×

zbMATH — the first resource for mathematics

Techniques for the application of the analytical solution to the multicylinder somatic shunt cable model for passive neurones. (English) Zbl 0819.92003
Summary: The general solution for the voltage response to a generic impulse current input in a multicylinder somatic shunt cable model for passive neurones has been developed by L. F. Abbott [Physica A 185, 343-356 (1992)]. We consider the application of the multicylinder solution to examples previously considered by other authors for the single cylinder case: long and short current input and synaptic input modeled by an alpha-function and a multi-exponential function.
Simple expansions appropriate for small and large times are found and efficient means of obtaining these expansions are clearly demonstrated. The dependence of the small and large time solutions upon the dimensionless parameters appearing in the conservation of current condition at the soma is investigated. Relevant limits of these dimensionless parameters which further simplify the small and large time solutions are related back to equivalent dimensional problems of interest to the practitioner. The well-posedness of the dimensionless inverse problem is investigated and a method proposed for the solution of the dimensional inverse problem for the somatic shunt.

MSC:
92C20 Neural biology
92-08 Computational methods for problems pertaining to biology
78A70 Biological applications of optics and electromagnetic theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abbott, L.F., Simple diagramatic rules for solving dendritic cable problems, Physica A, 185, 343-356, (1992)
[2] Abbott, L.F.; Farhi, E.; Gutmann, S., The path integral for dendrític trees, Biol. cybern., 66, 49-60, (1991) · Zbl 0743.92010
[3] (), Dover, NY
[4] Ashida, H., General solution of cable theory with both ends sealed, J. theor. biol., 112, 727-740, (1985)
[5] Bluman, G.; Tuckwell, H., Techniques for obtaining solution for Rall’s model neuron, J. neurosci. methods, 20, 151-166, (1987)
[6] Cao, B.J.; Abbott, L.F., A new computational method for cable theory problems, Biophys. J., 64, 303-315, (1993)
[7] Carslaw, H.S.; Jaeger, J.C., Conduction of heat in solids, (1959), O.U.P Oxford · Zbl 0029.37801
[8] Clements, J.D.; Redman, S.J., Cable properties of cat spinal motoneurones measured by combining voltage clamp, current clamp, and intracellular staining, J. physiol., 409, 63-87, (1989)
[9] Colquhoun, D.; Jonas, P.; Sakmann, B., Activation and desensitisation of glutamate receptor channels after short agonist pulses in isolated patches from hippocampal neurones, Plügers arch., 419, suppl. 1, (1991), R68(5)
[10] Durand, D., The somatic shunt cable model for neurons, Biophys. J., 46, 645-653, (1984)
[11] Evans, J.D.; Kember, G.C.; Major, G., Techniques for obtaining analytical solutions to the multicylinder somatic shunt cable model for passive neurones, Biophys. J., 63, 350-365, (1992)
[12] Holmes, W.R.; Rall, W., Electrotonic length estimates in neurones with dendritic tapering or somatic shunt, J. neurophysiol., 68, 4, 1421-1437, (1992)
[13] Holmes, W.R.; Rall, W., Estimating the electrotonic structure of neurones with compartmental models, J. neurophysiol., 68, 4, 1438-1452, (1992)
[14] Holmes, W.R.; Segev, I.; Rall, W., Interpretation of time constant and electrotonic length estimates in multicylinder or branched neuronal structures, J. neurophysiol., 68, 4, 1401-1419, (1992)
[15] 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)
[16] 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)
[17] Jack, J.J.B.; Noble, D.; Tsien, R.W., Electric current flow in excitable cells, (1975), Clarendon Press Oxford
[18] Jackson, M.B., Cable analysis with the whole-cell patch clamp, Biophys. J., 61, 756-766, (1992)
[19] Kawato, M., Cable properties of a neuron model with nonuniform membrane resistivity, J. theor. biol., 111, 149-169, (1994)
[20] Major, G.; Evans, J.D.; Jack, J.J.B., Solutions for transients in arbitrarily branching cables: II. voltage clamp theory, Biophys. J., 65, 450-468, (1993)
[21] Moore, J.A.; Appenteng, K., The morphology and electrical geometry of rat jaw-elevator motoneurones, J. physiol., 440, 325-343, (1991)
[22] Philips, G.M.; Taylor, P.J., Theory and applications of numerical analysis, (1973), Academic Press London · Zbl 0312.65002
[23] R. R. Poznanski, Techniques for obtaining analytical solutions for the somatic shunt cable model, Math. Biosci. 83:1-23 (19??). · Zbl 0622.92010
[24] Poznanski, R.R., Transient responses in a somatic shunt cable model for synaptic input activated at the terminal, J. theor. biol., 127, 31-50, (1987)
[25] Rall, W., Branching dendritic trees and motoneurone membrane resistivity, Exptl. neurol., 1, 491-527, (1959)
[26] Rall, W., Core conductor theory and cable properties of neurons, (), 37-97
[27] Rall, W., Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input, J. neurophysiol., 30, 1138-1168, (1967)
[28] Rall, W., Membrane potential transients and membrane time constant of motoneurons, Exptl. neurol., 2, 503-532, (1960)
[29] Rall, W., Theory of physiological properties of dendrites, Ann. NY acad. sci., 96, 1071-1092, (1962)
[30] Rall, W., Time constants and electronic length of membrane cylinders and neurons, Biophys. J., 9, 1483-1508, (1969)
[31] Redman, S.J., The attenuation of passively propagating dendritic potentials in a motoneurone cable model, J. physiol., 234, 637-664, (1973)
[32] Segev, I.; Rall, W., Theoretical analysis of neuron models with dendrites of unequal lengths, Soc. neurosci. abstr., 9, 341, (1983)
[33] Stratford, K.J.; Mason, A.J.R.; Larkman, A.U.; Major, G.; Jack, J.J.B., The modelling of pyramidal neurones in the visual cortex, (), 296-321
[34] Tuckwell, H.C., ()
[35] Van Dyke, M., Perturbation methods in fluid mechanics, (1969), Academic Press NY · Zbl 0136.45001
[36] White, J.A.; Manis, P.B.; Young, E.D., The parameter identification problem for the somatic shunt model, Biol. cybern., 66, 307-318, (1992)
[37] Wilson, M.A.; Bower, J.M., The simulation of large-scale neural networks, (), 291-333
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.