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Mean-field equations, bifurcation map and chaos in discrete time, continuous state, random neural networks

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Abstract

The dynamical behaviour of a very general model of neural networks with random asymmetric synaptic weights is investigated in the presence of random thresholds. Using mean-field equations, the bifurcations of the fixed points and the change of regime when varying control parameters are established. Different areas with various regimes are defined in the parameter space. Chaos arises generically by a quasi-periodicity route.

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References

  • Amari, S. (1972). Characteristics of random nets of analog neuron-like elements. IEEE Trans. Syst. Man. Cyb., Vol. SMC-2, N0 5.

  • Babloyantz, A., C. Nicolis and J.M. Salazar (1985). Evidence of chaotic dynamics of brain activity during the sleep cycle. Phys. Lett. 111A: 152–156.

    Google Scholar 

  • Cessac, B., B. Doyon, M. Quoy and M. Samuelides (1994). Mean-field equations, bifurcation map and route to chaos in discrete time neural networks. Physica D 74: 24–44.

    Google Scholar 

  • Doyon, B. (1992). On the existence and the role of chaotic processes in the nervous system. Acta Biotheoretica 40: 113–119.

    Google Scholar 

  • Doyon, B., B. Cessac, M. Quoy and M. Samuelides (1993). Control of the transition to chaos in neural networks with random connectivity. Int. J. Bifurc. Chaos. 3: 279–291.

    Google Scholar 

  • Doyon, B., B. Cessac, M. Quoy and M. Samuelides (1994). On bifurcations and chaos in random neural networks. Acta Biotheoretica 42: 215–225.

    Google Scholar 

  • Eckhorn R., R. Bauer, W. Jordan, M. Brosch, W. Kruse, M. Munk and H.J. Reitboeck (1988). Coherent oscillations: A mechanism of feature linking in the visual cortex? Multiple electrode and correlation analysis in the cat. Biol. Cybern. 60: 121–130.

    Google Scholar 

  • Gallez, D. and A. Babloyantz (1991). Predictability of human EEG: a dynamical approach. Biol. Cybern. 64: 381–392.

    Google Scholar 

  • Geman, S. (1982). Almost Sure Stable Oscillations In A Large System Of Randomly Coupled Equations. SIAM J. Appl. Math. 42: 695–703.

    Google Scholar 

  • Gray, C.M. and W. Singer (1987). Stimulus-dependent neuronal oscillations in the cat visual cortex area 17. 2nd IBRO-Congress, Neurosci. Suppl. 1301P.

  • Gray, C.M., P. Koenig, A.K. Engel and W. Singer (1989). Oscillatory responses in cat visual cortex exhibit intercolumnar synchronisation which reflects global stimulus properties. Nature 338: 334–337.

    Google Scholar 

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

    Google Scholar 

  • Sompolinsky, H., A. Crisanti and H.J. Sommers (1988). Chaos in random neural networks. Phys. Rev. Lett. 61: 259–262.

    Google Scholar 

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Doyon, B., Cessac, B., Quoy, M. et al. Mean-field equations, bifurcation map and chaos in discrete time, continuous state, random neural networks. Acta Biotheor 43, 169–175 (1995). https://doi.org/10.1007/BF00709441

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