Mathematical Modelling of Macroscopic Brain Phenomena

  • Robert M. Miura
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 71)


Theoretical studies of the brain have concentrated mainly on microscopic phenomena at the membrane and cellular levels. At the macroscopic level, there are “gross” brain phenomena, e.g., some kinds of epilepsy, that occur on space scales which are large compared to cell size and on time scales which are long compared to time constants associated with molecular and cellular events. Because of the large number of nerve and glial cells involved (on the order of 1011 (Hubel, 1979)), it is difficult to construct a mathematical model of such phenomena by starting with individual cells, connecting them into small circuits, and then combining these circuits into a coherent model for a particular brain structure.


Excitable Medium Cortical Spreading Depression Spreading Depression Intracellular Space Rigid Body Rotation 
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  1. BURES, J., BURESOVA, O., & KRIVANEK, J. (1974). The Mechanism and Applications of Leap’s Spreading Depression of Electroencephalographic Activity, Prague: Academia.Google Scholar
  2. GARDNER-MEDWIN, A.R. (1981). Possible roles of vertebrate neuroglia in potassium dynamics) spreading depression and migraine. J. Exp. Biol. 95, 111–187.Google Scholar
  3. GARDNER-MEDWIN, A.R. (1983). A study of the mechanisms by which potassium moves through brain tissue in the rat. J. Physiol. 335, 353–374.Google Scholar
  4. HUBEL, D.H. (1979). The brain. Scientific American 241, 45–53.Google Scholar
  5. KRAIG, R.P. & NICHOLSON, C. (1978). Extracellular ionic variations during spreading depression. Neurosci. 3, 1045–1059.CrossRefGoogle Scholar
  6. LEAO, A.A.P. (1944). Spreading depression of activity in the cerebral cortex. J. Neurophysiol. 7, 359–390.Google Scholar
  7. MIURA, R.M. (1981). Nonlinear waves in neuronal cortical structures. In Nonlinear Phenomena in Physics and Biology, eds. ENNS, R.H., Jones, B.L., Miura, R.M., & Rangnekar, S.S., pp. 369–400. New York: Plenum.Google Scholar
  8. MIURA, R.M. AND PLANT, R.E. (1981). Rotating waves in models of excitable media. In Differential Equations and Applications in Ecology, Epidemics, and Population Problems, eds. Busenberg, S.N. & Cooke, K.L., pp. 247–257. New York: Academic Press.Google Scholar
  9. NICHOLSON, C. & PHILLIPS, J.M. (1981). Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of the rat cerebellum. J. Physiol. 321, 225–257.Google Scholar
  10. NICHOLSON, C., TEN BRUGGENCATE, G., STOCKLE, H., & STEINBERG, R. (1978). Calcrium and potassium changes in extracellular microenvironment of cat cerebellar cortex- J. Neurophysiol. 41, 1026–1039.Google Scholar
  11. OLESEN, J. (1985). Migraine and regional cerebral blood flow. Trends in NeuroScience 8, 318–321.CrossRefGoogle Scholar
  12. ORKAND, R.K., NICHOLLS, J.G. & KUFFLER, S.W. (1966). Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia. J. Neurophysiol 29, 788–806.Google Scholar
  13. SHIBATA, M. & BURES, J. (1972). Reverberation of cortical spreading depression along closed-loop pathways in rat cerebral cortex. J. Neurophysiol. 35, 381–388.Google Scholar
  14. SHIBATA, M. & BURES, J. (1974). Optimal topographic conditions for reverberating cortical spreading depression in cats. J. Neurobiol. 5, 107–118.CrossRefGoogle Scholar
  15. SHIBATA, M. & BURES, J. (1975). Techniques for termination of reverberating spreading depression in rats. J. Neurophysiol. 38, 158–166.Google Scholar
  16. SUGAYA, E., TAKATO, M., & NODA, Y. (1975). Neuronal and glial activity during spreading depression in cerebral cortex of cat. J. Neurophysiol. 38, 822–841.Google Scholar
  17. TOBIASZ, C. & NICHOLSON, C. (1982). Tetrodotoxin resistant propagation and extracellular sodium changes during spreading depression in rat cerebellum. Brain Research 241, 329–333.CrossRefGoogle Scholar
  18. TUCKWELL, H.C. & MIURA, R.M. (1978). A mathematical model for spreading cortical depression. Biophys. J. 23, 257–276.CrossRefGoogle Scholar
  19. WEINER, N. & ROSENBLUETH, A. (1946). The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mex. 16, 205–265.Google Scholar
  20. WINFREE, A.T. (1974). Rotating solutions to reaction-diffusion equations in simply-connected media. SIAM-AMS Proceedings 8, 13–31.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Robert M. Miura
    • 1
  1. 1.Departments of Mathematics and Pharmacology & Therapeutics, Institute of Applied MathematicsUniversity of British ColumbiaVancouverCanada

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