Abstract

Recent cascaded-integrator models do not fit the sine-wave flicker thresholds as well as we might wish, but neither does the Ferry-Porter law. In fact, the Ferry-Porter function is not physically realizable as a linear model. By modifying it to yield realizable responses like those of the cascaded integrator, we obtain a much simpler model, which appears to be a special case of the photochemical diffusion mechanism proposed by Ives and more recently by Veringa. This model is a good fit, not only to the flicker data, but also to human phase-shift measurements obtained by the phosphene method. We infer that receptor-cell properties probably control the high-frequency linear filtering of flicker waveforms.

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  1. Of course, this also depends on the type of nonlinearity postulated. For a discussion of the required precision in the case of a logarithmic nonlinearity, see L. H. van der Tweel, Ann. N. Y. Acad. Sci. 89, 829 (1961).
  2. H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 254 (1922).
  3. H. deLange, Physica 18, 935 (1952).
  4. D. H. Kelly, J. Opt. Soc. Am. 51, 422 (1961).
  5. D. H. Kelly, Doc. Ophthalmol. 18, 16 (1964).
  6. J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).
  7. H. deLange, J. Opt. Soc. Am. 48, 777 (1958).
  8. D. H. Kelly, J. Opt. Soc. Am. 51, 917 (1961).
  9. H. deLange, J. Opt. Soc. Am. 44, 380 (1954).
  10. J. Levinson, Science 130, 919 (1959).
  11. K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).
  12. H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 343 (1922).
  13. D. H. Kelly, J. Opt. Soc. Am. 51, 747 (1961).
  14. M. G. F. Fuortes and A. L. Hodgkin, J. Physiol. (London) 172, 239 (1964).
  15. R. B. Pinter, J. Gen. Physiol. 49, 565 (1966).
  16. A. Troelstra, Non-linear Systems Analysis in Electro-retinograpity (Institute for Perception RVO-TNO, Soesterberg, 1964).
  17. R. B. Marimont, J. Physiol. (London) 179, 489 (1965).
  18. R. D. deVoe, in The Functional Organization of the Compound Eye (Pergamon Press, Ltd., Oxford, 1966), p. 309.
  19. R. D. deVoe, J. Gen. Physiol. 50, 1993 (1967).
  20. J. Levinson, J. Opt. Soc. Am. 56, 95 (1966).
  21. L. Matin, J. Opt. Soc. Am. 58, 404 (1968).
  22. G. Sperling and M. M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).
  23. F. Veringa, thesis, Amsterdam (1961).
  24. F. Veringa, Kon. Ned. Akad. Wetensch. Proc. Ser. B 64, 413 (1961).
  25. This function was suggested by one of the referees.
  26. See, e.g., S. J. Mason and H. J. Zimmerman, Electronic Circuits, Signals and Systems (John Wiley & Sons, Inc., New York, 1960), Sec. 7.14.
  27. This integral can be solved by contour integration in the complex plane, or the solution can be found in standard integral tables.
  28. G. S. Brindley, J. Physiol. (London) 164, 157 (1962).
  29. F. Veringa, Nature 197, 998 (1963).
  30. F. Veringa, Doc. Ophthalmol. 18, 72 (1964).
  31. F. Veringa and J. Roelofs, Nature 211, 321 (1966).
  32. See Fig. 2 of Ref. 23.
  33. D. H. Kelly, J. Opt. Soc. Am. 59, 1361 (1969).

Brindley, G. S.

G. S. Brindley, J. Physiol. (London) 164, 157 (1962).

deLange, H.

H. deLange, J. Opt. Soc. Am. 48, 777 (1958).

H. deLange, Physica 18, 935 (1952).

H. deLange, J. Opt. Soc. Am. 44, 380 (1954).

deVoe, R. D.

R. D. deVoe, J. Gen. Physiol. 50, 1993 (1967).

R. D. deVoe, in The Functional Organization of the Compound Eye (Pergamon Press, Ltd., Oxford, 1966), p. 309.

Fuortes, M. G. F.

M. G. F. Fuortes and A. L. Hodgkin, J. Physiol. (London) 172, 239 (1964).

Gibbins, K.

K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).

Harmon, L. D.

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).

Hodgkin, A. L.

M. G. F. Fuortes and A. L. Hodgkin, J. Physiol. (London) 172, 239 (1964).

Howarth, C. I.

K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).

Ives, H. E.

H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 254 (1922).

H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 343 (1922).

Kelly, D. H.

