, Volume 13, Issue 2, pp 55–80 | Cite as

A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue

  • H. R. Wilson
  • J. D. Cowan


It is proposed that distinct anatomical regions of cerebral cortex and of thalamic nuclei are functionally two-dimensional. On this view, the third (radial) dimension of cortical and thalamic structures is associated with a redundancy of circuits and functions so that reliable signal processing obtains in the presence of noisy or ambiguous stimuli.

A mathematical model of simple cortical and thalamic nervous tissue is consequently developed, comprising two types of neurons (excitatory and inhibitory), homogeneously distributed in planar sheets, and interacting by way of recurrent lateral connexions. Following a discussion of certain anatomical and physiological restrictions on such interactions, numerical solutions of the relevant non-linear integro-differential equations are obtained. The results fall conveniently into three categories, each of which is postulated to correspond to a distinct type of tissue: sensory neo-cortex, archior prefrontal cortex, and thalamus.

The different categories of solution are referred to as dynamical modes. The mode appropriate to thalamus involves a variety of non-linear oscillatory phenomena. That appropriate to archior prefrontal cortex is defined by the existence of spatially inhomogeneous stable steady states which retain contour information about prior stimuli. Finally, the mode appropriate to sensory neo-cortex involves active transient responses. It is shown that this particular mode reproduces some of the phenomenology of visual psychophysics, including spatial modulation transfer function determinations, certain metacontrast effects, and the spatial hysteresis phenomenon found in stereopsis.


Prefrontal Cortex Thalamic Nucleus Modulation Transfer Function Stable Steady State Hysteresis Phenomenon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols


Post-synaptic membrane potential (psp)


Maximum amplitude of psp




The neuronal membrane time constant


Threshold value of membrane potential


Absolute refractory period


Synaptic operating delay


Velocity of propagation of action potentil


Cartesian coordinate


The probability that cells of class j′ are connected with cells of class j a distance x away


The mean synaptic weight of synapses of the jj-th class at x


The space constant for connectivity


Surface density of excitatory neurons in a one-dimensional homogeneous and isotropic tissue


Surface density of inhibitory neurons in a one-dimensional homogeneous and isotropic tissue

E(x, t)

Excitatory Activity, proportion of excitatory cells becoming active per unit time at the instant t, at the point x

I(x, t)

Inhibitory Activity, proportion of inhibitory cells becoming active per unit time at the instant t, at the point x


A small segment of tissue


A small interval of time

P(x, t)

Afferent excitation or inhibition to excitatory neurons

Q(x, t)

Afferent excitation or inhibition to inhibitory neurons

Ne(x, t)

Mean integrated excitation generated within excitatory neurons at x

Ni(x, t)

Mean integrated excitation generated within inhibitory neurons at x


Expected proportion of excitatory neurons receiving at least threshold excitation per unit time, as a function of N e


Expected proportion of inhibitory neurons receiving at least threshold excitation per unit time, as a function of N i


Distribution function of excitatory neuronal thresholds


Distribution function of inhibitory neuronal thresholds


A fixed value of neuronal threshold

h(Ne; ρ1)

Proportion per unit time of excitatory neurons at x reaching ρ1 with a mean excitation N e

1[ ]

Heaviside's “step-function”

Re(x, t)

Number of excitatory neurons which are sensitive at the instant t

Ri(x, t)

