How do we quantify the relationship between the physical and the mental? An Introduction to Psychophysical Laws

Psychology and cognitive science have at their foundation the notion that we can somehow measure the mental world. But can we? It is unclear, at least to the naïve observer, how such a goal could be achieved given that the mental is unobserved and private to each individual that has a mental capacity. However, there is good reason to think that we can relate the mental to the physical in a meaningful way. The methods for doing so were first developed by Gustav Fechner in the 1800s and were called Psychophysics (psycho meaning mental and physics meaning physical). Psychophysics is still studied and used today and has served as the foundation for much of experimental psychology and cognitive science.

A basic psychophysics experiment is designed to determine the relationship between a particular sensation and a particular physical stimulus. Ernst Weber used this type of experiment to look at the relationship between the sensation of heaviness and the actual weight of a weight lifted by the observer. Weber varied the magnitude of standard and comparison weights held in turn by the observer, who made reports of whether the comparison was less than, equal to or greater than the standard weight. It was discovered that the change in the weights that was just noticeable by the observer (called the “just noticeable difference” or JND or “difference threshold”) was a constant fraction of the magnitude of the standard stimulus. This meant that for larger weights, more weight was required to get the observer to notice a change compared to when they were lifting smaller weights.

Weber investigated further and found that this relationship held for most human senses (e.g., sight, sound, taste, touch). Weber’s law states simply that the JND is proportional to stimulus magnitude. Gustav Fechner proposed another law, using Weber’s law for the foundational assumptions. Fechner’s law looks specifically at the relationship between stimulus intensity and perceived magnitude. It assumes that Weber’s law holds (i.e., JND is a constant fraction of magnitude) and also that the JND is the fundamental unit of perception, meaning that one JND in one sense is “perceptually equivalent” to one JND in another sense. With these assumptions in mind he hypothesised that the perceived magnitude of a stimulus can be calculated by adding up JNDs. Mathematically, this leads to the formula:

P = k.log.I

, where P is perceived magnitude, k is the Weber fraction and I is actual stimulus intensity/magnitude. Fechner’s law allows you to calculate whether a light that is twice as bright (in terms of physical magnitude) will appear so to the observer. However, work since Fechner has cast doubt on the second foundational assumption of the law (that the JND is the fundamental unit of perception), and this has led to the conceptualisation of Fechner’s law as a special case of the more general “Power law” or “Steven’s power law”, expressed mathematically as:

P = k.I^n

, where P is again perceived magnitude, k is again the Weber fraction and I is again stimulus intensity/magnitude. However, in this case, I is raised to a power n, which specifies the relationship between P and I — that is, does a doubling in P lead to a doubling in I (a linear relationship with n of 1) or is there more or less of a doubling in I for a doubling in P? Steven’s law has been found to better quantify the relationship between P and I for many senses, compared to Fechner’s law.

Above, I introduced three psychophysical laws which quantify the relationship between physical stimuli and mental sensations. These laws were derived from experiments that vary a physical stimulus on one dimension (e.g., brightness, heaviness) and ask the observer to somehow indicate the relative magnitude of each presentation of a stimulus lying somewhere along this uni-dimensional scale. The science of creating and carrying out such experiments is called psychophysics (psycho- meaning psychological/mental and physics meaning the physical). I will now look in more detail at the types of experiments that can be designed using this basic methodology and how these experiments have evolved.

As mentioned previously, Ernst Weber presented observers with pairs of weights of varying weight and observers were asked to make a relative judgement, indicating whether the second weight was lighter or heavier than the first. This type of experiment is equivalent to what is now called two-interval forced-choice (2IFC), where an observer is presented with two different stimuli sequentially and required to choose the ‘larger’ or ‘smaller’ one (or the ‘brightest’ or ‘dimmest’ one; ‘longest’ or ‘shortest’ one etc.). That is, they are forced to choose one of the two stimuli. Additionally, one of the stimuli has a fixed value (a standard) while the other one varies (a comparison). In such an experiment, it is clearly important to keep track not only of the observer’s choice on each trial, but also the levels used for the comparison stimulus (the standard remains fixed). But how do we go about selecting appropriate values for this stimulus?

Using the “method of constant stimuli” we would pre-set the different levels of each stimulus (generally at equally- or logarithmically- spaced intervals across the stimulus dimension) and have a number of trials for each level, but the trials would be presented in a random order. This enables us to plot the proportion correct data across the stimulus dimension. The random order helps us prevent effects due to learning or adaptation. Generally, the plotting of proportion correct using such a method will generate a psychometric function as shown below (taken from The psychometric function indicates that detection probability is very low for low values on the stimulus scale and rises sharply at a certain point, after which it is very high. This characteristic curve relates to the modern conception of the psychometric threshold, the point at which the stimulus changes from not-detectable to detectable. Clearly, from this graph, it can be seen that this is not a sharp, single-point, all-or-nothing threshold (or a Just Noticeable Difference, JND), but a more gradual transition of detection probability. However, many modern experiments will still report the stimulus value (on the x-axis) corresponding to 75% correct detection (on the y-axis) as a single point threshold, because it is exactly halfway between detection at chance (50%) and certain detection (100%).

Another type of psychophysical experiment is called “adaptive” because the stimulus values used on a given trial depend on the observer’s previous responses. This type of experiment can, with reasonable starting values set for the stimulus, effectively “zero in” on the threshold of the observer, by gradually adjusting the stimulus value until it is around the “sweet spot” of detectability where the transition happens from 0 to 100. The method of constant stimuli, however, needs to cover this sweet spot with its pre-set stimulus values a priori, and so needs additional guidance from say, previous research, experimenter knowledge, or another experiment. The last 30 years has seen a focus on adaptive methods, with some of the most efficient ones, which require the least number of trials to gain knowledge of the underlying psychometric function, leveraging Bayes’ rule to adaptively place stimulus values during an experiment.

In this article I have briefly looked at the main types of psychophysical experiment designed to quantify the relationship between physical stimuli and the sensations they illicit. It is important to note that these types of experiment can be used for any type of stimulus variation that can be described by a single numeric dimension. A method I did not cover is called the “adjustment method”, which involves the observer manually adjusting a comparison stimulus until it is perceptually equivalent to the standard (that is, they cannot distinguish between the two). In this case, the experimenter will present many trials with different starting stimulus values and take the average of the adjusted values. This method does not, however, yield knowledge of the underlying psychometric function, whereas the more modern methods I have discussed above, do. Knowledge of the underlying psychometric function for a given set of observers can help us understand what is common and what is different between observers when they make these sensory observations.

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