Both the neural signal and its noise have a nonlinear dependence on the contrast of the stimulus. The two separate nonlinearities may be measured psychophysically as well as in single cortical cells. The resulting contrast relationships represent a signature of the visual processing stage that limits the human observer’s performance. For both sustained and transient Gabor patches the predominant noise was found to be multiplicative with a power exponent of about 0.8. This noise source was preceded by an accelerating signal transducer with power 2 - 2.7 throughout the contrast range. These exponents combine to account for the classic compressive power of about 0.4 for the signal-to-noise ratio in contrast discrimination. The estimated transducer acceleration suggests that there is a direct computation of contrast energy in the visual cortex.

Introduction

Since the pioneering work of Tanner (1961) and Swets (1961), it has been widely recognized that discrimination between sensory stimuli of different strengths is limited by noise in the internal responses to those stimuli. It would be natural to expect, then, that independent measurement of the signal and noise-related components in the internal response would be a core issue in neurophysiology and psychophysics. Neurophysiology has made a significant effort in this direction, with numerous studies of the statistical variability of the signals from single neurons in primary cortex (Tolhurst et al., 1983; Vogels et al., 1989; Snowden et al.; 1992; Softky & Koch, 1993; Geisler & Albrecht, 1997) However, the neurophysiological results have not provided a decisive answer because signal and noise characteristics differ between the visual processing stages and it remains unclear which neural stage is critical to the observer’s performance. Determining this critical stage may be challenging since it may be different for different visual tasks. In psychophysics, theoretical analyses provided by Legge et al. (1987) and Ahumada (1987) have demonstrated that the effects of a transducer nonlinearity are psychophysically equivalent at a given contrast to the effects of an internal, signal-dependent noise. This equivalence led them to conclude that distinguishing between the two types of nonlinearity would be impossible.

Derivation of this theoretical equivalence rests on an assumption of Green & Swets (1966) that the noise levels are the same for both stimuli in the discrimination task. This assumption is reasonable when the differences in test level are a small fraction of the pedestal value; the assumption can easily fail, however, when the differences to be discriminated are large (Tyler & Liu, 1996). If the noise increases with signal strength, the larger signal will also have a higher noise level. Elaborating this notion, the present study introduces a novel approach for separate assessment of the nonlinearity in the internal response and the nonlinearity in its noise at the critical neural processing stage that limits psychophysical discrimination thresholds. This approach will be described in general terms and tested for the contrast dimension of visual stimulus strength.

The Model Observer

The observer model used in this study is shown in Fig. 1 and justified in the remainder of this section. In summary, the model is built around the concept of critical noise locus that limits performance of a particular psychophysical task. Within the model framework, there are three components that determine observer’s performance: nonlinearities preceding the critical noise infusion, the critical noise itself, and the decision stage. The nonlinearity is assumed to be monotonic. In a two-alternative forced-choice (2AFC) task the decision stage compares the two stimuli based on their responses and chooses the stronger one. Any nonlinearity following the critical stage is irrelevant because it has no effect on the observer’s performance. Other (non-critical) noise sources are also omitted because they are likely to have a negligible effect. It should be noted that the critical noise source may vary among different stimuli and tasks, and, therefore, the preceding nonlinearity would also vary. In particular, if the critical noise is external, such as quantal or added stimulus noise, there should be no nonlinear stage in the model.

 

Fig. 1. Block model of the detection process. The luminance profile is transduced to contrast information with a nonlinear output function f, after which noise is added with a nonlinear contrast response function v. The signal then passes to a decision stage that decides whether a stimulus event is present in the noisy signal.

 

The reader might notice that this specification does not incorporate a contrast gain control stage. This feature is omitted for application to the contrast discrimination task, where the mask has the identical spatial and temporal profile as the test increment. In this context, any contrast gain control can be treated as an intrinsic component of the signal transducer.

