Abstract

Photopic flicker data are explained in terms of a theoretical model of two retinal processes. The first is a linear diffusion process (presumably in the receptors), with a large dynamic range (∼105). The second is a nonlinear inhibiting network (neural feedback at the synapses of the plexiform layers) that adaptively controls the sensitivity and time constants of the model. The magnitude of its transfer function fits the flicker data quantitatively at all frequencies, over a wide range of adaptation levels. The corresponding small-signal impulse responses are also calculated: their latencies and leading edges (associated with receptor activity) are invariant with adaptation level; the remaining phases of these transient waveforms (associated with the graded potentials of secondary neurons) adapt strongly, in accord with current histology and micro-electrode findings.

© 1971 Optical Society of America

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References

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  1. D. H. Kelly, J. Opt. Soc. Am. 59, 1665 (1969).
    [Crossref] [PubMed]
  2. D. H. Kelly, J. Opt. Soc. Am. 59, 1361 (1969).
    [Crossref]
  3. H. E. Ives, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 343 (1922).
    [Crossref]
  4. F. Veringa, J. Opt. Soc. Am. 60, 286 (1970). This and Ref. 1 cite Veringa’s earlier papers.
    [Crossref]
  5. M. G. F. Fuortes and A. L. Hodgkin, J. Physiol. (London) 172, 239 (1964).
  6. D. H. Kelly, J. Opt. Soc. Am. 51, 422 (1961).
    [Crossref] [PubMed]
  7. D. H. Kelly, “Visual Flicker,” in Handbook of Sensory Physiology, Volume VII/4, Visual Psychophysics, edited by L. M. Hurvich and D. Jameson (Springer, Heidelberg, to be published).
  8. D. H. Kelly, J. Opt. Soc. Am. 51, 747 (1961).
    [Crossref]
  9. L. Walløe, Acta Ophthalmol. 47, 1163 (1969). See also Ref. 17.
    [Crossref]
  10. F. Ratliff, Mach Bands (Holden-Day, San Francisco, 1965), Ch. 3. See also Refs. 15–17.
  11. The theoretical illuminances used for the two middle levels are slightly different from the measured values given in Ref. 6, but they are within the probable calibration error. This adjustment changes the vertical position of two theoretical curves in Fig. 5, but it has no effect on the experimental data points. (Illuminance adjustments were not permitted with the new data of the second paper of this series, where the calibration is believed to be more accurate.)
  12. H. deLange, thesis, Technical University, Delft, 1957.See also J. Opt. Soc. Am. 48, 777 (1958).
  13. G. Sperling and M. M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).
    [Crossref] [PubMed]
  14. L. A. Riggs, in Vision and Visual Perception, edited by C. H. Graham (Wiley, New York, 1965), pp. 81 ff.
  15. K. T. Brown and K. Watanabe, Science 148, 1113 (1965).
    [Crossref] [PubMed]
  16. J. E. Dowling, Science 155, 273 (1967), reviews recent physiological evidence on the site of adaptation.
    [Crossref] [PubMed]
  17. J. E. Dowling and F. S. Werblin, J. Neurophysiol. 32, 315 (1969), discuss the functions of all types of retinal neurons.
    [PubMed]

1970 (1)

1969 (4)

D. H. Kelly, J. Opt. Soc. Am. 59, 1665 (1969).
[Crossref] [PubMed]

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

L. Walløe, Acta Ophthalmol. 47, 1163 (1969). See also Ref. 17.
[Crossref]

J. E. Dowling and F. S. Werblin, J. Neurophysiol. 32, 315 (1969), discuss the functions of all types of retinal neurons.
[PubMed]

1968 (1)

1967 (1)

J. E. Dowling, Science 155, 273 (1967), reviews recent physiological evidence on the site of adaptation.
[Crossref] [PubMed]

1965 (1)

K. T. Brown and K. Watanabe, Science 148, 1113 (1965).
[Crossref] [PubMed]

1964 (1)

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

1961 (2)

1922 (1)

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

Brown, K. T.

K. T. Brown and K. Watanabe, Science 148, 1113 (1965).
[Crossref] [PubMed]

deLange, H.

H. deLange, thesis, Technical University, Delft, 1957.See also J. Opt. Soc. Am. 48, 777 (1958).

Dowling, J. E.

J. E. Dowling and F. S. Werblin, J. Neurophysiol. 32, 315 (1969), discuss the functions of all types of retinal neurons.
[PubMed]

J. E. Dowling, Science 155, 273 (1967), reviews recent physiological evidence on the site of adaptation.
[Crossref] [PubMed]

Fuortes, M. G. F.

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

Hodgkin, A. L.

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

Ives, H. E.

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

Kelly, D. H.

D. H. Kelly, J. Opt. Soc. Am. 59, 1665 (1969).
[Crossref] [PubMed]

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

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

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

D. H. Kelly, “Visual Flicker,” in Handbook of Sensory Physiology, Volume VII/4, Visual Psychophysics, edited by L. M. Hurvich and D. Jameson (Springer, Heidelberg, to be published).

