Skip to main content
SpringerLink
Log in

Statistical Mechanical Foundation of Weber–Fechner Laws

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Weber–Fechner laws are phenomenological relations describing a logarithmic relation between perception and sensory stimulus in a great variety of organisms. While firmly established, a theoretical argument for those laws in terms of relevant models or from statistical physics is largely missing. We present such a discussion in terms of response theory for nonequilibrium systems, where the induced displacement or current, which stands for the perceived stimulus, crucially depends on the change in time-symmetric reactivities. Stationary nonequilibria may indeed generate extra currents by changing the dynamical activity. The argument finishes by understanding how the extra dynamical activity logarithmically encodes the actual stimulus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Weber, E.H.: De pulsu, resorptione, auditu et tactu [On stimulation, response, hearing and touch]. Annotationes, anatomical et physiological. Koehler, Leipzig (1834)

  2. Weber, E.H.: Der Tastsinn und das Gemeingefühl (1851)

  3. Fechner, G.T.: Elemente der psychophysik, vol. 1. Breitkopf und Harterl, Leipzig (1860)

    Google Scholar 

  4. von Helmholtz, H.: The Facts of Perception. Address given during the anniversary celebrations of the Friedrich Wilhelm University in Berlin in 1878. Reprinted in Vorträge und Reden, vol. II, pp. 215–247, 387–406. In: Selected writings of Hermann von Helmholtz, edited, with an introduction, by Russell Kahl. Wesleyan University Press, Middletown (1971)

  5. Methods and Models in Neurophysics: Volume 80, 1st Edition - Lecture Notes of the Les Houches Summer School 2003. Elsevier Science (2005)

  6. Stevens, S.S.: On the psychophysical law. Psychol. Rev. 64, 153–181 (1957)

    Article  Google Scholar 

  7. Stevens, S.S.: To honor Fechner and repeal his law. Science 133, 80–88 (1961)

    Article  ADS  Google Scholar 

  8. Goldstein, E.B.: Sensation and Perception, 3rd edn. Wadsworth, Belmont, CA (1989)

    Google Scholar 

  9. Examples and some elements of criticism are for instance collected on the webpages of psychology.wikia.org/wiki/Stevens’\_power\_law

  10. Quote associated to E.B. Titchener. Source: Encyclopaedia Britannica, online August 2020

  11. Cope, F.W.: Derivation of the Weber-Fechner law and Loewenstein equation as the steady-state response of an Elovich solid state biological system. Bull. Math. Biol. 38, 111–118 (1976)

    Article  Google Scholar 

  12. McLintock, I.: The Elovich equation in chemisorption kinetics. Nature 216, 1204–1205 (1967)

    Article  ADS  Google Scholar 

  13. Bhowmick, S., Shenoy, V.B.: Weber-Fechner type nonlinear behavior in zigzag edge graphene nanoribbons. Phys. Rev. B 82, 155448 (2010)

    Article  ADS  Google Scholar 

  14. Bhowmick, S., Medhi, A., Shenoy, V.B.: Sensory-organ-like response determines the magnetism of zigzag-edged honeycomb nanoribbons. Phys. Rev. B 87, 085412 (2013)

    Article  ADS  Google Scholar 

  15. Pardo-Vazquez, J.L., Castineiras-de Saa, J.R., Valente, M., Damiao, I., Costa, T., Vicente, M.I., Mendonca, A.G., Mainen, Z.F., Renart, A.: The mechanistic foundation of Weber’s law. Nat. Neurosci. 22, 1493–1502 (2019)

    Article  Google Scholar 

  16. Ratcliff, R., McKoon, G.: The diffusion decision model: theory and data for two-choice decision tasks. Neural Comput. 20, 873–922 (2008)

