Advertisement

Modelling of psychological behavior on the basis of ultrametric mental space: Encoding of categories by balls

  • Andrei Yu. KhrennikovEmail author
Review Articles

Abstract

In this paper we present a model of processing of mental information based on encoding by points of ultrametric space. Basic mental entities categories are encoded by ultrametric balls. Our model describes processes which take place in subconsciousness. It seems that ultrametric is a right tool for modeling of unconscious mental processes. Properties of ultrametric balls match well properties of unconscious representation of information which have been discussed in psychology.

ultrametric cognition consciousness and unconsciousness ultrametric balls ideas categories processing of mental information p-adic dynamical systems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Albeverio, A. Yu. Khrennikov and P. Kloeden, “Memory retrieval as a p-adic dynamical system,” Biosystems 49, 105–115 (1999).CrossRefGoogle Scholar
  2. 2.
    D. Amit, Modeling Brain Function (Cambridge Univ. Press, Cambridge, 1989).zbMATHGoogle Scholar
  3. 3.
    R. Ashby, Design of a brain (Chapman-Hall, London, 1952).Google Scholar
  4. 4.
    B. J. Baars, In the theater of consciousness, The workspace of mind (Oxford University Press, Oxford, 1997).CrossRefGoogle Scholar
  5. 5.
    M. Bar, “Top-down facilitation of visual object recognition,” in Neurobiology of attention, pp. 140–145, Eds. L. Itti, G. Rees and J. K. Tsotsos (Elsevier, Amsterdam, 2005).CrossRefGoogle Scholar
  6. 6.
    W. Bechtel and A. Abrahamsen, Connectionism and the mind (Basil Blackwell, Cambridge, 1991).Google Scholar
  7. 7.
    W. Bechterew, Die Funktionen der Nervencentra (Fischer, Jena, 1911).Google Scholar
  8. 8.
    J. Benois-Pineau, A. Yu. Khrennikov and N. V. Kotovich, “Segmentation of images in p-adic and Euclidean metrics,” Dokl. Akad. Nauk. 381(5), 604–609. English Translation: Doklady Mathematics 64 (3), 450–455 (2001).MathSciNetGoogle Scholar
  9. 9.
    C. Blomberg, H. Liljenstrym, B. I. B. Lindahl and P. Arhem, (Eds), “Mind and Matter: Essays from Biology, Physics and Philosophy: An Introduction,” J. Theor. Biol. 171 (1994).Google Scholar
  10. 10.
    N. Chomsky, “Formal properties of grammas,” in Handbook of Mathematical Psychology, pp. 323–418, Eds. R. D. Luce, R. R. Bush and E. Galanter (Wiley, New York, 1963).Google Scholar
  11. 11.
    P. S. Churchland and T. Sejnovski, The Computational Brain (MITP, Cambridge, 1992).Google Scholar
  12. 12.
    A. Clark, Psychological Models and Neural Mechanisms. An Examination of Reductionism in Psychology (Clarendon Press, Oxford, 1980).Google Scholar
  13. 13.
    H. Damasio and A. R. Damasio, Lesion Analysis in Neuropsychology (Oxford Univ. Press, New-York, 1989).Google Scholar
  14. 14.
    A. R. Damasio, Descartes’ Error: Emotion, Reason, and the Human Brain (Penguin, 2005).Google Scholar
  15. 15.
    B. Dragovich and A. Dragovich, “A p-adic model of DNA sequence and genetic code,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 34–41 (2009); arXiv:q-bio.GN/0607018v1 (2006).CrossRefGoogle Scholar
  16. 16.
    D. Dubischar, V. M. Gundlach, O. Steinkamp and A. Yu. Khrennikov, “A p-adic model for the process of thinking disturbed by physiological and information noise,” J. Theor. Biology 197, 451–467 (1999).CrossRefGoogle Scholar
