Abstract

Discrete and continuous linear models are defined to describe contrast phenomena such as Mach bands. Two-sided z transforms are used to describe the discrete systems and Fourier transforms are used to describe continuous systems. A psychological model is defined based on the work of v. Békésy on skin and vision which suggests a “neural unit” consisting of an area of sensation surrounded by an area of inhibition. A physiological model is also defined based on the experiments of Hartline et al., describing lateral neural interaction in the eye of Limulus. It is shown that the physiological model is related to the psychological model and that the form of the physiological inhibitory coefficients kp uniquely specify the form of the psychological neural unit hp, and vice versa. Also, by assuming a “blurring” of the stimulus spatial distribution as the excitation distribution of the receptors, a neural unit is obtained from the physiological model which is similar to the psychological neural unit suggested by the experiments of v. Békésy. Illustrative examples are given.

© 1961 Optical Society of America

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References

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  1. F. Ratliff, W. J. Miller, and H. K. Hartline, Anns. N. Y. Acad. Sci. 74, 210–222 (1958).
    [Crossref]
  2. G. v. Békésy, J. Opt. Soc. Am. 50, 1060–1070 (1960).
    [Crossref] [PubMed]
  3. S. J. Mason and H. J. Zimmermann, Electronic Circuits, Signals, and Systems (John Wiley & Sons, Inc., New York, 1960).
  4. J. R. Ragazzini and G. F. Franklin, Sampled-Data Control Systems, (McGraw-Hill Book Company, Inc., New York, 1958).
  5. R. W. Sittler, Pulsed-Data Systems, notes on a course given at the Massachusetts Institute of Technology (1959–60).
  6. L. V. Ahlfors, Complex Analysis (McGraw-Hill Book Company, Inc., New York, 1953).
  7. G. v. Békésy, Psychol. Revs. 66, 1–22 (1959).
    [Crossref]
  8. C. McCollough, J. Exptl. Psychol. 49, 141–153 (1955).
    [Crossref]

1960 (1)

1959 (1)

G. v. Békésy, Psychol. Revs. 66, 1–22 (1959).
[Crossref]

1958 (1)

F. Ratliff, W. J. Miller, and H. K. Hartline, Anns. N. Y. Acad. Sci. 74, 210–222 (1958).
[Crossref]

1955 (1)

C. McCollough, J. Exptl. Psychol. 49, 141–153 (1955).
[Crossref]

Ahlfors, L. V.

L. V. Ahlfors, Complex Analysis (McGraw-Hill Book Company, Inc., New York, 1953).

Békésy, G. v.

Franklin, G. F.

J. R. Ragazzini and G. F. Franklin, Sampled-Data Control Systems, (McGraw-Hill Book Company, Inc., New York, 1958).

Hartline, H. K.

F. Ratliff, W. J. Miller, and H. K. Hartline, Anns. N. Y. Acad. Sci. 74, 210–222 (1958).
[Crossref]

Mason, S. J.

S. J. Mason and H. J. Zimmermann, Electronic Circuits, Signals, and Systems (John Wiley & Sons, Inc., New York, 1960).

McCollough, C.

C. McCollough, J. Exptl. Psychol. 49, 141–153 (1955).
[Crossref]

Miller, W. J.

F. Ratliff, W. J. Miller, and H. K. Hartline, Anns. N. Y. Acad. Sci. 74, 210–222 (1958).
[Crossref]

Ragazzini, J. R.

J. R. Ragazzini and G. F. Franklin, Sampled-Data Control Systems, (McGraw-Hill Book Company, Inc., New York, 1958).

Ratliff, F.

F. Ratliff, W. J. Miller, and H. K. Hartline, Anns. N. Y. Acad. Sci. 74, 210–222 (1958).
[Crossref]

Sittler, R. W.

R. W. Sittler, Pulsed-Data Systems, notes on a course given at the Massachusetts Institute of Technology (1959–60).

Zimmermann, H. J.

