Abstract

Renewal theory is applied to a particle-counter model of visual discrimination in order to determine the effects of neural-impulse scaling and dead time on the detection of increment thresholds. Let the ratio of absorbed photons to neural spikes (scaling factor) be denoted by r and the dead time by τ. We show that the particle counter is equivalent to one with dead time τ* = τ/r and scaling factor r* = 1. Further, if τ = 0, the particle counter does not exhibit the Weber–Fechner behavior for high background luminances as predicted by Barlow. These are asymptotic results, valid for large observation times. For more general observation times, the performance of a particle-counter mechanism with r = 2 and τ = 0 is evaluated for different types of starting procedures.

© 1969 Optical Society of America

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References

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  1. M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc. Am. 53, 121 (1963).
    [Crossref] [PubMed]
  2. H. B. Barlow, Cold Spring Harbor Symp. Quant. Biol. 30, 539 (1965).
    [Crossref]
  3. Deterministic explanations, which do not attribute any significance to fluctuations, also exist. Rushton [W. A. H. Rushton, Proc. Roy. Soc. (London) 162B, 20 (1965)] exploits an analogy between a neural mechanism and a leaky electrical cable with feedback while van de Grind and Bouman [W. A. van de Grind and M. A. Bouman, Kybernetick 4, 136 (1968)] use a special type of adaptive-scaling mechanism but consider only mean values.
    [Crossref]
  4. E. A. Trabka, Vision Res. 8, 113 (1968).
    [Crossref] [PubMed]
  5. E. A. Trabka, Vision Res. 8, 613 (1968).
    [Crossref] [PubMed]
  6. D. R. Cox, Renewal Theory (John Wiley& Sons, Inc., New York, 1962).
  7. G. Sperling and M. Sondhi, J. Opt. Soc. Am. 58, 1133 (1968).
    [Crossref] [PubMed]
  8. D. M. MacKay, Science 159, 338 (1968).
    [Crossref]
  9. Sec. 5.5 of Ref. 6.
  10. Sec. 4.2 of Ref. 6.
  11. E. Parzen, Stochastic Processes (Holden-Day, Inc., San Francisco, 1962), p. 47.
  12. G. Wyszecki and W. S. Stiles, Color Science (John Wiley & Sons. Inc., New York, 1967), p. 569.
  13. These were obtained from the conditional moments about the origin, given t0, by taking expected values with respect to the assumed a priori density on t0. If we had instead started with the conditional moments about the mean, these would yield different expressions upon averaging although the numerical values plotted in Fig. 2 would not be significantly different.

1968 (4)

E. A. Trabka, Vision Res. 8, 113 (1968).
[Crossref] [PubMed]

E. A. Trabka, Vision Res. 8, 613 (1968).
[Crossref] [PubMed]

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

D. M. MacKay, Science 159, 338 (1968).
[Crossref]

1965 (2)

H. B. Barlow, Cold Spring Harbor Symp. Quant. Biol. 30, 539 (1965).
[Crossref]

Deterministic explanations, which do not attribute any significance to fluctuations, also exist. Rushton [W. A. H. Rushton, Proc. Roy. Soc. (London) 162B, 20 (1965)] exploits an analogy between a neural mechanism and a leaky electrical cable with feedback while van de Grind and Bouman [W. A. van de Grind and M. A. Bouman, Kybernetick 4, 136 (1968)] use a special type of adaptive-scaling mechanism but consider only mean values.
[Crossref]

1963 (1)

Barlow, H. B.

H. B. Barlow, Cold Spring Harbor Symp. Quant. Biol. 30, 539 (1965).
[Crossref]

Bouman, M. A.

Cox, D. R.

D. R. Cox, Renewal Theory (John Wiley& Sons, Inc., New York, 1962).

MacKay, D. M.

D. M. MacKay, Science 159, 338 (1968).
[Crossref]

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, Inc., San Francisco, 1962), p. 47.

Rushton, W. A. H.

Deterministic explanations, which do not attribute any significance to fluctuations, also exist. Rushton [W. A. H. Rushton, Proc. Roy. Soc. (London) 162B, 20 (1965)] exploits an analogy between a neural mechanism and a leaky electrical cable with feedback while van de Grind and Bouman [W. A. van de Grind and M. A. Bouman, Kybernetick 4, 136 (1968)] use a special type of adaptive-scaling mechanism but consider only mean values.
[Crossref]

Sondhi, M.

Sperling, G.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science (John Wiley & Sons. Inc., New York, 1967), p. 569.

Trabka, E. A.

E. A. Trabka, Vision Res. 8, 113 (1968).
[Crossref] [PubMed]

E. A. Trabka, Vision Res. 8, 613 (1968).
[Crossref] [PubMed]

Vos, J. J.

Walraven, P. L.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science (John Wiley & Sons. Inc., New York, 1967), p. 569.

