What is quantum efficiency? Following Rose's (1946) pioneering work, Horace Barlow provided a clear answer to this question a quarter of a century ago (Barlow, 1962) by defining quantum efficiency as:
Barlow stressed the importance of adhering to a strict definition of what is meant by the overall quantum efficiency of visual performance (Barlow, 1977). The emphasis is appropriate because his equation is more than a mere definition, it is a way of thinking about how real visual detectors behave:
Now imagine a human subject and an ideal device performing the same task: with no filter in front of it, the ideal device will of course perform better, but by interposing the appropriate filter its performance can be reduced until it matches that achieved by the subject. The fraction of light transmitted by this filter is then equal to the overall quantum efficiency, F, as defined above.(Barlow, 1962; pp. 155-6).
Barlow's idea was a major step forward for at least three reasons. First, it emphasized the importance of identifying the task. Second, it suggested the use of an absolute standard of comparison, the ideal detector, for assessing the performance of real visual systems. Third, it offered a specific method for making that comparison: handicap the ideal detector until its performance falls to the level of the inferior, real device. In so doing, one obtains a conceptual model of how the real system performs. A real detector acts like an ideal detector looking through a filter that passes only a fraction F of the incoming photons.
Beyond the specification of an absolute figure of merit for real visual detectors, Barlow emphasized that the concept of quantum efficiency is chiefly of interest in leading one to factors other than quantum fluctuations that limit performance. In this spirit, we propose to pursue the foregoing ideas by asking: how do we know that it will always be possible to match the performance of a real detector just by placing a neutral filter in front of the hypothetical ideal device? And if it is not possible, what have we learned about those other factors Barlow foresaw?
To tackle these questions requires that we be more specific about what is meant by 'performance'. Consider the following detection experiment. A steady light source emits P photons per unit time and the signal to be detected is a change of mP photons per unit time, where m is a modulation parameter within the range -1 to +1. On some trials just the steady light is present and on other trials the signal occurs. These two conditions are randomly presented with equal probability and the observer's task is to decide after each trial whether or not the stimulus condition had been presented. The observer's performance in this case is given by two numbers:
Y=proportion of trials on which the subject correctly said the signal was present,
X=proportion of trials on which the subject incorrectly said the signal was present.
A simplifying assumption is sometimes made that X, the rate of false-positive responses, is fixed in order that performance may be specified solely by Y, the frequency-of-seeing. In general, however, both X and Y will vary with the detector's decision criterion and so both must be included in a specification of performance. In the context of signal detection theory (Green & Swets, 1966; Egan, 1975), the covariation of Y with X is known as a receiver operating characteristic (ROC) and it is this curve which embodies the various modes in which a given performance is manifested. Assessment of ROC curves is aided by two fiducial lines: the positive diagonal, which indicates chance performance, and the negative diagonal, which is the axis for displaying a special measure of signal detectability called d'.
The design of an ideal detector for an incremental (m>0) or for a decremental (m<0) change of stimulus luminance is dictated by the statistical nature of light. For low levels of illumination, the arrival of photons may be described by the Poisson random process (Mandel & Wolf, 1965). It is known that the optimal method for detecting an increase in the rate of Poisson events is to count the events (Cox & Lewis, 1966) and decide in favor of the hypothesis of 'stimulus presented' if the number exceeds some criterion. Alternatively, to detect optimally a decrease in mean rate, one should say the stimulus occurred if the number of events is less than some criterion. These decision rules enable the ROC curves for an ideal detector to be determined theoretically from the probability distribution functions for Poisson random variables. It so happens that ideal photon detectors have ROC curves which are very nearly linear when plotted on Gaussian probability cordinates (Thibos, Levick & Cohn, 1979). This simplifies matters considerably because the entire performance curve of the ideal device can be summarized by just two numbers, slope and intercept of the ROC curve with the d' axis. The following approximate formulae for slope and d' in terms of known stimulus parameters P and m were derived:
 |
(1) |
 |
(2) |
Note that these equations reveal two Poisson signatures of quantal fluctuations. First, d' for a decrement stimulus is always greater than d' for an increment stimulus of the same magnitude (|m|). Second, ROC slope is greater than unity for decrements and less than unity for increments.
