Weber-Fechner law

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The Weber-Fechner law is the formulation of a psychophysical relationship in sensory physiology and states that a linear increase in the (psychologically) subjectively perceived strength of sensory impressions corresponds to the logarithm of the increase in the (physically) objectively measurable intensity of the stimulus .

With this, Weber's relationship - that the amount of difference sufficient for a barely perceptible difference in the intensity of stimuli is in a constant ratio to the stimulus strength - is extended to Fechner's relationship by the theoretical assumption that, due to the same relative thresholds of difference, a scale of sensitivity is related to be defined on the absolute threshold . This can be used for various sensory modalities in the range of medium stimulus intensities. However, the Weber-Fechner law does not apply to very low stimulus strengths close to the stimulus threshold or to very high stimulus strengths close to or above the saturation threshold .

Weber's law

The German anatomist and physiologist Ernst Heinrich Weber (1795–1878) examined, among other things, the relationship between sensory perceptions and the stimuli of different physical quality and quantity that evoke them through various experiments (such as the Weber experiment ). He also looked into the question of how much a stimulus has to be amplified in order to be perceived as stronger. Weber found that the minimum difference required for this - also known as the difference limit ( DL ) - is significantly lower for low stimulus intensities than for high stimulus intensities. In 1834 he noticed that for stimuli of the same type, these difference thresholds are almost the same if they are given as a proportion in relation to the respective intensity of the stimuli. Fechner called this relationship that the just perceptible difference  ΔR to the comparison  stimulus R has a certain constant ratio  k , formulated mathematically:

(1a)  Weber law.    

Reformulated, this means that the difference threshold .DELTA.R in proportional is related to the stimulus intensity: .DELTA.R = k · R . In the medium stimulus strengths this law is approximately valid for different sensory modalities and - qualities .

The Weber quotient ΔR / R represents the proportionality factor k as a dimensionless ratio specification ; this is not the same with different stimuli or different sensory organs .

Examples:

  • A relative weight difference of approximately 2% of an object held in the resting hand is recognized. An object's weight increase of 50 g is only noticed when the weight has increased by 1 gram to 51 g. Correspondingly, 500 g weight has to increase by 10 g to appear heavier. Here the difference threshold to the stimulus intensity behaves like ΔR / R = 1/50 = 10/500 = 0.02 = 2%.
  • In terms of the sense of touch , the required difference  threshold ΔR, according to Weber's experiments, is around 3% of the stimulus R applied as pressure on the skin : ΔR / R ≈ 0.03.
  • When seeing brightness , the quotient ΔR / R is lowest at around 0.01–0.02 with medium ambient brightness, the difference threshold is then 1 to 2% of the light intensity ; with decreasing brightness, the quotient increases to over 0.1 in twilight; with very weak stimuli near the absolute threshold , it is even higher.
  • In the case of taste , the concentration must increase by 10 to 20% in order to be perceived as stronger.

Weber-Fechner law

The physicist, psychologist and philosopher Gustav Theodor Fechner (1801–1887) is considered to be the founder of psychophysics . He formally extended Weber's law in 1860 by integration under the assumption that k is constant and independent of  R :

(1b)  Fechner's law    
(2)  Weber-Fechner's law    

Here c is a constant characteristic of the respective type of stimulus; R 0 is an integration constant that usually defines the intensity at the stimulus threshold as a threshold stimulus . The formula (2) states that with an exponential increase in stimulus intensity, only a linear increase in sensitivity E can be expected.

With the sense of sight , brightness sensations are possible in a wide range of stimulus intensity, but this requires the eye to adapt to the respective light conditions. For the transition from a bright environment to the darkness of a moonless night, the adaptation time is around half an hour. With regard to the physical luminance , the absolute threshold, as the only just perceptible lowest stimulus intensity (≈ 10 −6 cd / m²), differs by more than ten powers of ten from the saturation threshold (≈ 10 6 cd / m²), from which the intensity of sensation can hardly be increased by increasing stimulus intensity is.

The apparent brightness of a freiäugig barely visible star 6. Size (6 mag) is trillion times weaker than that of the sun (about -27 mag). There is a logarithmic relationship between the classification of stars according to their perceived brightness - according to their perceived brightness - and their objectively measured luminosity L , based on the Weber-Fechner law: m = -5/2 log 10 ( L  /  L 0 ) . The magnitude is a size indication using a scale, in which every step from one level to the next (1 mag) corresponds to a luminosity difference by a factor of 10 2/5 (≈ 2.51) - with steps of 5 mag by 10 2 (= 100). To estimate fine differences in brightness between two stars, astronomers use special methods in visual photometry ( Argelander's level estimation method ).

In the sense of hearing , the perceived pitch of a musical tone depends exponentially on the basic frequency : doubling the basic frequency changes the pitch by exactly one octave , while quadrupling it is two octaves.

With the temperature sense, on the other hand, the thermoreceptors respond approximately linearly to changes in the size of the stimulus. Psychophysical relationships cannot be formulated generally valid for the perception of pain ; Depending on the pain memory , an equally strong stimulus can be felt differently within the individual.

Stevens power function

The experimental psychologist Stanley Smith Stevens (1906–1974) found in 1957 that the extension of Weber's law (1) to relationship (2) was too general. If one takes into account the dependence of the reaction strength  E on the size of the stimulus, it follows from (1):

(3) 

The integration of this relationship leads to the Stevens power function:

(4) 

The constant c arises from the two integration constants  . For k  <1 it resembles the logarithmic Weber-Fechner law. For the perception of brightness, k  ≈ 0.33.

The perceived volume, the loudness , does not follow the Weber-Fechner law for medium and high sound pressures , but rather the Stevens power law with k  ≈ 0.6: an increase in the sound pressure by a factor (10  dB ) results in a doubling of the loudness.

Microeconomics

In microeconomics , the phenomenon can be found again as a tangible threshold . It is of interest when examining indifference curves for continuity and transitivity . Microeconomic theory assumes that indifference curves are continuous, falling and convexly curved .

The practical problem that a person cannot perceive marginal differences in color or temperature etc. means that the transitivity assumption has to be formulated a little more loosely.

See also

Individual evidence

  1. ↑ H. Handwerker: Allgemeine Sinnesphysiologie. In: R. Schmidt, G. Thews, F. Lang (Hrsg.): Physiologie des Menschen. 28th edition, 2013, p. 210 limited preview in the Google book search.
  2. ^ Rainer Klinke, Stefan Silbernagl: Physiology . Ed .: Hans-Christian Pape, Armin Kurtz, Stefan Silbernagl. 7th edition. Thieme, Stuttgart 2014, ISBN 978-3-13-796007-2 , p. 942 .
  3. ↑ Böventer, Illing: Introduction to Microeconomics. Oldenbourg Wissenschaftsverlag, ISBN 3486242482 , p. 64f.

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