Mathematical modeling of the transmission of electrical signals in neurons
Keywords:
review, neuron, computer modeling, cable theory, passive membrane, synaptic signal, non-linear mechanisms
Abstract
The article is a brief overview to get acquainted with the methods of mathematical modeling of the
transmission of electrical signals in biological neurons. The neuronal doctrine and modern method of
obtaining morphological data of nerve cells are described. It is the computer reconstruction of in vivo
stained neurons with branched dendritic processes. The cable model of transmission of electrical signals
in the dendrites of neurons is overviewed. The use of cable theory for simulations in simplified and
realistic models, which are built using the data of reconstructed neurons, is discussed. Described the
methods of inclusion in the model the non-linear factors such as non-linear summation of synaptic input
multiple-static signals and nonlinear membrane conductance - ion channels (for example, mo-Delhi
Hodgkin-Huxley).
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References
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46. Synaptic background activity influences spatiotemporal integration in single pyramidal neurons in vivo / Ö. Bernander, J. R. Douglas, K. A. C. Martin, C. Koch // PNAS – 1991. – V. 88. – P. 11569–11573.
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48. Korogod S. M. Electrical dynamics of the dendritic space / S. M. Korogod, S. Tyč-Dumont. – Cambridge University Press, 2009.
49. Active properties of neuronal dendrites / D. Johnston, J. C. Magee, C. M. Colbert, B. R. Christie // Annu. Rev. Neurosci. – 1996. – V. 19. – P. 165–186.
50. Mel B. W. Information processing in dendritic trees / B. W. Mel // Neural Comput. – 1994. – V. 6. – P. 1031–1085.
51. Leuchtag H. R. Voltage-Sensitive Ion Channels. Biophysics of Molecular Excitability / H. R. Leuchtag. – Springer, 2008.
52. Safronov B. V. Electrical activity of individual neurons: patch-clamp techniques / B. V. Safronov, W. Vogel – Springer-Verlag, – 1999 // in Modern Techniques in Neuroscience Research. – Ch. 6. – P. 173–192.
53. Hodgkin A. A quantitative description of membrane current and its application to conduction and excitation in nerve / A. Hodgkin, A. Huxley // J. Physiol. – 1952. – V. 117. – P. 500–544.
54. Markram H. The Blue Brain Project / H. Markram // Nat. Rev. Neurosci. – 2006. – V. 7. – P. 153–160.
2. Modern Techniques in Neuroscience Research. / Ed. by U. Windhorst, H. Johansson. – Springer-Verlag, 1999.
3. Ling G. The normal membrane potential of frog sartorius fibers. / G. Ling, R. W. Gerard // J. Cell Physiol. – 1949. – Vol. 34, № 3. – P. 383–396.
4. Halliwell J. Using microelectrodes. / J. Halliwell, M. Whitaker, D. Ogden. – Cambridge: Company of Biologists Limited, 1994. – in Microelectrode Techniques, The Plymouth workshop handbook, 2nd ed., P. 1–15.
5. Brown A. G. Direct observations on the contacts made between Ia afferent fibers and alpha motoneurones in the cat’s lumbosacral spinal cord. / A. G. Brown, R. E. W. Fyffe // J. Physiol. – 1981. – V. 313. – P. 121–140.
6. Stewart W. W. Lucifer dyes – highly fluorescent dyes for biological tracing. / W. W. Stewart // Nature. – 1981. – V. 292. – P. 17–21.
7. Glaser E. M. A semi-automatic computer-microscope for the analysis of neuronal morphology. / E. M. Glaser, H. Van der Loos // IEEE Trans. Biomed. Engin. 1965. – V. 12. – P. 22–31.
8. Mannen H. Contribution to the quantitative study of the nervous tissue. A new method for measurement of the volume and surface area of neurons. / H. Mannen // J. Comp. Neurol. – 1966. – V. 126. – P. 75–90.
9. Capowski J. J. Computer Techniques in Neuroanatomy. / ed. by J. J. Capowski. – Plenum Press, New York. – 1989. – P. 264.
10. Levinthal C. Three dimensional reconstruction from serial sections / C. Levinthal, R. Ware. // Nature. – 1972. – V. 236. – P. 207–210.
11. Bras H. The dendrites of single brain-stem motoneurons intracellularly labelled with horseradish peroxidise in the cat. Morphological and electrical differences. / H. Bras, P. Gogan, S. Tyč-Dumont // Neuroscience. 1987. – V. 22, №.3. – P. 947–970.
12. Stochastic geometry and electrotonic architecture of dendritic arborisation of brain stem motoneuron. / H. Bras, S. Korogod, Y. Driencourt [at al.] // Eur. J. Neurosci. 1993. – V. 5. – P. 1485–1493.