 

The Critical Noise in the Psychophysical Task

A general model for observer performance is shown in Fig. 2. The response evoked by a visual stimulus passes through a number of stages. Any -th stage is specified by its (possibly nonlinear) transducer function and injected noise . Each noise component in this scheme may be contrast-dependent (as in Fig. 1). External noise in this scheme derives from environmental sources such as quantal noise in the light signals, and precedes all the neural processing stages. The internal response arriving at the decision stage (i.e., after all transducer stages) accumulates the distortions from all previous stages:

, (1)

^{where is the stimulus contrast and is the total number of stages. In contrast discrimination tasks at high contrast levels, the noise standard deviation is typically small relative to the response. This property allows a first-order approximation, i.e., the first term of the Taylor expansion to be used.}

 

^{Fig. 2. The sequence of neural processing stages and noise sources relevant to a psychophysical task. The signal passes through of a number of the stages specified by its signal transducer function (c) and injected noise ., with its separate transducer v(.(c)). Each signal transducer can be nonlinear and each noise component in this scheme may have a separate nonlinear contrast-dependence. External noise in this scheme precedes all the processing stages. The decision stage receives a signal that combines all the transducers and noise sources. If one of the noise sources is predominant, other noise sources can be neglected and the transducers can be collapsed into one before an one after the critical noise stage. According to Birdsall’s Theorem, the second transducer does not alter the decision in 2AFC trial and can be removed, as in Fig. 1.}

 

After applying these linear approximations to Eq. 1, we arrive at the following approximation formula for the response :

, (2)

where is the contrast-dependent gain for the -th noise component. Assuming statistical independence between the noise sources, the standard deviation of the response at the decision stage is

(3)

The quadratic sum in the right-hand side of the Eq. 3 will tend to be dominated by the largest component. For example, if one noise source has a standard deviation only twice that of another, the ratio of their contributions is about 8.5:1. There, indeed, remains some possibility that many smaller noise sources could overwhelm the strongest one, but there is a high probability that the noise in the internal response at the decision stage is dominated across the full contrast range by only one noise source, which may be called the critical noise.

^{Fig. 3. If the transducer in the visual system were not monotonic, as shown by the curvy line, there should exist different contrasts ( and ), which produce the same responses and, therefore, indistinguishable. Such behavior is not typically reported.}

 

Given this assumption, the other (non-critical) noise sources may be neglected and all nonlinear stages can be collapsed into two (one before and the other after the critical noise source). An input signal, which is set by the stimulus contrast, passes through the first nonlinearity, after which noise is introduced into the signal, which then passes through the second nonlinear stage. It should be noted that the nonlinear stages implicitly incorporate spatial and temporal integration of the stimulus, which determine the values of the weight coefficients in the Eq. 2.

The nonlinear transducers in this model are assumed to be monotonic, that is, increase of input contrast should always lead to increase of the internal response at intermediate and decision stages. If this were not the case, stimuli of different contrasts could produce the same internal response and would be indiscriminable as shown in Fig. 3. Such a result is atypical for the psychophysical literature on contrast detection and discrimination, however.

 

 

An important property of monotonic transducers is that performance in a psychophysical experiment does not depend on the second transducer of the model depicted in Fig. 4). This notion, known as Birdsall’s Theorem (Lasley & Cohn, 1981; Pelli, 1991), can be readily justified. Consider a two-alternative forced choice (2AFC) trial where an observer has to compare two stimuli with contrasts and . The responses at the output of the first nonlinear transducer are two values determined by the stimuli contrasts: and . Then, after noise infusion, the responses become and , where and are two instances of the noise random variable . These noisy responses pass through the second nonlinear transducer , producing the output responses to be compared by the decision stage are given by the following expressions: and . The judgment made at the decision stage is based on the values of these responses: if the first response is greater than the second then the first stimulus will be selected as having higher contrast, and vice versa. Since transducer is monotonic, the order of its output values is identical to the order of its input values and . Therefore, if the decision stage were located right after the critical noise infusion and compared and , it would produce exactly the same judgment as one located after the second (monotonic) transducer. As a result, the second nonlinear transducer is transparent to a psychophysical experiment assessing probabilities of the correct responses and can be safely removed from the model, which leads to the final version shown in Fig. 1.

To summarize, the components of the model that may affect the outcome of a 2AFC experiment are the critical noise and the nonlinearity preceding the critical noise infusion. The second nonlinearity is transparent for the 2AFC task and can be ignored in this application (although it would affect the perceived contrast of the target).

