韦伯定理

1860年由德国心理学家的古斯塔夫·费希纳提出费希纳定律，定义于绝对阈限上。

定义

阈限：是物理刺激能量可以被个人觉察的临界点。

绝对阈限：是个体对单一刺激引起感觉经验时，所需最低的刺激强度，分为上绝对阈及下绝对阈。

差异阈限：又称恰辨差，指辨别两个刺激之间的差异时，这两种刺激强度最低的差异量。

人类五官重要的感觉绝对阈限（采自Galanter, 1962）

以数学方式表示的话： Δ p=k(Δ S)/S (韦伯定律) Δ S 为某量实质变化而； Δ p 为某量的主观的感觉变化。

• 用 dp,dS 取代 Δ p,Δ S, 作积分: dp=kdS/S
• ∫ ∫ ₀^(p)dp=k∫ _(S)₀^(S)dS/S
• （公式略） (费希纳定律)

延伸

韦伯定律：在同类刺激之下，其差异阈限的大小是随着标准刺激强弱而成一定比例关系的，K=ΔI/I K为常数。

费希纳定律：在绝对阈限之上，主观的感觉强度与刺激强度的改变，两者间呈对数的关系，亦即，刺激强度如果按几何级数增加，而引起的感觉强度却只按算术级数增加。

相较于一般人，调音师对音调的高低敏感度具有较低的差异阈

The Weber–Fechner laws are two related scientific laws in the field of psychophysics, known as Weber's law and Fechner's law. Both relate to human perception, more specifically the relation between the actual change in a physical stimulus and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell.

Ernst Heinrich Weber states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is proportional to the pre-existent stimulus," while Gustav Fechner's law is an inference from Weber's law (with additional assumptions) which states that the intensity of our sensation increases as the logarithm of an increase in energy rather than as rapidly as the increase.

History and formulation of the laws

Both Weber's law and Fechner's law were formulated by Gustav Theodor Fechner (1801–1887). They were first published in 1860 in the work Elemente der Psychophysik (Elements of Psychophysics). This publication was the first work ever in this field, and where Fechner coined the term psychophysics to describe the interdisciplinary study of how humans perceive physical magnitudes. He made the claim that "...psycho-physics is an exact doctrine of the relation of function or dependence between body and soul."

Weber's law

Ernst Heinrich Weber (1795–1878) was one of the first persons to approach the study of the human response to a physical stimulus in a quantitative fashion. Fechner was a student of Weber and named his first law in honor of his mentor, since it was Weber who had conducted the experiments needed to formulate the law.

In sensation and contrast, change is defined as "contrast over time". The senses detect the change; in Weber's law, specifically, it is relative change, not absolute change, which is detected.

Types of perception

Weber and Fechner conducted research on differences in light intensity and the perceived difference in weight. Other sense modalities provide only mixed support for either Weber's law or Fechner's law.

Weight perception

Weber found that the just noticeable difference (JND) between two weights was approximately proportional to the weights. Thus, if the weight of 105 g can (only just) be distinguished from that of 100 g, the JND (or differential threshold) is 5 g. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight (any weight) seems to have to increase by 5% for someone to be able to reliably detect the increase, and this minimum required fractional increase (of 5/100 of the original weight) is referred to as the "Weber fraction" for detecting changes in weight.

Logarithmic coding schemes for neurons

Lognormal distributions

Activation of neurons by sensory stimuli in many parts of the brain is by a proportional law: neurons change their spike rate by about 10–30%, when a stimulus (e.g. a natural scene for vision) has been applied. However, as Scheler (2017) showed, the population distribution of the intrinsic excitability or gain of a neuron is a heavy tail distribution, more precisely a lognormal shape, which is equivalent to a logarithmic coding scheme. Neurons may therefore spike with 5–10 fold different mean rates. Obviously, this increases the dynamic range of a neuronal population, while stimulus-derived changes remain small and linear proportional.

Other applications

The Weber–Fechner law has been applied in other fields of research than just the human senses.

Numerical cognition

Psychological studies show that it becomes increasingly difficult to discriminate between two numbers as the difference between them decreases. This is called the distance effect. This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances. It may also play a role in explaining why consumers neglect to shop around to save a small percentage on a large purchase, but will shop around to save a large percentage on a small purchase which represents a much smaller absolute dollar amount.

Pharmacology

It has been hypothesized that dose-response relationships can follow Weber's Law which suggests this law – which is often applied at the sensory level – originates from underlying chemoreceptor responses to cellular signaling dose relationships within the body. Dose response can be related to the Hill equation, which is closer to a power law.