13. Heterogeneous synaptic covering and differential charge transfer sensitivity among the dendrites of a reconstructed abducens motor neurone: correlations between electron microscopic and computer simulation data. / H. Bras, F. Lahjouji, S. M. Korogod [et al.] // J. Neurocitol. – 2003. – V. 32. – P. 5–24.
14. The problem of morphological noise in reconstructed dendritic arborisation. / G. Horcholle-Bossavit, P. Gogan, Y. Ivanov [at al.] // J. Neurosci. Methods. – 2000. – V. 95. – P. 83–93.
15. Neuronal morphology data bases: morphological noise and assessment of data quality. / A. V. Kaspirzhny, P. Gogan, G. Horcholle-Bossavit, S. Tyč-Dumont // Network: Comput. Neural Syst. – 2002. – V. 13. – P. 357–380.
16. The Human Brain Project: neuroinformatics tools for integrating, searching and modelling multidisciplinaty neuroscience data. / G. M. Shepherd, J. S. Mirsky, M. D. Healy [et al.] // TINS. – 1998. – V. 21. №.11. – P. 460–468.
17. An on-line archive of reconstructed hippocampal neurons. / R. C. Cannon, D. A. Turner, G. K. Pyapali, H. V. Weal // J. Neurosci. Methods. – 1998. – V. 84. – P. 49–54.
18. Kötter R. Neuroscience databases: tools for exploring brain structure-functional relationships. / R. Kötter // Phil. Trans. R. Soc. Lond. – 2001. – V. 356. – P. 1111–1120.
19. Generation, description and storage of dendritic morphology data. / G. A. Ascoli, J. L. Krichmar, S. J. Nasuto, S. L. Senft // Phil. Trans. R. Soc. Lond. B. – 2001. – V. 356. – P. 1131–1145.
20. Ascoli G. A. NeuroMorpho.org: a central resource for neuronal morphologies. / G. A. Ascoli, D. E. Donohue, M. Halavi // J. Neurosci. – 2007. – V. 27, №.35. – P. 9247–9251.
21. Dendrites. / Ed. by G. Stuart, N. Spruston, M. Häusser. – New York: Oxford. 1999.
22. Jack J. J. B. Electric current flow in excitable cells. / J. J. B. Jack, D. Noble, R. W. Tsien. – Oxford: Clarendon, 1975.
23. Rall W. Core conductor theory and cable properties of neurons. / W. Rall // Handbook of physiology. The nervous system. Cellular Biology of Neurons. / Ed. by E. Kandel. – Bethesda., MD: Am. Physiol. Soc., 1977. – Sect.1. – V. 1. – P. 39–98.
24. Koch C. Biophysics of computation: information processing in single neurons. / C. Koch. Computational Neuroscience. – New York, Oxford: Oxford university press, 1999.
25. Barrett J. N. Influence of dendritic location and membrane properties on the effectiveness of synapses of cat motoneurones. / J. N. Barrett, W. E. Crill // J. Physiol. – 1974. – V. 239. – P.325–345.
26. Schierwagen A. K. Exploring the computational capabilities of single neurons by continuous cable modeling. / A. K. Schierwagen // Prog. Brain Res. – 1994. – V. 102. – P. 151–167.
27. Korogod S. M. Parameter sensitivity of distributed transfer properties of neuronal dendrites: a passive cable approximation. / S. M. Korogod, A. V. Kaspirzhny // Biol. Cybern. – 2008. – V. 98, №. 2. – P.87–100.
28. Korogod S. M. Spatial heterogeneity of passive electrical transfer properties of neuronal dendrites due to their metrical asymmetry. / S. M. Korogod, A. V. Kaspirzhny // Biol. Cybern. – 2011. – V. 105, №. 5-6. – P. 305–317.
29. Coombs J. S. The electrical constants of the motoneurone membrane. / J. S. Coombs, D. R. Curtis, J. C. Eccles // J. Physiol. – 1959. – V. 145. – P. 505–528.
30. Rall W. Branching dendritic trees and motoneuron membrane resistivity. / W. Rall // Exp. Neurol. – 1959. – V. 1. – P. 491–527.
31. Lux H. D. Direct matching of morphological and electrophysiological data in cat spinal motoneurones. / H. D. Lux, P. Schubert, G. W. Kreutzberg – Oslo: Universitetsforlaget, – 1970. // in Excitatory synaptic mechanisms, P. 189–198.
32. Iansek R. An analysis of the cable properties of spinal motoneurones using a brief intracellular current pulse. / R. Iansek, S. J. Redman // J. Physiol. – 1973. – V. 234. – P. 613–636.