This result has a corollary that may be employed for the analysis of the equivalent noise paradigm. When external (pattern) noise dominates over the internal (neural) noise, the critical stage shifts to the noise-infusion point , i.e., to the very beginning of the processing chain of Fig. 2. In this situation, the outcome of a trial in a contrast discrimination experiment becomes independent of the nonlinear properties of the transducer and of any noise in the visual system.

 

Measuring Contrast Transduction Throughout its Range.

The measurement of response nonlinearities will be described with specific reference to the domain of the contrast of sinusoidal stimuli. The range over which we need to specify the transduction behavior for contrast stimuli is all the way from the detection threshold for contrast to the maximum contrast of 1.0 (as for a sinusoid from zero to some maximum light value). Detection threshold may be as low as 0.001 under optimal conditions, implying a maximum of three log units of usable contrast range. This range may be addressed with the contrast discrimination paradigm of measuring the just-discriminable increment in contrast for base contrasts throughout the contrast range. For a noisy linear system (that is, the signal is directly proportional to stimulus contrast and the noise is independent of both contrast and internal signal strength), the just-discriminable increment would be independent of contrast throughout the range. Any contrast dependence of the contrast discrimination function reflects some nonlinearity in the signal transducer of its noise dependence. We develop a model for contrast discrimination thresholds in the framework of the 2AFC paradigm (Legge & Foley, 1980; Legge, 1981; Greenlee & Heitger, 1988; Ross & Speed, 1991; Wilson & Humanski, 1993; Foley, 1994). This derivation differs from the previous ones in considering the separate noise levels for the base alone and test+base components of the test trial.

When the input stimulus has contrast , the response at the critical stage is a random variable, which can be expressed as

, (4)

where is the transducer preceding the critical stage and is the noise at the critical stage. If, in a 2AFC trial, two stimuli with contrasts (reference stimulus) and (test stimulus) are compared, the probability for the test stimulus to appear stronger than the reference can be expressed as

(5)

or, substituting for in Eq. 5 from Eq. 4 and expressing the result in terms of the cumulative density function (CDF) of the difference between noise components,

(6)

where stands for the CDF of the random variable shown in the subscript.

As in previous analyses of the 2AFC paradigm (Green & Swets, 1966; Foley & Legge, 1981) the random variable is postulated to have a Gaussian distribution. There are two factors that make this assumption highly plausible. First, as was noted by Green & Swets (1966), if any (even very limited) pooling at the critical stage is present, the independent noise instances from different spatial/temporal loci are pooled together and tend to a normal distribution, as required by Central Limit Theorem (Bain & Engelhardt, 1987). Second, if the noise distribution is skewed, the subtraction operation in cancels the skewness of the components making the resultant distribution more symmetrical and, therefore, more similar to Gaussian. (When the distributions for and are identical, the difference distribution is inherently symmetrical.)

Incorporating the Gaussian assumption, Eq. 6 can be re-written in the following form:

, (7)

where stands for the Gaussian CDF with unity variance (which introduces the normalization term in the denominator) and is the standard deviation of noise . The argument of the CDF on the right of the Eq. 7 is conventionally expressed in terms of the discriminability parameter , which is traditionally defined as signal-to-noise ratio for the additive noise case (Tanner & Birdsall, 1958; Green & Swets, 1966). For arbitrary noise we define discriminability as

, (8)

(note that there are two variances under the square root sign, which necessitates the multiplier in the definition). For the experimental data given, discriminability can be computed as

, (9)

thus providing a link between model and experiment.

We are now ready to derive a formula for discrimination threshold. Let the discrimination threshold be defined by , which corresponds to , which, in turn, translates to the 76% correct level:

(10)

For arbitrary functions and there is no closed-form solution for . However, when the reference contrast exceeds a few detection thresholds, the experimental discrimination thresholds are typically much smaller than the reference contrast, and the first-order approximation of Eq. 8 can be used

. (11)

This approximation leads to a closed-form solution for the threshold value :

(12)

Thus, the discrimination threshold is proportional to the standard deviation of the noise distribution and is reciprocal to the first derivative (instantaneous gain) of the transducer at the reference contrast level. Note that this expression requires explicit knowledge of both the response and the noise, and cannot be derived solely from the response-to-noise ratio .

