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ANALYTICAL SOLUTIONS OF THE FRANKENHAEUSER-HUXLEY EQUATIONS I: MINIMAL MODEL FOR BACKPROPAGATION OF ACTION POTENTIALS IN SPARSELY EXCITABLE DENDRITES

2004; Imperial College Press; Volume: 03; Issue: 03 Linguagem: Inglês

10.1142/s0219635204000439

1757-448X

Roman R. Poznański,

Photoreceptor and optogenetics research

Journal of Integrative NeuroscienceVol. 03, No. 03, pp. 267-299 (2004) Research ReportsNo AccessANALYTICAL SOLUTIONS OF THE FRANKENHAEUSER-HUXLEY EQUATIONS I: MINIMAL MODEL FOR BACKPROPAGATION OF ACTION POTENTIALS IN SPARSELY EXCITABLE DENDRITESROMAN R. POZNANSKIROMAN R. POZNANSKIDepartment of Psychology, Indiana University, Bloomington, IN 47405, USA Search for more papers by this author https://doi.org/10.1142/S0219635204000439Cited by:13 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractHodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na+ channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na+ ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na+ and K+. In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Φ) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na+ activation (μ) and a state-dependent inactivation (η) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na+ channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.Keywords:Cable theoryanalytical solutionssodium channelsaction potentialdendrites References R. W. Aldrich, D. P. Corey and C. F. Stevens, Nature 306, 436 (1983), DOI: 10.1038/306436a0. Crossref, Medline, ISI, Google ScholarS. D. Antic, J. Physiol. (Lond.) 550, 35 (2003), DOI: 10.1113/jphysiol.2002.033746. Crossref, Google ScholarC. M. Armstrong and F. Bezanilla, J. Gen. Physiol. 70, 567 (1977), DOI: 10.1085/jgp.70.5.567. Crossref, Medline, ISI, Google ScholarD. Attwell and J. J. B. Jack, Prog. Biophys. Mol. Bio. 34, 81 (1978), DOI: 10.1016/0079-6107(79)90015-4. Crossref, Medline, ISI, Google ScholarJ. M. Bekkers, J. Physiol. (Lond.) 525, 611 (2000), DOI: 10.1111/j.1469-7793.2000.t01-2-00611.x. Crossref, Google ScholarJ. Bell and L. M. Cook, SIAM J. Appl. Math. 35, 678 (1978), DOI: 10.1137/0135056. Crossref, ISI, Google ScholarJ. Bell and L. M. Cook, Math. Biosci. 46, 11 (1979), DOI: 10.1016/0025-5564(79)90012-9. Crossref, ISI, Google ScholarC. Bernard and D. Johnston, J. Neurophysiol. 90, 1807 (2003), DOI: 10.1152/jn.00286.2003. Crossref, Medline, ISI, Google ScholarF. Bezanilla and C. M. Armstrong, J. Gen. Physiol. 70, 549 (1977), DOI: 10.1085/jgp.70.5.549. Crossref, Medline, ISI, Google ScholarJ. Bischofberger and P. Jonas, J. Physiol. (Lond.) 504, 359 (1997), DOI: 10.1111/j.1469-7793.1997.359be.x. Crossref, Google Scholar N. F. Britton , Reaction-Diffusion Equations & Their Applications to Biology ( Academic Press , London , 1986 ) . Google ScholarT. Brismar, J. Physiol. (Lond.) 298, 171 (1980). Crossref, Google ScholarG. Buzsakiet al., Proc. Natl. Acad. Sci. USA 93, 9921 (1996), DOI: 10.1073/pnas.93.18.9921. Crossref, Medline, Google ScholarJ. H. Caldwellet al., Proc. Natl. Acad. Sci. USA 97, 5616 (2000), DOI: 10.1073/pnas.090034797. Crossref, Medline, ISI, Google ScholarJ. C. Callaway and W. N. Ross, J. Neurophysiol. 74, 1395 (1995). Crossref, Medline, ISI, Google ScholarR. G. Casten, H. Cohen and P. A. Lagerstrom, Q. Appl. Math. 32, 365 (1975). Crossref, ISI, Google ScholarW. R. Chenet al., J. Neurophysiol. 