A Fractal Model of the Stimulus-Response Relationship

The models relating the intensity of perception to the stimulus are of great importance in sensorial analysis and practical applications. For example, in olfactometry they have been employed to determine: the odour threshold and suprathreshold, hedonic scale assessment and the degree of appreciation. The perceived intensity of a stimuli should be expressed in the form of function depending on a physico-chemical variable, eg concentration of the substance in fresh air or clean water. It must take into account a characteristic threshold of its perception, whereas at higher concentrations the perceived intensity should reach a plateau. Those requirements well satisfy the sigmoidal Gompertz function, which depends on the concentration variable and three parameters representing the initial perceived intensity, retardation constant and the initial increase rate constant. The application of the mapping procedure converts this function into power law formula k(C)C including the signal-dependent scaling factor k(C) and exponent n(C). Because the Gompertz model describes the self-similar and allometric processes of the fractal nature, the mapping procedure transfers those peculiar properties from the Gompertz function onto power law one. Hence, the phenomena described by the latter are endowed with the typical fractal properties; in particular n(C) is interpreted as C-dependent fractal dimension of the stimulus-response process. The formula obtained will be applied in modelling the odour intensity-concentration relationship for acetone, ethyl acrylate, pyridine, vanillin and then in calculating the scaling factor k(C) and exponent n(C), representing the C-dependent fractal dimension of the stimulus-response relationship.