The mental and physical worlds were considered to be distinct for centuries. Although it was possible to predict the movements of inanimate objects using mathematics, organisms’ behavior, which is how they move, could not be predicted. However, these forces can be controlled by the will.
Around 200 years ago, Ernst Heinrich Weber, a German physician, made an apparently innocuous observation that led to the creation of Psychophysics. Psychophysics is the science of connecting physical stimuli in the environment and the sensations they cause in the brain of subjects. Weber had subjects choose which of two weights they preferred. Weber discovered from these experiments that the ratio of the weights determines the likelihood that subjects will make the correct choice.
If a subject is right 75% of time when comparing 1 Kg to 1.1 Kg in weight, she will be correct 75% when comparing 2 Kg and 2.25 Kg. This is a pair of weights that are 10% heavier than each other. This simple, but exact rule allowed for the quantification and application of behavior to mathematical “laws”.
Weber’s observations were subsequently extended to all sensory modalities across many animal types, resulting in what is now called Weber’s Law. It is the most established and oldest law in psychophysics. Psychophysical laws are precise rules of perception. They can be used to calculate brain processes and provide mathematical explanations for behavior. This is similar to how the patterns of motion of the planets in space were helpful in understanding gravitation.
Weber’s law has been the subject of many explanations over the years. Weber’s findings can be explained by all of them, but no experimental test has been conducted to determine which model is correct. Weber’s law remains a puzzle.
A team of researchers from the Champalimaud Centre for the Unknown, Lisbon, Portugal has found that Weber’s Law is the result of a new psychophysical rule that involves the time it takes to make a decision, and not the outcome. This new rule was sufficient to create a mathematical model that accurately describes the cognitive process behind Weber’s Law. The team’s results were published in Nature Neuroscience.
It is crucial to have enough time
Alfonso Renart and his team trained rats to distinguish between sounds with slightly different intensities in this new study. They made tiny headphones that were specifically sized for the ears of rats and delivered sounds simultaneously to both their ears.
Each trial would have a slightly different sound. The rat was asked to identify which speaker produced the loudest sound, and orient his head towards that side. Jose Pardo-Vazquez (one of the article’s co-authors) says that rats naturally orient their heads toward the sound source. To make a decision, the rats could listen to the sound for as long as it took. Each attempt offered a choice and a time limit.
Pardo-Vazquez says, “Our experiments confirmed the animals’ behavior matched Weber’s Law.” Their ability to distinguish between the two sounds depended only on their intensities. The accuracy of a rat’s ability to distinguish between the intensities of two sounds played softly was the same as if it had to do the same thing with two loud sounds.
The team then began to examine in detail the time taken by the rats to make decisions. This step proved to be crucial. Pardo-Vazquez says that Weber’s law studies were typically focused on the accuracy and discrimination. Weber described this as well. Surprisingly, little attention has been paid to the time it takes to decide. The team discovered that the loudness of two sounds and decision times were closely linked. The longer the decision time, the more loud the sounds. They proved that this link was mathematically unique. The decision times in discrimination between quiet sounds and loud sounds were, in fact, proportional to the decision time measured by the subject when they discriminated between the two loud sounds. This is as long as their relative intensities remained constant.
Beyond Weber’s Law
In fact, the team discovered a new “psychophysical” law, which they called “Time-Intensity Equation in Discrimination” (TIED). It linked the intensity of a pair and the time it took for them to discriminate. Because it links pairs of discriminations with their associated decision times, the TIED is stricter than Weber’s Law. Pardo-Vazquez says, “The precision of the relationship between the decision time in our experiments is incredible”. “It is rare that the behavior of animals be described mathematically with such mathematical precision.”
The team conducted the same experiment with human subjects to see if the TIED would hold in other conditions. They also obtained similar results. The team also examined other experiments in which rats were trained to detect odor combinations. Again, the results were similar. Pardo-Vazquez stated that although it is too early to know if the TIED works as well as Weber’s law but that the identical results were obtained in two species and two sensory modalities was an encouraging start.
Finding the right model
Weber’s law has been explained by dozens of mathematical models over the years, but no one was able to identify them. Researchers concluded that the TIED was a promising way forward. The researchers discovered that a mathematical model of discrimination must satisfy a strict set of conditions in order to be compatible with the TIED. Juan Castineiras, another coauthor of the study, said that “this was amazing”. The TIED contained all possible explanations and resolved the ambiguity in Weber’s law. Stephen Link, a psychologist, proposed a model that was close to finding the answer. However, it missed an important condition which describes how sensory neurons encode the intensity of stimuli.
Next, we took this set of conditions and created a model to see how it accurately accounted for the behavior. Castineiras explains that they analyzed the simplest model, with the least number of parameters. Castineiras discovered that the model fitted remarkable well when the parameters were selected to be the most similar to the behavior of rats. “Even the most basic model accurately captured all we could measure with virtually no error. Renart says that this greatly increased our confidence in the model’s ability to accurately describe perception.
Experiments and theories that are precise lead to tangible progress
These results are unique in their field due to the precision of both the psychophysical rule and the mathematical model that describes them. Renart says that although it is less common, precise experimental results in biology and the study of behavior — just like in physics — allow precise explanations which resolve past ambiguities and thus constitute progress. Their results, for example, suggest that the main theory in psychophysics wasn’t sufficient to explain the TIED. Castineiras notes that it is rare to find mathematical explanations for neuroscience that can rule out competing theories. This is because one model could be slightly modified to match the experimental data. We showed that Signal Detection Theory, a highly influential theory in psychophysics, did not model decision time and therefore could not describe the TIED. It failed to capture the essence of Weber’s law.
Renart states that one of their next goals is to discover how the brain implements the mathematical model they identified.