  17. 17.
    C. Eliasmith, “The third contender: a critical examination of the dynamicist theory of cognition,” Phil. Psychology 9(4), 441–463 (1996).CrossRefGoogle Scholar
  18. 18.
    J. A. Fodor and Z. W. Pylyshyn, “Connectionism and cognitive architecture: a critical analysis,” Cognition 280, 3–17 (1988).CrossRefGoogle Scholar
  19. 19.
    S. Freud, The Interpretation of Dreams (Standard Edition, 4 and 5, 1900).Google Scholar
  20. 20.
    S. Freud, New Introductory Lectures on Psychoanalysis (Norton, New York, 1933).Google Scholar
  21. 21.
    J. M. D. Fuster, The Prefrontal Cortex: Anatomy, Physiology, and Neuropsychology of the Frontal Lobe (Lippincott-Raven, Philadelphia, 1997).Google Scholar
  22. 22.
    J. F. Herbart, The Application of Psychology to the Science of Education (Charles Scribners Sons, 1898).Google Scholar
  23. 23.
    J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 1554–2558 (1982).CrossRefMathSciNetGoogle Scholar
  24. 24.
    F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons: Modeling in the Frequency Domain (Cambridge Univ. Press, New York, 1997).zbMATHGoogle Scholar
  25. 25.
    D. Hubel and T. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” J. Physiol. 160, 106–154 (1962).Google Scholar
  26. 26.
    Ivanitsky, A. M.: “Brain’s physiology and the origin of the human’s subjective world,” J. High Nerves Activity 49,(5), 707–713 (1999).Google Scholar
  27. 27.
    A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).zbMATHGoogle Scholar
  28. 28.
    A. Yu. Khrennikov, “Human subconscious as the p-adic dynamical system,” J. Theor. Biology 193, 179–196 (1998).CrossRefGoogle Scholar
  29. 29.
    A. Yu. Khrennikov, “p-Adic dynamical systems: description of concurrent struggle in biological population with limited growth,” Dokl. Akad. Nauk 361, 752–763 (1998).zbMATHMathSciNetGoogle Scholar
  30. 30.
    A. Yu. Khrennikov, “Description of the operation of the human subconscious by means of p-adic dynamical systems,” Dokl. Akad. Nauk 365, 458–460 (1999).zbMATHMathSciNetGoogle Scholar
  31. 31.
    A. Yu. Khrennikov, “p-Adic discrete dynamical systems and collective behaviour of information states in cognitive models,” Discrete Dynamics in Nature and Society 5, 59–69 (2000).CrossRefGoogle Scholar
  32. 32.
    A. Yu. Khrennikov, “Classical and quantum mechanics on p-adic trees of ideas,” Bio Systems 56, 95–120 (2000).Google Scholar
  33. 33.
    A. Yu. Khrennikov, Classical and Quantum Mental Models and Freud’s Theory of Unconscious Mind, Series Math. Modelling in Phys., Engineering and Cognitive Sciences, 1 (Växjö Univ. Press, Växjö, 2002).Google Scholar
  34. 34.
    A. Yu. Khrennikov and N. V. Kotovich, “Representation and compression of images with the aid of the m-adic coordinate system,” Dokl. Akad. Nauk. 387(2), 159–163 (2002).MathSciNetGoogle Scholar
  35. 35.
    A. Yu. Khrennikov, Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena (Kluwer, Dordreht, 2004).zbMATHGoogle Scholar
  36. 36.
    A. Yu. Khrennikov, “Probabilistic pathway representation of cognitive information,” J. Theor. Biology 231, 597–613 (2004).CrossRefGoogle Scholar
  37. 37.
    A. Yu. Khrennikov, N. V. Kotovich and E. L. Borzistaya, “Compression of images with the aid of representation by p-adic maps and approximation by Mahler’s polynomials,” Dokl. Akad. Nauk 396(3), 305–308. English Translation: Doklady Mathematics 69 (3), 373–377 (2004).MathSciNetGoogle Scholar