To be a useful model for real visual neurons, the ideal detector must be handicapped in some way so that it performs at the same level. A general way of thinking of this handicap is to envision the stimulus photons traversing a hypothetical 'black box' en route to the ideal detector. Inside the box is some mechanism which hampers the detector by changing the mean rate or modulation of the photon stream. Later on we will be more specific about what might be inside the black box, but for now we need make only one key restriction: that the black box does not alter the essential Poisson nature of quantal fluctuations. Accordingly, we will be considering as a class all those handicap mechanisms for which the stream of output particles emerging from the box remains a Poisson random process. To remind us of this key assumption, we will refer to this kind of handicap as a 'Poisson' box.
Because the output of a Poisson box is a Poisson random process, a simple counter remains the ideal detector of the photic input and its ROC will be a straight line. If P' is the mean rate of events emerging from the Poisson box, and m' is the effective modulation of the output stream caused by the stimulus, then by equations 1, 2, above, the ROC will have intercept and slope:
 |
(3) |
 |
(4) |
Barlow's proposition was that the black box contains a filter. We conceive of an optical filter as a device which randomly deletes photons and it is known that random deletion of events of a Poisson process preserves the Poisson nature of the process (Parzen, 1962). Therefore, Barlow's handicap qualifies as a Poisson box. Since the filter will reduce the steady light P and the signal mP by the same fraction F, signal modulation is not affected by a filter. Accordingly, the parameters of the ROC curve will be given by equations 3, 4 when we make the substitutions:
| P'=FP | (5) |
| m'=m | (6) |
Note that to close approximation, ROC slope for Poisson signals is independent of the mean rate of the steady light and depends only on the amount of signal modulation. Therefore, the handicap imposed by a filter is manifest graphically by a parallel shifting of the ROC curve towards the chance line in accordance with the reduced d'. An example of this behavior is shown in Fig. 8.1A for incremental stimuli and Fig. 8.1B for decremental stimuli of fixed modulation (|m|=67%) In the next section we will compare these theoretical curves against experimental ROCs obtained from retinal neurons.
In a neurophysiological experiment, detection performance depends not only on the visual neuron being recorded, but also on the fidelity of equipment used to monitor the cell, the measure of neural response extracted by the experimenter and the experimenter's strategy for deciding whether or not the stimulus occurred based solely on the observed response. Since retinal ganglion cells signal visual information to the brain by all-or-none action potentials, the fidelity requirement is met by accurately measuring the time of occurrence of each neural event. What is not so obvious is the best measure of response, which may be different for different types of ganglion cells. The time course of response is perhaps simplest in sustained ganglion cells, which respond to changes in stimulus luminance by a more or less sustained change in their rate of discharge. A natural measure of response for this cell type is the number of nerve impulses occurring during the interval over which the discharge is displaced from that due to the steady light. The decision strategy for an on-type cell (one which increases its discharge rate in response to an incremental stimulus) would then be to say the stimulus occurred if the number of nerve impulses counted is greater than some criterion. Alternatively, if the stimulus to be detected is a decrement, the experimenter should say the stimulus occurred if the number of nerve impulses counted is less than some criterion. If the discharge pattern of a ganglion cell were a Poisson random process, then the foregoing strategy would be the best one. However, this is not a good statistical model for cat retinal ganglion cells (Barlow & Levick, 1969; Barlow, Levick & Yoon, 1971) so it is possible that the above experimental method imposes an additional handicap upon the preparation of which we are not aware.
