The limits of counting accuracy in distributed neural representations.

Learning about a causal or statistical association depends on comparing frequencies of joint occurrence with frequencies expected from separate occurrences, and to do this, events must somehow be counted. Physiological mechanisms can easily generate the necessary measures if there is a direct, one-to-one relationship between significant events and neural activity, but if the events are represented across cell populations in a distributed manner, the counting of one event will be interfered with by the occurrence of others. Although the mean interference can be allowed for, there is inevitably an increase in the variance of frequency estimates that results in the need for extra data to achieve reliable learning. This lowering of statistical efficiency (Fisher, 1925) is calculated as the ratio of the minimum to actual variance of the estimates. We define two neural models, based on presynaptic and Hebbian synaptic modification, and explore the effects of sparse coding and the relative frequencies of events on the efficiency of frequency estimates. High counting efficiency must be a desirable feature of biological representations, but the results show that the number of events that can be counted simultaneously with 50% efficiency is fewer than the number of cells or 0.1-0.25 of the number of synapses (on the two models) - many fewer than can be unambiguously represented. Direct representations would lead to greater counting efficiency, but distributed representations have the versatility of detecting and counting many unforeseen or rare events. Efficient counting of rare but important events requires that they engage more active cells than common or unimportant ones. The results suggest reasons that representations in the cerebral cortex appear to use extravagant numbers of cells and modular organization, and they emphasize the importance of neuronal trigger features and the phenomena of habituation and attention.

Detecting collinear dots in noise.

We estimated the sensitivity for detecting a row of collinear target elements (usually dots) by measuring the maximum density of randomly positioned noise elements that allowed 75% correct detection of the orientation of alignment (binary choice: horizontal versus vertical) of the target elements. We varied the number of target elements, their mode of generation, and their accuracy of positioning. As reported previously (Moulden (1994) Higher-order processing in the visual system. Ciba Foundation Symposium 184. Chichester: Wiley), target detection improved rapidly until the number of target elements reached about seven, and then improved more slowly beyond this point. However, this break was reduced (and often removed entirely) when the target array was formed by repositioning pre-existing noise elements lying close to the target location, rather than by superimposition of additional target elements onto the noise array. This almost linear slope of improvement, coupled with the observation that target detection was disrupted more by random jitter of target elements at right angles to their axis of alignment than by jittering along this axis, argues against a two-stage process of perceptual grouping (Moulden, 1994) and supports instead an explanation based on the operation of a single mechanism. This single mechanism explanation is further supported by the observation that intrinsic positional uncertainty (estimated from the results of jitter experiments) was independent of target element number. Additional experiments showed that target detection is facilitated by aperiodic noise dots that fall close to the target axis. The results are discussed in relation to alternative explanations of perceptual grouping.

The knowledge used in vision and where it comes from.

Knowledge is often thought to be something brought from outside to act upon the visual messages received from the eye in a 'top-down' fashion, but this is a misleadingly narrow view. First, the visual system is a multilevel heterarchy with connections acting in all directions so it has no 'top'; and second, knowledge is provided through innately determined structure and by analysis of the redundancy in sensory messages themselves, as well as from outside. This paper gives evidence about mechanisms analysing sensory redundancy in biological vision. Automatic gain controls for luminance and contrast depend upon feedback from the input, and there are strong indications that the autocorrelation function, and other associations between input variables, affect the contrast sensitivity function and our subjective experience of the world. The associative structure of sensory message can provide much knowledge about the world we live in, and neural mechanisms that discount established associative structure in the input messages by recoding them can improve survival by making new structure more easily detectable. These mechanisms may be responsible for illusions, such as those produced by a concave face-mask, that are classically attributed to top-down influences.

Adaptation to contingencies in macaque primary visual cortex.

We tested the hypothesis that neurons in the primary visual cortex (V1) adapt selectively to contingencies in the attributes of visual stimuli. We recorded from single neurons in macaque V1 and measured the effects of adaptation either to the sum of two gratings (compound stimulus) or to the individual gratings. According to our hypothesis, there would be a component of adaptation that is specific to the compound stimulus. In a first series of experiments, the two gratings differed in orientation. One grating had optimal orientation and the other was orthogonal to it, and therefore did not activate the neuron under study. These experiments provided evidence in favour of our hypothesis. In most cells adaptation to the compound stimulus reduced responses to the compound stimulus more than it reduced responses to the optimal grating, and the responses to the compound stimulus were reduced more by adaptation to the compound stimulus than by adaptation to the individual gratings. This suggests that a component of adaptation was specific to (and caused by) the simultaneous presence of the two orientations in the compound stimulus. To test whether V1 neurons could adapt to other contingencies in the stimulus attributes, we performed a second series of experiments, in which the component gratings were parallel but differed in spatial frequency, and were both effective in activating the neuron under study. These experiments failed to reveal convincing contingent effects of adaptation, suggesting that neurons cannot adapt equally well to all types of contingency.