Number of inhibitory neurons which are sensitive at the instant t


Refractory period of excitatory neurons


Refractory period of inhibitory neurons

E(x, t)〉

Time coarse-grained excitatory activity

I(x, t)〉

Time coarse-grained inhibitory activity

Spatial convolution


Threshold of a neuronal aggregate


Sensitivity coefficient of response of a neuronal aggregate


Time coarse-grained spatially localised excitatory activity


Time coarse-grained spatially localised inhibitory activity


See § 2.2.1, § 2.2.7, § 3.1


Velocity with which retinal images are moved apart


Stimulus width

Eo〉, 〈Io

Spatially homogeneous steady states of neuronal activity


See § 5.1


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allison, A.C.: Biol. Rev. 28, 195 (1953)Google Scholar
  2. Andersen, P., Andersson, S.A.: Physiological basis of the alpha rhythm. New York: Appleton-Century-Crofts 1968Google Scholar
  3. Andersen, P., Eccles, J.C.: Nature (London) 196, 645 (1962)Google Scholar
  4. Beurle, R.L.: Phil. Trans. Roy. Soc. B, 240, 55 (1956)Google Scholar
  5. Beurle, R.L.: Phil. Trans. Roy. Soc. B, 240, 55 (1956)Google Scholar
  6. Beurle, R.L.: In: Foerster, H.v., Zopf, G.W. (Eds.): Principles of selforganization, 291. Pergamon Press 1962Google Scholar
  7. Bishop, P.O., Coombs, J.S., Henry, G.H.: J. Physiol. 219, 625, 659 (1971)Google Scholar
  8. Bishop, P.O., Kozak, W., Levick, W.R., Vakkur, G.J.: J. Physiol. (London) 163, 503 (1962)Google Scholar
  9. Blinkov, S.M., Glezer, I.I.: The human brain in figures and tables. New York: Plenum Press 1968Google Scholar
  10. Brindley, G.S.: Physiology of the retina and visual pathway. London: Edward Arnold 1970Google Scholar
  11. Burns, B.D.: J. Physiol. 112, 156 (1951)Google Scholar
  12. Burns, B.D.: The mammalian cerebral cortex. London: Edward Arnold 1958Google Scholar
  13. Burns, B.D.: The uncertain nervous system. London: Edward Arnold 1968Google Scholar
  14. Burns, B.D., Heron, W., Pritchard, R.: J. Neurophysiol. 25, 165 (1962)Google Scholar
  15. Campbell, F.W., Green, D.G.: J. Physiol. 181, 576 (1965)Google Scholar
  16. Colonnier, M.L.: In: Eccles, J.C. (Ed.): Brain and conscious experience. Berlin-Heidelberg-New York: Springer 1965Google Scholar
  17. Cowan, J.D.: In: Caianiello, E.R. (Ed.): Neural networks. Berlin-Heidelberg-Nw York: Springer 1968Google Scholar
  18. Cowan, J.D.: In: Gerstenhaber, M. (Ed.): Mathematical problems in the life sciences. I. pp. 1–57. Providence, R. I., American Mathematical Society 1970Google Scholar
  19. Cowan, J.D.: In: Rice, S.A., Freed, K.F., Light, J.C. (Eds.): Statistical mechanics — new concepts, new problems, new applications. Chicago: University of Chicago Press 1972Google Scholar
  20. Cragg, B.G., Temperley, H.N.V.: Brain, 78, 304 (1955)Google Scholar
  21. Creuzfeldt, O., Ito, M.: Exp. Brain Res. 6, 324 (1968)Google Scholar
  22. Daniel, P.M., Whitteridge, D.: J. Physiol. 159, 203 (1961)Google Scholar
  23. Demetrescu, M., Demetrescu, M., Iosif, G.: Electroenceph. clin. Neurophysiol. 18, 1–24 (1965)Google Scholar
  24. Dewan, E.M.: J. Theoretical Biology 7, 141 (1964)Google Scholar
  25. Ditchburn, R.W., Ginsborg, B.L.: Nature, 170, 36 (1952)Google Scholar
  26. Donchin, E.: Vision Res. 7, 79 (1967)Google Scholar
  27. Eccles, J.C.: The physiology of synapses. New York: Academic Press 1964Google Scholar
  28. Eccles, J.C.: Epilepsia, 6, 89 (1965)Google Scholar
  29. Farley, B., Clark, W.A.: In: Cherry, C. (Ed.): Information theory (p. 242) (Fourth London Symposium). London: Butterworth and Co. 1961Google Scholar
  30. Fender, D., Julesz, B.: J. Opt. Soc. Am. 57, 819 (1967)Google Scholar
  31. Freeman, W.J.: Logistics Review 3, 5 (1967)Google Scholar
  32. Freeman, W.J.: Math. Biosci. 2, 181 (1968a)Google Scholar
  33. Freeman, W.J.: J. Neurophysiol. 31, 337 (1968b)Google Scholar
  34. Fuster, J.M., Alexander, G.E.: Science 173, 652 (1971)Google Scholar
  35. Griffith, J.S.: Bull. Math. Biophys. 25, 111 (1963)Google Scholar
  36. Griffith, J.S.: Bull. Math. Biophys. 27, 187 (1965)Google Scholar