Note that the standard analysis of discrimination thresholds measured with the 2AFC paradigm (e.g., Foley & Legge, 1981) restricts the test-reference difference to non-negative values because the probability measure of correct responses adopted in this paradigm cannot be less than 0.5. The equations derived (Eqs. 5-9) avoid such a constraint because the probability measure varies across the full range from 0 to 1: for positive and for negative . The measured probability range, therefore, doubles that measured by the standard one for the 2AFC procedure, providing additional information about underlying processes.

 

Parametric Model

The major unknown of the model proposed is how the signal and noise are related to the stimulus contrast at high contrast levels. For simplicity, both are assumed to be power functions of the stimulus contrast (Stevens, 1957; Gottesman et al., 1981; Tolhurst et al., 1983; Geisler & Albrecht, 1997).

We are free to choose the response units such that

, (13)

with no gain-related multiplier in this formula. The noise power function is defined by a separate exponent relative to the response

, (14)

where sets the same units for the noise standard deviation as for the response. Recall that corresponds to additive noise and corresponds to Poisson noise. Given these parameterizations Eq. 10 can be re-written as

(15)

and Eq. 12 as

. (16)

Eq. 16 provides a stringent test for the feasibility of the power approximations postulated by Eqs. 13 and 14, requiring a power function relationship between discrimination thresholds and reference contrast. Such a relationship, indeed, was empirically discovered by Legge (1981) and later confirmed in numerous studies. The power function exponent, i.e., in terms of the model, varies from 0.2-0.3 (Wilson et al., 1983; Greenlee & Heitger, 1988; Wilson & Humanski, 1993) to 1 (Kulikowski & Gorea, 1978; Bradley & Ohzawa, 1986) and even higher (Greenlee & Heitger, 1988), with typical values clustering at around 0.5-0.7 (Nachmias & Sansbury, 1974; Legge & Foley, 1980; Legge, 1981; Wilson et al., 1983; Ross et al., 1993).

Eq. 16 reveals that discrimination thresholds impose two constraints on the three free parameters of the model, providing values for the response-to-noise ratio exponent and the fraction . The values of the individual parameters, however, cannot be obtained with the threshold data alone. To resolve the separate signal and noise transducers, therefore, we need to find an additional constraint to resolve the individual parameter values. This may be provided by a new view of the effects of added external noise.

 

Analysis of the Equivalent Noise Paradigm

In assessing the properties of the signal and noise processing in the brain, one approach that has been introduced is the equivalent noise paradigm, where a controlled amount of external noise is introduced to titrate the level at which it overcomes the noise inherent to the signal (Pelli, 1985, 1991). This equivalent noise paradigm is widely considered a powerful tool for assessing internal noise in the visual system. If this were correct, we could get a direct measure for the noise exponent and then derive the transducer exponent from the discrimination threshold exponent. To do so, however, the result has to be interpreted through the assumption that the noise is additive, or independent of stimulus strength. Unfortunately, the equivalent noise measure depends simultaneously on the noise in the system and the transducer nonlinearity (Legge, Kersten & Burgess, 1987). A nonlinear system with external additive noise will behave exactly the same way (within the accuracy of the first-order approximation) as a linear system with internal signal-dependent noise. If the noise cannot be assumed to be independent of the signal, the separate signal and noise functions cannot be disentangled (Kontsevich, Chen & Tyler, 1999).

 

An Additional Constraint For Model Parameters

So far, attempts to constrain signal and noise nonlinearities have been limited to the range where the difference between test and reference is much smaller than the reference () as in most contrast discrimination experiments. Observer performance for this range can be faithfully described by first-order approximations for the model expressed by Eqs. 11 and 12 The failure to resolve signal and noise within the linear range suggests that we should assess observer performance at larger differences between test and reference, where the model behavior becomes nonlinear. To test the feasibility of this approach, we evaluated the nonlinear behavior of the model by means of Eq. 8. The linear component of the model determined by Eq. 11 predicts a linear relationship presented between discriminability and contrast difference . Any deviation from the straight line, thus, represents a nonlinear component of the model which may produce the desired constraint.

