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.

© 1969 Optical Society of America

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References

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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).
    [Crossref] [PubMed]
  2. H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 254 (1922).
    [Crossref]
  3. H. deLange, Physica 18, 935 (1952).
    [Crossref]
  4. D. H. Kelly, J. Opt. Soc. Am. 51, 422 (1961).
    [Crossref] [PubMed]
  5. D. H. Kelly, Doc. Ophthalmol. 18, 16 (1964).
    [Crossref]
  6. J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).
    [Crossref] [PubMed]
  7. H. deLange, J. Opt. Soc. Am. 48, 777 (1958).
    [Crossref]
  8. D. H. Kelly, J. Opt. Soc. Am. 51, 917 (1961).
    [Crossref] [PubMed]
  9. H. deLange, J. Opt. Soc. Am. 44, 380 (1954).
    [Crossref]
  10. J. Levinson, Science 130, 919 (1959).
    [Crossref] [PubMed]
  11. K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).
    [Crossref] [PubMed]
  12. H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 343 (1922).
    [Crossref]
  13. D. H. Kelly, J. Opt. Soc. Am. 51, 747 (1961).
    [Crossref]
  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-retinography (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).
    [Crossref] [PubMed]
  21. L. Matin, J. Opt. Soc. Am. 58, 404 (1968).
    [Crossref] [PubMed]
  22. G. Sperling and M. M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  30. F. Veringa, Doc. Ophthalmol. 18, 72 (1964).
    [Crossref]
  31. F. Veringa and J. Roelofs, Nature 211, 321 (1966).
    [Crossref] [PubMed]
  32. See Fig. 2 of Ref. 23.
  33. D. H. Kelly, J. Opt. Soc. Am. 59, 1361 (1969).
    [Crossref]

1969 (1)

1968 (2)

1967 (1)

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

1966 (3)

J. Levinson, J. Opt. Soc. Am. 56, 95 (1966).
[Crossref] [PubMed]

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

F. Veringa and J. Roelofs, Nature 211, 321 (1966).
[Crossref] [PubMed]

1965 (1)

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

1964 (3)

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

F. Veringa, Doc. Ophthalmol. 18, 72 (1964).
[Crossref]

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

1963 (1)

F. Veringa, Nature 197, 998 (1963).
[Crossref] [PubMed]

1962 (1)

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

1961 (7)

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

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).
[Crossref] [PubMed]

D. H. Kelly, J. Opt. Soc. Am. 51, 917 (1961).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

D. H. Kelly, J. Opt. Soc. Am. 51, 422 (1961).
[Crossref] [PubMed]

K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).
[Crossref] [PubMed]

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

1959 (1)

J. Levinson, Science 130, 919 (1959).
[Crossref] [PubMed]

1958 (1)

1954 (1)

1952 (1)

H. deLange, Physica 18, 935 (1952).
[Crossref]

1922 (2)

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

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

Brindley, G. S.

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

deLange, H.

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).
[Crossref] [PubMed]

Harmon, L. D.

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Ives, H. E.

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

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

Kelly, D. H.

Levinson, J.

J. Levinson, J. Opt. Soc. Am. 56, 95 (1966).
[Crossref] [PubMed]

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).
[Crossref] [PubMed]

J. Levinson, Science 130, 919 (1959).
[Crossref] [PubMed]

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.

Pinter, R. B.

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

Roelofs, J.

F. Veringa and J. Roelofs, Nature 211, 321 (1966).
[Crossref] [PubMed]

Sondhi, M. M.

Sperling, G.

Troelstra, A.

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

van der Tweel, L. H.

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).
[Crossref] [PubMed]

Veringa, F.

F. Veringa and J. Roelofs, Nature 211, 321 (1966).
[Crossref] [PubMed]

F. Veringa, Doc. Ophthalmol. 18, 72 (1964).
[Crossref]

F. Veringa, Nature 197, 998 (1963).
[Crossref] [PubMed]

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

F. Veringa, thesis, Amsterdam (1961).

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.

