Logarithmic and Power Law Input-Output Relations in Sensory Systems with Fold-Change Detection

Two central biophysical laws describe sensory responses to input signals. One is a logarithmic relationship between input and output, and the other is a power law relationship. These laws are sometimes called the Weber-Fechner law and the Stevens power law, respectively. The two laws are found in a wide variety of human sensory systems including hearing, vision, taste, and weight perception; they also occur in the responses of cells to stimuli. However the mechanistic origin of these laws is not fully understood. To address this, we consider a class of biological circuits exhibiting a property called fold-change detection (FCD). In these circuits the response dynamics depend only on the relative change in input signal and not its absolute level, a property which applies to many physiological and cellular sensory systems. We show analytically that by changing a single parameter in the FCD circuits, both logarithmic and power-law relationships emerge; these laws are modified versions of the Weber-Fechner and Stevens laws. The parameter that determines which law is found is the steepness (effective Hill coefficient) of the effect of the internal variable on the output. This finding applies to major circuit architectures found in biological systems, including the incoherent feed-forward loop and nonlinear integral feedback loops. Therefore, if one measures the response to different fold changes in input signal and observes a logarithmic or power law, the present theory can be used to rule out certain FCD mechanisms, and to predict their cooperativity parameter. We demonstrate this approach using data from eukaryotic chemotaxis signaling.Author Summary: One of the first measurements an experimentalist makes to understand a sensory system is to explore the relation between input signal and the systems response amplitude. Here, we show using mathematical models that this measurement can give important clues about the possible mechanism of sensing. We use models that incorporate the nearly-universal features of sensory systems, including hearing and vision, and the sensing pathways of individual cells. These nearly-universal features include exact adaptation-the ability to ignore prolonged input stimuli and return to basal activity, and fold-change detection- response to relative changes in input, not absolute changes. Together with information on the input-output relationship-e.g. is it a logarithmic or a power law relationship-we show that these conditions provide enough constraints to allow the researcher to reject certain circuit designs; it also predicts, if one assumes a given design, one of its key parameters. This study can thus help unify our understanding of sensory systems, and help pinpoint the possible biological circuits based on physiological measurements.