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A dendritic cable model for the amplification of synaptic potentials by an ensemble average of persistent sodium channels. (English) Zbl 0978.92005
Summary: The persistent sodium current density \((I_{\text{NaP} }\)) at the soma measured with the ‘whole-cell’ patch-clamp recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called ‘hot-spots’) is used to obtain conclusions on the density and distribution of persistent sodium ion channels.
It is shown that the excitatory postsynaptic potentials (EPSPs) are amplified if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of \(I_{\text{NaP}}\) with respect to the recording site shown to specifically increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution of \(I_{\text{NaP}}\) serves to broaden the time course of the amplified EPSP. However, in the case of a distally positioned synaptic input, both local and nonlocal densities yield an approximately identical enhancement of EPSPs in contradiction to the computer simulations performed by R. Lipowsky et al. [J Neurophysiol. 76, 2181ff (1996)].
The results indicate that persistent sodium channels produce EPSP amplification even when their distribution is relatively sparse (i.e., approximately \(1-2\%\) of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables.

92C20 Neural biology
92C05 Biophysics
Full Text: DOI
[1] S.J. Redman, A quantitative approach to integrative function of dendrites, in: R. Porter (Ed.), International Review of Physiology: Neurophysiology II, vol. 10, University Park Press, Baltimore, MD, 1976
[2] Urban, N.N.; Barrionuevo, G., Active summation of excitatory postsynaptic potentials in hippocampal CA3 pyramidal neurons, Proc. nat. acad. sci. USA, 95, 11450, (1998)
[3] Huguenard, J.R.; Hamill, O.P.; Prince, D.A., Sodium channels in dendrites of rat cortical pyramidal neurons, Proc. nat. acad. sci. USA, 86, 2473, (1989)
[4] Stuart, G.; Spruston, N.; Sakmann, B.; Hausser, M., Action potential initiation and backpropagation in neurons of the Mammalian CNS, Trends neurosci., 20, 125, (1997)
[5] Jung, H.-Y.; Mickus, T.; Spruston, N., Prolonged sodium channel inactivation contributes to dendritic action potential attenuation in hippocampal pyramidal neurons, J. neurosci., 17, 6639, (1997)
[6] Colbert, C.M.; Magee, J.C.; Hoffman, D.A.; Johnston, D., Slow recovery from inactivation of na^{+} channels underlies the activity-dependent attenuation of dendritic action potentials in hippocampal CA1 pyramidal neurons, J. neurophysiol., 17, 6512, (1997)
[7] Regehr, W.; Kehoe, J.; Ascher, P.; Armstrong, C., Synaptically triggered action potentials in dendrites, Neuron, 11, 145, (1993)
[8] Colling, S.B.; Wheal, H.V., Fast sodium action potentials are generated in the distal apical dendrites of rat hippocampal CA1 pyramidal cells, Neurosci. lett., 172, 73, (1994)
[9] Andreasen, M.; Lambert, J.D.C., Regenerative properties of pyramidal cell dendrites in area CA1 of rat hippocampus, J. physiol. (lond.), 483, 421, (1995)
[10] Stuart, G.; Schiller, J.; Sakmann, B., Action potential initiation and propagation in rat neocortical pyramidal neurons, J. physiol. (lond.), 505, 617, (1997)
[11] Golding, N.L.; Spruston, N., Dendritic sodium spikes are variable triggers of axonal action potentials in hippocampal CA1 pyramidal neurons, Neuron, 21, 1189, (1998)
[12] Stafstrom, C.E.; Schwindt, P.C.; Chubb, M.C.; Crill, W.E., Properties of persistent sodium conductance and calcium conductance of layer V neurons from cat sensorimotor cortex in vitro, J. neurophysiol., 53, 153, (1985)
