A neural-counting model incorporating refractoriness and spread of excitation. II. Application to loudness estimation

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.