    Article  Google Scholar 

  17. Olsman, N., Groentoro, L.: Allosteric proteins as logarithmic sensors. PNAS 113, E4423–30 (2016)

    Article  Google Scholar 

  18. Adler, M., Alon, U.: Fold-change detection in biological systems. Curr. Opin. Syst. Biol. 8, 81–89 (2018)

    Article  Google Scholar 

  19. Kubo, R., Toda, M., Hashitsume, N.: Nonequilibrium Statistical Mechanics (1985 2nd edit. 1991)

  20. Kubo, R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957)

  21. Chandler, D.: Introduction to Modern Statistical Mechanics, 1st edn. Oxford University Press, Oxford (1987)

    Google Scholar 

  22. Maggi, C., Paoluzzi, M., Angelani, L., Di Leonardo, R.: Memory-less response and violation of the fluctuation-dissipation theorem in colloids suspended in an active bath. Sci. Rep. 7, 17588 (2017)

    Article  ADS  Google Scholar 

  23. Roldán, É., Barral, J., Martin, P., Parrondo, J.M.R., Jülicher, F.: Arrow of time in active fluctuations. arXiv:1803.04743v3 [cond-mat.stat-mech]

  24. Maes, C.: Response theory: a trajectory-based approach. Front. Phys. section Interdisciplinary Physics (2020)

  25. Baiesi, M., Maes, C., Wynants, B.: Fluctuations and response of nonequilibrium states. Phys. Rev. Lett. 103, 010602 (2009)

    Article  ADS  Google Scholar 

  26. Basu, U., Maes, C.: Nonequilibrium response and Frenesy. J. Phys. 638, 012001 (2015)

    Google Scholar 

  27. Baiesi, M., Maes, C., Wynants, B.: Nonequilibrium linear response for Markov dynamics, I: jump processes and overdamped diffusions. J. Stat. Phys. 137, 1094–1116 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  28. Baerts, P., Basu, U., Maes, C., Safaverdi, S.: The frenetic origin of negative differential response. Phys. Rev. E 88, 052109 (2013)

    Article  ADS  Google Scholar 

  29. Falasco, G., Cossetto, T., Penocchio, E., Esposito, M.: Negative differential response in chemical reactions. New J. Phys. 21, 073005 (2019)

    Article  ADS  Google Scholar 

  30. Maes, C., Netočný, K.: Nonequilibrium corrections to gradient flow. Chaos 29, 073109 (2019)

    Article  MathSciNet  ADS  Google Scholar 

  31. Fodor, É.: Marchetti, M.C.: The statistical physics of active matter: from self-catalytic colloids to living cells. Lecture notes for the international summer school “Fundamental Problems in Statistical Physics” 2017 in Bruneck. arXiv:1708.08652v3 [cond-mat.soft]

  32. Maes, C.: Frenesy: Time-symmetric dynamical activity in nonequilibria. Phys. Rep. 850, 1–33 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  33. Maes, C.: Non-Dissipative Effects in Nonequilibrium Systems. SpringerBriefs in Complexity, ISBN 978-3-319-67780-4 (2018)

  34. Maes, C., van Wieren, M.H.: Time-symmetric fluctuations in nonequilibrium systems. Phys. Rev. Lett. 96, 240601 (2006)

    Article  ADS  Google Scholar 

  35. Sartori, P., Granger, L., Lee, C.F., Horowitz, J.M.: Thermodynamic costs of information processing in sensory adaptation. PLoS Comput. Biol. 10, 1003974 (2014)

    Article  ADS  Google Scholar 

  36. Nissen, M.J.: Stimulus intensity and information processing. Percept. Psychophys. 22, 338–352 (1977)

    Article  Google Scholar 

  37. Portugal, R.D., Svaite, B.F.: Weber-Fechner law and the optimality of the logarithmic scale. Minds Mach. 21, 73–81 (2011)

    Article  Google Scholar 

  38. Scheler, G.: Logarithmic distributions prove that intrinsic learning is Hebbian. F1000Research 6, 1222 (2017)

  39. Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituut voor Theoretische Fysica, KU Leuven, Leuven, Belgium

    Christian Maes

Authors
  1. Christian Maes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Maes.

Additional information

Communicated by Udo Seifert.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maes, C. Statistical Mechanical Foundation of Weber–Fechner Laws. J Stat Phys 182, 49 (2021). https://doi.org/10.1007/s10955-021-02726-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-021-02726-0

Search

Navigation