Skip to main content
SpringerLink
Log in

A Complex Systems Approach to an Interpretation of Dynamic Brain Activity I: Chaotic Itinerancy Can Provide a Mathematical Basis for Information Processing in Cortical Transitory and Nonstationary Dynamics

  • Conference paper
Book cover Computational Neuroscience: Cortical Dynamics (NN 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3146))

Included in the following conference series:

Abstract

The transitory activity of neuron assemblies has been observed in various areas of animal and human brains. We here highlight some typical transitory dynamics observed in laboratory experiments and provide a dynamical systems interpretation of such behaviors. Using the information theory of chaos, it is shown that a certain type of chaos is capable of dynamically maintaining the input information rather than destroying it. Taking account of the fact that the brain works in a noisy environment, the hypothesis can be proposed that chaos exhibiting noise-induced order is appropriate for the representation of the dynamics concerned. The transitory dynamics typically observed in the brain seems to appear in high-dimensional systems. A new dynamical systems interpretation for the cortical dynamics is reviewed, cast in terms of high-dimensional transitory dynamics. This interpretation differs from the conventional one, which is usually cast in terms of low-dimensional attractors. We focus our attention on, in particular, chaotic itinerancy, a dynamic concept describing transitory dynamics among “exotic attractors”, or “attractor ruins”. We also emphasize the functional significance of chaotic itinerancy.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Tsuda, I.: A hermeneutic process of the brain. Prog. Theor. Phys. Suppl. 79, 241–251 (1984)

    Article  Google Scholar 

  2. Arbib, M.A., Hesse, M.B.: The construction of reality. Cambridge University Press, London (1986)

    Book  Google Scholar 

  3. Tsuda, I.: Chaotic itinerancy as a dynamical basis of Hermeneutics of brain and mind. World Futures 32, 167–185 (1991)

    Article  Google Scholar 

  4. Tsuda, I.: Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behavioral and Brain Sciences 24, 793–847 (2001)

    Article  Google Scholar 

  5. Erdi, P.: The brain as a hermeneutic device. Biophysics 38, 179–189 (1996)

    Google Scholar 

  6. Erdi, P., Tsuda, I.: Hermeneutic approach to the brain: Process versus Device? Theoria et Historia Scientiarum 6, 307–321 (2002)

    Google Scholar 

  7. Matsumoto, K., Tsuda, I.: Noise-induced order. J. Stat. Phys. 31, 87–106 (1983)

    Article  MathSciNet  Google Scholar 

  8. Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A14, L453–457 (1981)

    Google Scholar 

  9. Freeman, W.J.: A proposed name for aperiodic brain activity: stochastic chaos. Neural Networks 13, 11–13 (2000)

    Article  MathSciNet  Google Scholar 

  10. Freeman, W.J.: Simulation of chaotic EEG patterns with a dynamic model of the olfactory system. Biol. Cybern. 56, 139–150 (1987)

    Article  Google Scholar 

  11. Skarda, C.A., Freeman, W.J.: How brains make chaos in order to make sense of the world. Behavioral and Brain Sciences 10, 161–195 (1987)

    Article  Google Scholar 

  12. Freeman, W.J.: Societies of Brains – A Study in the Neuroscience of Love and Hate. Lawrence Erlbaum Associates, Inc., Hillsdale (1995)

    Google Scholar 

  13. Freeman, W.J.: How Brains Make up Their Minds. Weidenfeld & Nicolson, London (1999)

    Google Scholar 

  14. Gray, C., Engel, A.K., Koenig, P., Singer, W.: Synchronization of oscillatory neuronal responses in cat striate cortex: Temporal properties. Visual Neuroscience 8, 337–347 (1992)

    Article  Google Scholar 

  15. Kay, L., Shimoide, K., Freeman, W.J.: Comparison of EEG time series from rat olfactory system with model composed of nonlinear coupled oscillators. Int. J. Bifurcation and Chaos 5, 849–858 (1995)

    Article  MATH  Google Scholar 

  16. Kay, L., Lancaster, L.R., Freeman, W.J.: Reafference and attractors in the olfactory system during odor recognition. Int. J. Neural Systems 7, 489–495 (1996)

    Article  Google Scholar 

  17. Freeman, W.J.: Evidence from human scalp EEG of global chaotic itinerancy. Chaos 13, 1067–1077 (2003)

    Article  Google Scholar 

  18. Ikeda, K., Otsuka, K., Matsumoto, K.: Maxwell-Bloch turbulence. Prog. Theor. Phys. 99(suppl.), 295–324 (1989)

    Google Scholar 

  19. Kaneko, K.: Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements. Physica D 41, 137–172 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tsuda, I.: Chaotic neural networks and thesaurus. In: Holden, A.V., Kryukov, V.I. (eds.) Neurocomputers and Attention I, pp. 405–424. Manchester University Press, Manchester (1991)