Ratliff, F.

F. Ratliff, Mach Bands (Holden-Day, San Francisco, 1965), Ch. 3. See also Refs. 15–17.

Riggs, L. A.

L. A. Riggs, in Vision and Visual Perception, edited by C. H. Graham (Wiley, New York, 1965), pp. 81 ff.

Sondhi, M. M.

Sperling, G.

Veringa, F.

Walløe, L.

L. Walløe, Acta Ophthalmol. 47, 1163 (1969). See also Ref. 17.
[Crossref]

Watanabe, K.

K. T. Brown and K. Watanabe, Science 148, 1113 (1965).
[Crossref] [PubMed]

Werblin, F. S.

J. E. Dowling and F. S. Werblin, J. Neurophysiol. 32, 315 (1969), discuss the functions of all types of retinal neurons.
[PubMed]

Acta Ophthalmol. (1)

L. Walløe, Acta Ophthalmol. 47, 1163 (1969). See also Ref. 17.
[Crossref]

J. Neurophysiol. (1)

J. E. Dowling and F. S. Werblin, J. Neurophysiol. 32, 315 (1969), discuss the functions of all types of retinal neurons.
[PubMed]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. and Rev. Sci. Instr. (1)

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

J. Physiol. (London) (1)

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

Science (2)

K. T. Brown and K. Watanabe, Science 148, 1113 (1965).
[Crossref] [PubMed]

J. E. Dowling, Science 155, 273 (1967), reviews recent physiological evidence on the site of adaptation.
[Crossref] [PubMed]

Other (5)

L. A. Riggs, in Vision and Visual Perception, edited by C. H. Graham (Wiley, New York, 1965), pp. 81 ff.

D. H. Kelly, “Visual Flicker,” in Handbook of Sensory Physiology, Volume VII/4, Visual Psychophysics, edited by L. M. Hurvich and D. Jameson (Springer, Heidelberg, to be published).

F. Ratliff, Mach Bands (Holden-Day, San Francisco, 1965), Ch. 3. See also Refs. 15–17.

The theoretical illuminances used for the two middle levels are slightly different from the measured values given in Ref. 6, but they are within the probable calibration error. This adjustment changes the vertical position of two theoretical curves in Fig. 5, but it has no effect on the experimental data points. (Illuminance adjustments were not permitted with the new data of the second paper of this series, where the calibration is believed to be more accurate.)

H. deLange, thesis, Technical University, Delft, 1957.See also J. Opt. Soc. Am. 48, 777 (1958).

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Figures (11)

F. 1
F. 1

Sine-wave flicker thresholds plotted in terms of luminous amplitude vs flicker frequency, for the six adaptation levels given in Table I. The points are experimental data for a wide (65°), uniformly flickering field, taken from Ref. 6. The curves are all graphs of Eq. (10), the theoretical magnitude-response function derived in the text.

F. 2
F. 2

Schematic diagram of the proposed two-stage model.

F. 3
F. 3

Curves represent Eq. (8), the theoretical magnitude response of the inhibition stage, with the parameter values given in Table I. Points represent the amplitude sensitivity data of Fig. 1, divided by Eq. (3), the theoretical magnitude response of the diffusion stage.

F. 4
F. 4

Low-frequency inhibiting network containing a variable-gain integrator in a feedback loop (we use a simple form of analog-computer notation, instead of the more cumbersome electrical symbols). More elaborate networks are assembled from these components in Appendix B.

F. 5
F. 5

Amplitude thresholds from Fig. 1, divided by the corresponding adaptation levels and replotted in terms of modulation. The curves are theoretical, derived from the model as discussed in the text, with the parameter values given in Table I. (Data for the highest and lowest adaptation levels of Fig. 1 have been omitted here, to avoid confusion.)

F. 6
F. 6

Impulse responses of the model, calculated from Eq. (9) by means of a fast-Fourier-transform computer program, for the same parameter values used in Fig. 5. Note that the vertical scales are different for each waveform. Because their peak values vary by more than 80 to 1, this was done to show all four responses in one figure.

F. 7
F. 7

General cascaded-feedback-loop network, considered (and rejected) in attempting to fit the data of Fig. 3.

F. 8
F. 8

Pole-zero configuration for network of Fig. 7, showing conditional instability when α < K.

F. 9
F. 9

Pole-zero configuration with conjugate pair adjusted for maximal flatness and unconditional stability, chosen to represent the inhibition stage of the model.

F. 10
F. 10

Graph of Eq. (B3) with α = 0. The shape of its upper corner is essentially independent of the value of α.

F. 11
F. 11

Physical realizations of Eq. (B2) and Figs. 9 and 10. When r = 2, network (a) has the transfer function given by Eq. (B2) for any value of α. When α = 0 (and r = 2), this simplifies to network (b). Note that although network (b) uses only two integrators, it contains three interacting feedback paths.