Many factors can act to reduce ganglion cell performance to a level below the theoretical maximum. The best performance occurs when these biological and experimental factors are minimized and the cell is limited mainly by the quantal fluctuations in the stimulus itself. Such conditions are difficult to achieve experimentally and cells which show evidence of being limited by quantal fluctuations are relatively rare. One such cell provided the data of Fig. 8.1, taken from the study of Levick, Thibos, Cohn, Catanzaro & Barlow (1983). This was a dark adapted, retinal ganglion cell of the on-center, brisk-sustained class. The stimulus configuration was a small spot of light contained within the center component of the receptive field. The spot was on continuously at a level dim enough to avoid light adapting the cell (P=88 photons/0.1 s) and the stimulus to be detected was a brief (0.1 s) change (|m|=0.67) of the steady level which occurred once every 0.6 s. The cell's ROC curves for the incremental stimulus (Fig. 8.1A) and for the decremental stimulus (Fig. 8.1B) both show the Poisson signature expected when quantal fluctuations are a major factor limiting performance. First, these ROC curves were closely fitted by straight lines when plotted on Gaussian coordinates. Second, least-squares regression of the data indicated that the detectability (d'=1.25) of the increment stimulus was less than the detectability (d'=1.59) of the decrement stimulus of the same modulation magnitude. Third, ROC slope (0.97) for the increment was less than unity and slope (1.19) for the decrement was greater than unity.
Fig. 8.1. ROC curves for a retinal ganglion cell (symbols) compared to those of an ideal detector handicapped by an attenuating filter (lines). Stimulus was a small spot of light (0.21 deg) centered on the receptive field of a cell (G-9-8) in the dark-adapted cat retina. The signal to be detected was a brief (0.1 s), 67% increment (A) or 67%, decrement (B) of a steady light with mean luminance 87.5 photon (507nm) per stimulus duration (600 trials). Inset shows model of retinal ganglion cell as a Poisson box. Inside the box is a filter of transmission factor F. ROC curves for the model are a family of parallel lines, corresponding to various values of filter factor F, as indicated by number near each line.
We are now in a position to consider the question posed in the introduction: can the ROC curves for a retinal ganglion cell be matched by the ROC curves for an ideal detector handicapped by a filter? The cell of Fig. 8.1 is a good candidate for a match since it shows signs of quantum limited behavior just as the ideal detector does. Comparison of the ganglion cell data with the family of theoretical curves for an increment stimulus (Fig. 8.1A) indicates a reasonable fit for the curve corresponding to the filter value F=5.0%, although the slope is not quite right. The more troublesome result, however, is that the data for the decrement stimulus are not well matched by the corresponding theoretical curve (Fig. 8. lB) but instead are fitted best by a curve for the lower value of F=3.7%. This discrepancy is too large to be ignored.
The difficulties encountered in matching the ROC data for this ganglion cell are indicative of a fundamental limitation of the theoretical model. In principle, the slope of an empirically obtained ROC may not be the same as that of an ideal detector with various levels of neutral filter handicap. In such a case, an empirical ROC is free to cut across the parallel family of theoretical ROC curves so that each point of the empirical ROC corresponds to a different value of F. Consequently, no single choice of filter can produce a match between the full ROC curves of real and filter-handicapped, ideal detectors. The problem becomes acute when the stimulus is a decrement of 100% modulation. If the steady light provides even a few photons per trial, an ideal detector can perform without error by always saying the signal is present when zero photons are caught and saying there is no signal when it catches one or more photons. In this case, signal detectability is indefinitely large, the ROC curve is indeterminate, and introducing a filter poses no handicap. For real ganglion cells, however, removing all of the light does not necessarily remove all of the variability of the cell's discharge and so the cell cannot attain perfect performance. Thus, the model of an ideal detector handicapped by a filter is certainly inadequate, but through this failure comes the kind of success forecast by Barlow since it shows that real detectors are limited by more than an inability to catch every quantum.
Since an attenuating filter by itself is insufficient to mimic real detectors, it becomes necessary to model the equivalently handicapped, ideal detector in another way. Barlow solved a similar problem in the analysis of frequency-of-seeing curves for human subjects by suggesting that there exists a source of biological noise which produces neural events indistinguishable from the natural response to light (Barlow, 1956). Since these events occur even in the dark, they have been called 'dark light events' or 'eigengrau'. Unfortunately, the first phrase carries with it an apparent contradiction of terms which can be unsettling for the student, and the second does not convey the essential idea of discrete events. Therefore, we propose to coin a new word to stand for those hypothetical events which are confusable with the absorption of photons but which are due to internal, biological causes and may therefore occur even in complete darkness. As they are complementary to photons, the elementary particles of light, let these particles of darkness be known as 'scotons', from the Greek root Scotos(skotos) meaning darkness.
For present purposes we make the simplifying assumption that these hypothetical scotons have the statistical properties of a Poisson random process. Since the superposition of two independent trains of Poisson events is another Poisson train (Cox & Lewis, 1966), a black box containing an additive source of Poisson noise of mean rate S scotons/unit fime qualifies as a Poisson box. Since photons and scotons are assumed to be physiologically indistinguishable, the effective rate of background events is
| P'=P+S | (7) |
| m'=mP/(P+S) | (8) |
Applying these results to the general expressions for d' and slope given by equations 3 and 4, we see that the weakness of the filter model has been averted by the use of the noise model. By increasing S, ROC slope approaches unity and d' approaches 0 as required. Thus it is conceivable that a particular value of S could produce a match of both slope and intercept of ROCs for real and ideal detectors. Further, the noise model avoids the critical weakness of the original filter model because any positive value of S will keep d' finite even for 100% decrements.
A test of the noise model is presented in Fig. 8.2A, which illustrates the performance of another retinal ganglion cell for the detection of 100% decrements of a steady light emitting on average P=61 photons/unit time. The ROC curve has slope 1.20 and crosses the d' axis at 2.01. To reduce d' of the ideal detector to that of the ganglion cell requires S=860 scotons/unit time, which implies an effective modulation of -6.6%. But the slope of the theoretical ROC curve (solid line) is then 1.01, which is too far from the experimental value to be explained on the basis of experimental variability. A second test of the model also suggests the noise model is inadequate. Since the model assumes that the internal noise is independent of the light stimulus, the amount of noise present should be the same regardless of whether the stimulus is an increment or a decrement. Therefore, by calculating the amount of noise necessary to account for decrement performance, it should be possible to predict ganglion cell performance for an increment. As shown in Fig. 8.2A, however, the prediction (dashed line) clearly fails to match the ganglion cell data (open squares).
The reasons the noise model fails can be understood as follows. In order to reduce the infinite detectability of the 100% decrement stimulus, the ideal detector must be handicapped with a source of noise events which has a mean rate much higher than the steady level of the light stimulus. Consequently, the effective modulation provided by the stimulus is very low. When the modulation is low, increments and decrements are about equally detectable. Thus the two theoretical ROC curves in Fig. 8.2 are nearly identical. For the retinal ganglion cell, however, the Poisson signature of the light source is clearly evident in the ROC curves as signal detectability is significantly higher for the decrement stimulus than for an increment of the same magnitude. This is an indication that the model has the wrong proportions of photons and scotons. Evidently the quantal fluctuations of the stimulus are of much greater importance for the ganglion cell than for the model and this is why the model fails.
From the preceding it is clear that our hypothetical Poisson box must contain more than just either a filter or a source of noise. The obvious next step is to incorporate both components into the model, which is equivalent to an arbitrary sequence of noise and filter stages. It is convenient, but not essential, to suppose the filter acts after the superposition of photons and scotons. The only effect of this order is to measure scotons in the same units as the input photons. Let the transmission factor of the filter be U, which stands for the fraction of quantal events, both photons and scotons, utilized by the ideal detector. The effective, steady rate of events seen by the ideal detector is
| P'=U(P+S) | (9) |
| m' =mP/(P+S) | (10) |
In principle, the two parameters d' and slope are sufficient to determine the two unknowns of the model, U and S. A straightforward approach would be to estimate slope and d' of an empirical ROC by least-squares regression and then use equations 3 and 4, in conjunction with 9 and 10, to determine the model's parameters. This method guarantees that an empirical ROC will be well described by the model since the ROC of the model is in fact the best fitting straight line to the data. To test the model, it could be used to predict the results of other experiments, say with different modulation strengths or different intensity levels.
Fig. 8.2. ROC curves for 100% increment (open symbols and dashed lines) and decrement (closed symbols and solid lines) of a spot of light (0.43 deg) supplying 61 photons (507 nm) per stimulus duration (0.5 s). Cell H-l-8, 1000 trials. Theoretical curves in (A) are for the noise-only model (see inset), where the ideal detector is handicapped by an added noise source of rate S = 860 scotons per stimulus duration. Curves in (B) are for noise + filter model (see inset), where the ideal detector is handicapped by a noise source of rate S = 82 scotons per stimulus duration and by a filter which passes U = 12% of photons and scotons.
In practice, the above method of parameter estimation is often unsatisfactory
because it yields results which have a large statistical uncertainty. Consider,
for instance, the ROC curve for an incremental stimulus. According to equation
2, the range of possible slope values is 1.0 to 0.89, which corresponds
to the modulation range of 0 to 100%. Suppose the empirical ROC slope is B
and we ask the question, what is the 95% confidence interval for this value?
To answer this question, we need to know the statistical properties of ROC slope
for Poisson signals. These are not known theoretically, but computer simulation
for typical stimulus parameters has indicated that ROC slope has a Gaussian
distribution with standard deviation given approximately by SD= 
(3/N),
where N is the number of stimulus trials used to generate the ROC curve
(Thibos et al., 1979). Under the best of conditions,
N=1000 is about the most that could be expected of a physiological
experiment so the 95% confidence interval for slope of an increment ROC would
be B±SD*t(0.05,k-2) where t(0.05,k-2) is Student's
t-statistic for k-2 degrees of freedom and k is the
number of ROC points used to determine slope. The smallest confidence interval
occurs when k is large, and under this assumption the interval is B-0.1
to B+0.1. Unfortunately, even this minimum range of uncertainty for
B completely overlaps the full range of possible slope values expected
for a detector of Poisson signals. In other words, it appears that we will learn
nothing about the parameters of the model from measurements of ROC slope for
an increment. The situation is not quite so hopeless for a decrement stimulus,
as the permissible range of slopes is somewhat larger. Nevertheless, these practical
problems suggest an alternative approach should be developed.
A better method for estimating the parameters of the model is based on detectability measurements for an increment (d'+) and a decrement (d'_ ) stimulus of the same magnitude m. Applying equation 3, the ratio r=d'_ / d'+ yields an estimate of the effective modulation magnitude m':
| m'=(r4-1)/(r4+1) | (11) |
| S=P(m/m'-1) | (12) |
| U=d'+ d'_ (1-m' 2)1/4/Pmm' | (13) |
Applying these equations to the data of retinal ganglion cell of Fig. 8.2A indicates that this cell performs the same as the ideal detector model with U=12% and S=82. The theoretical ROC curves for the model, shown by the lines in Fig. 8.2B, appear to fit the data reasonably well. More searching tests of the model would involve predictipn of results for other stimulus conditions based on the model parameters determined from the initial experiment. This requires long periods of stable physiological behavior which has been achieved for only a small number of cells. Nevertheless, on those occasions it was possible to verify that a model with fixed parameters could adequately describe the performance of retinal ganglion cells over the full range of modulations at a fixed background level and over about 1 log unit range of backgrounds above absolute threshold (Levick et al., 1983).
In summary, the performance of the brisk-sustained type of cat retinal ganglion cell can be adequately matched by a theoretical model consisting of an ideal detector handicapped by a source of added noise and an attenuating filter. Simpler models consisting of either one or other of these handicaps can be ruled out, since neither is sufficient to account for ganglion cell performance over a range of stimulus conditions. The proposed model seems irreducible for other reasons as well. It is certain that ganglion cells cannot utilize all incident quanta since some are undoubtedly reflected, scattered and absorbed by the ocular media. Therefore, some kind of attenuating filter seems essential to the model. Similarly, the presence of retinal noise cannot be denied since retinal ganglion cells typically have a variable maintained discharge in the dark. This noise can only come from intrinsic, biological sources. The challenge for the future will be to determine how useful this simple model will be in describing the performance of other cell types in the dark-adapted condition and in accounting for the changes of performance brought on by light adaptation.
Two problems have been identified with Barlow's original formulation of quantum efficiency. First, in general the ROC curve for real detectors of light may cut across the family of theoretical ROC curves for an ideal detector, each curve corresponding to a different value of quantum efficiency, so that no single value of F will produce a match between empirical and theoretical curves. Second, the quantum efficiency cannot be determined when the stimulus is a decrement of 100% modulation. Although there is no satisfactory resolution of the latter problem, it would be possible to resolve the former by adopting a convention which specifies which performance point on the experimental ROC curve is to be matched by the theoretical model. A natural choice is to require the theoretical curve to intersect the empirical ROC at the negative diagonal. In other words, the suggested convention is to say that the real and ideal detectors have equal performance when they have equal d'. For the model, d' is known from equations 3, 5 and 6:
| d'= |m|(FP)1/2(1+m)1/4 | (14) |
Setting d' of this equation to the experimentally determined value for the real detector, and solving for F, gives a simple formula for the real detector in terms of its own performance and the given signal parameters:
| F=d' 2(1+m)1/2/Pm2 | (15) |
A general expression for the efficiency of an arbitrary Poisson box may be obtained by equating d' for an ideal detector handicapped by a Poisson box (equation 3) with d' for an ideal detector handicapped by a filter (equation 14) and solving for F. This gives the desired result in terms of the signal parameters before and after the box:
|
F=(P'/P)(m'/m)2[(1+m)/(1+m')]1/2 |
(16) |
Explicit formulae for special cases of interest follow immediately from this general solution. When the box contains a filter of transmission U, then m=m' and P= UP, so quantum efficiency is just the filter's transmission factor:
| Ffilter=U | (17) |
Unlike the case of a filter handicap, when the box contains a source of added noise, quantum efficiency is not fixed but depends upon stimulus parameters P and m. This is shown by substituting the expressions of equations 7 and 8 into equation 16:
| Fnoise=(P/P+S)[(1+m)/(1+mP/(P+S))]1/2 | (18) |
Finally, when the box contains both a filter and a source of added noise, quantum efficiency is found from equations 9, 10 and 16 to be:
|
Fnoise+filter=(UP/P+S)[(1 + m)/( 1 + mP/(P+ S))]1/2 |
(19) |
The product rule for a sequence of inefficient stages in the visual system was one of the attractive features of Barlow's original formulation of quantum efficiency. By this rule, the overall quantum efficiency of series of concatenated filters is the product of their individual efficiencies. Given the above development, it is now possible to verify the more general product rule for the quantum efficiency of a series of Poisson boxes. According to the convention suggested above, each box in the sequence will have the same effect on d' as does some filter, the value of which is given by equation 19, and so its efficiency is equal to the filter's transmission factor. By extension, a series of Poisson boxes will have the same overall effect on d' as a series of filters. But a series of filters is equivalent to a single filter which reduces d' by the same amount. The transmission of this single filter is, by definition, equal to the overall quantum efficiency of the series of Poisson boxes. It is also, by computation, equal to the product of the transmission factors of the individual filters and thus the efficiencies of the individual Poisson boxes. It should be kept in mind that in the general case the efficiency of a Poisson box depends upon signal parameters at its own input. Therefore, the order of the boxes is important and changing the order will in general change the overall quantum efficiency.
The foregoing analysis provides a tool for assessing the relative importance of the noise component and filter component of the Poisson box used by Levick et al. (1983) to model the behaviour of cat refinal ganglion cells. Factoring out the two components is a specific application of product rule, for it is evident from equations 17, 18 and 19 that:
| Fnoise+filter=Ffilter * Fnoise | (20) |
The quantum efficiency of eight retinal ganglion cells studied by Levick et al. (1983) ranged from 2% to 11% and averaged 7%. ROC curves for these cells were matched by ROC curves for the filter+noise model presented above and the parameters of the model were presented in their Table 1. The range of quantum efficiencies for the filter component for the eight cells was 3% to 29% with an average of 14%. The range of efficiencies for the noise component was 35% to 81% and the mean was 55%. This comparison reveals that the inability of the ganglion cell to utilize all of the incident quanta of light had a much greater effect on the cell's overall quantum efficiency than did the internal source of biological noise.
