Science 3 min read

Newly-Derived Mathematical Model Explains Weber's Law

A new psychophysical law helped scientists derive a unique and accurate mathematical model that could explain Weber's Law.

Image courtesy of Shutterstuck

Image courtesy of Shutterstuck

Portuguese researchers have derived a unique mathematical model to describe the cognitive process underlying Weber’s Law.

The German physiologist Ernst Heinrich Weber postulated Weber’s Law in 1834. It is a principle of perception which states that the size of the just noticeable difference is a constant proportion of the original stimulus value.

For instance, if you’re already holding a tennis ball on your right hand and another one is placed on it, you will notice the difference in weight quickly. However, if you’re holding a bowling ball and a tennis ball is placed on top of it, you won’t notice the change in weight.

Gustav Fechner later used Weber’s Law to develop psychophysics, the science that deals with the relationship between physical stimuli and mental phenomena. Since its inception, Weber’s observations have been applied by neuroscientists to all sensory modalities across many animal species.

Over the years, many explanations about the said law have been proposed. But, no experimental or mathematical model has been presented to support any of the proposals — until now.

The Mathematical Model to Explain Weber’s Law

Neuroscientists at the Champalimaud Centre for the Unknown in Lisbon found that Weber’s Law could be the result of a new psychophysical law. One that also considers the time it takes to make a choice and not just the decision’s outcome.

According to the researchers, the new law is enough for them to derive a “unique and accurate mathematical model” to describe Weber’s Law’s underlying cognitive process.

The team, led by Alfonso Renart, trained rats to discriminate between two sounds of slightly different intensities. To deliver the sounds simultaneously to the rats’ ears, they developed mini-headphones that would fit the head of the rodents.

During the experiments, the researchers delivered slightly louder sound on one of the two headsets’ speakers. Then, the rats reported which of the speaker sides played louder by orienting towards it.

José Pardo-Vazquez, one of the article’s co-authors, reported:

“Our experiments confirmed that the animals’ behavior matched Weber’s Law. Their ability to tell which of the two sounds was louder only depended on the ratio between the sounds’ intensities. If the rat had to compare the intensities of two sounds that were played softly, its accuracy was just as good as with a pair of sounds that were played loudly, as long as both pairs had the same intensity ratio.”

The scientists then analyzed the time it took the rats to make their decisions. By doing so, the team discovered that the rats’ decision times and the loudness of the sounds were related – “the louder the sounds, the shorter the decision time.”

The team noted that the nature of the link they observed was, in fact, mathematically accurate.

For instance, the decision times in discriminating between two quiet sounds were precisely proportional to the decision times observed when the rats discriminated between two loud sounds; given that the sound intensities were constant.

The team referred to their newly-postulated psychophysical law as Time-Intensity Equivalence in Discrimination (TIED). Pardo-Vasquez added:

“The precision of this relationship between the decision times in our experiments is amazing. It is unusual that the behavior of animals can be described with such mathematical precision.”

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Rechelle is an SEO content producer, technical writer, researcher, social media manager, and visual artist. She enjoys traveling and spending time anywhere near the sea with family and friends.

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