Biological Cybernetics

, Volume 68, Issue 4, pp 363–374 | Cite as

A biologically motivated and analytically soluble model of collective oscillations in the cortex

I. Theory of weak locking
  • Wulfram Gerstner
  • Raphael Ritz
  • J. Leo van Hemmen


A model of an associative network of spiking neurons with stationary states, globally locked oscillations, and weakly locked oscillatory states is presented and analyzed. The network is close to biology in the following sense. First, the neurons spike and our model includes an absolute refractory period after each spike. Second, we consider a distribution of axonal delay times. Finally, we describe synaptic signal transmission by excitatory and inhibitory potentials (EPSP and IPSP) with a realistic shape, that is, through a response kernel. During retrieval of a pattern, all active neurons exhibit periodic spike bursts which may or may not be synchronized (‘locked’) into a coherent oscillation. We derive an analytical condition of locking and calculate the period of collective activity during oscillatory retrieval. In a stationary retrieval state, the overlap assumes a constant value proportional to the mean firing rate of the neurons. It is argued that in a biological network an intermediate scenario of “weak locking” is most likely.


Active Neuron Firing Rate Collective Activity Signal Transmission Refractory Period 
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. Abbott LF (1991) Realistic synaptic inputs for model neural networks. Network 2:245–258Google Scholar
  2. Amit DJ (1989) Modeling brain function. Cambridge University Press, CambridgeGoogle Scholar
  3. Amit DJ, Gutfreund H, Sompolinsky H (1985) Spin-glass models of neural networks. Phys Rev A 32:1007–1032Google Scholar
  4. Amit DJ, Gutfreund H, Sompolinsky H (1987) Statistical mechanics of neural networks near saturation. Ann Phys (NY) 173:30–67Google Scholar
  5. Baird B (1986) Nonlinear dynamics of pattern formation and pattern recognition in the rabbit olfactory bulb. Physica D 22:150–175Google Scholar
  6. Bindman L, Christofi G, Murphy K, Nowicky A (1991) Long-term potentiation (LTP) and depression (LTD) in the neocortex and hippocampus: an overview. In: Stone TW (ed), Aspects of synaptic transmission, vol 1. Taylor & Francis, LondonGoogle Scholar
  7. Brown TH, Ganong AH, Kairiss EW, Keenan CL, Kelso SR (1989) Long-term potentation in two synaptic systems of the hippocampal brain slice. In: Byrne JH, Berry WO (eds) Neural models of plasticity. Academic Press, San Diego, pp 266–306Google Scholar
  8. Buhmann J, Schulten K (1986) Association, recognition and storage in a model network with physiological neurons. Biol Cybern 54:319–335Google Scholar
  9. Buhmann J (1989) Oscillations and low firing rates in associative memory neural networks. Phys Rev A 40:4145–4148Google Scholar
  10. Bush PC, Douglas RJ (1991) Synchronization of bursting action potential discharge in a model network of neocortical neurons. Neural Comput 3:19–30Google Scholar
  11. Domany E, vanHemmen JL, Schulten K (eds) (1991) Models of neural networks. Springer, Berlin Heidelberg New YorkGoogle Scholar
  12. Eckhorn R, Bauer R, Jordan W, Brosch M, Kruse W, Munk M, Reitboeck HJ (1988) Coherent oscillations: A mechanism of feature linking in the visual cortex? Biol Cybern 60:121–130Google Scholar
  13. Ekeberg Ö, Wallen P, Lansner A, Traven H, Brodin L, Grillner S (1991) A computer based model for realistic simulations of neural networks. Biol Cybern 65:81–90Google Scholar
  14. Engel AK, König P, Singer W (1991) Direct physiological evidence for scene segmentation by temporal coding. Proc Natl Acad Sci USA 88:9136–9140Google Scholar
  15. Gerstner W, van Hemmen JL (1992a) Associative memory in a network of ‘spiking’ neurons. Network 3:139–164Google Scholar
  16. Gerstner W, van Hemmen JL (1992b) Universality in neural networks: The importance of the mean firing rate. Biol Cybern 67:195–205Google Scholar
  17. Gray CM, Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc Natl Acad Sci USA 86:1698–1702Google Scholar
  18. Gray CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar syncronization which reflects global stimulus properties. Nature 338:334–337Google Scholar
  19. Hansel D, Sompolinski H (1992) Synchronization and computation in a chaotic neural network. Phys Rev Lett 68:718–721Google Scholar
  20. Hebb DO (1949) The organization behavior. Wiley, New YorkGoogle Scholar
  21. Hemmen JL van, Kühn R (1986) Nonlinear neural networks. Phys Rev Lett 57:913–916Google Scholar
  22. Hemmen JL van, Grensing D, Huber A, Kühn R (1986) Elementary solution of classical spin glass models. Z Phys B-Condensed Matter 65:53–63Google Scholar
  23. Hemmen JL van, Grensing D, Huber A, Kühn R (1988) Nonlinear neural networks I and II. J Stat Phys 50:231–257 and 259–293Google Scholar
  24. Hemmen JL van, Gerstner W, Herz AVM, Kühn R, Sulzer B, Vaas M (1990) Encoding and decoding of patterns which are correlated in space and time. In: Dorffner G (ed) Konnektionismus in Artificial Intelligence und Kognitionsforschung, Springer, Berlin Heidelberg New YorkGoogle Scholar
  25. Hemmen JL van, Gerstner W, Ritz R (1992) A ‘microscopic’ model of collective oscillations in the cortex. In: Taylor JG, Caianiello EK, Cotterill RNJ, Clark JW (eds) Neural network dynamics. Springer, Berlin Heidelberg New York, pp 250–257Google Scholar
