The Weber-Fechner law is among the oldest quantitative relationships in experimental psychology. Neural codes with Weber-Fechner spacing are widely-observed in the nervous system, most famously extrafoveal retinal position. We show new evidence suggesting that a neural representation for time in the rodent hippocampus obeys Weber-Fechner spacing; preliminary evidence suggests a similar relationship may also hold in the mPFC and striatum. We describe a neural mechanism for constructing Weber-Fechner scales. The mechanism relies on taking the Laplace transform of incoming experience and can be applied to generate scale-invariant representations of time, space and number. The apparent ubiquity of Weber-Fechner scales in the brain and the Laplace method for constructing representations of time, space, and number suggest a general framework for cognitive computation. Operations such as translation, convolution, and cross-correlation can be efficiently computed in the Laplace domain, enabling flexible computation on scale-invariant representations. Circuits constructed in this way obey properties such as compositionality that are challenging for traditional connectionist models. A simple circuit for performing subtraction is demonstrated.
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