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.
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 |
Keywords:
Green’s function; dendritic cable; somatic shunt neuron model; voltage response; synaptic input; conductance change; Neumann expansionReferences:
| [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, (Stats. Res. Rep., No. 127 (1986), Dept. of Mathematics, Monash Univ: Dept. of Mathematics, Monash Univ Australia) |
| [3] | Doetsch, G., (Guide to the Applications of the Laplace and Z Transforms (1971), Van Nostrand Reinhold: Van Nostrand Reinhold London), 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, (Heim, R.; Palm, G., Theoretical Approaches to Complex Systems. Theoretical Approaches to Complex Systems, Lecture Notes in Biomathematics, Vol. 21 (1978), Springer: Springer Heidelberg), 89-115 · Zbl 0398.92015 |
| [10] | Poggio, T.; Torre, V., A theory of synaptic interactions, (Reichardt, W.; Poggio, T., Theoretical Approaches in Neurobiology (1981), MIT Press: MIT Press Cambridge, Mass), 28-38 · Zbl 0398.92015 |
| [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, (Kandel, E. R., Handbook of Physiology, Section 1, The Nervous System, Vol. 1 (1977), Amer. Physiol. Soc: Amer. Physiol. Soc Bethesda, Md), 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: 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 |
| [23] | Tuckwell, H. C., On Shunting Inhibition, Biol. Cybernet., 55, 83-90 (1986) · Zbl 0598.92006 |
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