  26. Herz A, Sulzer B, Kühn R, Hemmen JL van (1988) The Hebb rule: Representation of static and dynamic objects in neural nets. Europhys Lett 7:663–669 (1989) Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets. Biol Cybern 60:457–467Google Scholar
  27. Hodgkin AL, Huxley AF (1952) A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. J Physiol (London) 117:500–544Google Scholar
  28. Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA 79:2554–2558Google Scholar
  29. Hopfield JJ (1984) Neurons with graded response have computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092Google Scholar
  30. Horn D, Usher M (1989) Neural networks with dynamical thresholds. Phys Rev A 40:1036–1040Google Scholar
  31. Horn D, Sagi D, Usher M (1991) Segmentation, binding and illusory conjunctions. Neural Comput 3:510–525Google Scholar
  32. Jack JJB, Noble D, Tsien RW (1975) Electric current flow in excitable cells, Clarendon Press, OxfordGoogle Scholar
  33. Kitajima T, Hara K (1990) A model of the mechanisms of long-term potentiation in the hippocampus. Biol Cybern 64:33–39Google Scholar
  34. König P, Schillen TB (1991) Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization. Neural Comput 3:155–166Google Scholar
  35. Kreiter AK, Singer W (1992) Oscillatory neuronal response in the visual cortex of the awake macaque monkey. Eur J Neurosci 4:369–375Google Scholar
  36. Kuffler SW, Nicholls JG, Martin AR (1984) From neuron to brain, 2nd Ed. Sinauer, SunderlandGoogle Scholar
  37. Kuramoto Y, Nishikawa I (1987) Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillatory communities. J Stat Phys 49:569–605Google Scholar
  38. Kurrer C, Nieswand B, Schulten K (1990) A model for synchroneous activity in the visual cortex. In: Babloyantz A (ed) Self-organization, emerging properties and learning. Plenum Press, New YorkGoogle Scholar
  39. Larson J, Lynch G (1986) Induction of synaptic potentiation in Hippocampus by patterned stimulation involves two events. Science 232:985–988Google Scholar
  40. Lisman J (1989) A mechanism for Hebb and anti-Hebb processes underlying learning and memory. Proc Natl Acad Sci USA 86:9574–9578Google Scholar
  41. Malinow R, Miller JP (1986) Synaptic hyperpolarization during conditioning reversibly blocks induction of long-term potentiation. Nature 320:529–530Google Scholar
  42. Malsburg C von der, Schneider W (1986) A neural cocktail-party processor. Biol Cybern 54:29–40Google Scholar
  43. McCormick DA (1990) Membrane properties and neurotransmitter actions. In: Sheperd GM (ed) The synaptic organization of the brain, 3rd Ed. Oxford University Press, OxfordGoogle Scholar
  44. Pawelzik K (1991) Nichtlineare Dynamik und Hirnaktivität. Verlag Harri Deutsch, FrankfurtGoogle Scholar
  45. Rall W (1964) Theoretical significance of dendritic trees for neuronal input-output relations. In: Reiss RF (ed) Neural theory and modeling. Stanford University Press, Stanford, pp 73–97Google Scholar
  46. Ritz R (1991) Kollektive Oszillationen in neuronalen Netzen. Diplomarbeit, Technische Universität MünchenGoogle Scholar
  47. Ritz R, Gerstner W, Hemmen JL van (1993) A biologically motivated and analytically soluble model of collective oscillations in the cortex: II. Association, segmentation, and binding. IV. Columnar organization (in preparation)Google Scholar
  48. Sompolinsky H, Golomb D, Kleinfeld D (1990) Global processing of visual stimuli in a neural network of coupled oscillators. Proc Natl Acad Sci USA 87:7200–7204Google Scholar
  49. Schuster HG, Wagner P (1990a) A model for neuronal oscillations in the visual cortex 1. Mean-field theory and derivation of the phase equations. Biol Cybern 64:77–82Google Scholar
  50. Schuster HG, Wagner P (1990b) A model for neuronal oscillations in the visual cortex 2. Phase description and feature dependent synchronization. Biol Cybern 64:83–85Google Scholar
  51. Sporns O, Gally JA, Reeke GN, Edelman GM (1989) Reentrant signaling among simulated neuronal groups leads to coherency in their oscillatory activity. Proc Natl Acad Sci USA 86:7265–7269Google Scholar
  52. Sporns O, Tononi G, Edelman GM (1991) Modeling perceptual grouping and figure-ground segregation by means of active reentrant connections. Proc Natl Acad Sci USA 88:129–133Google Scholar
  53. Trefz T (1991) Oszillationen im Cortex. Diplomarbeit, Technische Universität MünchenGoogle Scholar
  54. Wang D, Buhmann J, von der Malsburg C (1990) Pattern segmentation in associative memory. Neural Comput 2:94–106Google Scholar
  55. Wilson AM, Bower JM (1991) A computer simulations of oscillatory behavior in primary visual cortex. Neural Comput 3:498–509Google Scholar
  56. Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24Google Scholar
  57. Yamaguchi Y, Shimizu H (1984) Theory of selfsynchronization in the presence of native frequency distribution and external noises. Physica D 11:212–226Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Wulfram Gerstner
    • 1
  • Raphael Ritz
    • 1
  • J. Leo van Hemmen
    • 1
  1. 1.Institut für Physik der Technischen Universität MünchenGarching bei MünchenGermany

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