88, 2755 (2002), DOI: 10.1152/jn.00057.2002. Crossref, Medline, ISI, Google ScholarC. Chiang, B. Math. Biol. 40, 247 (1978). Crossref, Medline, ISI, Google ScholarS. Y. Chiuet al., J. Physiol. (Lond.) 292, 149 (1979). Crossref, Google ScholarE. D. Cohen, Visual Neurosci. 18, 799 (2001). Crossref, Medline, ISI, Google ScholarK. Cooper, E. Jakobsson and P. Wolynes, Prog. Biophys. Mol. Bio. 46, 51 (1985), DOI: 10.1016/0079-6107(85)90012-4. Crossref, Medline, ISI, Google ScholarC. Colbertet al., J. Neurosci. 17, 6512 (1997). Crossref, Medline, ISI, Google ScholarK. S. Cole, R. Guttman and F. Bezanilla, Proc. Natl. Acad. Sci. USA 65, 884 (1970), DOI: 10.1073/pnas.65.4.884. Crossref, Medline, ISI, Google ScholarJ. W. Cooley and F. A. Dodge, Biophys. J. 6, 583 (1966), DOI: 10.1016/S0006-3495(66)86679-1. Crossref, Medline, ISI, Google Scholar J. Cronin , Mathematical Aspects of Hodgkin-Huxley Neural Theory ( CUP , Cambridge , 1987 ) . Crossref, Google ScholarF. Debarbieux, E. Andinat and S. Charpak, J. Neurosci. 23, 5553 (2003). Crossref, Medline, ISI, Google ScholarA. Destexhe, M. Rudolph and D. Pare, Nat. Rev. Neurosci. 4, 739 (2003), DOI: 10.1038/nrn1198. Crossref, Medline, ISI, Google ScholarN. A. Dimitrova and G. V. Dimitrov, Biol. Cybern. 66, 185 (1991), DOI: 10.1007/BF00243294. Crossref, Medline, ISI, Google Scholar F. A. Dodge , The Neurosciences Fourth Study Progam , The nonuniform excitability of central neurons as exemplified by a model of the spinal motoneuron , eds. F. O. Schmidt and F. G. Worden ( MIT Press , Cambridge, MA , 1979 ) . Google ScholarF. A. Dodge and B. Frankenhaeuser, J. Physiol. (Lond.) 148, 188 (1959). Crossref, Google ScholarB. Doironet al., J. Neurophysiol. 86, 1523 (2001). Crossref, Medline, ISI, Google ScholarG. Ehrenstein and H. Lecar, Annu. Rev. Biophys. Bio. 1, 347 (1972), DOI: 10.1146/annurev.bb.01.060172.002023. Crossref, Medline, Google ScholarT. Euler, P. B. Detwiler and W. Denk, Nature 418, 845 (2002), DOI: 10.1038/nature00931. Crossref, Medline, ISI, Google ScholarJ. Evans and N. Shenk, Biophys. J. 10, 1090 (1970), DOI: 10.1016/S0006-3495(70)86355-X. Crossref, Medline, ISI, Google ScholarR. FitzHugh, J. Gen. Physiol. 43, 867 (1960), DOI: 10.1085/jgp.43.5.867. Crossref, Medline, ISI, Google ScholarR. FitzHugh, J. Gen. Physiol. 49, 989 (1966), DOI: 10.1085/jgp.49.5.989. Crossref, Medline, ISI, Google ScholarR. FitzHugh, J. Theor. Biol. 40, 517 (1973), DOI: 10.1016/0022-5193(73)90008-8. Crossref, Medline, ISI, Google ScholarB. Frankenhaeuser and A. F. Huxley, J. Physiol. (Lond.) 171, 302 (1964). Crossref, Google ScholarS. I. Fried, T. A. Munch and F. S. Werblin, Nature 420, 411 (2002), DOI: 10.1038/nature01179. Crossref, Medline, ISI, Google ScholarA. Frick, J. Magee and D. Johnson, Nat. Neutrosci. 7, 126 (2004), DOI: 10.1038/nn1178. Crossref, Medline, ISI, Google ScholarA. Frick, W. Zieglgansberger and H.-U. Dodt, J. Neurophysiol. 86, 1412 (2001). Crossref, Medline, ISI, Google ScholarJ. H. Goldberg, G. Tamas and R. Yuste, J. Physiol. (Lond.) 551, 49 (2003), DOI: 10.1113/jphysiol.2003.042580. Crossref, Google ScholarN. L. Golding, W. L. Kath and N. Spruston, J. Neurophysiol. 