To honor Fechner and obey Stevens: relationships between psychophysical and neural nonlinearities

G. T. Fechner (1860/1966) famously described two kinds of psychophysics: Outer psychophysics captures the black box relationship between sensory inputs and perceptual magnitudes, whereas inner psychophysics contains the neural transformations that Fechner's outer psychophysics elided. The relationship between the two has never been clear. Moreover, psychophysical power laws are found in almost every sensory system, yet the vast majority of neurons show sigmoid nonlinearities. Here, we selectively review the literatures on psychophysical and physiological nonlinearities and show how they can be placed within a framework for understanding the relationship between inner and outer psychophysics: a neural organization with a logical structure commensurate to outer psychophysical theory. In theoretical treatments of Stevens's law, the power law is a consequence of combining a Weber's law scaling of inputs with a Weber's law-like scaling of sensation magnitudes, yielding an exponent that is the ratio of the Weber constants. A neural derivation using physiological sigmoid nonlinearities should be commensurate to this internal logic. There is a class of models in which two nonlinear neural mechanisms (e.g., a sensory channel and the cortical numerosity mechanism tapped by magnitude estimation) are coupled through feedback, yielding power law behavior as an emergent property of the system, with an exponent that is a ratio of neural coupling strengths. Rather than a discrepancy between psychophysics and physiology, these models suggest complementarity between inner and outer psychophysics, because the Weber constants required for outer psychophysics modeling can be derived from the sigmoid nonlinearities of inner psychophysics.