  37. Harth, E.M., Csermely, T.J., Beek, B., Lindsay, R.D.: J. Theor. Biol. 26, 93 (1970)Google Scholar
  38. Hartline, H.K.: Am. J. Physiol. 121, 400 (1938)Google Scholar
  39. Hartline, H.K., Ratliff, F.: J. Gen. Physiol. 41, 1049 (1958)Google Scholar
  40. Hebb, D.O.: The organization of behavior. New York: John Wiley 1949Google Scholar
  41. Hollander, H.: Exp. Brain Res. 10, 219 (1970)Google Scholar
  42. Hubel, D.H., Wiesel, T.N.: J. Physiol. 165, 559 (1963)Google Scholar
  43. Hubel, D.H., Wiesel, T.N.: J. Neurophysiol. 28, 229 (1965)Google Scholar
  44. Hubel, D.H., Wiesel, T.N.: J. Physiol. 195, 215 (1968)Google Scholar
  45. Johannesma, P.I.M.: In: Caianiello, E.R. (Ed.): Neural networks, p. 116. Berlin-Heidelberg-New York: Springer 1968Google Scholar
  46. Julesz, B.: Foundations of cyclopean perception. Chicago: University of Chicago Press 1971Google Scholar
  47. Kahneman, D.: Quart. J. exp. Psychol. 17, 308 (1965)Google Scholar
  48. Kahneman, D.: Psychol. Bull. 70, 404 (1968)Google Scholar
  49. Kalil, R.E., Chase, R.: J. Neurophysiol. 33, 459 (1970)Google Scholar
  50. Kirkwood, J.G.: J. Chem. Phys. 14, 180 (1946)Google Scholar
  51. Kohlers, P.A.: Vision Res. 2, 277 (1962)Google Scholar
  52. Kohlers, P.A., Rosner, B.S.: Am. J. Psychol. 73, 2 (1960)Google Scholar
  53. LeGrand, Y.: Light, color, and vision. New York: John Wiley 1957Google Scholar
  54. Lorente de Nó, R.: In: Fulton, J.F. (Ed.): Physiology of the nervous system, p. 288. New York: Oxford University Press 1949Google Scholar
  55. MacKay, D.M.: In: Schmitt, F. O. (Ed.): Neurosciences research symposium summaries, p. 397. Cambridge, Massachusetts: M.I.T. Press 1970Google Scholar
  56. Mountcastle, V.B.: J. Neurophysiol. 20, 408 (1957)Google Scholar
  57. Oshima, T.: In: Jasper, H. (Ed.): Basic mechanisms of the epilepsies, p. 253. Boston: Little, Brown & Co. 1969Google Scholar
  58. Poggio, G.F., Viernstein, L.J.: J. Neurophysiol. 27, 517 (1964)Google Scholar
  59. Polyak, S.L.: The vertebrate visual system. Chicago: University of Chicago Press 1957Google Scholar
  60. Purpura, D.R.: In: Schmitt, F.O. (Ed.): The neurosciences: Second study program. New York: Rockefeller University Press 1970Google Scholar
  61. Rall, W.: J. Cell. Comp. Physiol. 46, 413 (1955)Google Scholar
  62. Ratliff, F.: Mach bands. London: Holden-Day 1965Google Scholar
  63. Roy, B.K., Smith, D.R.: Bull. Math. Biophys. 31, 341 (1969)Google Scholar
  64. Rushton, W.H.: Proc. Roy. Soc. London, B, 162, 20 (1965)Google Scholar
  65. Sanderson, K.J.: Exp. Brain Res. 13, 159 (1971)Google Scholar
  66. Scheibel, M.E., Scheibel, A. B.: In: Schmitt, F.O. (Ed.): The neurosciences — second study program. New York: Rockefeller University Press 1970Google Scholar
  67. Sholl, D.A.: The organization of the cerebral cortex. London: Methuen 1956Google Scholar
  68. Spehlmann, R.: Electroenceph. clin. Neurophysiol. 19, 560 (1965)Google Scholar
  69. Sperling, G.: J. Am. Psychol. 83, 461 (1970)Google Scholar
  70. Sperling, G.: Perception and Psychophysics 8, 143 (1970)Google Scholar
  71. Szentagothai, J.: In: Lissak, K. (Ed.): Recent development of neurobiology in Hungary 1, 9 (1967)Google Scholar
  72. Tasaki, I.: In: Field, J., Magoun, H.W., Hall, V.E. (Eds.): Handbook of physiology, Section 1: Neurophysiology, p. 75, 1950Google Scholar
  73. Uttley, A.M.: Proc. Roy. Soc. B, 144, 229 (1955)Google Scholar
  74. von Neumann, J.: In: Shannon, C., McCarthy, J. (Eds.): Automata studies, p. 43. Princeton: Princeton University Press 1956Google Scholar
  75. Wilson, H.R., Cowan, J.D.: Biophys. J. 12, 1 (1972)Google Scholar
  76. Winograd, S., Cowan, J.D.: Reliable computation in the presence of noise. Cambridge, Mass.; M.I.T. Press 1963Google Scholar
  77. Zusne, L.: Visual perception of form. New York: Academic Press 1971Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • H. R. Wilson
    • 1
  • J. D. Cowan
    • 1
  1. 1.Department of Theoretical BiologyThe University of ChicagoChicagoUSA

Personalised recommendations