D. H. Kelly, Doc. Ophthalmol. 18, 16 (1964).

D. H. Kelly, J. Opt. Soc. Am. 51, 422 (1961).

D. H. Kelly, J. Opt. Soc. Am. 51, 747 (1961).

D. H. Kelly, J. Opt. Soc. Am. 51, 917 (1961).

D. H. Kelly, J. Opt. Soc. Am. 59, 1361 (1969).

Levinson, J.

J. Levinson, J. Opt. Soc. Am. 56, 95 (1966).

J. Levinson, Science 130, 919 (1959).

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).

Marimont, R. B.

R. B. Marimont, J. Physiol. (London) 179, 489 (1965).

Mason, S. J.

See, e.g., S. J. Mason and H. J. Zimmerman, Electronic Circuits, Signals and Systems (John Wiley & Sons, Inc., New York, 1960), Sec. 7.14.

Matin, L.

L. Matin, J. Opt. Soc. Am. 58, 404 (1968).

Pinter, R. B.

R. B. Pinter, J. Gen. Physiol. 49, 565 (1966).

Roelofs, J.

F. Veringa and J. Roelofs, Nature 211, 321 (1966).

Sondhi, M. M.

G. Sperling and M. M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).

Sperling, G.

G. Sperling and M. M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).

Troelstra, A.

A. Troelstra, Non-linear Systems Analysis in Electro-retinograpity (Institute for Perception RVO-TNO, Soesterberg, 1964).

Veringa, F.

F. Veringa, Doc. Ophthalmol. 18, 72 (1964).

F. Veringa, thesis, Amsterdam (1961).

F. Veringa and J. Roelofs, Nature 211, 321 (1966).

F. Veringa, Kon. Ned. Akad. Wetensch. Proc. Ser. B 64, 413 (1961).

F. Veringa, Nature 197, 998 (1963).

Zimmerman, H. J.

See, e.g., S. J. Mason and H. J. Zimmerman, Electronic Circuits, Signals and Systems (John Wiley & Sons, Inc., New York, 1960), Sec. 7.14.

Other

Of course, this also depends on the type of nonlinearity postulated. For a discussion of the required precision in the case of a logarithmic nonlinearity, see L. H. van der Tweel, Ann. N. Y. Acad. Sci. 89, 829 (1961).

H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 254 (1922).

H. deLange, Physica 18, 935 (1952).

D. H. Kelly, J. Opt. Soc. Am. 51, 422 (1961).

D. H. Kelly, Doc. Ophthalmol. 18, 16 (1964).

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).

H. deLange, J. Opt. Soc. Am. 48, 777 (1958).

D. H. Kelly, J. Opt. Soc. Am. 51, 917 (1961).

H. deLange, J. Opt. Soc. Am. 44, 380 (1954).

J. Levinson, Science 130, 919 (1959).

K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).

H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 343 (1922).

D. H. Kelly, J. Opt. Soc. Am. 51, 747 (1961).

M. G. F. Fuortes and A. L. Hodgkin, J. Physiol. (London) 172, 239 (1964).

R. B. Pinter, J. Gen. Physiol. 49, 565 (1966).

A. Troelstra, Non-linear Systems Analysis in Electro-retinograpity (Institute for Perception RVO-TNO, Soesterberg, 1964).

R. B. Marimont, J. Physiol. (London) 179, 489 (1965).

R. D. deVoe, in The Functional Organization of the Compound Eye (Pergamon Press, Ltd., Oxford, 1966), p. 309.

R. D. deVoe, J. Gen. Physiol. 50, 1993 (1967).

J. Levinson, J. Opt. Soc. Am. 56, 95 (1966).

L. Matin, J. Opt. Soc. Am. 58, 404 (1968).

G. Sperling and M. M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).

F. Veringa, thesis, Amsterdam (1961).

F. Veringa, Kon. Ned. Akad. Wetensch. Proc. Ser. B 64, 413 (1961).

This function was suggested by one of the referees.

See, e.g., S. J. Mason and H. J. Zimmerman, Electronic Circuits, Signals and Systems (John Wiley & Sons, Inc., New York, 1960), Sec. 7.14.

This integral can be solved by contour integration in the complex plane, or the solution can be found in standard integral tables.

G. S. Brindley, J. Physiol. (London) 164, 157 (1962).

F. Veringa, Nature 197, 998 (1963).

F. Veringa, Doc. Ophthalmol. 18, 72 (1964).

F. Veringa and J. Roelofs, Nature 211, 321 (1966).

See Fig. 2 of Ref. 23.

D. H. Kelly, J. Opt. Soc. Am. 59, 1361 (1969).

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