Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Mathematical study of a nonlinear neuron model with active dendrites

1 Computational Physiology Lab, Cornell University, Ithaca, NY 14850, USA
2 Advances Applied Mathematics and Statistical Science Center, University of Milan, via Celoria 2 20133 Milan, Italy

Topical Section: Mathematical modeling

In this work, we have studied an extended version of the cable equation that includes both active and passive membrane properties, under the so-called sealed-end boundary condition. We have thus proved the existence and uniqueness of the weak solution, and defined a novel mathematical form of the somatic cable equation. In particular, we have manipulated the equation set to demonstrate that the diffusion term in the somatic equation is equivalent to the first-order space derivative of the membrane potential in the proximal dendrites. Our conclusion therefore clues how the somatic potential depends on the dynamic of the proximal dendritic segments, and provides the basis for the morphological reduction of neurons without any significant loss of computational properties.
  Figure/Table
  Supplementary
  Article Metrics

References

1.G. M. Shepherd, The Synaptic Organization of the Brain 5th Edition, Oxford Press, 2004.

2.A. L. Hodgkin and A. F. Huxley, A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes, J. Physiol., (1952), 500-544.

3.P. Poirazi, T. Brannon, B. W. Mel, Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell, Neuron, 37 (2003), 977-987.    

4.P. Poirazi, T. Brannon, B. W. Mel, Pyramidal neuron as two-layer neural network, Neuron, 37 (2003), 989-999.    

5.C. Koch and I. Segev, Methods in Neuronal Modeling: From Ions to Networks, 2 eds. Cambridge: MIT Press, 1998.

6.E. R. de Moraes, C. Kushmerick, L. A. Naves, Morphological and functional diversity of first-order somatosensory neurons, Biophysical reviews, 9 (2017), 847-856.    

7.P. Fromherz, C. O. Müller, Cable properties of a straight neurite of a leech neuron probed by a voltage-sensitive dye, P. Natl. Acad. Sci. USA, 91 (1994), 4604-4608.    

8.B. J. Zandt, M. L. Veruki, E. Hartveit, Electrotonic signal processing in AII amacrine cells: compartmental models and passive membrane properties for a gap junction-coupled retinal neuron, Brain Struct. Funct., 223 (2018), 3383-3410.    

9.R. R. Llinas, I of the Vortex: From Neurons to Self, MIT Press, 2002.

10.R. R. Llinas, The Workings of the Brain: Development, Memory, and Perception (Readings from Scientific American), MIT Press, 1989.

11.H. Bokil, N. Laaris, K. Blinder, et al. Ephaptic interactions in the mammalian olfactory system, J. Neurosci., 21 (2001), RC173.

12.C. A. Anastassiou, R. Perin, H. Markram, et al. Ephaptic coupling of cortical neurons, Nat. Neurosci., 14 (2011), 217-223.    

13.M. Martinez-Banaclochar, Ephaptic Coupling of Cortical Neurons: Possible Contribution of Astroglial Magnetic Fields? Neuroscience, 370 (2018), 37-45.

14.J. G. Jefferys, P. Jiruska, M. de Curtis, et al. Limbic Network Synchronization and Temporal Lobe Epilepsy, National Center for Biotechnology Information (US), 2012.

15.A. A. Fingelkurts, A. A. Fingelkurts, S. Bagnato, et al. The value of spontaneous EEG oscillations in distinguishing patients in vegetative and minimally conscious states, Supplements to Clinical neurophysiology, 62 (2013), 81-99.    

16.K. Mori, H. Manabe, K. Narikiyo, et al. Olfactory consciousness and gamma oscillation couplings across the olfactory bulb, olfactory cortex, and orbitofrontal cortex, Frontiers in psychology, 4 (2013), 743.

17.L. Lamberti, Solutions to the Hodgkin-Huxley equations, Appl. Math. Comput., (1986), 43-70.

18.M. Mascagni, An initial-boundary value problem of physiological significance for equations of nerve conduction, Comm. Pure Appl. Math., (1989), 213-227.

19.M. Mascagni, The backward Euler method for numerical solution of the Hodgkin-Huxley equations of nerve conduction, SIAM J. Numer. Anal., (1990), 941-962.

20.H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, New York: Springer-Verlag, 2011.

21.J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, Vol. 1, Dunod, Paris, 1968.

22.L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 2010.

23.J. D. Evans, Analysis of a multiple equivalent cylinder model with generalized taper, Mathematical Medicine and Biology: A Journal of the IMA, (2000), 347-377.

24.W. Rall, Perspectives on neuron modeling, In: M. D. Binder and L. M. Mendell, The Segmental Motor System, Oxford: Oxford University Press, 1990.

25.M. Ohme and A. Schierwagen, A reduced model for dendritic trees with active membrane, In: C. von der Malsburg, et al. Artificial Neural Networks - ICANN96, LNCS vol. 1112, Berlin: Springer Verlag, 1996.

26.H. C. Tuckwell, Introduction to Theoretical Neurobiology, Volume 1. Linear Cable Theory and Dendritic Structure, New York: Cambridge University Press, 1988.

27.P. Colli, Mathematical study of a nonlinear neuron multi-dendritic model, Q. Appl. Math., 52 (1994), 689-706.    

28.V. Comincioli, D. Funaro, A. Torelli, et al. A mathematical model of potential spreading along neuron dendrites of cerebellar granule cells, Appl. Math. Comput.,59 (1993), 73-87.

29.M. London and M. Häusser,Dendritic Computation, Annu. Rev. Neurosci., 28 (2005), 503-532.    

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved