Chaos and Neural Networks

  • E. Labos
Part of the NATO ASI Series book series (NSSA, volume 138)


Nervous systems are strongly connected networks of building modules the neurons. The irregular or periodic neural autoactivity might take its origin either from units or networks.

The formal concept of network presented here is a separable system of a finite number of units or component variables (cells). These ‘state-variables’ interact, similarly to the nerve cells. This means that their future is determined by the past history of a set of other variables. In models the different sets of formal variables may correspond either to real units (cells) or real networks of neurons.

In specific cases regular (stable) units may become irregular or unstable when coupled into nets. In other examples unstable or irregular unit activities may turn into stable or periodic functioning. The prediction of the fate of the interconnected units in a network in some cases is possible. Nevertheless, no general theory of the possible consequences of interconnections is now available.


Threshold Gate Columnar Permutation Spike Sequence Silent Unit Building Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    T.H. Bullock, The origins of patterned nervous discharges, Behaviour 17: 48–59 (1961).CrossRefGoogle Scholar
  2. [2]
    G.J. Chaitin, Randomness and Mathematical Proof, Sci. Am. May: 47–52 (1975).Google Scholar
  3. [3]
    P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progr. Physics 1, Birkhauser, Boston (1980).Google Scholar
  4. [4]
    M.R. Carey and D.S. Johnson, “Computers and Intractability”, Freeman and Co., San Francisco (1979).Google Scholar
  5. [5]
    A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117: 500–544 (1952).PubMedGoogle Scholar
  6. [6]
    A.V. Holden, Why is the nervous system not as chaotic as it should be? in “Dynamic phenomena in Neurochemistry and Neurophysics: Theoretical Aspects”, Proceedings of a Workshop, ed. P`. Erdi, pp 6–11, Press of KFKI, Budapest 1984.Google Scholar
  7. [7]
    S.C. Kleene, Representation of events in nerve nets and finite automata, in “Automata Studies”, eds. C.E. Shannon and J. McCarthy, 34–41 (1956).Google Scholar
  8. [8]
    A.N. Kolmogoroff, Three approaches for defining the concept of information quantity, Information Transmission 1: 3–11 (1965).Google Scholar
  9. [9]
    E. Labos, Periodic and non-periodic motions in different classes of formal neuronal networks and chaotic spike generators, in “Cybernetics and System Research 2”, ed. R. Trappl, 237–243, Elsevier, Amsterdam (1984).Google Scholar
  10. [10]
    E. Labos, Optimal Design of Neuronal Networks, in “Neural Communication and Control”, eds. Gy. Szekely et al., Adv. Physiol. Sci. 30:127–153 (1980a).Google Scholar
  11. [11]
    E. Labos, Effective Extraction of Information Included in Network Descriptions and Neural Spike Records, in “Biomathematics in 1980”, eds. L.M. Ricciardi and A.C. Scott, North-Holland, Amsterdam (1980b).Google Scholar
  12. [12]
    E. Labos, A Model of Dynamic Behaviour of Neurons and Networks, Lecture Abstracts of Annual Meeting of Hungarian Physiological Society, Budapest, 1.S4: 117 (in Hungarian), (1981).Google Scholar
  13. [13]
    E. Labos, Spike Generating Dynamical Systems and Networks, Proceedings of a Symposium on The Mathematics of Dynamic Processes held in Sopron (Hungary), 1985 (in press).Google Scholar
  14. [14]
    E. Labos and E. Nogradi, Examples of computer-aided exploration and design of dynamical systems in neurosciences: design of optimal nets, Proceedings of a Workshop on Dynamical Systems and Environmental Models held in Wartburg (DDR), 1986 (in press).Google Scholar
  15. [15]
    E. Labos, Self-dual and other structures of optimal cycles in neural network behaviour, in “Cybernetics and Systems ‘86” ed. R. Trappl, pp 367–374, Reidel, Dordrech (1986).Google Scholar
  16. [16]
    R. May, Simple mathematical models with very complicated dynamics, Nature 261: 459–467 (1976).PubMedCrossRefGoogle Scholar
  17. [17]
    W.S. McCulloch and W.H. Pitts, A Logical Calculus of the Ideas Imminent in Nervous Activity, Bull. Math. Biophys. 5: 115–133 (1943).CrossRefGoogle Scholar
  18. [18]
    M.L. Minsky, “Computation: Finite and Infinite Machines” Englewood Cliffs, N.J., Prentice Hall (1967).Google Scholar
  19. [19]
    S. Muroga, “Threshold Logics and Its Application”, WileyInterscience, New York (1971).Google Scholar
  20. [20]
    E. Nogradi and E. Labos, Simulations of Spontaneous Neuronal Activity by Pseudo-Random Functions, Lecture Abstracts of the Ann. Meeting of Hungarian Phys. Soc., Budapest I.P. 74: 151 (1981).Google Scholar
  21. [21]
    E. Nogradi and E. Labos, Pseudo-Random Interval Maps for Simulation of Normal and Exotic Neuronal Activities, in “Cybernetics and Systems ‘86”, Reidel, Dordrecht, pp 427–434 (1986).Google Scholar
  22. [22]
    E. Nogradi, personal communication.Google Scholar
  23. [23]
    L.F. Olsen and H. Degn, Chaos in Biological Systems, Quarterly Review Of Biophysics 18: 165–225 (1985).CrossRefGoogle Scholar
  24. [24]
    E. Ott, Strange attractors and chaotic motion of dynamical systems, Rev. Mod. Phys. 53: 655–671 (1981).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • E. Labos
    • 1
  1. 1.1st Dept. of AnatomySemmelweis Medical SchoolBudapestHungary

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