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.
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References
Tsuda, I.: A hermeneutic process of the brain. Prog. Theor. Phys. Suppl. 79, 241–251 (1984)
Arbib, M.A., Hesse, M.B.: The construction of reality. Cambridge University Press, London (1986)
Tsuda, I.: Chaotic itinerancy as a dynamical basis of Hermeneutics of brain and mind. World Futures 32, 167–185 (1991)
Tsuda, I.: Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behavioral and Brain Sciences 24, 793–847 (2001)
Erdi, P.: The brain as a hermeneutic device. Biophysics 38, 179–189 (1996)
Erdi, P., Tsuda, I.: Hermeneutic approach to the brain: Process versus Device? Theoria et Historia Scientiarum 6, 307–321 (2002)
Matsumoto, K., Tsuda, I.: Noise-induced order. J. Stat. Phys. 31, 87–106 (1983)
Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A14, L453–457 (1981)
Freeman, W.J.: A proposed name for aperiodic brain activity: stochastic chaos. Neural Networks 13, 11–13 (2000)
Freeman, W.J.: Simulation of chaotic EEG patterns with a dynamic model of the olfactory system. Biol. Cybern. 56, 139–150 (1987)
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)
Freeman, W.J.: Societies of Brains – A Study in the Neuroscience of Love and Hate. Lawrence Erlbaum Associates, Inc., Hillsdale (1995)
Freeman, W.J.: How Brains Make up Their Minds. Weidenfeld & Nicolson, London (1999)
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)
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)
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)
Freeman, W.J.: Evidence from human scalp EEG of global chaotic itinerancy. Chaos 13, 1067–1077 (2003)
Ikeda, K., Otsuka, K., Matsumoto, K.: Maxwell-Bloch turbulence. Prog. Theor. Phys. 99(suppl.), 295–324 (1989)
Kaneko, K.: Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements. Physica D 41, 137–172 (1990)
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)
Tsuda, I.: Dynamic link of memories–chaotic memory map in nonequilibrium neural networks. Neural Networks 5, 313–326 (1992)
For example, see Kaneko, K., Tsuda, I.: Complex Systems: Chaos and Beyond – A Constructive Approach with Applications in Life Sciences. Springer, Heidelberg (2001)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)
Rössler, O.E.: The chaotic hierarchy. Zeit. für Naturf. 38a, 788–801 (1983)
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)
Kaneko, K., Tsuda, I.: Focus Issue on Chaotic Itinerancy. Chaos 13, 926–1164 (2003)
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)
Kozma, R.: On the constructive role of noise in stabilizing itinerant trajectories in chaotic dynamical systems. Chaos 13, 1078–1089 (2003)
Milnor, J.: On the concept of attractor. Comm. Math. Phys. 99, 177–195 (1985)
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)
Tsuda, I., Umemura, T.: Chaotic itinerancy generated by coupling of Milnor attractors. Chaos 13, 926–936 (2003)
Matsumoto, K., Tsuda, I.: Information theoretical approach to noisy dynamics. J. Phys. A 18, 3561–3566 (1985)
Matsumoto, K., Tsuda, I.: Extended information in one-dimensional maps. Physica D 26, 347–357 (1987)
Matsumoto, K., Tsuda, I.: Calculation of information flow rate from mutual information. J. Phys. A 21, 1405–1414 (1988)
Oono, Y.: Kolmogorov-Sinai entropy as disorder parameter for chaos. Prog. Theor. Phys. 60, 1944–1947 (1978)
Shaw, R.: Strange attractors, chaotic behavior, and information flow. Zeit. für Naturf. 36a, 80 (1981)
Crutchfield, J.: Inferring statistical complexity. Phys. Rev. Lett. 63, 105–108 (1989)
Nicolis, J.S.: Should a reliable information processor be chaotic? Kybernet 11, 393–396 (1982)
Nicolis, J.: Chaos and Information Processings. World Scientific, Singapore (1991)
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)
Nicolis, J., Tsuda, I.: Mathematical description of brain dynamics in perception and action. J. Consc. Studies 6, 215–228 (1999)
Miller, G.A.: The Psychology of Communication. Penguin, Harmondsworth (1974)
Tsuda, I., Kor̈ner, E., Shimizu, H.: Memory dynamics in asynchronous neural networks. Prog. Theor. Phys. 78, 51–71 (1987)
Szentágothai, J.: The ‘module-concept’ in cerebral cortex architecture. Brain Res. 95, 475–496 (1975)
Szentágothai, J.: The neuron network of the cerebral cortex: a functional interpretation. Proc. Roy. Soc. Lond (B) 20, 219–248 (1978)
Szentágothai, J.: The modular architectonic principle of neural centers. Rev. Phys. Biochem. Pharma. 98, 11–61 (1983)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of 1/f noise. Phy. Rev. Lett. 59, 381 (1987)
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)
Buescu, J.: Exotic attractors: from Liapunov stability to riddled basins. Birkhäuser, Basel (1997)
Guckenheimer, J., Holmes, P.: Structurally stable heteroclinic cycles. Math. Proc. Camb. Phil. Soc. 103, 189–192 (1988)
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)
Tsuda, I., Kuroda, S.: Cantor coding in the hippocampus. Japan J. Indus. Appl. Math. 18, 249–258 (2001)
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)
Aihara, K., Takabe, T., Toyoda, M.: Chaotic neural networks. Phys. Lett. A 144, 333–340 (1990)
Nozawa, H.: Solution of the optimization problem using the neural network model as a globally coupled map. Physica D 75, 179–189 (1994)
Nara, S., Davis, P.: Chaotic wandering and search in a cycle-memory neural network. Prog. Theor. Phys. 88, 845–855 (1992)
Han, S.K., Kurrer, C., Kuramoto, Y.: Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995)
Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifur. Chaos 10, 1171–1266 (2000)
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)
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)
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)
Fujii, H., Tsuda, I.: Itinerant dynamics of class I* neurons coupled by gap junctions (in this issue)
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)
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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
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DOI: https://doi.org/10.1007/978-3-540-27862-7_6
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