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Analytical solutions to the multicylinder somatic shunt cable model for passive neurones with differing dendritic electrical parameters. (English) Zbl 0807.92005
Summary: The multicylinder somatic shunt cable model for passive neurones with differing time constants in each cylinder is considered. The solution to the model with general inputs is developed, and the parametric dependence of the voltage response is investigated. The method of analysis is straightforward and follows that laid out by the authors and G. C. Kember in Biophys. J. 63, 350-365 (1992) and in Math. Biosci. 125, No. 1, 1-50 (1995):
(i) The dimensional problem is stated with general boundary and initial conditions. (ii) The model is fully non-dimensionalised, and a dimensionless parameter family which uniquely governs the behaviour of the dimensionless voltage response is obtained. (iii) The fundamental unit impulse problem is solved, and the solutions to problems involving general inputs are written in terms of the unit impulse solution. (iv) The large and small time behaviour of the unit impulse solution is examined. (v) The parametric dependence of the unit impulse upon the dimensionless parameter family is explored for two limits of practical interest.
A simple expression for the principle relationship between the dimensionless parameter family is derived and provides insight into the interaction between soma and cylinders. A well-posed method for the solution of the dimensional inverse problem is presented.

MSC:
92C20 Neural biology
78A70 Biological applications of optics and electromagnetic theory
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[1] Barrett JN, Crill WE (1974) Specific membrane properties of cat motoneurones. J Physiol 239:301–324
[2] Clements JD, Redman SJ (1989) Cable properties of cat spinal motoneurones measured by combining voltage clamp, current clamp and intracellular staining. J Physiol 409:63–87
[3] Douglas RJ, Martin KAC (1992) Exploring cortical microcircuits: a combined anatomical, physiological, and computational approach. In: McKenna T, Davis J, Zornetzer SF (eds) Single Neuron Computation. Academic Press, San Diego, 381–412
[4] Evans JD, Kember GC, Major G (1992) Techniques for obtaining analytical solutions to the multi-cylinder somatic shunt cable model for passive neurones. Biophys J 63:350–365 · doi:10.1016/S0006-3495(92)81631-4
[5] Evans JD, Kember GC, Major G (1994) Techniques for the application of the analytical solutions to the multi-cylinder somatic shunt cable model for passive neurones. Math Biosci (in press)
[6] Hillman D, Chen S, Aung TT, Cherksey B, Sugimori M, Llinas RR (1991) Localization of P-type calcium channels in the central nervous system. Proc Natl Acad Sci USA 88:7076–7080 · doi:10.1073/pnas.88.16.7076
[7] Holmes WR, Rall W (1992a) Electronic length estimates in neurones with dendritic tapering or somatic shunt. J Neurophysiol 68:1421–1437
[8] Holmes WR, Rall W (1992b) Estimating the electronic structure of neurones with compartmental models. J Neurophysiol 68:1438–1452
[9] Holmes WR, Segev I, Rall W (1992) Interpretation of time constants and electronic length estimates in multicylinder or branched neuronal structures. J Neurophysiol 68:1401–1419
[10] Huguenard JR, Hamill OP, Prince DA (1989) Sodium channels in dendrites of rat cortical pyramidal neurones. Proc Natl Acad Sci USA 86:2473–2477 · doi:10.1073/pnas.86.7.2473
[11] Jonas P, Major G, Sakmann B (1993) Quantal analysis of unitary EPSCs at the mossy fibre synapse on CA3 pyramidal cells of rat hippocampus. J Physiol (in press)
[12] Larkman AU (1991) Dendritic morphology of pyramidal neurones of the visual cortex of the rat. III. Spine distributions. J Comp Neurol 306:332–343 · doi:10.1002/cne.903060209
[13] Major G, Evans JD (1994) Solutions for transients in arbitrarily branching cables. IV. Non-uniform electrical parameters. Biophys J (in press)
[14] Major G, Evans JD, Jack JJB (1993) Solutions for transients in arbitrarily branching cables. I. Voltage recording with a somatic shunt. Biophys J 65:423–449 · doi:10.1016/S0006-3495(93)81037-3
[15] Rapp M, Yarom Y, Segev I (1992) The impact of parallel fiber back-ground activity on the cable properties of cerebeller purkinje cells. Neural Comput 4:518–533 · doi:10.1162/neco.1992.4.4.518
[16] Shelton DP (1985) Membrane resistivity estimated for the Purkinje neuron by means of a passive computer model. Neuroscience 14:111–131 · doi:10.1016/0306-4522(85)90168-X
[17] Stratford KJ, Mason AJR, Larkman AU, Major G, Jack JJB (1989) The modelling of pyramidal neurones in the visual cortex. In: Durbin R, Miall C, Mitchison G (eds) The computing neuron. Addison-Wesley, Reading, pp 296–321
[18] Trimmer JS (1991) Immunological identification and characterisation of a delayed rectifier K+ channel polypeptide in rat brain. Proc Natl Acad Sci USA 88:10764–10768 · doi:10.1073/pnas.88.23.10764
[19] Westenbroek RE, Ahlijanian MK, Catterall WA (1990) Clustering of L-type Ca2+ channels at the base of major dendrites in hippocampal pyramidal neurones. Nature 347:281–284 · doi:10.1038/347281a0
[20] Westenbroek RK, Hell JW, Warner C, Dubel SJ, Snutch TP, Catterall WA (1992) Biochemical properties and subcellular distribution of an N-type calcium channel \(\alpha\)1 subunit. Neuron 9:1099–1115 · doi:10.1016/0896-6273(92)90069-P
[21] White JA, Manis PB, Young ED (1992) The parameter identification problem for the somatic shunt model. Biol Cybern 66:307–318 · doi:10.1007/BF00203667
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