The series of a four-node motif and coherent type-1 feed-forward loop can provide sensitive detection over arbitrary range of input signal, thereby explain Weber's law in higher-order sensory processes, and compute logarithm.

Weber's law states that the ratio of the smallest perceptual change in an input signal and the background signal is constant. The law is observed across the perception of weight, light intensity, and sound intensity and pitch. To explain Weber's law observed in steady-state responses, two models of perception have been proposed, namely the logarithmic and the linear model. This paper argues in favour of the linear model, which requires the sensory system to generate linear input-output relationship over several orders of magnitude. To this end, a four-node motif (FNM) is constructed from first principles whose series provides almost linear relationship between input signal and the output over arbitrary range of input signal. Mathematical analysis into the origin of this quasi-linear relationship shows that the series of coherent type-1 feed-forward loop (C1-FFL) is able to provide perfectly linear input-output relationship over arbitrary range of input signal. FNM also reproduces the neuronal data of numerosity detection study on the monkey. The series of FNM also provides a mechanism for sensitive detection over arbitrary range of input signal when the output has an upper limit. Further, the series of FNM provides a general basis for a class of bow-tie architecture where the number of receptors is much lower than the range of input signal and the "decoded output". Besides (quasi-)linear input-output relationship, another example of this class of bow-tie architecture that the series of FNM is able to produce is absorption spectra of cone opsins of humans. Further, the series of FNM and C1-FFL, both, can compute logarithm over arbitrary range of input signal.