86, 2998 (2001). Crossref, Medline, ISI, Google ScholarL. Goldman, Biophys. J. 15, 119 (1975), DOI: 10.1016/S0006-3495(75)85796-1. Crossref, Medline, ISI, Google ScholarS. S. Goldstein and W. Rall, Biophys. J. 14, 731 (1974), DOI: 10.1016/S0006-3495(74)85947-3. Crossref, Medline, ISI, Google Scholar I. S. Gradshteyn and I. M. Ryzhik , Table of Integrals, Series, and Products ( Academic Press , New York , 1980 ) . Google ScholarJ. E. Hanson, Y. Smith and D. Jaeger, J. Neurosci. 24, 329 (2004), DOI: 10.1523/JNEUROSCI.3937-03.2004. Crossref, Medline, ISI, Google Scholar B. Hille , Ionic Channels of Excitable Membranes , 3rd edn. ( Sinauer , Sunderland, MA , 2001 ) . Google ScholarA. L. Hodgkin, Philos. T. Roy. Soc. Lond. B 270, 297 (1975), DOI: 10.1098/rstb.1975.0010. Crossref, Medline, ISI, Google ScholarA. L. Hodgkin and A. F. Huxley, J. Physiol. (Lond.) 117, 500 (1952). Crossref, Google ScholarD. A. Hoffmanet al., Nature 387, 869 (1997), DOI: 10.1038/42571. Crossref, Medline, ISI, Google ScholarA. V. Holden, Biol. Cybern. 38, 1 (1980), DOI: 10.1007/BF00337395. Crossref, Medline, ISI, Google Scholar A. V. Holden , Advances in Physiological Sciences, vol. 30: Neural Communication and Control , Membrane current fluctuations and neuronal information processing , ed. G. Szekely ( Pergamon , Oxford , 1981 ) . Google ScholarY. Horikawa, Biol. Cybern. 79, 251 (1998), DOI: 10.1007/s004220050475. Crossref, Medline, ISI, Google ScholarR. C. Hoyt, Biophys. J. 45, 55 (1984), DOI: 10.1016/S0006-3495(84)84106-5. Crossref, Medline, ISI, Google ScholarJ. R. Huguenard, O. P. Hamill and D. A. Prince, Proc. Natl. Acad. Sci. USA 86, 2473 (1989), DOI: 10.1073/pnas.86.7.2473. Crossref, Medline, ISI, Google Scholar J. J. B. Jack , D. Noble and R. W. Tsien , Electric Current Flow in Excitable Cells ( Clarendon Press , Oxford , 1983 ) . Google ScholarD. Johnstonet al., Annu. Rev. Neurosci. 19, 165 (1996), DOI: 10.1146/annurev.ne.19.030196.001121. Crossref, Medline, ISI, Google ScholarH.-Y. Jung, T. Mickus and N. Spruston, J. Neurosci. 17, 6639 (1997). Crossref, Medline, ISI, Google ScholarT. B. Kepler, L. F. Abbott and E. Marder, Biol. Cybern. 66, 381 (1992), DOI: 10.1007/BF00197717. Crossref, Medline, ISI, Google ScholarV. I. Krinskii and Yu. M. Kokoz, Biophysics-USSR 18, 533 (1973). Google Scholar K. N. Leibovic , Nervous System Theory: An Introductory Study ( Academic Press , New York , 1972 ) . Google Scholar K. N. Leibovic and N. H. Sabah , Information Processing in the Nervous System , On synaptic transmission, neural signals and psychophysiological phenomena , ed. K. N. Leibovic ( Springer-Verlag , Berlin , 1969 ) . Google ScholarN. Lemon and R. W. Turner, J. Neurophysiol. 84, 1519 (2000). Crossref, Medline, ISI, Google ScholarH.-R. Lüscher and M. E. Larkum, J. Neurophysiol. 80, 715 (1998). Crossref, Medline, Google ScholarM. Ma and J. Koester, J. Neurosci. 16, 4089 (1996). Crossref, Medline, ISI, Google ScholarJ. Magee and D. Johnston, J. Physiol. (Lond.) 487, 67 (1995). Crossref, Google ScholarZ. F. Mainenet al., Neuron 15, 1427 (1995), DOI: 10.1016/0896-6273(95)90020-9. Crossref, Medline, ISI, Google ScholarR. Marcus, T. Am. Math. Soc. 198, 177 (1974), DOI: 10.2307/1996753. ISI, Google ScholarS. Marom and L. F. Abbott, Biophys. J. 67, 575 (1994). Google ScholarM. Mascagni, Commun Pur. Appl. Math. 42, 213 (1989), DOI: 10.1002/cpa.3160420206. Crossref, ISI, Google Scholar P. L. McGeer , J. C. Eccles and E. G. McGeer , Molecular Neurobiology of the Mammalian Brain , 2nd edn. ( Plenum Press , New York , 1987 ) . Crossref, Google ScholarB. W. Mel, J. Neurophsyiol. 