    Google Scholar 

  21. Tsuda, I.: Dynamic link of memories–chaotic memory map in nonequilibrium neural networks. Neural Networks 5, 313–326 (1992)

    Article  MathSciNet  Google Scholar 

  22. For example, see Kaneko, K., Tsuda, I.: Complex Systems: Chaos and Beyond – A Constructive Approach with Applications in Life Sciences. Springer, Heidelberg (2001)

    Google Scholar 

  23. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rössler, O.E.: The chaotic hierarchy. Zeit. für Naturf. 38a, 788–801 (1983)

    Google Scholar 

  25. Kaneko, K.: Dominance of Milnor attractors in globally coupled dynamical systems with more than 7 ± 2 degrees of freedom. Phys. Rev. E 66, 055201(R) (2002)

    Google Scholar 

  26. Kaneko, K., Tsuda, I.: Focus Issue on Chaotic Itinerancy. Chaos 13, 926–1164 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Freeman, W.J.: Taming chaos: Stabilization of aperiodic attractors by noise. IEEE Trans. on Circuits and Sys.: Fundamental Theor. and Appl. 44, 989–996 (1997)

    Article  MathSciNet  Google Scholar 

  28. Kozma, R.: On the constructive role of noise in stabilizing itinerant trajectories in chaotic dynamical systems. Chaos 13, 1078–1089 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  29. Milnor, J.: On the concept of attractor. Comm. Math. Phys. 99, 177–195 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: A source of nonhyperbolicity in chaotic systems. Physica D 109, 81–90 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tsuda, I., Umemura, T.: Chaotic itinerancy generated by coupling of Milnor attractors. Chaos 13, 926–936 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Matsumoto, K., Tsuda, I.: Information theoretical approach to noisy dynamics. J. Phys. A 18, 3561–3566 (1985)

    Article  MathSciNet  Google Scholar 

  33. Matsumoto, K., Tsuda, I.: Extended information in one-dimensional maps. Physica D 26, 347–357 (1987)

    Article  MATH  Google Scholar 

  34. Matsumoto, K., Tsuda, I.: Calculation of information flow rate from mutual information. J. Phys. A 21, 1405–1414 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  35. Oono, Y.: Kolmogorov-Sinai entropy as disorder parameter for chaos. Prog. Theor. Phys. 60, 1944–1947 (1978)

    Article  Google Scholar 

  36. Shaw, R.: Strange attractors, chaotic behavior, and information flow. Zeit. für Naturf. 36a, 80 (1981)

    Google Scholar 

  37. Crutchfield, J.: Inferring statistical complexity. Phys. Rev. Lett. 63, 105–108 (1989)

    Article  MathSciNet  Google Scholar 

  38. Nicolis, J.S.: Should a reliable information processor be chaotic? Kybernet 11, 393–396 (1982)

    MathSciNet  Google Scholar 

  39. Nicolis, J.: Chaos and Information Processings. World Scientific, Singapore (1991)

    Google Scholar 

  40. Nicolis, J.S., Tsuda, I.: Chaotic dynamics of information processing: The “magic number seven plus-minus two” revisited. Bull. Math. Biol. 47, 343–365 (1985)

    MATH  MathSciNet  Google Scholar 

  41. Nicolis, J., Tsuda, I.: Mathematical description of brain dynamics in perception and action. J. Consc. Studies 6, 215–228 (1999)

    Google Scholar 

  42. Miller, G.A.: The Psychology of Communication. Penguin, Harmondsworth (1974)

    Google Scholar 

  43. Tsuda, I., Kor̈ner, E., Shimizu, H.: Memory dynamics in asynchronous neural networks. Prog. Theor. Phys. 78, 51–71 (1987)

    Article  Google Scholar 

  44. Szentágothai, J.: The ‘module-concept’ in cerebral cortex architecture. Brain Res. 95, 475–496 (1975)

    Article  Google Scholar 

  45. Szentágothai, J.: The neuron network of the cerebral cortex: a functional interpretation. Proc. Roy. Soc. Lond (B) 20, 219–248 (1978)

    Article  Google Scholar 

  46. Szentágothai, J.: The modular architectonic principle of neural centers. Rev. Phys. Biochem. Pharma. 98, 11–61 (1983)

    Article  Google Scholar 

  47. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of 1/f noise. Phy. Rev. Lett. 59, 381 (1987)