Cold Spring Harbor Symp. Quant. Biol. (1)

H. B. Barlow, Cold Spring Harbor Symp. Quant. Biol. 30, 539 (1965).
[Crossref]

J. Opt. Soc. Am. (2)

Proc. Roy. Soc. (London) (1)

Deterministic explanations, which do not attribute any significance to fluctuations, also exist. Rushton [W. A. H. Rushton, Proc. Roy. Soc. (London) 162B, 20 (1965)] exploits an analogy between a neural mechanism and a leaky electrical cable with feedback while van de Grind and Bouman [W. A. van de Grind and M. A. Bouman, Kybernetick 4, 136 (1968)] use a special type of adaptive-scaling mechanism but consider only mean values.
[Crossref]

Science (1)

D. M. MacKay, Science 159, 338 (1968).
[Crossref]

Vision Res. (2)

E. A. Trabka, Vision Res. 8, 113 (1968).
[Crossref] [PubMed]

E. A. Trabka, Vision Res. 8, 613 (1968).
[Crossref] [PubMed]

Other (6)

D. R. Cox, Renewal Theory (John Wiley& Sons, Inc., New York, 1962).

Sec. 5.5 of Ref. 6.

Sec. 4.2 of Ref. 6.

E. Parzen, Stochastic Processes (Holden-Day, Inc., San Francisco, 1962), p. 47.

G. Wyszecki and W. S. Stiles, Color Science (John Wiley & Sons. Inc., New York, 1967), p. 569.

These were obtained from the conditional moments about the origin, given t0, by taking expected values with respect to the assumed a priori density on t0. If we had instead started with the conditional moments about the mean, these would yield different expressions upon averaging although the numerical values plotted in Fig. 2 would not be significantly different.

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Figures (2)

Fig. 1
Fig. 1

Particle-counter model of the visual mechanism. Photons of wavelength λ are incident at a rate nλ/sec/mm2. All photons absorbed within an area A are pooled into a single channel. The fraction of A occupied by receptors is ρ. The probability that an incident photon will be absorbed is sλ, producing an average absorption rate m. Spontaneous neural activity at a rate m0 adds to the firing rate, but can be neglected at high irradiances. A scaling mechanism reduces the average firing rate by the factor r. A type I particle counter introduces saturation at high irradiances and the average firing rate m(+r)−1τ−1 as m → ∞.

Fig. 2
Fig. 2

Normalized signal-to-noise ratio (S/N)2 between the outputs of a pair of particle counters determining increment thresholds with a scaling mechanism but no dead time. The average number of photons (due to the background radiance) absorbed in an interval of length t is given by the abscissa mt. Curve A is for a quantum-to-spike ratio r = 1, whereas B and C are for r = 2. Curve B corresponds to synchronous counting, whereas C is for random-asynchronous counting with random starting time (following a photon absorption) having an average vasue of 0.1 of the total counting time t. Vertical line D defines a region to its left in which asymptotic relations are considered to hold.

Equations (26)

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m = n λ ρ A s λ ,
H ( t ) ~ μ c - 1 t
V ( t ) ~ σ c 2 μ c - 3 t ,
t > μ c 3 σ 2 - 2 .
f c ( T ) = { f r ( T - τ ) ; T > τ 0 ; T < τ
μ c = r m - 1 + τ
σ c 2 = r m - 2 .
H ( t ) ~ [ m / ( r + m τ ) ] t
V ( t ) ~ [ r m / ( r + m τ ) 3 ] t .
t > ( r + m τ ) 3 / r m .
d H ~ [ r t / ( r + m τ ) 2 ] d m .
( S / N ) 2 = [ d H ( t ) ] 2 / V ( t )
( S N ) 2 ~ t ( 1 + m τ * ) · ( d m ) 2 m ,
τ * = τ / r
( d m ) 2 = K 2 m ( 1 + m τ * ) t - 1 .
( S / N ) 2 = [ ( d m ) 2 / m ] t .
H ( t ) = 1 2 m t - 1 4 e - 2 m t 0 ( 1 - e - 2 m t )
V ( t ) = 1 4 m t + 1 8 ( 1 - e - 2 m t ) - 1 16 e - 4 m t 0 ( 1 - e - 2 m t ) 2 - 1 2 m t exp [ - 2 m ( t + t 0 ) ]
( S N ) 2 = ( d m ) 2 m + 1 6 ( r 2 - 1 ) ,
( S / N ) 2 = ( d m ) 2 / ( m + 1 2 ) .
Ĥ ( t ) = 1 2 x - 1 4 ( 1 - e - 2 x ) ( 1 + 2 a x )
V ˆ ( t ) = 1 4 x + 1 8 ( 1 - e - 2 x ) - 1 16 ( 1 - e - 2 x ) 2 ( 1 + 2 a x ) 2 - 1 2 x e - 2 x ( 1 + 2 a x ) .
( S N ) NORM 2 = ( S / N ) 2 t [ ( d m ) 2 / m ] = g 2 W ,
g ( x ) = 1 - e - 2 x 1 + 2 a x + a ( 1 - e - 2 x ) ( 1 + 2 a x ) 2 ,
W ( x ) = 1 - 2 e - 2 x 1 + 2 a x + 1 2 Q ( x ) { 2 - 1 - e - 2 x ( 1 + 2 a x ) 2 } ,
Q ( x ) = ( 1 - e - 2 x ) / 2 x .