Tables (1)

Tables Icon

Table I Theoretical values used in plotting Figs. 1, 3, 5, and 6 (1/C = 0.00018 td, τ = 0.5 s, and α = 11 rad/s).

Equations (40)

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G V ( s ) = C 1 sech { [ 2 τ ( s + ρ ) ] 1 2 } ,
G 1 ( s ) = C 2 exp { [ 2 τ ( s + ρ ) ] 1 2 } ,
G V ( s ) = C 1 sech [ ( 2 τ s ) 1 2 ] ,
G 1 ( s ) = C 2 exp [ ( 2 τ s ) 1 2 ] .
| G 1 ( j ω ) | = C 2 exp ( | ω τ | 1 2 ) .
| G | data / | G 1 | theory = | G 2 | data ,
H = K / ( s + α ) ,
G H ( s ) = 1 1 + H = s + α s + α + K ,
| G H ( j ω ) | = [ ω 2 + α 2 ω 2 + ( α + K ) 2 ] 1 2 .
G 2 ( s ) = [ ( s + α ) 2 ( s + K ) 2 + K 2 ] r / 2 ,
| G 2 ( j ω ) | = [ ( ω 2 + α 2 ) 2 ( ω 2 + K 2 ) 2 + 2 K 2 ( K 2 ω 2 ) + K 4 ] r / 4 .
G ( s ) = G 2 G 1 = C [ ( s + α ) 2 ( s + K ) 2 + K 2 ] r / 2 exp [ ( 2 τ s ) 1 2 ]
| G ( j ω ) | = C [ ( ω 2 + α 2 ) 2 ( ω 2 + K 2 ) 2 + 2 K 2 ( K 2 ω 2 ) + K 4 ] r / 4 × exp ( | ω τ | 1 2 ) .
1 / m = B | G ( j ω ) | .
g ( t ) = g 1 ( t ) g 2 ( t ) .
g 1 ( t ) = C ( τ / 2 π ) 1 2 ( t ) 3 2 e τ / 2 t .
k 2 f x 2 = f t + ρ f ,
f ( x , 0 ) = 0 .
k 2 x 2 F ( x , s ) = ( s + ρ ) F ( x , s ) .
p 2 F ¯ ( p , s ) p F ( 0 , s ) x F ( 0 , s ) = a 2 F ¯ ( p , s )
F ¯ ( p , s ) = F ( 0 , s ) p p 2 a 2 + [ x F ( 0 , s ) ] 1 p 2 a 2 .
F ( x , s ) = F ( 0 , s ) cosh a x + 1 a [ x F ( 0 , s ) ] sinh a x .
1 a x F ( 0 , s ) = F ( 0 , s ) coth a x 0 .
F ( x , s ) / F ( 0 , s ) = cosh a x coth a x 0 sinh a x .
G ( x , x 0 , s ) = x F ( x , s ) x F ( 0 , s ) = cosh a x tanh a x 0 sinh a x .
G ( x , , s ) = cosh a x sinh a x = e a x .
G ( x 0 , x 0 , s ) = G ( x , x , s ) = sech a x = 2 / ( e a x + e a x ) .
G ( 1 , , s ) = exp { [ 2 τ ( s + ρ ) ] 1 2 }
G ( 1 , 1 , s ) = sech { [ 2 τ ( s + ρ ) ] 1 2 } ,
e a x < sech a x < 2 e a x ,
G ( j ω ) G ( 1 , , j ω ) = M ( ω ) e j φ ( ω ) ,
j φ + log M = [ 2 τ ( ρ + j ω ) ] 1 2 .
( R cos θ + j R sin θ ) 1 2 = R 1 2 cos ( θ / 2 ) + j R 1 2 sin ( θ / 2 ) = ( 1 2 R + ρ τ ) 1 2 + j ( 1 2 R ρ τ ) 1 2 .
( log M ) 2 = R cos 2 ( θ / 2 ) = τ ( ω 2 + ρ 2 ) 1 2 + ρ τ ,
φ 2 = R sin 2 ( θ / 2 ) = τ ( ω 2 + ρ 2 ) 1 2 ρ τ ,
G ( j ω ) = exp { [ τ ( ω 2 + ρ 2 ) 1 2 + ρ τ ] 1 2 ± j [ τ ( ω 2 + ρ 2 ) 1 2 ρ τ ] 1 2 } ,
G ( j ω ) = exp [ ( ω τ ) 1 2 ( 1 ± j ) ] = exp [ ( 2 j ω τ ) 1 2 ] ,
G T ( s ) = [ 1 1 + [ K / ( s + α ) ] n ] m = [ ( s + α ) n ( s + α ) n + K n ] m ,
G 2 ( s ) = [ ( s + α ) 2 ( s + K ) 2 + K 2 ] r / 2 ,
| G 2 ( j ω ) | = ( ω 2 + α 2 ) r / 2 [ ( ω 2 + K 2 ) 2 + 2 K 2 ( K 2 ω 2 ) + K 4 ] r / 4 .