Ann. N. Y. Acad. Sci. (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).
[Crossref] [PubMed]

Doc. Ophthalmol. (2)

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

F. Veringa, Doc. Ophthalmol. 18, 72 (1964).
[Crossref]

J. Gen. Physiol. (2)

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

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

J. Opt. Soc. Am. (9)

J. Opt. Soc. Am. and Rev. Sci. Instr. (2)

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

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

J. Physiol. (London) (3)

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

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

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

Kon. Ned. Akad. Wetensch. Proc. Ser. B (1)

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

Kybernetik (1)

J. Levinson and L. D. Harmon, Kybernetik 1, 107 (1961).
[Crossref] [PubMed]

Nature (3)

K. Gibbins and C. I. Howarth, Nature 190, 330 (1961).
[Crossref] [PubMed]

F. Veringa, Nature 197, 998 (1963).
[Crossref] [PubMed]

F. Veringa and J. Roelofs, Nature 211, 321 (1966).
[Crossref] [PubMed]

Physica (1)

H. deLange, Physica 18, 935 (1952).
[Crossref]

Science (1)

J. Levinson, Science 130, 919 (1959).
[Crossref] [PubMed]

Other (7)

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

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

See Fig. 2 of Ref. 23.

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.

F. Veringa, thesis, Amsterdam (1961).

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Figures (2)

F. 1
F. 1

Theoretical amplitude-sensitivity functions compared with experimental flicker data replotted from Refs. 4 and 5. Open symbols represent variable-modulation, constant-illuminance experiments. Open circles: 850 td; triangles: 77 td; diamonds: 7.1 td; squares: 0.65 td. Filled circles: periodic flash experiments, constant (200%) modulation, variable illuminance. The dashed curve represents the Ferry–Porter law; the dotted curve represents the cascaded-integrator model (with N = 5); both have arbitrarily been set tangent to the data at f = 30 Hz. The solid curve is a plot of Eq. (1″).

F. 2
F. 2

Theoretical impulse response given by Eq. (7). Both ordinate and abscissa scales depend on the value of τ, but the peak always occurs at t = τ/3.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

| G ( ω ) | = exp ( k | ω | a )
g F ( t ) = 2 k / ( k 2 / l 2 )
P ( ω ) = 0 log | G | 1 + ω 2 d ω
P ( ω ) = k 0 ω a 1 + ω 2 d ω = k π 2 cos ( a π / 2 ) ,
φ ( ω ) = 2 ω π 0 log | G ( ω ) | log | G ( α ) | ω 2 α 2 d α .
φ ( ω ) = 2 k ω π 0 ω a α a ω 2 α 2 d α = k ω a tan ( a π / 2 )
G ( ω ) = exp { k ω a [ 1 + j tan ( π a / 2 ) ] } .
G ( ω ) = exp { k ( ω ) a [ 1 j tan ( π a / 2 ) ] }
| G ( ω ) | = exp ( | ω τ | 1 2 ) ,
φ ( ω ) = ( ω τ ) 1 2
G ( ω ) = exp [ ( 2 j ω τ ) 1 2 ] .
| G | = 6 × 10 3 exp [ ( π f ) 1 2 ] ,
φ ( ω ) = ( π f ) 1 2 ,
g ( t ) = 1 2 π G ( ω ) e j ω t d ω .
g ( t ) = ( τ / 2 π ) 1 2 ( t ) 3 2 e τ / 2 t
f t = c 2 f x 2 ρ f ,
g υ ( t ) = ( 2 τ / π ) 1 2 ( t ) 3 2 e τ / 2 t e ρ t .
g υ ( t ) = 2 e ρ t g ( t ) .
2 f x 2 = 2 τ f t ,
f ( x , t ) = e j ω t α x ,
G ( ω ) = f ( x + 1 , t ) / f ( x , t ) = e α .
α 2 e j ω t α x = 2 j ω τ e j ω t α x ;
α = ( 2 j ω τ ) 1 2 .
G ( ω ) = exp [ ( 2 j ω τ ) 1 2 ] .