Limit to the detection of Glass patterns in the presence of noise.

A method is developed for quantifying the strength of the moiré effects known as Glass patterns. Unpaired randomly placed dots are added to the pattern while the discriminability d' of the degraded pattern is determined in a yes-no test. For a given discriminability the number of pairs required increases in direct proportion to the number of random dots. A model is developed based on the ideal discrimination of an excess of oriented pairs. Results conforming to Weber's law are predicted; the dependence of d' on the amount of noise and the number of point pairs is also predicted. A numerical constant derived from the model provides a measure of the strength of the moiré effect of a chosen pattern. Note that, for this task, statistical considerations predict Weber's law, not the square-root law, as a limit, and this result holds whenever second-order structure is detected in the image.

Human contrast discrimination and the threshold of cortical neurons.

The human contrast-discrimination function has a curious shape: In addition to rising for increasing contrasts, both positive and negative, it also rises for very low contrasts on either side of zero. It is shown that this rise near zero contrast is not much affected by procedures that increase or decrease the subject's knowledge of the stimulus; this counts against uncertainty as the immediate cause of the elevation near zero contrast. The alternative explanation in terms of a genuine response threshold is shown to be promising when measurements of human contrast discrimination are compared with values calculated from records of neurons in monkey primary visual cortex. The comparison also suggests that the dynamic range of single neurons is lower than that shown psychophysically. It is suggested that having a cortical threshold may be the visual system's way of preventing false positives when there is much stimulus uncertainty. This threshold may also help to explain why quantum efficiencies calculated from detection thresholds are so poor compared with those estimated from the visual system's susceptibility to added noise.

Why have multiple cortical areas?

Image processing requires free access to information about all parts of an image, but a nerve cell in V1 can only interact directly with a tiny fraction of the other cells in V1. The problem this poses might be alleviated by forming secondary "neural images" in which information is re-arranged, and some possible rules of projection for forming such images are explored. It is also suggested that all parts of the cerebral cortex detect, and subsequently signal, suspicious coincidences in their inputs. Acquiring knowledge of the associative structure of sensory messages, in the form of the unexpected coincidences that occur, may represent the beginning of the formation of a working model, or cognitive map, of the environment.

Localization of function in the cerebral cortex. Past, present and future.

At a famous meeting of the International Medical Congress held in London on August 4, 1881 Goltz of Strassburg (as it was then spelt) confronted Ferrier of London on the subject of the localization of function in the cerebral cortex. In the first part of this paper the events of that meeting are recalled. Goltz was reluctant to accept the idea of localization because of the restitution of function after injury to the cortex, and because of the general rather than specific residual disabilities of his lesioned dogs. On the other hand, Ferrier's monkeys with cortical lesions demonstrated convincingly that local lesions can produce loss of specific functions. One hundred years later a meeting was held in Oxford on the same topic, and the discussions that took place are summarized in the second part of this paper. No-one doubted the doctrine of localization, namely that different parts of the cerebral cortex normally perform different specialized roles. However, there was no unanimity about how to separate or count the number of different parts of the cortex, nor about the nature of the specialized roles of the parts, nor about any common characteristics of the functions of different parts. In other words, though localization was agreed upon, precisely what the functions are that are localized remained obscure. The third section of this paper advances some speculations on this point. Is a theory of cortical function that would encompass the diverse roles of different parts perhaps within sight, which might even explain the plasticity that must underlie the restitution of function that so impressed Goltz one hundred years ago?

Performance of cat retinal ganglion cells at low light levels.

