A nonlinear systems approach to Fechner's paradox

It is possible to predict the topology of isointensity plots under conditions of extreme imbalance of the stimulus inputs, without making any assumptions specific to the circumstances in which Fechner's Paradox is sometimes observed. This is done by extending a nonlinear model for a sensory channel, by postulating a form of cross-coupling or interference between two channels which represents other phenomena in psychophysics. It is noted that the form in which data are usually reported is not an adequate basis for testing all the predictions of a nonlinear model in sensory psychophysics. The physiologist Panum (1958), and later Fechner (1860) reported that the apparent brightness of an object viewed binocularly could, under conditions where the input to one eye was diminished by filtering, be less than its brightness viewed monocularly by the unfiltered eye. To a first approximation, binocular brightness is more like an averaging of two monocular inputs than a summation of those same inputs. For over 120 years this phenomenon, which came to be called "Fechner's Paradox", though Panum should perhaps have had some credit, has been the subject of experimental investigation and associated mathematical modelling. If one consults a dictionary of psychological terms, for example Evans (1978), one may read something like Fechner's Paradox: The name give to the observation that something [which is] viewed binocularly seems to increase in brightness when one eye is closed. And yet we now know that this definition is misleading, because the same phenomenon in pooling two sensory inputs has its analogues in audition (Lehky 1983) and in olfaction (Gregson 1986). Gilchrist and McIver (1985) have now shown an analogue of the paradox exists in ocular contrast sensitivity. The definition also goes awry when the input luminance to one eye is zero, or when the luminance and ocular adaptation are closely matched for the two eyes. It is wiser, in the light of results reporting individual differences in the existence and extent of the paradox, and its sensitivity to stimulus conditions, to side with Blake and Fox (1973) when they observed that it is not unreasonable to suppose that various stimulus conditions might yield varying amounts of summation or even inhibition. Empirical reviews of relevant data in vision have been given by Roelofs and Zeeman (1914), Blake and Fox (1973), and Blake et al. (1981), but a theoretical model of interest as a starting point is that of Lehky (1983).