[13] Thomson, A.M.; Girdlestone, D.; West, D.C., Voltage-dependent currents prolong single-axon postsynaptic potentials in layer III pyramidal neurons in rat neocortical slices, J. neurophysiol., 60, 1896, (1988)
[14] Sutor, B.; Hablitz, J.J., EPSPs in rat neocortical neurons in vitro. II involvement of NMDA receptors in the generation of epsps, J. neurophysiol., 61, 621, (1989)
[15] Deisz, R.A.; Fortin, G.; Zieglgansberger, W., Voltage dependence of excitatory postsynaptic potentials of rat neocortical neurons, J. neurophysiol., 65, 371, (1991)
[16] Nicoll, A.; Larkman, A.; Blakemore, C., Modulation of EPSP shape and efficacy by intrinsic membrane conductances in rat neocortical pyramidal neurons in vitro, J. physiol. (lond.), 468, 693, (1993)
[17] Stuart, G.; Sakmann, B., Amplification of EPSPs by axosomatic sodium channels in neocortical pyramidal neurons., Neuron, 15, 1065, (1995)
[18] P.C. Schwindt, W.E. Crill, Membrane properties of cat spinal motoneurons, in: R.A. Davidoff (Ed.), Handbook of the Spinal Cord: Anatomy and Physiology, vols. 2 & 3, Marcel Dekker, New York, 1984
[19] Magee, J.C.; Johnston, D., Synaptic activation of voltage-gated channels in the dendrites of hippocampal pyramidal neurons, Science, 268, 301, (1995)
[20] Vallet, A.M.; Coles, J.A.; Eilbeck, J.C.; Scott, A.C., Membrane conductances involved in amplification of small signals by sodium channels in photoreceptors of drone honey bee, J. physiol. (lond.), 456, 303, (1992)
[21] Hidaka, S.; Ishida, A.T., Voltage-gated na^{+} current availability after step- and spike-shaped conditioning depolarizations of retinal ganglion cells, Pflugers arch., 436, 497, (1998)
[22] Stafstrom, C.E.; Schwindt, P.C.; Crill, W.E., Negative slope conductance due to a persistent subthreshold sodium current in cat neocortical neurons in vitro, Brain res., 236, 221, (1982)
[23] Schwindt, P.C.; Crill, W.E., Amplification of synaptic current by persistent sodium conductance in apical dendrite of neocortical neurons, J. neurophysiol., 74, 2220, (1995)
[24] MacVicar, B.A., Depolarizing prepotentials are na^{+} dependent in CA1 pyramidal neurons, Brain res., 333, 378, (1985)
[25] Taylor, C.P., Na^{+} currents that fail to inactiviate, Trends neurosci., 16, 455, (1993)
[26] Crill, W.E., Persistent sodium current in Mammalian central neurons, Annu. rev. physiol., 58, 349, (1996)
[27] Gillessen, T.; Alzheimer, C., Amplification of EPSPs by low ni^{2+} and amiloride-sensitive ca2+ channels in apical dendrites of rat CA1 pyramidal neurons, J. neurophysiol., 77, 1639, (1997)
[28] Urban, N.N.; Henze, D.A.; Barrionuevo, G., Amplifcation of perforant-path EPSPs in CA3 pyramidal cells by LVA calcium and sodium channels, J. neurophysiol., 80, 1558, (1998)
[29] W. Rall, Core conductor theory and cable properties of neurons, in: E.R. Kandel (Eds.), Handbook of Physiology, American Physiological Society, Bethesda, MD, 1977, pp. 39 (Chapter 3)
[30] Byzov, A.L.; Trifonov, A.Yu.; Chailahian, L.M.; Golubtzov, K.W., Amplification of graded potentials in horizontal cells of the retina, Vis. res., 17, 265, (1977)
[31] Taylor, G.C.; Coles, J.A.; Eilbeck, J.C., Conditions under which na^{+} channels can boost conduction of small graded potentials, J. theoret. biol., 172, 379, (1995)
[32] Bernander, O.; Koch, C.; Douglas, R.J., Amplification and linearization of distal synaptic input to cortical pyramidal cells, J. neurophysiol., 72, 2743, (1994)
[33] De Schutter, E.; Bower, J.M., Simulated responses of cerebellar purkinje cells are independent of the dendritic location of granule cell synaptic inputs, Proc. nat. acad. sci. USA, 91, 4736, (1994)
