Some of the statistical properties of a mode of propagation for an electromagnetic field which is a superposition of thermal and coherent radiation are derived. It is found that the electric field and the magnetic field have Gaussian probability densities. The variance of the mixed field is the same as the variance of the thermal part of the field alone, while the average of the mixed field is the same as the average of the coherent field alone. It is pointed out that this result differs from the classical theory in that the zero-point field appears only once in the variance for the mixed field while it appears once in the variance of each of the constituent fields. This result is extended to a more general class of fields, and for that class it is shown that except for the zero-point field, the quantum properties of superposition are the same as the classical properties of superposition of noisy fields. The probability distribution for the number of photons in a mode of mixed radiation is also derived. The results shows that there are fluctuations in the number of photons that arise, because of interference effects.

We consider in detail a new mathematical neural-counting model that is remarkably successful in predicting the correct detection law for pure-tone intensity discrimination, while leaving Weber's law intact for other commonly encountered stimuli. It incorporates, in rather simple form, two well-known effects that become more marked in the peripheral auditory system as stimulus intensity is increased: (1) the spread of excitation along the basilar membrane arising from the tuned-filter characteristics of individual primary afferent fibers and (2) the saturation of neural counts due to refractoriness. For sufficiently high values of intensity, the slope of the intensity-discrimination curve is calculated from a simplified (crude saturation) model to be 1-1/4N, where N is the number of poles associated with the tuned-filter characteristic of the individual neural channels. Since 1 less than or equal to N less than infinity, the slope of this curve is bounded by 3/4 and 1 and provides a theoretical basis for the "near miss" to Weber's law.

In previous paper [Teich and Lachs, J. Acoust. Soc. Am 66, 1738--1749 (1979)] we demonstrated that an energy-based neural counting model incorporating refractoriness and spread of excitation satisfactorily described the results of pure-tone intensity discrimination experiments. In this paper, we show that the identical linear filter refractoriness model (LFRM) also provides proper results for pure-tone loudness estimation experiments at all stimulus levels. In particular, as the stimulus intensity increases from very low to moderate values, the model predicts that the slope of the intensity discrimination curve will climb from 1/2 toward 1, whereas the slope of the loudness function will gradually decline below 1 in this same region. For sufficiently high values of the stimulus intensity, the slopes calculated from a simplified (crude saturation) version of the model are found to be 1--1/4N for the intensity discrimination curve and 1/2N for the loudness function. The quantity N is the number of poles associated with the tuned-filter characteristic of the individual neural channels; it is the only important free parameter in the model. Appropriate values for N appear to lie between 2 and 4, providing an asymptotic slope for the intensity discrimination curve bounded by 7/8 and 15/16 (the near miss to Weber's Law), and an asymptotic slope for the loudness function bounded by 1/4 and 1/8. The results follow from the assumption that the neural concomitant of loudness is the number of impulses observed on a collection of parallel neural channels during a fixed observation time. Our calculations are supported by Hellman and Zwislocki's [J. Acoust. Soc. Am. 33, 687--694 (1961)] observation of unit slope for the loudness function at low intensities and provide a theoretical foundation, based on spread of excitation, for Stevens' power law at high intensities.

An energy-based neural-counting model, incorporating refractoriness and spread of excitation, has recently been applied to intensity discrimination and loudness estimation for pure-tone stimuli [M. C. Teich and G. Lachs, J. Acoust. Soc. Am. 66, 1738–1749 (1979); G. Lachs and M. C. Teich, J. Acoust. Soc. Am. 69, 774 (1981)]. We now examine the behavior of this model when the stimulus is variable-bandwidth noise rather than a pure tone. The theoretical predictions are in good agreement with psychophysical data for intensity discrimination [C. E. Bos and E. deBoer, J. Acoust. Soc. Am. 39, 708–715 (1966)] and loudness estimation [B. Scharf, “Loudness,” in Handbook of Perception, Vol. IV, Hearing, edited by E. C. Carterette and M. P. Friedman (Academic, New York, 1981)], pp. 187–242. The functional dependence of the theoretical intensity discrimination and loudness curves on the noise bandwidth is established by the excitation pattern along the basilar membrane. This pattern, in turn, is principally determined by the tuned-filter characteristics of the neural channels in conjunction with the spectral properties of the stimulus. We appeal neither to stimulus intensity fluctuations nor to critical bands in carrying out our analysis. [Work supported by NSF.]

s-The psychophysical properties of a multiple-channel neuralcounting model are investigated. Each channel represents. a peripheral afferent fiber (or a group of such fibers) and consists of a cascade of signal-processing transformations, each of which has a physiological correlate in the auditory system. The acoustic signal (which may be a pure tone or Gaussian noise) is passed by our mathematical construct through the following series of transformations: an outer-and middle-ear transmission function, an inner-ear multiple-pole linear-filter tuning mechanism, a nonlinear receptor saturation function, and a refractoriness-modified Poisson transduction mechanism (which leads to a sub-Poisson neural spike count). Spontaneous neural activity is independently incorporated into each channel by means of an additive refractoriness-modified Poisson process. A union process at a more distal center in the nervous system is generated by a parallel collection of such channels with a density (in frequency) determined by the cochlear mapping function. The statistics of the union count (in a fixed time) are then processed at a decision center in a manner that depends on the psychophysical paradigm under consideration. This random count number is assumed to contain all of the information for the examples we consider. Our model has been used to calculate psychophysical functions for the following paradigms: pure-tone loudness estimation, pure-tone and variable-bandwidth noise intensity discrimination, and variable-bandwidth noise loudness summation. The theoretical results, which are determined in large part by spread of excitation, are in good agreement with human psychophysical data, provided that the parameters of the theoretical model are appropriately chosen. It has been found that a suitable choice of parameters is both physiologically sensible and self-consistent. As a further indication of the consistency of the model, the same general parametric dependencies as neurophysiological isointensity contours for peripheral afferent fibers in the squirrel monkey are exhibited by the single-channel theoretical count mean, which is calculated as a function Manuscript .of stimulus level and frequency. The single-channel count mean-to-variance ratio is in accord with laboratory data. Finally, the roles of the various components comprising our theoretical system are discussed, and our model is compared with related constructs.