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Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling. (English) Zbl 1071.92004
Summary: A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solutions of the cable equation with synaptic reversal potentials are considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.

MSC:
 92C20 Neural biology 78A70 Biological applications of optics and electromagnetic theory 45D05 Volterra integral equations 45A05 Linear integral equations
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 [1] Abramowitz, M.; Stegun, I., Handbook of mathematical functions, (1965), Dover New York [2] Amitai, Y.; Gibson, J.R.; Beierlein, M.; Patrick, S.L.; Ho, A.M.; Connors, B.W.; Golomb, D., The spatial dimensions of electrically coupled networks of interneurons in the neocortex, J. neurosci., 22, 4142, (2002) [3] Baer, S.M.; Tier, C., An analysis of a dendritic neuron model with an active membrane site, J. math. biol., 23, 137, (1986) · Zbl 0587.92012 [4] Evans, J.D., Analysis of a multiple equivalent cylinder model with generalized taper, IMA J. math. med. biol., 17, 347, (2000) · Zbl 0970.92004 [5] Evans, J.D., A cable model for coupled neurons with somatic gap-junctions, Biol. cybern., 92, 164, (2005) · Zbl 1101.92008 [6] Evans, J.D., Multicylinder models for synaptic and gap-junctional integration, (), 117, (Chapter 6) · Zbl 1091.92015 [7] Evans, J.D.; Kember, G.C.; Major, G., Techniques for the application of the analytical solutions to the multi-cylinder somatic shunt cable model for passive neurons, Math. biosci., 125, 1, (1995) · Zbl 0819.92003 [8] Getting, P.A., Modification of neuron properties by electrotonic synapses. I. input resistance time constant, and integration, J. neurophysiol., 37, 846, (1974) [9] Golomb, D.; Hansel, D.; Mato, G., Theory of synchrony of neuronal activity, (), 887 [10] Haag, J.; Borst, A., Dendrodendritic interactions between motion-sensitive large-field neurons in the fly, J. neurosci., 22, 3227, (2002) [11] Hidaka, S.; Christensen, B.N.; Naka, K., The synaptic ultrastructure in the outer plexiform layer of the catfish retina: A three-dimensional study with HVEM and conventional EM of golgi-impregnated bipolar and horizontal cells, J. comp. neurol., 272, 181, (1986) [12] Hidaka, S.; Akahori, Y.; Kurosawa, Y., Dendrodendritic electrical synapses between Mammalian retinal ganglion cells, J. neurosci., 24, 10553, (2004) [13] MacVicar, B.A.; Dudek, F.E., Local synaptic circuits in rat hippocampus: interactions between pyramidal cells, Brain res., 184, 220, (1980) [14] MacVicar, B.A.; Dudek, F.E., Dye-coupling between CA3 pyramidal cells in slices of rat hippocampus, Brain res., 196, 494, (1980) [15] MacVicar, B.A.; Dudek, F.E., Electrotonic coupling between pyramidal cells: A direct demonstration in rat hippocampal slices, Science, 213, 782, (1981) [16] MacVicar, B.A.; Ropert, N.; Krnjevic, K., Dye-coupling between pyramidal cells of rat hippocampus in vivo, Brain res., 238, 239, (1982) [17] Pfeuty, B.; Mato, G.; Golomb, D.; Hansel, D., Electrical synapses and synchrony: the role of intrinsic currents, J. neurosci., 23, 6280, (2003) [18] Poznanski, R.R., Analysis of a postsynaptic scheme based on a tapering equivalent cable model, IMA J. math. appl. med. biol., 7, 175, (1990) · Zbl 0715.92004 [19] Poznanski, R.R., Electrophysiology of a leaky cable model for coupled neurons, J. austral. math. soc. ser. B, 40, 59, (1998) · Zbl 0907.92006 [20] Poznanski, R.R., On recent cable models in neurophysiology, Math. scientist, 26, 74, (2001) · Zbl 0999.92009 [21] Poznanski, R.R., Analytical solutions of the frankenhaeuser-Huxley equations. I. minimal model for backpropagation of action potentials in sparsely excitable dendrites, J. integr. neurosci., 3, 267, (2004) [22] Poznanski, R.R.; Bell, J., A dendritic cable model for the amplification of synaptic potentials by an ensemble average of persistent sodium channels, Math. biosci., 166, 101, (2000) · Zbl 0978.92005 [23] Poznanski, R.R.; Umino, O., Syncytial integration by a network of coupled bipolar cells in the retina, Prog. neurobiol., 53, 273, (1997) [24] Poznanski, R.R.; Gibson, W.G.; Bennett, M.R., Electrotonic coupling between two CA3 hippocampal pyramidal neurons: A distributed cable model with somatic gap-junction, Bull. math. biol., 57, 6, 865, (1995) · Zbl 0835.92009 [25] Publicover, N.G., Mathematical models of intercellular communication, (), 183 [26] Rall, W., Core conductor theory and cable properties of neurons, (), 39 [27] Rall, W., Functional aspects of neuronal geometry, () [28] Schmalbruch, H.; Jahnsen, H., Gap-junctions on CA3 pyramidal cells of guinea pig hippocampus shown by freeze-fracture, Brain res., 217, 175, (1981) [29] Schuster, T.H., Gap junctions and electrotonic coupling between hippocampal neurons: recent evidence and possible significance - A review, Concepts neurosci., 3, 135, (1992) [30] Segev, I.; Parnas, I., Synaptic integration mechanisms: A theoretical and experimental investigation of temporal postsynaptic interaction between excitatory and inhibitory inputs, Biophys. J., 41, 41, (1983) [31] Skrzypek, J., Electrical coupling between horizontal cell bodies in the tiger salamander retina, Vision res., 24, 701, (1984) [32] Tuckwell, H.C., Some aspects of cable theory with synaptic reversal potentials, J. theoret. neurobiol., 4, 113, (1985) [33] Tuckwell, H.C., On shunting inhibition, Biol. cybern., 55, 83, (1986) · Zbl 0598.92006 [34] Turner, D.; Schwartzkroin, P.A., Steady-state electrotonic analysis of intracellularly stained hippocampal neurons, J. neurophysiol., 44, 184, (1980) [35] Turner, D.; Schwartzkroin, P.A., Electrical characteristics of dendrites and dendritic spines in intracellularly stained CA3 and dentate hippocampal neurons, J. neurosci., 3, 2381, (1983)
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