[34] White, J.A.; Sekar, N.S.; Kay, A.R., Errors in persistent inward currents generated by space-clamp errors: a modeling study, J. neurophysiol., 73, 2369, (1995)
[35] Lipowsky, R.; Gillessen, T.; Alzheimer, C., Dendritic na^{+} channels amplify EPSPs in hippocampal CA1 pyramidal cells, J. neurophysiol., 76, 2181, (1996)
[36] Cook, E.P.; Johnston, D., Active dendrites reduce location-dependent variability of synaptic input trains, J. neurophysiol., 78, 2116, (1997)
[37] Cook, E.P.; Johnston, D., Voltage-dependent properties of dendrites that eliminate location-dependent variability of synaptic input, J. neurophysiol., 81, 535, (1999)
[38] De Schutter, E., Dendritic voltage and calcium-gated channels amplify the variability of postsynaptic responses in purkinje cell model, J. neurophysiol., 80, 504, (1998)
[39] Takagi, H.; Sato, R.; Mori, M.; Ito, E.; Suzuki, H., Roles of A- and D-type K channels in EPSP integration at a model dendrite, Neurosci. lett., 254, 165, (1998)
[40] Mozrzymas, J.W.; Bartoszkiewicz, M., The discrete nature of biological membrane conductance, channel interaction through electrolyte layers and the cable equation, J. theoret. biol., 162, 371, (1993)
[41] Larsson, H.P.; Kleene, S.J.; Lecar, H., Noise analysis of ion channels in non-space-clamped cables: estimates of channel parameters in olfactory cilia, Biophys. J., 72, 1193, (1997)
[42] Andrietti, F.; Bernardini, G., Segmented and ‘equivalent’ representation of the cable equation, Biophys. J., 46, 615, (1984)
[43] 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
[44] Baer, S.M.; Rinzel, J., Propagation of dendritic spikes mediated by excitable spines: a continuum theory, J. neurophysiol., 65, 874, (1991)
[45] Bell, J.; Holmes, M., Model of the dynamics of receptor potential in a mechano-receptor, Math. biosci., 110, 139, (1992) · Zbl 0761.92011
[46] Toth, T.I.; Crunelli, V., Effects of tapering geometry and inhomogeneous ion channel distribution in a neuron model, Neurosci., 84, 1223, (1998)
[47] Magee, J.; Johnston, D., Characterization of single voltage-gated na^{+} and ca2+ channels in apical dendrites of rat CA1 pyramidal neurons, J. physiol. (lond.), 487, 67, (1995)
[48] Sabah, N.H.; Leibovic, K.N., Subthreshold oscillatory responses of the hodgkin – huxley cable model for the squid gaint axon, Biophys. J., 9, 1206, (1969)
[49] Llinas, R.; Nicholson, C., Electrophysiological properties of dendrites and somata in alligator purkinje cells, J. neurophysiol., 34, 532, (1971)
[50] FitzHugh, R., Computation of impulse initiation and saltatory conduction in a myelinated nerve fiber, Biophys. J., 2, 11, (1962)
[51] Hodgkin, A.L.; Huxley, A.F., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol. (lond.), 117, 500, (1952)
[52] MacGregor, R.J., A model for responses to activation by axodendritic synapses, Biophys. J., 8, 305, (1968)
[53] Madsen, E.L., Theory for a test of the electric cable model of myelinated axon and saltatory conduction, J. theoret. biol., 67, 203, (1977)
[54] Basser, P.J., Cable equation for a myelinated axon derived from its microstructure, Med. biol. eng. comput., 31, S87, (1993)
[55] Keizer, J.; Smith, G.D.; Ponce-Dawson, S.; Pearson, J.E., Saltatory propagation of ca^{2+} waves by ca2+ sparks, Biophys. J., 75, 595, (1998)
[56] Bennett, M.R.; Gibson, W.G.; Poznanski, R.R., Extracellular current flow and potential during quantal transmission from varicosities in a smooth muscle syncytium, Philos. trans. R. soc., 342, 89, (1993)
[57] R.R. Poznanski, J. Bell, Math. Biosci., this issue, p. 123