  38. 38.
    A. Yu. Khrennikov, “p-Adic information space and gene expression,” in Integrative Approaches to Brain Complexity, p. 14, Eds. S. Grant, N. Heintz and J. Noebels (Wellcome Trust Publ., 2006).Google Scholar
  39. 39.
    A. Yu. Khrennikov and S. V. Kozyrev, “Genetic code on the dyadic plane,” Physica A: Stat. Mech. Appl. 381, 265–272 (2007); arXiv:q-bio/0701007.CrossRefGoogle Scholar
  40. 40.
    A. Yu. Khrennikov, “Toward an adequate mathematical model of mental space: Conscious/unconscious dynamics on m-adic trees,” Biosystems 90(3), 656–675 (2007).CrossRefGoogle Scholar
  41. 41.
    R. Lauro-Grotto, “The unconscious as an ultrametric set,” American Imago 64, 535–543 (2007).CrossRefGoogle Scholar
  42. 42.
    A. Luczak, P. Bartho, S. L. Marguet, G. Buzsaki and K. D. Hariis, “Neocortical spontaneous activity in vivo: cellular heterogeneity and sequential structure,” Preprint of CMBN, Rutgers Univ. (2007).Google Scholar
  43. 43.
    F. Murtagh, “On ultrametricity, data coding, and computation,” J. Classification 21, 167–184 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    M. Pitkänen, TGD Inspired Theory of Consciousness with Applications to Biosystems, Electronic book, http://www.physics.helsinki.fi/matpitka/cbookI.html (1998).
  45. 45.
    M. Pitkänen, “Could genetic code be understood number theoretically?” Electronic preprint, www.helsinki.fi/matpitka/pdfpool/genenumber.pdf (2006).Google Scholar
  46. 46.
    A. Revonsuo and J. Newman, “Binding and consciousness,” Consciousness and Cognition 8, 123–127 (1999).CrossRefGoogle Scholar
  47. 47.
    E. Rosch, “Cognitive reference points,” Cognitive Psychology 7, 532–547 (1975).CrossRefGoogle Scholar
  48. 48.
    J. R. Smythies, Brain Mechanisms and Behaviour (Blackwell Sc. Publ., Oxford, 1970).Google Scholar
  49. 49.
    S. M. Stringer and E. Rolls, “Invariant object recognition in the visual system with novel views of 3D objects,” Neural. Comput. 14, 2585–2596 (2002).zbMATHCrossRefGoogle Scholar
  50. 50.
    S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (Addison Wesley, 1994).Google Scholar
  51. 51.
    A. Thiele and G. Stoner, “Neuronal synchrony does not correlate with motion coherence in cortical area MT,” Nature 421, 366–370 (2003).CrossRefGoogle Scholar
  52. 52.
    T. van Gelder and R. Port, “It’s about time: Overview of the dynamical approach to cognition,” in Mind as motion: Explorations in the dynamics of cognition, pp. 1–43, Eds. T. van Gelder and R. Port, (MITP, Cambridge, Mass, 1995).Google Scholar
  53. 53.
    T. van Gelder, “What might cognition be, if not computation?” J. Philosophy 91, 345–381 (1995).CrossRefGoogle Scholar
  54. 54.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sc. Publ., Singapore, 1994).Google Scholar
  55. 55.
    R. J. Watt and W. A. Phillips, “The function of dynamical grouping in vision,” Trends Cogn. Sc. 4, 447–454 (2000).CrossRefGoogle Scholar
  56. 56.
    H. Zimmer, A. Mecklinger, and U. Lindenberger, Handbook of Binding and Memory. Perspectives from Cognitive Neuroscience (Oxford Univ. Press, Oxford, 2006).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Center forMathematical Modeling in Physics and Cognitive SciencesUniversity of VäxjöVaxjoSweden

Personalised recommendations