S. J. Mason and H. J. Zimmermann, Electronic Circuits, Signals, and Systems (John Wiley & Sons, Inc., New York, 1960).

Anns. N. Y. Acad. Sci. (1)

F. Ratliff, W. J. Miller, and H. K. Hartline, Anns. N. Y. Acad. Sci. 74, 210–222 (1958).
[Crossref]

J. Exptl. Psychol. (1)

C. McCollough, J. Exptl. Psychol. 49, 141–153 (1955).
[Crossref]

J. Opt. Soc. Am. (1)

Psychol. Revs. (1)

G. v. Békésy, Psychol. Revs. 66, 1–22 (1959).
[Crossref]

Other (4)

S. J. Mason and H. J. Zimmermann, Electronic Circuits, Signals, and Systems (John Wiley & Sons, Inc., New York, 1960).

J. R. Ragazzini and G. F. Franklin, Sampled-Data Control Systems, (McGraw-Hill Book Company, Inc., New York, 1958).

R. W. Sittler, Pulsed-Data Systems, notes on a course given at the Massachusetts Institute of Technology (1959–60).

L. V. Ahlfors, Complex Analysis (McGraw-Hill Book Company, Inc., New York, 1953).

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Figures (10)

Fig. 1
Fig. 1

The pressure of a single point on the surface of the skin produces (a) an area of sensation and around it a refractory area in which the neighboring stimulus is inhibited; in (b) the pattern is simplified to a rectangular shape; v. Békésy, Fig. 3.2

Fig. 2
Fig. 2

The physiological model for lateral inhibition. The figure illustrates that the response rp is equal to the excitation ep inhibited by the responses of neighboring receptors. The figure is drawn for kj=0, |j|>2.

Fig. 3
Fig. 3

The psychological model for a neural unit. The figure illustrates how a stimulus fp contributes, via its neural unit hj, directly to neighboring responses. The figure is drawn for hj=0, |j|>2.

Fig. 4
Fig. 4

A neural unit for the discrete linear model. The unit is derived from a geometric lateral inhibition kp=b+|p| and represents the response due to a unit excitation of one receptor.

Fig. 5
Fig. 5

A five-receptor neural unit and its corresponding inhibition function. The inhibition function is a sum of two geometric series.

Fig. 6
Fig. 6

Step excitation (a) of the five-receptor neural unit of Fig. 5(a) yields response (b).

Fig. 7
Fig. 7

A finite approximation to the spatial impulse response function for the continuous model. This function is derived from an exponential lateral inhibition k(x)=ea|x|.

Fig. 8
Fig. 8

An excitation function e(x)=−∞x exp(−t2/2)dt+1 (a) and the corresponding response functions predicted by (b) Mach’s equation r(x)=1.44 log[2e(x)−4e″(x)/e(x)] and (c) the physiological model r(x)=e(x)−2e″(x).

Fig. 9
Fig. 9

Signal flow graphs illustrating the addition of a “blurring function,” L(ω), to (a) the physiological model and (b) the psychological model. The graphs display the analytical relations

Fig. 10
Fig. 10

Neural unit for the continuous model. This unit is derived from a Gaussian-shaped “blurring function,” l(x)=Bebx2 and an exponential inhibition function,

Equations (49)