    Article  MathSciNet  Google Scholar 

  48. Sauer, T.: Abstracts for SIAM Pacific Rim Dynamical Systems Conference, August 9-13, Hawaii, Maui, 51; Chaotic itinerancy based on attractors of onedimensional maps. Chaos 13, 947–952 (2000)

    Google Scholar 

  49. Buescu, J.: Exotic attractors: from Liapunov stability to riddled basins. Birkhäuser, Basel (1997)

    MATH  Google Scholar 

  50. Guckenheimer, J., Holmes, P.: Structurally stable heteroclinic cycles. Math. Proc. Camb. Phil. Soc. 103, 189–192 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  51. Feudel, U., Grebogi, C., Poon, L., Yorke, J.A.: Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors. Chaos, Solitons & Fractals 9, 171–180 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  52. Tsuda, I., Kuroda, S.: Cantor coding in the hippocampus. Japan J. Indus. Appl. Math. 18, 249–258 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  53. Tsuda, I., Kuroda, S.: A Complex Systems Approach to an Interpretation of Dynamic Brain Activity, Part II: Does Cantor coding provide a dynamic model for the formation of episodic memory? In: Érdi, P., Esposito, A., Marinaro, M., Scarpetta, S. (eds.) Computational Neuroscience: Cortical Dynamics. LNCS, vol. 3146, pp. 129–139. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  54. Aihara, K., Takabe, T., Toyoda, M.: Chaotic neural networks. Phys. Lett. A 144, 333–340 (1990)

    Article  MathSciNet  Google Scholar 

  55. Nozawa, H.: Solution of the optimization problem using the neural network model as a globally coupled map. Physica D 75, 179–189 (1994)

    Article  MATH  Google Scholar 

  56. Nara, S., Davis, P.: Chaotic wandering and search in a cycle-memory neural network. Prog. Theor. Phys. 88, 845–855 (1992)

    Article  Google Scholar 

  57. Han, S.K., Kurrer, C., Kuramoto, Y.: Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995)

    Article  Google Scholar 

  58. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifur. Chaos 10, 1171–1266 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  59. Rose, R.M., Hindmarsh, J.L.: The assembly of ionic currents in a thalamic neuron I. The three-dimensional model. Proc. Roy. Soc.Lond. B 237, 267–288 (1989)

    Article  Google Scholar 

  60. Fujii, H., Tsuda, I., Nakano, M.: Spatio-temporal Chaos in Gap Junction- Coupled Class I Neurons Exhibiting Saddle-Node Bifurcations (In Japanese, to appear in Electric Journal, Japan SIAM 2003)

    Google Scholar 

  61. Fujii, H., Tsuda, I.: Neocortical gap junction-coupled interneuron systems may induce chaotic behavior itinerant among quasi-attractors exhibiting transient synchrony. To appear in Neurocomputing (2004)

    Google Scholar 

  62. Fujii, H., Tsuda, I.: Itinerant dynamics of class I* neurons coupled by gap junctions (in this issue)

    Google Scholar 

  63. Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. Roy. Soc. Lond. B221, 87–102 (1984)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Ichiro Tsuda

  2. Department of Information and Communication Sciences, Kyoto Sangyo University, Kyoto, 603-8555, Japan

    Hiroshi Fujii

Authors
  1. Ichiro Tsuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroshi Fujii
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Center for Complex Systems Studies, Kalamazoo College, Kalamazoo, and, Department of Biophysics, Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences, MI 49006, U.S.A.

    Péter Érdi

  2. Department of Psychology, Second University of Naples, and IIASS, Via Pellegrino 19, 84019, Vietri sul Mare (SA), Italy

    Anna Esposito

  3. Dipartimento di Fisica “E.R. Caianiello”, Università degli Studi di Salerno, Via S. Allende, 84081, Baronissi (SA), Italy

    Maria Marinaro

  4. Dipartimento di Fisica “E.R. Caianiello”, Università degli Studi di Salernopar, Via S. Allende, 84081, Baronissi (SA), Italy

    Silvia Scarpetta

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Tsuda, I., Fujii, H. (2004). A Complex Systems Approach to an Interpretation of Dynamic Brain Activity I: Chaotic Itinerancy Can Provide a Mathematical Basis for Information Processing in Cortical Transitory and Nonstationary Dynamics. In: Érdi, P., Esposito, A., Marinaro, M., Scarpetta, S. (eds) Computational Neuroscience: Cortical Dynamics. NN 2003. Lecture Notes in Computer Science, vol 3146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27862-7_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-27862-7_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22566-9

  • Online ISBN: 978-3-540-27862-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation