Experimental statistical studies

With this theoretical background, the history of experimental studies of single neurons can be resumed. Among the earliest uses of signal detection theory in visual neurophysiology were FitzHugh's studies (1957, 1958) of the threshold responses of cat retinal ganglion cells. These and other studies showed how the maintained discharge of neurons constitutes noise limiting what can be detected (Barlow and Levick, 1969; Kuffler et al., 1957; Werner and Mountcastle, 1963). Good neurophysiological SNRs can be obtained for stimuli that are of the same order of magnitude as behavioral or psychophysical absolute thresholds (Barlow et al., 1971). The power of using the approach at several successive levels in the same visual pathway was shown by Laughlin (1973, 1974a, 1974b), who demonstrated the improvement that occurs as information from several insect photoreceptors is brought together on a single higher level neuron.

Note that noise may be represented in other than ways as a variable neural discharge rate. Intracellular biophysical variables often depend on rather small numbers of molecules, implying a high level of random variability which is certainly liable to interfere with the processing of sensory information. On the other hand, a high level of variability does not necessarily limit performance: this will occur only when the intrinsic variability is comparable to or greater than the variability arising from the extrinsic noise which enters with the sensory stimulus.

I learned this lesson in Kuffler's lab. Before going there, I had been struggling to establish that noise was important in human vision (Barlow, 1956, 1957), and I was expecting it to be similarly inconspicuous neurophysiologically. I could not believe my ears when I heard the dreadful irregular cacophony produced by a cat's retinal ganglion cell, and it took some time for Steve and Dick to convince me that this noisiness was not an artifact (Kuffler et al., 1957). The apparent paradox became clear when we measured the ganglion cell's sensitivity to light (Barlow et al., 1957), for like a good radar operator or ham radio enthusiast, the retina turns up the gain far enough to make the noise easily detectable but not so far that it swamps the signal. I strongly suspect that MT does a similar job with correspondence noise.

Using Both Neural and Behavioral Responses Mountcastle (1984) defined three stages of analysis of sensation, in the last of which single units were recorded in alert animals while doing sensory discriminations. The studies of this sort that I think show most promise for sorting out the roles of single units in vision are those on the detection of coherent motion by neurons in MT of the monkey, and I shall concentrate on the following five papers: those of Britten et al., (1992, 1993, 1996), Celebrini and Newsome (1994), and Shadlen et al. (1996). These are referred to collectively as "work from Newsome's lab."

Using alert, trained macaques, these authors recorded the responses of single neurons in MT while the monkey was performing a behavioral discrimination task and signaling its apparent result. The task chosen was to discriminate between two opposite directions of motion of a random dot kinematogram in which the proportion of dots moving coherently in one direction could be varied. The recordings from the isolated neurons gave the distributions in numbers of impulses for different coherence levels for movements in both preferred and null directions, while the behavioral responses gave the proportions with which the monkey correctly identified the direction of motion in their forced-choice task.

The authors plotted the behavioral results as the percentage correct at the varying coherence levels and found that, like humans, the monkeys could perform this task very well, achieving 82% correct responses at coherences of 10% or less. The criterion of 82% they chose for threshold corresponds to a ¿-prime value of 1.13 and can be obtained directly from the percentage correct in a two-alternative forced-choice task (e.g., from Elliott's tables in Swets, 1964).

To obtain comparable curves for the neural responses, they used the following method: they assumed that there was another neuron, which they called the anti-neuron, with exactly the same properties as the one they were recording from but having a reversed directional preference; they then took the number of impulses in each trial, and said that it signaled the preferred direction if it was a number more likely to have been generated by preferred direction trials at that particular coherence level and null if it was more likely to have been generated by null direction trials.

There are two slight problems with this way of making unit responses comparable with psychometric responses. First, there is little if any evidence that the monkey brain actually uses an anti-neuron in making its decisions, and it is a pity for such an uncertain feature to be so deeply embedded in the comparisons and models made. Second, the authors' neurometric curve shows the performance expected of two neurons, the hypothetical anti-neuron as well as the one they actually recorded from, and this is, of course, better than that expected from a single neuron, provided that, as they assume, the neurons give independent information about the signal.

There seems no good reason to avoid expressing a single neuron's performance directly as an SNR in the way an engineer would naturally express measurements of a signal variable, that is, as the difference between the mean responses at a given coherence and at zero coherence, divided by the standard deviation of the response at zero coherence. This is merely a step in thinking clearly about what can and cannot be done with signals such as those that have been recorded, and it does not necessarily entail the assumption that the brain actually uses the expected response to zero coherence in making its decisions. The unit's SNR at the coherence required for an 82% correct behavioral threshold can be compared directly with 1.13, the value of ¿-prime for their psychometric measure at 82% correct.

It is worth pointing out that the two-alternative forced-choice task the monkeys performed is a method devised by psychophysicists to determine the ratio of signal response on the decision variable to the noisiness of that repre-sentation—that is, its standard deviation. The neuron/ anti-neuron artifice seems to be a way of doing this in reverse, going from recorded signal responses back to expected psychophysical-like performance, but why do this? The neurophysiologist has, or hopes he is going to have, access to the signals from which decision variables are constructed, so why go backward? Why use psychophysical performance as the currency for comparisons?

The main result of their study was to show an astonishing agreement between the psychometric curves obtained behaviorally and those obtained from single units using the anti-neuron method. It is true that there was a large range in sensitivities, the ratio of neurometric to psychometric threshold varying over a 10-fold range, with many neuro-metric sensitivities exceeding the psychometric sensitivity, but the mean ratio was close to 1. Furthermore, the agreement extended to the slopes of the curves as well as to their means. Note, however, that these exact agreements depend in part on the assumption that the decision is based on the use of all of the 2000-msec responses collected from the neuron; this is implausible, for the monkey's life must often depend on much more rapid responses to moving stimuli. Departing from this assumption is likely to spoil the exactness of the agreement, but the change would probably not be large enough to alter the main conclusion— that MT neurons can signal the direction of coherent motion in a random dot kinematogram with a sensitivity comparable to that attainable by the whole animal. This is in agreement with a good body of other work showing that single neurons are very sensitive, and I personally think it is about as far as their results take us at present. But there is hope of further progress from measurements of choice probabilities (called the sender operating characteristics in Celebrini and Newsome, 1994, and the predictive index in Shadlen and Newsome, 2001).

Choice probability is a measure of the way the monkey's behavioral responses covary with the neural responses of a recorded neuron, and it is related to the area under the receiver operating characteristic (ROC) curve in signal detection theory. It represents the probability of determining the behavioral response correctly from an optimal analysis of a particular single unit's responses. If the neural response fully determined the behavioral response, choice probability would have a value of 1, whereas if there was only a chance relationship, it would have a value of 0.5. In fact, the authors found that the average value for all neurons analyzed was significantly above 0.5, though not by very much. A few neurons seem to have had considerably higher values, but they imply that these were untrustworthy chance results. Higher average values have been reported from other experiments on MT (Dodd et al., 2001), in which many neurons seemed to have values above 0.8, and Shadlen and Newsome (2001) and Horwitz and Newsome (2001a, 2001b) have also obtained much higher values in parietal cortex and superior colliculus, where they actually used a much shorter analysis period.

From the point of view of SNR analysis, the choice probability is a measure of the amount of noise added between unit response and behavioral choice: high choice probabilities near 1 are evidence of little added noise, whereas low probabilities near 0.5 are evidence of much added noise.

The trouble is that a high choice probability does not prove that a neuron did actually determine the behavioral response; it only proves that it could have done so, and the possibility that other neurons actually did so is not excluded. Similarly, a low choice probability near 0.5 does not prove that the neuron played no role at all in the behavioral response, for it may have stepped in and caused a correct response on rare occasions when no other neuron was able to do so. What a low choice probability does prove is that that particular neuron, by itself, could not have been responsible for all the correct decisions made behaviorally, and this remains true even if the neuron's SNR was as high as or higher than that given by the psychometric results. It is an important additional measure that has to be taken into account in figuring out how single units control behavior.