[58] FitzHugh, R., Dimensional analysis of nerve models, J. theoret. biol., 40, 517, (1973)
[59] Poznanski, R.R., Membrane voltage changes in passive dendritic trees: a tapering equivalent cylinder model, IMA J. math. appl. med. biol., 5, 113, (1988) · Zbl 0651.92010
[60] Evans, J.D.; Kember, G.C., Analytical solutions to a tapering multicylinder somatic shunt cable model for passive neurones, Math. biosci., 149, 137, (1998) · Zbl 0943.92011
[61] Sabah, N.H.; Spangler, R.A., Repetitive response of the hodgkin – huxley model for the squid gaint axon, J. theoret. biol., 29, 155, (1970)
[62] Mauro, A.; Conti, F.; Dodge, F.; Schor, R., Subthreshold behavior and phenomenological impedance of the squid giant axon, J. gen. physiol., 55, 497, (1970)
[63] Schiller, J.; Major, G.; Koester, H.J.; Schiller, Y., NMDA spikes in basel dendrites of cortical pyramidal neurons, Nature, 404, 285, (2000)
[64] French, C.R.; Sah, P.; Buckett, K.J.; Gage, P.W., A voltage-dependent persistent sodium current in Mammalian hippocampal neurons, J. gen. physiol., 95, 1139, (1990)
[65] Hodgkin, A.L., The optimum density of sodium channels in an unmyelinated nerve, Philos. trans. R. soc. lond. B, 270, 297, (1975)
[66] Sigworth, F.J.; Neher, E., Single sodium channel currents observed in cultured rat muscle cells, Nature, 287, 497, (1980)
[67] Stuhmer, W.; Methfessel, B.; Sakmann, B.; Noda, M.; Numa, S., Patch clamp characterization of sodium channels expressed from rat brain cdna, Eur. biophys. J., 14, 131, (1987)
[68] Sabah, N.H.; Leibovic, K.N., The effect of membrane parameters on the properties of the nerve impulse, Biophys. J., 12, 1132, (1972)
[69] Hayashi, H.; Ishizuka, S., Chaotic nature of bursting discharge in the onchidium pacemaker neuron, J. theoret. biol., 156, 269, (1992)
[70] Gabel, L.A.; Nisenbaum, E.S., Biophysical characterization and functional consequences of a slowly inactivating potassium current in neostriatal neurons, J. neurophysiol., 79, 1989, (1998)
[71] Magee, J.C., Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal CA1 pyramaidal neurons, J. neurosci., 18, 7613, (1998)
[72] H.C. Tuckwell, Instruction to Theoretical Neurobiology. Linear Cable Theory and Dendritic Structure, vol. 1, Cambridge University, Cambridge, 1988
[73] Durand, D., The somatic shunt cable model for neurons, Biophys. J., 46, 645, (1984)
[74] Jackson, M.B., Cable analysis with the whole-cell patch clamp: theory and experiment, Biophys. J., 61, 756, (1992)
[75] 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, (1992)
[76] J.D. Evans, The multiple equivalent cylinder model, in: R.R. Poznanski (Ed.), Modeling in the Neurosciences: From Ionic Channels to Neural Networks, Harwood Academic Publishers, Amsterdam, 1999 (Chapter 5)
[77] Major, G.; Evans, J.D.; Jack, J.J.B., Solutions for transients in arbitrarily branching cables, Biophys. J., 65, 423, (1993)
[78] W. Pogorzelski, Integral Equations and Their Applications, vol. 1, Pergamon, Oxford, 1966 · Zbl 0137.30502
[79] 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
[80] Hoffman, D.A.; Magee, J.C.; Colbert, C.M.; Johnston, D., K^{+} channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons, Nature, 387, 869, (1997)
[81] Nicoll, A.; Larkman, A.; Blakemore, C., EPSPs in rat neocortical pyramidal neurones in vitro are prolonged by NMDA receptor-mediated currents, Neurosci. lett., 143, 5, (1992)
[82] Alzheimer, C.; Schwindt, P.C.; Crill, W.E., Model gating of na^{+} channels as a mechanism of persistent na+ current in pyramidal neurons from rat and cat sensorimotor cortex, J. neurosci., 13, 660, (1993)
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