Mathematical study of a nonlinear neuron multi-dendritic model
Author:
Pierluigi Colli
Journal:
Quart. Appl. Math. 52 (1994), 689-706
MSC:
Primary 92C20; Secondary 35K60, 35Q80
DOI:
https://doi.org/10.1090/qam/1306044
MathSciNet review:
MR1306044
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Abstract: A mathematical model of potential spreading along neuron dendrites is proposed to describe the synaptic transmissions in the so-called cerebellar granule cells, which consist of a nearly spherical soma emitting a finite number of dendrites. The model accounts for the nonlinear dependence of the NMDA receptors, located at the virtual ends of any dendrite, upon the voltage. The corresponding initial-boundary value problem is formulated in the framework of Sobolev spaces. Existence and uniqueness of a weak solution are proved along with regularity results ensuring that the solution is classical.
P. Asher and L. Nowak, The role of divalent cations in the N-methyl-D-aspartate responses of mouse central neurones in culture, J. Physiol. 399, 247–266 (1988)
R. FitzHugh, Dimensional analysis of nerve models, J. Theoret. Biol. 40, 517–541 (1973)
V. Comincioli, E. D’Angelo, D. Funaro, P. Rossi, and A. Torelli, A mathematical model of potential spreading along neuron dendrites of cerebellar granule cells, Appl. Math. Comput. 59, 73–87 (1993)
E. D’Angelo, P. Rossi, and J. Garthwaite, Dual-component NMDA receptor currents at a single central synapse, Nature 346, 467–469 (1990)
D. Funaro, Approximation by the Legendre collocation method of a model problem in electrophysiology, J. Comput. Appl. Math. 43, 261–271 (1992)
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969
M. L. Mayer, G. L. Westbrook, and P. B. Guthrie, Voltage dependent block by Mg$^{2+}$ of NMDA responses in spinal cord neurons, Nature 309, 261–263 (1984)
L. Nowak, P. Bregestowsky, P. Asher, A. Herbet, and A. Prochiantz, Magnesium gates glutamate-activated channels in mouse central neurones, Nature 307, 462–465 (1984)
S. Palay and V. Chan-Palay, Cerebellar cortex, Springer-Verlag, Berlin, 1974
M. Palkovits, P. Magyar, and J. Szentagothai, Quantitative hystological analysis of the cerebellar cortex in the cat. IV. Mossy fiber-Purkinje cell numerical transfer, J. Brain Res. 45, 15–29 (1972)
W. Rall, Core conductor theory and cable properties of neurons, Handbook of Physiology. The Nervous System. Cellular Biology of Neurons, Amer. Physiol. Soc., Bethesda, MD, 1981, pp. 39–97
P. Asher and L. Nowak, The role of divalent cations in the N-methyl-D-aspartate responses of mouse central neurones in culture, J. Physiol. 399, 247–266 (1988)
R. FitzHugh, Dimensional analysis of nerve models, J. Theoret. Biol. 40, 517–541 (1973)
V. Comincioli, E. D’Angelo, D. Funaro, P. Rossi, and A. Torelli, A mathematical model of potential spreading along neuron dendrites of cerebellar granule cells, Appl. Math. Comput. 59, 73–87 (1993)
E. D’Angelo, P. Rossi, and J. Garthwaite, Dual-component NMDA receptor currents at a single central synapse, Nature 346, 467–469 (1990)
D. Funaro, Approximation by the Legendre collocation method of a model problem in electrophysiology, J. Comput. Appl. Math. 43, 261–271 (1992)
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969
M. L. Mayer, G. L. Westbrook, and P. B. Guthrie, Voltage dependent block by Mg$^{2+}$ of NMDA responses in spinal cord neurons, Nature 309, 261–263 (1984)
L. Nowak, P. Bregestowsky, P. Asher, A. Herbet, and A. Prochiantz, Magnesium gates glutamate-activated channels in mouse central neurones, Nature 307, 462–465 (1984)
S. Palay and V. Chan-Palay, Cerebellar cortex, Springer-Verlag, Berlin, 1974
M. Palkovits, P. Magyar, and J. Szentagothai, Quantitative hystological analysis of the cerebellar cortex in the cat. IV. Mossy fiber-Purkinje cell numerical transfer, J. Brain Res. 45, 15–29 (1972)
W. Rall, Core conductor theory and cable properties of neurons, Handbook of Physiology. The Nervous System. Cellular Biology of Neurons, Amer. Physiol. Soc., Bethesda, MD, 1981, pp. 39–97
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Article copyright:
© Copyright 1994
American Mathematical Society
