Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 35)


We now present mathematical theories for describing dendrites. Dendrites are very important for many reasons. Indeed, the majority of the total membrane area of many neurons is occupied by the dendritic tree. Dendrites enable neurons to connect to thousands of other cells, far more than would be possible with just a soma, as there is a huge membrane area to make connections. Dendrites may direct many subthreshold postsynaptic potentials toward the soma, which summates these inputs and determines if the neuron will fire an action potential. In addition to the treelike structure of dendrites, many dendrites have additional fine structures at the ends of the branches called spines. During development, animals that are raised in rich sensory environments have more extensive dendritic trees and more spines.


Input Resistance Dendritic Tree Cable Equation Input Conductance Dendritic Spike 
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  1. 1.
    H. Agmon-Snir, C. E. Carr, and J. Rinzel. A case study for dendritic function: Improving the performance of auditory coincidence detectors. Nature, 393:268–272, 1998.CrossRefGoogle Scholar
  2. 10.
    H. B. Barlow, R. M. Hill, and W. Levick. Retinal ganglion cells responding selectively to direction and speed of image motion in the rabbit. J. Physiol., 173:377–407, 1964.Google Scholar
  3. 33.
    N. Carnevale and M. Hines. The NEURON Book. Cambridge University Press, Cambridge, UK, 2006.CrossRefGoogle Scholar
  4. 96.
    C. W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, volume 13 of Springer Series in Synergetics. Springer, Berlin, third edition, 2004.MATHGoogle Scholar
  5. 135.
    E. M. Izhikevich. Dynamical Systems in Neuroscience. MIT, Cambridge, MA, 2007.Google Scholar
  6. 137.
    J. J. B. Jack, D. Noble, and R. W. Tsien. Electrical Current Flow in Excitable Cells. Clarendon Press, Oxford, 1975.Google Scholar
  7. 154.
    P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics (New York). Springer, Berlin, 1992.MATHGoogle Scholar
  8. 155.
    H. Kluver. Mescal and the Mechanisms of Hallucination. University of Chicago Press, Chicago, IL, 1969.Google Scholar
  9. 182.
    J. E. Lisman, J. M. Fellous, and X. J. Wang. A role for NMDA-receptor channels in working memory. Nat. Neurosci., 1:273–275, 1998.CrossRefGoogle Scholar
  10. 189.
    K. Maginu. Geometrical characteristics associated with stability and bifurcations of periodic traveling waves in reaction-diffusion systems. SIAM J. Appl. Math., 45:750–774, 1985.MathSciNetMATHCrossRefGoogle Scholar
  11. 211.
    D. H. Perkel and B. Mulloney. Motor pattern production in reciprocally inhibitory neurons exhibiting postsynaptic rebound. Science, 145:61–63, 1974.CrossRefGoogle Scholar
  12. 216.
    D. J. Pinto and G. B. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses. SIAM J. Appl. Math., 62(1):206–225 (electronic), 2001.MathSciNetMATHCrossRefGoogle Scholar
  13. 217.
    D. J. Pinto, J. C. Brumberg, D. J. Simons, and G. B. Ermentrout. A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations. J. Comput. Neurosci., 3:247–264, 1996.CrossRefGoogle Scholar
  14. 237.
    M. Rudolph and A. Destexhe. Characterization of subthreshold voltage fluctuations in neuronal membranes. Neural Comput., 15:2577–2618, 2003.MATHCrossRefGoogle Scholar
  15. 263.
    D. Terman. The transition from bursting to continuous spiking in an excitable membrane model. J. Nonlinear Sci., 2:133–182, 1992.MathSciNetCrossRefGoogle Scholar
  16. 267.
    D. Terman, S. Ahn, X. Wang, and W. Just. Reducing neuronal networks to discrete dynamics. Phys. D, 237:324–338, 2008.MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Dept. MathematicsOhio State UniversityColumbusUSA

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