Sample functions were computed with the formula of Eq. 8 for the psychometric function, measured for the contrast increment Æc in the contrast discrimination paradigm. Fig. 5 shows three kinds of the transducers: compressive (), linear () and expansive (), and for three kinds of the Gaussian noise: additive (), Poisson-like multiplicative () and linearly-multiplicative (). The noise gain was set at a realistic value of 0.25; the reference had a contrast as in the experiments to be described below. The plots in Fig. 5 clearly indicate that the shapes of the -vs.- curves depend measurably on the values of the parameters and , which may provide the missing constraint providing the values of the transducer nonlinearity. It is important to mention that the curve shapes are distinguishable within the range of between –3 and 3, which corresponds to the range from 0.017 to 0.983 for the experimentally measured probability . These probabilities are assessable in an experiment consisting of a few hundreds of trials per condition.

^{Fig. 5. Examples of possible contrast transducer functions, expressed in terms of the d' discriminability function of the contrast increment Æc in the contrast discrimination paradigm. These -vs.- curves were computed for a range of the transducer and noise power exponents and . These curves exhibit curvature whereas the first-order approximations analyzed so far predict straight lines. This curvature provides nonlinear constraint critical to resolving the parameter values from the data.}

 

One potential cause for discrepancy between the data and the model could be a deviation of the noise distribution from normality. In comparisons of the observer’s performance with Gaussian internal noise and Poisson noise, which is typical for neural spike generation processes, Monte-Carlo simulations showed that these distributions produce indistinguishable results (Kontsevich & Tyler, 1999).

To conclude the analysis, discrimination performance in the nonlinear regime (medium to large values) provides a new constraint onto the model parameters. This constraint can be separated from the effects caused by ‘finger errors’ and attention lapses. Deviation of the noise distribution from normal to Poisson has minor, if any, effect on the results.

 

Experimental Evidence

The stimuli for this experiment (Kontsevich & Tyler, 1999) were 2º grating patches of spatial frequency 3 cy/deg with phase randomized on each trial to reduce local luminance adaptation and eliminate local luminance cues. The experiments employed the constant stimulus method combined with a . On each trial of the 2AFC paradigm, the observer’s task was to choose the stimulus that had higher contrast by pressing one of two keys. Contrast discrimination performance was estimated for three reference contrasts: 0.15, 0.3 and 0.6. For the reference contrasts of 0.15 and 0.6, the test stimuli always had a higher contrast than the reference. For the 0.3 reference contrast, the test stimuli had both higher and lower contrasts to reveal an even nonlinear component in the psychometric function.

 

Fig. 6. Sample results of the probability transducers for three base contrasts under sustained stimulus presentation conditions. Functions were measured over the fill range of contrast increments for the center level of reference contrast, only for positive increments for the upper and lower levels of reference contrast. Lines represent model fits.

 

Sample results are shown in Fig. 6 on separate panels for each reference contrast. For both observers, the psychometric functions become shallower as reference contrast increases, with a corresponding threshold increase. The data were fit by the model of Eqs. 15 and 16 with 4 free parameters: response and noise exponents and , noise gain and finger error . Multi-parameter optimization procedure for these parameters minimized the error between the experimentally measured and predicted by the model probabilities of correct (Kontsevich & Tyler, 1999). The most remarkable result was that the power exponent for the response transducer varied from 2 to 2.7 for both observers and both stimuli. This high exponent value would not be surprising if it were estimated for near-detection contrast levels (Stromeyer & Klein, 1974; Foley & Legge, 1981; Legge, 1984). This experiment evaluated the response transducer at higher contrasts where it is widely believed to be saturating, which implies a transducer exponent of less than one (Wilson, 1980; Gottesman et al., 1981). At the same time, numerous studies suggest a stage in the visual processing pathway that computes contrast energy (e.g., Adelson & Bergen, 1985; Manahilov & Simpson, 1999; Thomas & Olzak, 1997; Watson & Solomon, 1999) across the whole range of contrasts. Current models of contrast gain control also assume the exponent of contrast transducer to be in the range between 2 (Heeger, 1992) and 2.4 (Foley, 1994). The full range data confirm the existence of such a stage and indicates that the observers’ performance (for the tasks studied) is limited by noise infused after the contrast energy is computed. There was no significant difference in either the transducer nonlinearity or the multiplicative noise gain between sustained and transient conditions (Kontsevich & Tyler, 1999).

