev, I.; Fleshman, J.W.; Miller, J.P.; Bunow, B., Modeling the electrical behavior of anatomically complex neurons using a network analysis program: passive membrane, Biol. cybernet., 53, 27-40, (1985) · Zbl 0565.92009 [22] H.C. Tuckwell, Introduction to Mathematical Neurobiology, Longman, London, to appear. [23] Tuckwell, H.C., On shunting inhibition, Biol. cybernet., 55, 83-90, (1986) · Zbl 0598.92006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.