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

Abstract

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.

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Abbreviations

α(t):

Post-synaptic membrane potential (psp)

α :

Maximum amplitude of psp

t :

Time

μ :

The neuronal membrane time constant

ϑ :

Threshold value of membrane potential

r :

Absolute refractory period

τ :

Synaptic operating delay

v :

Velocity of propagation of action potentil

x :

Cartesian coordinate

β jj′ (x):

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

b jj′ :

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

σ jj′ :

The space constant for connectivity

ϱ e :

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

ϱ i :

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

δx :

A small segment of tissue

δt :

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

N e (x, t):

Mean integrated excitation generated within excitatory neurons at x

N i (x, t):

Mean integrated excitation generated within inhibitory neurons at x

e [N e ]:

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

i [N i ]:

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

G(ρ e ):

Distribution function of excitatory neuronal thresholds

G(ρ 1 ):

Distribution function of inhibitory neuronal thresholds

ρ 1 :

A fixed value of neuronal threshold

h(N e ; ρ 1):

Proportion per unit time of excitatory neurons at x reaching ρ 1 with a mean excitation N e

1[ ]:

Heaviside's “step-function”

R e (x, t):

Number of excitatory neurons which are sensitive at the instant t

R i (x, t):

Number of inhibitory neurons which are sensitive at the instant t

R e :

Refractory period of excitatory neurons

r i :

Refractory period of inhibitory neurons

E(x, t)〉:

Time coarse-grained excitatory activity

I(x, t)〉:

Time coarse-grained inhibitory activity

⊗:

Spatial convolution

θ :

Threshold of a neuronal aggregate

v :

Sensitivity coefficient of response of a neuronal aggregate

E(t)〉:

Time coarse-grained spatially localised excitatory activity

I(t)> :

Time coarse-grained spatially localised inhibitory activity

L 1,L 2,L,Q :

See § 2.2.1, § 2.2.7, § 3.1

υ :

Velocity with which retinal images are moved apart

σ :

Stimulus width

E o〉, 〈I o〉:

Spatially homogeneous steady states of neuronal activity

k e ,k ij S e S ij :

See § 5.1

References

  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 1968

    Google 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 1962

  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 1968

    Google Scholar 

  10. Brindley, G.S.: Physiology of the retina and visual pathway. London: Edward Arnold 1970

    Google Scholar 

  11. Burns, B.D.: J. Physiol. 112, 156 (1951)

    Google Scholar 

  12. Burns, B.D.: The mammalian cerebral cortex. London: Edward Arnold 1958

    Google Scholar 

  13. Burns, B.D.: The uncertain nervous system. London: Edward Arnold 1968

    Google 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 1965

    Google Scholar 

  17. Cowan, J.D.: In: Caianiello, E.R. (Ed.): Neural networks. Berlin-Heidelberg-Nw York: Springer 1968

    Google 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 1970

    Google 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 1972

    Google 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 1964

    Google 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. 1961

    Google 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 1949

    Google 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 1968

    Google Scholar 

  46. Julesz, B.: Foundations of cyclopean perception. Chicago: University of Chicago Press 1971

    Google 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 1957

    Google Scholar 

  54. Lorente de Nó, R.: In: Fulton, J.F. (Ed.): Physiology of the nervous system, p. 288. New York: Oxford University Press 1949

    Google Scholar 

  55. MacKay, D.M.: In: Schmitt, F. O. (Ed.): Neurosciences research symposium summaries, p. 397. Cambridge, Massachusetts: M.I.T. Press 1970

    Google 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. 1969

    Google 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 1957

    Google Scholar 

  60. Purpura, D.R.: In: Schmitt, F.O. (Ed.): The neurosciences: Second study program. New York: Rockefeller University Press 1970

    Google Scholar 

  61. Rall, W.: J. Cell. Comp. Physiol. 46, 413 (1955)

    Google Scholar 

  62. Ratliff, F.: Mach bands. London: Holden-Day 1965

    Google 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 1970

    Google Scholar 

  67. Sholl, D.A.: The organization of the cerebral cortex. London: Methuen 1956

    Google 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)

  72. Tasaki, I.: In: Field, J., Magoun, H.W., Hall, V.E. (Eds.): Handbook of physiology, Section 1: Neurophysiology, p. 75, 1950

  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 1956

    Google 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 1963

    Google Scholar 

  77. Zusne, L.: Visual perception of form. New York: Academic Press 1971

    Google Scholar 

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Wilson, H.R., Cowan, J.D. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1973). https://doi.org/10.1007/BF00288786

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Keywords

  • Prefrontal Cortex
  • Thalamic Nucleus
  • Modulation Transfer Function
  • Stable Steady State
  • Hysteresis Phenomenon