In the final paper of the series, Newsome's group tried to model how the brain derives the decision variable underlying the behavioral responses from a set of neural responses and choice probabilities such as those they recorded, but although I admire the attempt, I feel it is not the final solution for the following five reasons:

1. It fails to take into account the fact that, if the conclusion reached by Barlow and Tripathy (1997) is accepted, correspondence noise, which is extrinsic, is a major factor limiting the performance of the MT system. Correspondence noise arises from the correspondence problem—that is, from the lack of information about which dots are to be considered as pairs in successive frames of a random dot kine-matogram. If we know how such stimuli have been generated, its magnitude can be calculated (Barlow and Tripathy, 1997). Human coherence thresholds behave as if they were limited by it over wide ranges of variation of the principal parameters of the target stimulus, and under optimum conditions, statistical efficiencies for detecting coherent motion are quite high. Newsome's group does recognize the problem of correlations between responses of MT neurons, but this measure does not seem to differentiate between correlations due to the signal being shared and correlations due to the noise being shared, yet these have very different implications.

If correspondence noise is the major natural problem in detecting and analyzing motion signals, one is unlikely to understand how the visual analysis of motion is organized without taking it into account. It is also a pity not to exploit the possibility of estimating SNRs on the stimulus as well as on the responses, for this enables one to follow the losses of efficiency directly. It is these losses of efficiency that are of most practical importance to the monkey, since they have the greatest effect on the utility and survival value of the system for detecting coherent visual motion.

2. One can expect to be able to derive the behavioral decision variable from the neural responses only if the measure of these responses that is used for the analysis actually corresponds to the measure that the brain uses. The only measure of the neural responses Newsome's group used was the total number of impulses over 2000 msec, but it seems unlikely that an animal that spends its life jumping around in the treetops uses only 2000-msec totals for controlling its responses to motion signals that might save its life if reacted to quickly. Bair and Koch (1996) have shown that the temporal frequency response of the motion detection system peaks at around 3 Hz, so perhaps the monkey makes a decision every 300 msec, and, if forced to wait before responding, bases its delayed response on some kind of probability summation among the stored results of these previous decisions, or perhaps simply uses the most recent, up-to-date one. The decrease in efficiency of psychophysical performance for durations above about 400 msec (Barlow and Tripathy, 1997), together with the margin by which the performance of many neurons exceed their behavioral performance, suggests that it would be hard to exclude such a possibility.

3. The neuronal data the authors used to model the formation of the final decision were collected for stimuli that were only approximately optimal for each of the individual neurons. No signal can be optimal for all the different neurons, even those in a single column, but the extent of the loss of performance from mismatching is obviously hard to estimate. Note particularly that the importance of disparity selectivity was not fully recognized when the data were collected, so this parameter was not optimized.

4. I think that at this stage of our understanding of the problem, we should use statistical arguments to derive limits on what is possible, and it is premature to attempt a direct model. The three rules stated earlier make it fairly easy to derive such limits. Note that if the statistical efficiency of the early stages is as high as some estimates indicate (Mogi and Barlow, 1999), the limits become more stringent and are more informative about which models are viable and which can be excluded.

5. At present, we have very little understanding of the way the brain does its statistical computations, important though these are for all decisions. Advancing knowledge of the cell biology of cortical neurons suggests that they may be able to do computations we have not hitherto suspected, as described in the final section of this chapter. It is not clear how this will affect the problem, but it seems unwise to pin one's faith (or one's model) on the assumption that they can do little more than compute weighted sums of impulse numbers, even though this is a crucial step in collecting the evidence needed for good decision making.

It's evident that the last two objections are swayed by prejudice and stylistic preference. Further analysis (Gold and Shadlen, 2000, 2001) shows interesting convergence with independent psychophysical measurements (Carpenter and Williams, 1995; Reddi and Carpenter, 2000), and may well prove these and my other doubts to be unfounded. Whatever happens in the future, Newsome's group has already shown that statistical analysis of single-unit responses greatly improves our understanding of behavioral decisions; including the role of extrinsic noise in this analysis will surely bring further insight into this crucial aspect of perceptual decision making.

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