70, 1086 (1993). Crossref, Medline, ISI, Google ScholarC. Meunier, Biol. Cybern. 67, 461 (1992), DOI: 10.1007/BF00200990. Crossref, Medline, ISI, Google ScholarT. Mickus, H. Jung and N. Spruston, Biophys. J. 76, 846 (1999), DOI: 10.1016/S0006-3495(99)77248-6. Crossref, Medline, ISI, Google ScholarM. Migliore, Biophys. J. 71, 2394 (1996), DOI: 10.1016/S0006-3495(96)79433-X. Crossref, Medline, ISI, Google ScholarM. Migliore and G. M. Shepherd, Nat. Neurosci. 3, 362 (2002). Crossref, ISI, Google ScholarR. F. Miller and R. Dacheux, Brain Res. 104, 157 (1976), DOI: 10.1016/0006-8993(76)90657-0. Crossref, Medline, ISI, Google ScholarJ. W. Moore and W. J. Adelman, J. Gen. Physiol. 45, 77 (1961), DOI: 10.1085/jgp.45.1.77. Crossref, Medline, ISI, Google ScholarJ. W. Mozrzymas and M. Bartoszkiewicz, J. Theor. Biol. 162, 371 (1993), DOI: 10.1006/jtbi.1993.1093. Crossref, Medline, ISI, Google ScholarC. B. Muratov, Biophys. J. 79, 2893 (2000), DOI: 10.1016/S0006-3495(00)76526-X. Crossref, Medline, ISI, Google ScholarJ. Patlak, Physiol. Rev. 71, 1047 (1991). Crossref, Medline, ISI, Google ScholarM. S. Penney and N. F. Britton, J. Theor. Biol. 219, 207 (2002), DOI: 10.1006/jtbi.2002.3116. Crossref, Medline, ISI, Google ScholarW. F. Pickard, Math. Biosci. 20, 75 (1974), DOI: 10.1016/0025-5564(74)90069-8. Crossref, Google ScholarP. Poirazi, T. Brannon and B. W. Mel, Neuron 37, 989 (2003), DOI: 10.1016/S0896-6273(03)00149-1. Crossref, Medline, ISI, Google ScholarR. R. Poznanski and J. Bell, Math. Biosci. 166, 101 (2000), DOI: 10.1016/S0025-5564(00)00031-6. Crossref, Medline, ISI, Google ScholarR. R. Poznanski and J. Bell, Math. Biosci. 166, 123 (2000), DOI: 10.1016/S0025-5564(00)00032-8. Crossref, Medline, ISI, Google ScholarN. Qian and T. J. Sejnowski, Biol. Cybern. 62, 1 (1989), DOI: 10.1007/BF00217656. Crossref, ISI, Google ScholarM. Rapp, Y. Yarom and I. Segev, Proc. Natl. Acad. Sci. USA 93, 11985 (1996), DOI: 10.1073/pnas.93.21.11985. Crossref, Medline, ISI, Google ScholarW. Regehret al., Neuron 11, 145 (1993), DOI: 10.1016/0896-6273(93)90278-Y. Crossref, Medline, ISI, Google ScholarA. Reyes, Annu. Rev. Neurosci. 24, 653 (2001), DOI: 10.1146/annurev.neuro.24.1.653. Crossref, Medline, ISI, Google ScholarJ. Rinzel, Fed. Proc. 44, 2944 (1985). Medline, Google ScholarJ. Rinzel and R. N. Miller, Math. Biosci. 49, 27 (1980), DOI: 10.1016/0025-5564(80)90109-1. Crossref, ISI, Google ScholarB. Rozsaet al., J. Neurosci. 24, 661 (2004), DOI: 10.1523/JNEUROSCI.3906-03.2004. Crossref, Medline, ISI, Google ScholarN. H. Sabah and K. N. Leibovic, Biophys. J. 9, 1206 (1969), DOI: 10.1016/S0006-3495(69)86446-5. Crossref, Medline, ISI, Google ScholarN. H. Sabah and K. N. Leibovic, Biophys. J. 12, 1132 (1972), DOI: 10.1016/S0006-3495(72)86150-2. Crossref, Medline, ISI, Google ScholarA. T. Schaeferet al., J. Neurophysiol. 89, 3143 (2003), DOI: 10.1152/jn.00046.2003. Crossref, Medline, ISI, Google Scholar A. C. Scott , Neuroscience: A Mathematical Primer ( Springer-Verlag , Berlin , 2002 ) . Google ScholarL.