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Human monochroma+c light discrimina+on explained by op+mal decoding of cone absorp+ons Li Zhaoping, Wilson S. Geisler, and Keith A. May (Paper in PLoS One 2011) Presented on June, 9, 2011, at the Gatsby external seminar.
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Human monochromac light discriminaon explained by opmal ...

Feb 17, 2022

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Page 1: Human monochromac light discriminaon explained by opmal ...

Human monochroma+c light discrimina+on  explained by op+mal decoding of cone absorp+ons 

Li Zhaoping,      Wilson S. Geisler,   and Keith A. May   (Paper in PLoS One 2011) 

Presented on June, 9, 2011, at the Gatsby external seminar. 

Page 2: Human monochromac light discriminaon explained by opmal ...

Motivation: Decoding from real physiological signals for real behavioral data

Sensory signal: S, neural responses r, perception: S’ Encoding: P(r|S) --- likelihood Decoding: P(S’|r) ~ P(r|S) P(S) Prior: P(S).

In most decoding work so far, one or more of the followings apply: (1) P(r|S) is assumed, not quantitatively known (e.g., noise is unknown) (2) r is not neural response, but artificial by experimental design (3) S’ is not the behavioral perception, but modeler’s toy.

e.g., Pillow et al, decoding visual inputs from retinal ganglion responses. many parameters in P(r|S) are assumed, S’ is not for behavioral

e.g., Koerding & Wolpert, inferring motor target position S’ from noisy position seen r, with prior P(S).

S: actual target position invisible to subjects r: not neural response, but fake target position shown to subjects with likelihood

P(r|S), --- typical of many behavioral studies of Bayesian inference. P(S): prior, controlled by experimenter, experienced by subjects.

S’: Subject’s estimate of target position, manifested in their motor responses

Page 3: Human monochromac light discriminaon explained by opmal ...

Sensory signal: S, neural responses r, perception: S’ Encoding: P(r|S) --- likelihood Decoding: P(S’|r) ~ P(r|S) P(S) Prior: P(S).

Motivation: Decoding from real physiological signals for real behavioral data

Some exceptions: Paradiso (1988),human orientation

discrimination from V1 neural responses.

Fisher Information

IF = dri∫i∑ P(ri | S)[−∂ 2 lnP(ri | S) /∂S2]

∝number of neurons

However, used several free parameters

3 parameters: Orientation tuning function and response amplitude,

2 parameters: Neural noise 1 parameter: total number of neurons.

Page 4: Human monochromac light discriminaon explained by opmal ...

Motivation: Decoding from real physiological signals for real behavioral data

Current work: Color perception from cone responses, only one free parameter (input intensity)

Sensory signal: S, neural responses r, perception: S’ Encoding: P(r|S) --- likelihood Decoding: P(S’|r) ~ P(r|S) P(S) Prior: P(S).

Behavior

λ

λ + dλ

400 500 600 70010−6

10−4

10−2

100

Wavelength ! (nm)

Cone Spectrum Sensitivity fa(! )

Physiology P(r|S), Poisson noise. Number of cones.

Page 5: Human monochromac light discriminaon explained by opmal ...

Current work: Color perception from cone responses, only one free parameter (input intensity)

Behavior

λ

λ + dλ

400 500 600 70010−6

10−4

10−2

100

Wavelength ! (nm)

Cone Spectrum Sensitivity fa(! )

Physiology P(r|S), Poisson noise. Number of cones.

(2) Na red cones, each Poisson, each with tuning function fa(λ), equivalent to one single giant red cone, Poisson, with tuning function Nafa(λ).

(1) Other than different sensitivities to wavelength, different cones give equal electrical responses to light after photon absorption--- physiologically known.

Hence, different amplitudes of cone tuning curves reflect different cone densities and pre-receptor optical absorption

Page 6: Human monochromac light discriminaon explained by opmal ...

Monochroma+c discrimina+on threshold depends on wavelength (Pokorny and Smith, 1970) 

λ

λ + dλ

Subject adjust the intensity of the test field to make it appear iden+cal to the standard field, threshold  is reached when this is impossible 

Why?  Can we understand this from the informa+on in the cone absorp+ons regardless of the post‐receptor mechanisms (like an ideal observer’s approach) 

Page 7: Human monochromac light discriminaon explained by opmal ...

P(rL ,rM ,rS | λ)

Due to noise, cone response (absorp+on) as a probability distribu+on ‐‐‐ finite discrimina+on threshold 

P(λ | rL ,rM ,rS )

λ

λ + dλ

P(λ | rL ,rM ,rS ) = P(rL ,rM ,rS | λ)P(λ) /P(rL ,rM ,rS )

P(λ | rL ,rM ,rS )∝P(rL ,rM ,rS | λ)

By Bayesian: 

Maximum likelihood: 

P(rL ,rM ,rS | λ)

P(λ | rL ,rM ,rS )

Page 8: Human monochromac light discriminaon explained by opmal ...

P(rL ,rM ,rS | λ)Cone response noise is Poisson noise 

r a = I • fa (λ)

P(rL ,rM ,rS | λ) = P(rL | λ)P(rM | λ)P(rS | λ)

P(ra | λ) =r a

ra

ra!exp(−r a )

Mean cone response

Spectral sensitivity

Page 9: Human monochromac light discriminaon explained by opmal ...

An example of maximum likelihood decoding, given get  

P(λ | rL ,rM ,rS ) ≈ exp[−(λ − λ )2

2σ 2(λ )]€

(rL ,rM ,rS )

Responses generated by wavelength at 550 nm 

σ(λ )Threshold value  

P(λ | rL ,rM ,rS )∝ P(rL ,rM ,rS | λ)

r a = I • fa (λ)

Fisher Information IF (λ) = I[ f 'aa∑ (λ)]2 / fa (λ)

threshold σ (λ) = IF−1/ 2

λ

Page 10: Human monochromac light discriminaon explained by opmal ...

