Mathematical Psychology and Psychophysiology

By Stephen Grossberg | Go to book overview

need more flexible and insightful and convenient ways of handling nonlinear systems.

Psychophysics. We have already discussed above the fact that probability summation is necessary to accurately account for the thresholds of patterns. That is, the threshold for a pattern is not determined by the one channel that is most sensitive to the pattern (the one channel that gives, on the average, the largest response to the pattern), but the threshold is always determined by nonlinear pooling of the sensitivities of all the channels that respond at all to the pattern. (Of course, channels that are very insensitive contribute negligibly, but all channels that are almost as sensitive as the most sensitive channel contribute substantially.) Thus, one is always looking at the action of a group of channels rather than a single channel.

Further, even if each receptive field of the channels operative in the pschophysical phenomena does turn out to be a linear system, as is currently assumed, the output of a channel (the output of a whole array of receptive fields of the same size) seems to be the resultant of probability summation across space or some similar kind of nonlinear pooling. So the response of a channel cannot be taken to be the response of a completely linear system.

Although the Quick pooling model presented above does a very good job at accounting for threshold data, we know it cannot be completely correct.

Physiology. If we limit our interest to single neurons, then it is easy to look at only one at a time. Then the only question is whether they are linear or not. Well, many are, or at least linear enough that Fourier analysis works, even in the cortex. But many are not. At all levels in the visual system, a major distinction is being made now between x and y, or sustained and transient, or linear and nonlinear cells. The nonlinear cells are, of course, much more difficult to figure out. Some progress is being made using newly-developed versions of Wiener analysis ( Shapely and Victor [ 1979], Victor and Knight [ 1979]).


REFERENCES

Bergen J. R, H. R. Wilson and J. D. Cowan, 1979. "Further evidence for four mechanisms mediating vision at threshold: Sensitivities to complex gratings and aperiodic stimuli", J. Opt. Soc. Amer. 69, 1580- 1587.

Bryngdahl O., 1964. "Characteristics of the visual system: Psychophysical measurements of the response to spatial sine-wave stimuli in the mesopic region", J. Opt. Soc. Amer. 54, 1152- 1160.

Campbell F. W. and J. G. Robson, 1968. "Application of Fourier analysis to the visibility of gratings", J. Physiol. 197, 551-566.

DePalma J. J. and E. M. Lowry, 1962. "Sine-wave response of the visual system. II. Sine-wave and square-wave contrast sensitivity", J. Opt. Soc. Amer. 52, 328-335.

DeValois R., D. G. Albrecht and L. Thorell, 1977. "Spatial contrast" ( H. Spekreijse and L. H. van der Tweel , eds.), North-Holland, Amsterdam, pp. 60-63.

Graham N., 1977. "Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels", Vision Res. 17, 637-652.

_____, 1980. "Spatial frequency channels in human vision: Detecting edges without edge detectors", Visual Coding and Adaptability ( C. Harris, ed.), Erlbaum, Hillsdale, N. J.

_____, 1981. "Psychophysics of spatial-frequency channels," Perceptual Organization ( M. Kubovy and J. Pomerantz, eds.), Erlbaum, Hillsdale, N. J.

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