33. Barrett J. N. Specific membrane properties of cat motoneurones. / J. N. Barrett, W. E. Crill // J. Physiol. – 1974. – V. 239. – P. 301–324.
34. Detailed passive cable models of whole-cell recorded CA3 pyramidal neurons in rat hippocampal slices. / G. Major, A. U. Larkman, P. Jonas [et al.] // J. Neurosci. – 1994. – V. 14, №. 8. – P. 4613–4638.
35. Passive electrical properties of ventral horn neurons in rat spinal cord slices. / D. Thurbon, H.-R. Lüscher, T. Hofstetter, S. J. Redman // J. Neurophysiol. – 1998. – V. 79. – P. 2485–2502.
36. Schierwagen A. Segmental cable modeling of electrotonic transfer properties of deep superior colliculus neurons in the cat. / A. Schierwagen // J. Hirnforsch. – 1986. – V. 27, №. 6. – P. 679–690.
37. Ohme M. An equivalent cable model for neuronal trees with active membrane / M. Ohme, A. Schierwagen // Biol. Cybern. – 1998. – V. 78. – P. 227–243.
38. D’Aguanno A. A. system model for investigating passive electrical properties of neurons / A. D’Aguanno, B. L. Bardakjian, P.L. Carlen // Biophys. J. – 1989. – V. 55. – P. 1169–1182.
39. Bower J. M. The book of GENESIS. / J. M. Bower, D. Beeman. – Springer-Verlag, New-York, 1998.
40. Carnevale N. T. The NEURON Book / N. T. Carnevale, M. L. Hines. – Cambridge University Press, 2006.
41. Rall W. Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input / W. Rall // J. Neurophysiol. – 1967. – V. 30. – P. 1138–1168.
42. Perkel D. H. Quantitative methods for predicting neuronal behavior / D. H. Perkel, B. Mulloney, R. W. Budelli // Neuroscience. – 1981. – V. 6, №.5. – P. 823–837.
43. Destexhe A. Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism / A. Destexhe, Z. F. Mainen, T. J. Sejnowski // J. Comput. Neurosci. – 1994. – V. 1. – P. 195–230.
44. Abbott L. F. Realistic synaptic inputs for model neural networks / L. F. Abbott // Network: Comp. Neural Syst. – 1991. – V. 2. – P. 245–258.
45. Kuno M. Non-linear summation of unit synaptic potentials in spinal motoneurones of the cat / M. Kuno, J. T. Miyahara // J. Physiol. – 1969. – V. 201. – P. 465–477.
46. Synaptic background activity influences spatiotemporal integration in single pyramidal neurons in vivo / Ö. Bernander, J. R. Douglas, K. A. C. Martin, C. Koch // PNAS – 1991. – V. 88. – P. 11569–11573.
47. Destexhe A. The high-conductance state of neocortical neurons in vivo / A. Destexhe, M. Rudolph, D. Paré // Nat. Rev. Neurosci. – 2003. – V. 4. – P. 739–751.
48. Korogod S. M. Electrical dynamics of the dendritic space / S. M. Korogod, S. Tyč-Dumont. – Cambridge University Press, 2009.
49. Active properties of neuronal dendrites / D. Johnston, J. C. Magee, C. M. Colbert, B. R. Christie // Annu. Rev. Neurosci. – 1996. – V. 19. – P. 165–186.
50. Mel B. W. Information processing in dendritic trees / B. W. Mel // Neural Comput. – 1994. – V. 6. – P. 1031–1085.
51. Leuchtag H. R. Voltage-Sensitive Ion Channels. Biophysics of Molecular Excitability / H. R. Leuchtag. – Springer, 2008.
52. Safronov B. V. Electrical activity of individual neurons: patch-clamp techniques / B. V. Safronov, W. Vogel – Springer-Verlag, – 1999 // in Modern Techniques in Neuroscience Research. – Ch. 6. – P. 173–192.
53. Hodgkin A. A quantitative description of membrane current and its application to conduction and excitation in nerve / A. Hodgkin, A. Huxley // J. Physiol. – 1952. – V. 117. – P. 500–544.
54. Markram H. The Blue Brain Project / H. Markram // Nat. Rev. Neurosci. – 2006. – V. 7. – P. 153–160.
Cited
How to Cite
Kaspirzhnyy, A. V. (1). Mathematical modeling of the transmission of electrical signals in neurons. Biophysical Bulletin, 1(29). Retrieved from https://periodicals.karazin.ua/biophysvisnyk/article/view/2347
Section
Biophysics of complex systems
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