Repeat this for all wavelength to find the wavelength discrimina+on threshold 

Model predic+on 

           Threshold σ (nm)                

Why large discrepancy    ? 

Data  

χ2 =

[σmodel(λ) −σ data (λ)]2

[Δσ data (λ)]2

λ

∑1

λ

σdata (λ)

σmodel(λ)

σ(λ) = { I[ f 'aa∑ (λ)]2 / fa (λ)}

−1/ 2

I chosen to minimize fitting error

χ

χ

Page 11: Human monochromac light discriminaon explained by opmal ...

Back to the original measurements (Pokorny and Smith, 1970) 

λ

λ + dλ

Subject adjust the test field by intensity I to make it appear iden+cal to the standard field, threshold is reached when this is impossible 

Both input intensity I and input wavelength λ are changed in matching. 

P(λ | rL ,rM ,rS )

P(λ,I | rL ,rM ,rS )

Page 12: Human monochromac light discriminaon explained by opmal ...

Back to the original measurements (Pokorny and Smith, 1970) 

λ

λ + dλ

Subject adjust the test field by intensity I to make it appear iden+cal to the standard field, threshold is reached when this is impossible 

Both input intensity I and input wavelength λ are changed in matching. 

P(λ,I | rL ,rM ,rS )

λ€

IContour plot

Page 13: Human monochromac light discriminaon explained by opmal ...

Back to the original measurements (Pokorny and Smith, 1970) Both input intensity I and input wavelength λ are changed in matching. 

P(λ,I | rL ,rM ,rS )

λ€

IContour plot

P(λ | rL ,rM ,rS )

Threshold value when I is fixed

Threshold when I can change

Page 14: Human monochromac light discriminaon explained by opmal ...

Back to the original measurements (Pokorny and Smith, 1970) 

P(λ,I | rL ,rM ,rS )

λ€

IContour plot

Threshold when I can change

Wavelength-intensity confound! Is the right patch redder or darker?

λ

λ + dλ

Page 15: Human monochromac light discriminaon explained by opmal ...

Back to the original measurements (Pokorny and Smith, 1970) 

Wavelength-intensity confound! Is the right patch redder or darker?

λ

λ + dλ

400 500 600 70010−6

10−4

10−2

100

Wavelength ! (nm)

Cone Spectrum Sensitivity fa(! )At long λ, increasing λ decreases responses from all 3 cones, difficult to tell whether the input is redder or darker, the confound is stronger, hence larger threshold.

At medium λ, increasing λ increases response from some cone and decreases response from other cones, easier to tell wavelength change. The confound is weaker, hence smaller threshold.

Page 16: Human monochromac light discriminaon explained by opmal ...

P(λ,I | rL ,rM ,rS )

λ€

IContour plot

Threshold when I can change

IF (λ,I) = −

∂ 2 lnP(r | λ,I)∂λ2

∂ 2 lnP(r | λ,I)∂λ∂I

∂ 2 lnP(r | λ,I)∂λ∂I

∂ 2 lnP(r | λ,I)∂I2

2-d Fisher information formulation

P(λ,I | r) ≈exp{−[IF ,11(λ − λ )2 + 2IF ,12(λ − λ )(I − I ) + IF ,22(I − I )2]/2}

σ(λ) =1I

fa (λ)a∑

[ f 'b (λ)]2

fb (λ)fc (λ) − [ f 'd (λ)]

2

d∑

c∑

b∑

1/ 2

Result:

Page 17: Human monochromac light discriminaon explained by opmal ...

Be^er explana+on of data! 

When input intensity  is adjusted in color matching 

otherwise 

σ(λ) =1I

fa (λ)a∑

[ f 'b (λ)]2

fb (λ)fc (λ) − [ f 'd (λ)]

2

d∑

c∑

b∑

1/ 2

Page 18: Human monochromac light discriminaon explained by opmal ...

Monochroma+c light wavelength discrimina+on explained by  op+mal decoding based on signals in the cones. 

Suggest that efficiency in informa+on processing efficiency in post‐receptoral mechanisms is a constant regardless of wavelength. 

Model has to match with experimental methods to account for data. 

Predic+on ‐‐‐ smaller threshold when input intensity is fixed in threshold  assessments 

Page 19: Human monochromac light discriminaon explained by opmal ...

Relative cone densities for L, M, S cones influence model prediction accuracy

normal amount of S cones S cones too numerous

Page 20: Human monochromac light discriminaon explained by opmal ...

When S cones are too few …

An extra peak… as seen in some data (Bedford and Wyszecki 1958)when the input field is too small, too few S cones

L & M cone co-vary here … becoming color blind.

Page 21: Human monochromac light discriminaon explained by opmal ...

Importance of proper experimental procedures:

λ€

I

Wavelength-intensity confound means that it is difficult to ask subjects to match the brightness of two color fields while checking whether they differ in hue.

λ

λ + dλPokorny and Smith (1970): Subject adjust the test field by intensity I to make it appear iden+cal to the standard field, threshold is reached when this is impossible 

Two kinds of procedures in the literature:

Bedford and Wyszecki (1958): Subject adjust the test field by intensity I to match the brightness of the two fields, and then see if there is a hue difference.  Threshold is reached when there is a hue difference. 

Page 22: Human monochromac light discriminaon explained by opmal ...

Summary:

Human wavelength discrimination can be understood as optimal decoding from cone absorptions (with constant efficiency)

This model reveals the reliability of data from different experimental procedures.