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r ( x ) = α log [ e ( x ) β ± γ e ( x ) d 2 e ( x ) d x 2 ] ,
r p = e p - j = 1 n k p , j ( r j - r p , j 0 ) ,
r ( x ) = - f ( λ ) h ( x - λ ) d λ ,
e p = j = 1 n k p , j r j ,
e p = j = - k p - j r j ,
r p = j = - h p - j f j ,
F T ( ω ) = - A u 0 ( t - T ) e - j ω t d t = A e - j ω T .
F X ( ω ) = A e - j ω X
F n ( ω ) = f n e - j ω n w
G n ( z ) = f n z n ,
G n ( e - j ω w ) = F n ( ω ) .
G N ( z ) = g - N z - N + + g N z N = n = - N N g N z n .
G ( z ) = - g n z n
G ( z ) = n = 0 g n z n
E ( z ) = - e p z p ,
R ( z ) = - r p z p ,
K ( z ) = - k p z p ,
H ( z ) = - h p z p .
E ( z ) = p = - ( j = - k p - j r j ) z p = p = - j = - k p - j z p - j r j z j = u = - j = - k u z u r j z j ,
E ( z ) = u = - k u z u j = - r j z j = K ( z ) R ( z ) .
R ( z ) = H ( z ) E ( z ) .
H ( z ) = 1 / K ( z ) .
k p = e - a w p
K ( z ) = ( 1 + b z + b 2 z 2 + ) + ( b 1 z + b 2 1 z 2 + ) = 1 1 - b z + 1 1 - b z - 1 - 1 = 1 - b 2 1 + b 2 - b [ z + z - 1 ]
H ( z ) = - b 1 - b 2 1 z + 1 + b 2 1 - b 2 + - b 1 - b 2 z
h p = { 1 + b 2 1 - b 2 p = 0 - b 1 - b 2 p = 1 0 p > 1.
R ( z ) = E ( z ) [ 1 + b 2 1 - b 2 - b 1 - b 2 ( 1 z + z ) ] ,
R ( z ) = 1 - b 1 + b E ( z ) - b 1 - b 2 [ E ( z ) z - 2 E ( z ) + z E ( z ) ] .
r p = 1 - b 1 + b e p - b 1 - b 2 [ ( e p + 1 - e p ) - ( e p - e p - 1 ) ] = 1 - b 1 + b e p - b 1 - b 2 Δ 2 e p ,
h p = { 1 p = 0 - 1 / 10 p = ± 1 , ± 2 0 p > 2
k p = 5 7 3 ( 3 - 2 ) p + 10 7 3 ( 3 - 5 ) p p = 0 , ± 1 ,
e ( x ) = - k ( x - λ ) r ( λ ) d λ .
k ( x ) = e - a x ,
e ( x ) = - e - a x - λ r ( λ ) d λ = x e a ( x - λ ) r ( λ ) d λ + - x e - a ( x - λ ) r ( λ ) d λ .
e ( x ) = - 0 e a u r ( x - u ) d u + 0 e - a u r ( x - u ) d u .
E ( ω ) = - - 0 e a u r ( x - u ) e - j ω x d u d x = - 0 e - a u γ ( x - u ) e - j ω x d u d x .
E ( ω ) = - 0 e a u d u - r ( x - u ) e - j ω x d u d x + 0 e - a u d u - r ( x - u ) e - j ω x d u d x = - 0 e ( a - j ω ) u d u - r ( λ ) e - j ω λ d λ + 0 e - ( a + j ω ) u d u - r ( λ ) e - j ω λ d λ = [ 1 / ( a - j ω ) + 1 / ( a + j ω ) ] R ( ω ) = 2 a a 2 + ω 2 R ( ω )
R ( ω ) = a 2 + ω 2 2 a E ( ω ) ,
e ( x ) = u 0 ( x ) ; E ( ω ) = 1 r ( x ) = h ( x ) ; R ( ω ) = H ( ω )
H ( ω ) = ( a 2 + ω 2 ) / 2 a .
h ( x ) = 1 2 a u 0 ( x ) - ( 1 / 2 a ) u 2 ( x ) ,
r ( x ) = a 2 e ( x ) - 1 2 a d 2 e ( x ) d x 2 .
e ( x ) = - f ( ξ ) l ( x - ξ ) d ξ = - k ( x - λ ) r ( λ ) d λ ,
H ( ω ) = R ( ω ) / F ( ω )
H ( ω ) = L ( ω ) G ( ω ) = L ( ω ) / K ( ω ) .
g ( x ) = 1 2 a u 0 ( x ) - ( 1 / 2 a ) u 2 ( x )
H ( ω ) = ( a 2 + ω 2 ) L ( ω ) 2 a
h ( x ) = a l ( x ) 2 - 1 2 a [ d 2 l ( x ) / d x 2 ] .
k ( x ) = exp ( - a x ) .