Given a contrast response transducer that is approximately quadratic at high contrasts, parsimony suggests that this property could hold over the whole range of contrasts including the lowest. This suggestion is consistent with the data on the exponent of the d'-vs.-contrast relationship measured in detection tasks. Under theses conditions, any response-dependent (multiplicative) component noise is likely to be buried in the spontaneous neural activity, which constitutes a response-independent (additive) component, and the contrast response exponent would solely determine the -vs.-contrast relationship, which is known to be close to a value of two (Stromeyer & Klein, 1974; Foley & Legge, 1981; Legge, 1984). The notion of a square law for the response transducer also gains support from a binocular contrast summation study (Legge, 1984) and from modeling (Watson, 2000) of the detection thresholds for a representative set of stimuli (Carney et al., in press).

The estimate of the power exponent for the noise transducer fell in close to 0.8 across observers and the stimuli (Kontsevich & Tyler, 1999). This result implies that the noise critical for contrast discrimination is infused after the quantal (Poisson) noise, which has an exponent of 0.5 and which has been shown to be essential for the detection of flashes (Reeves et al., 1998). Such a high exponent also places the critical stage after the ganglion cell level, where the neural noise has been shown to be additive relative to contrast (Croner et al., 1993). The noise exponent, though, are in line with the experimental measurements of fluctuations in neural activity (Cohn et al., 1975; Tolhurst et al., 1983; Vogels et al., 1989; Snowden et al., 1992; Softky & Koch, 1993; Geisler & Albrecht, 1997), which place the multiplicative noise exponents in the range between 0.5 and 0.75. The psychophysical estimate of 0.8 is near the upper bound of this range. The implication is that the noise limiting the psychophysical tasks has cortical nature and its source is located after the simple cells because simple cells appear to have linear transducer in the studied range of contrasts (e.g., Kontsevich, 1995) whereas the psychophysically-defined transducer is quadratic.

The exponents for the -vs.-contrast functions with sustained stimulation (0.44 and 0.60) are close to those measured by Legge (1981) under similar conditions. Those for the transient stimuli (0.58 and 0.70), however, were discrepant from the unity slope obtained by Kulikowski & Gorea (1978) for similar conditions. This discrepancy may be attributed to the lack of the long adaptation period (2 min) preceding each trial, which they found necessary to achieve the steeper slope.

Fig. 7. Contrast discrimination at a contrast of 0.3 for two observers, showing the particular form expected from Fig. 5 with an accelerating signal transducer and root-multiplicative noise. The crosses represent the raw experimental data, the circles show the data corrected for the finger errors, whose values are given in Table 1. This correction, as evident from the graphs, in most cases has little or no effect. The curves represent fits of the optimized model. The fits should be compared with the circles since the model optimizes the finger error value.

 

To illustrate how the model optimization tightly constrained the parameter values, Fig. 7 plots the optimization fit (smooth curves) to the contrast increment data for pedestals =0.3. The data points follow a similar pattern to the model predictions for =2 and 0.5 in the rightmost panel of Fig. 5 and obviously deviate from the predictions for smaller values of .

 

Conclusions

The technique for functional isolation of the separate signal and noise transducer behaviors for suprathreshold stimuli provides consistent evidence of a strong multiplicative nonlinearity controlling the noise behavior. The signal transducer, rather than being estimated as compressive, is shown to exhibit an accelerating form. The accelerating power exponent of the signal transducer for these suprathreshold conditions is about 2, suggesting that the visual system accurately computes contrast energy at a certain stage of visual processing. It should be emphasized that attributing the near-threshold nonlinearity solely to the transducer nonlinearity, thus, leaving no room for channel uncertainty effects (Pelli, 1985; Tyler & Chen, 2000) in typical detection tasks.

The analysis shows that the predominant noise for the contrast discrimination task is then infused into the contrast energy signal by some contrast-dependent process with an exponent of about 0.8, not far from proportionality with the stimulus contrast. Without the accelerating signal transducer, the high noise exponent would render the observers almost incapable of any contrast discrimination performance (Tyler & Chen, 2000). For a quadratic contrast transducer, this exponent implies a dependence on the internal signal with an exponent of about 0.4, bringing the signal/noise transducer that has been measured in previous studies into an acceptable range.

 

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