-R. Shaoet al., J. Physiol. (Lond.) 521, 135 (1999), DOI: 10.1111/j.1469-7793.1999.00135.x. Crossref, Google ScholarF. J. Sigworth and E. Neher, Nature 287, 497 (1980), DOI: 10.1038/287447a0. Google ScholarG. J. Stuart and M. Häusser, Neuron 13, 703 (1994), DOI: 10.1016/0896-6273(94)90037-X. Crossref, Medline, ISI, Google ScholarG. J. Stuart and M. Häusser, Nat. Neurosci. 4, 63 (2001), DOI: 10.1038/82910. Crossref, Medline, ISI, Google ScholarG. J. Stuart and B. Sakmann, Nature 367, 69 (1994), DOI: 10.1038/367069a0. Crossref, Medline, ISI, Google ScholarG. J. Stuart and N. Spruston, Curr. Opin. Neurobiol. 5, 389 (1995), DOI: 10.1016/0959-4388(95)80053-0. Crossref, Medline, ISI, Google ScholarG. J. Stuart and N. Spruston, J. Neurosci. 18, 3501 (1998). Crossref, Medline, ISI, Google ScholarG. J. Stuartet al., Trends Neurosci. 20, 125 (1997), DOI: 10.1016/S0166-2236(96)10075-8. Crossref, Medline, ISI, Google ScholarW. Stühmeret al., Biophys. J. 14, 131 (1987). Google Scholar H. C. Tuckwell , Introduction to Theoretical Neurobiology. Vol. 1 Linear Cable Theory & Dendritic Structure ( CUP , Cambridge , 1988 ) . Crossref, Google Scholar H. C. Tuckwell and J. Feng , Computational Neuroscience: A Comprehensive Approach , A theoretical overview , ed. J. Feng ( Chapman & Hall/CRC , Boca Raton, FL , 2004 ) . Google ScholarC. A. Vandenberg and F. Bezanilla, Biophys. J. 60, 1499 (1991), DOI: 10.1016/S0006-3495(91)82185-3. Crossref, Medline, ISI, Google ScholarA. Van Ooyenet al., Network-Comp. Neural 13, 311 (2002), DOI: 10.1088/0954-898X/13/3/304. Crossref, Medline, ISI, Google ScholarP. Vetter, A. Roth and M. Häusser, J. Neurophysiol. 85, 926 (2001). Crossref, Medline, ISI, Google ScholarA. Vermeulenet al., B. Math. Biol. 58, 493 (1996), DOI: 10.1007/BF02460594. Crossref, Medline, ISI, Google ScholarS. R. Williams and G. J. Stuart, J. Neurosci. 20, 1307 (2000). Crossref, Medline, ISI, Google ScholarH. R. Wilson, J. Theor. Biol. 200, 375 (1999), DOI: 10.1006/jtbi.1999.1002. Crossref, Medline, ISI, Google Scholar H. R. Wilson , Spikes, Decisions, and Actions: The Dynamical Foundations of Neuroscience ( Oxford University Press , New York , 1999 ) . Google ScholarW. Xiong and W. R. Chen, Neuron 34, 115 (2002), DOI: 10.1016/S0896-6273(02)00628-1. Crossref, Medline, ISI, Google ScholarY. Yamadaet al., J. Neurophysiol. 87, 2858 (2002). Crossref, Medline, ISI, Google Scholar FiguresReferencesRelatedDetailsCited By 13On a Model for Nerve Impulse Generation Mediated by Electromechanical ProcessesAlain M. Dikandé25 January 2022 | Brazilian Journal of Physics, Vol. 52, No. 2Exact solutions to cable equations in branching neurons with tapering dendritesLu Yihe and Yulia Timofeeva28 January 2020 | The Journal of Mathematical Neuroscience, Vol. 10, No. 1Action potential propagation and synchronisation in myelinated axonsHelmut Schmidt, Thomas R. Knösche and Boris S. Gutkin17 October 2019 | PLOS Computational Biology, Vol. 15, No. 10Induced mitochondrial membrane potential for modeling solitonic conduction of electrotonic signalsR. R. Poznanski, L. A. Cacha, J. Ali, Z. H. Rizvi and P. 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GOLDFINGER21 November 2011 | Journal of Integrative Neuroscience, Vol. 04, No. 02 Recommended Vol. 03, No. 03 Metrics History Received 16 September 2003 Accepted 11 December 2003 KeywordsCable theoryanalytical solutionssodium channelsaction potentialdendritesPDF download