Responses of brisk-sustained cat retinal ganglion cells were examined using receiver operating characteristic (ROC) analysis. Stimuli were brief luminance changes superimposed upon a weak steady pedestal ranging from 27 to 47,000 quanta (507 nm) per second at the cornea. Overall quantum efficiencies of cells ranged up to approximately 13% and were compatible with previous estimates at absolute threshold. The main work was done on on-center cells, but a small sample of off-center units behaved similarly. Experimental ROC curves verified a set of qualitative predictions based on a theoretical treatment of performance, assuming that response variability resulted solely from quantum fluctuations. However, quantitative predictions were not fulfilled. The discrepancy could be resolved by postulating a source of added internal variance, R, the value of which could then be deduced from the experimental measurements. A ganglion cell model limited by a fixed amount of added variance from physiological sources and having access to a fixed fraction of incident quanta can account quantitatively for (a) slopes of ROC curves, (b) variation of detectability with magnitude of both increments and decrements, and (c) performance over a range of pedestal intensities. Estimates of the proportion of incident quanta used ranged up to 29% under some conditions, a figure approximately matching estimates of the fraction of corneal quanta that isomerize rhodopsin in the cat.

Intelligence, guesswork, language.

A satisfactory definition of intelligence has never been found, and as a result it means different things to different people. What it is may remain too complex for a succinct definition, but the theory and practice of information handling have clarified what it does for us: it enables us to guess better, and the discovery of unexpected orderliness is the chief means of doing this.

What does the eye see best?

Our eyes see so much in such varied conditions that one might consider the question posed in the title to be meaningless, but we show here that, within the range that we have been able to test, there is a particular spatiotemporal pattern of light that is detected better than any other. At least two plausible theories of visual detection predict that a stimulus will be seen best (will have greatest quantum efficiency) when it matches the weighting function of the most efficient detector. We have measured quantum efficiency for detecting a wide variety of spatiotemporal patterns using foveal vision in bright light. The best stimulus found so far is a small, briefly exposed circular patch of sinusoidal grating having a spatial frequency of approximately 7 c deg-1, drifting at approximately 4 Hz. We propose that this is the weighting function of the most efficient human contrast detector. We believe this answer to the question is unexpected and may have fundamental implications with regard to the mechanisms of visual perception.

The precision of numerosity discrimination in arrays of random dots.

The precision of making discriminations between the numbers of dots in a pair of irregular arrays was measured. The results fit the assumption that the observer adds intrinsic variance to whatever variance is present in the numbers displayed, the errors depending upon the sum of the two. We found no evidence for incomplete use of the sample of information presented, other than this observer variance. Its value increases as about the 0.75 power of the mean number of dots in a display, except for numbers up to about 20 where it changes much more rapidly. Decreased irregularity in the arrangement of the dots decreases observer variance, but it is little affected by large variations in average density of dots per unit area, and is also little changed by making the dots vary irregularly in brightness.

What causes trichromacy? A theoretical analysis using comb-filtered spectra.

For colour vision, the task of the eye is to discriminate different distributions of energy over the spectrum. This is usually treated as a problem in the wavelength domain, analogous to treating spatial resolution in terms of spatial positions in the image. What is attempted here is a treatment of colour vision in terms of the system's responses to spectral energy distributions that are sinusoidal functions of wavelength. These are called comb-filtered spectra, and the treatment is analogous to that of spatial vision in terms of spatial sinusoids. This gives some insight into the reasons for trichromacy, the advantages of oil droplets, and the narrow separation of the red and green mechanisms. It is also shown that the absorption spectra of photosensitive pigments are superimposable if plotted as a function of the fourth root of wavelength.

A model for the economical encoding of the visual image in cerebral cortex.

We propose a model for the first stage of the cortical transformation of the visual image based on the principle that the cortex encodes the information with the minimum number of channels mathematically needed. We restrict our model to be consistent with the data on size adaptations, the known relationships of acuity and the inverse of magnification factor with eccentricity, and the electrophysiological findings on the physiological uniformity of the striate cortex. Assuming that each hypercolumn analyzes a limited spatial domain, we apply the sampling theorem to show that only 16 channels, composed of 4 sizes, are needed for dimension. The extension to 2 dimensions leads to a possible scheme for the number, spacing, and orientational disposition of the elements, together with predictions about the number of inputs from the eyes and the total number of hypercolumns. Since all these predictions are consistent with physical and neural estimates, we conclude that the cortex may analyze the image along the lines we suggest.

Efficiency of human visual signal discrimination.

We have measured the overall statistical efficiency of human subjects discriminating the amplitude of visual pattern signals added to noisy backgrounds. By changing the noise amplitude, the amount of intrinsic noise can be estimated and allowed for. For a target containing a few cycles of a spatial sinusoid of about 5 cycles per degree, the overall statistical efficiency is as high as 0.7 +/- 0.07, and after correction for intrinsic noise, efficiency reaches 0.83 +/- 0.15. Such a high figure leaves little room for residual inefficiencies in the neural mechanisms that handle these patterns.