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From neuron to neural networks dynamics

  • B. Cessac
  • M. Samuelides
Article

Abstract.

This paper presents an overview of some techniques and concepts coming from dynamical system theory and used for the analysis of dynamical neural networks models. In a first section, we describe the dynamics of the neuron, starting from the Hodgkin-Huxley description, which is somehow the canonical description for the “biological neuron”. We discuss some models reducing the Hodgkin-Huxley model to a two dimensional dynamical system, keeping one of the main feature of the neuron: its excitability. We present then examples of phase diagram and bifurcation analysis for the Hodgin-Huxley equations. Finally, we end this section by a dynamical system analysis for the nervous flux propagation along the axon. We then consider neuron couplings, with a brief description of synapses, synaptic plasticity and learning, in a second section. We also briefly discuss the delicate issue of causal action from one neuron to another when complex feedback effects and non linear dynamics are involved. The third section presents the limit of weak coupling and the use of normal forms technics to handle this situation. We consider then several examples of recurrent models with different type of synaptic interactions (symmetric, cooperative, random). We introduce various techniques coming from statistical physics and dynamical systems theory. A last section is devoted to a detailed example of recurrent model where we go in deep in the analysis of the dynamics and discuss the effect of learning on the neuron dynamics. We also present recent methods allowing the analysis of the non linear effects of the neural dynamics on signal propagation and causal action. An appendix, presenting the main notions of dynamical systems theory useful for the comprehension of the chapter, has been added for the convenience of the reader.

Keywords

Lyapunov Exponent Hopf Bifurcation European Physical Journal Special Topic Bifurcation Point Recurrent Neural Network 
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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References

  1. M. Abeles, Firing rates and well-timed events, in Models of Neural Networks II, edited by E. Domany, K. Schulten, J.L. van Hemmen (Springer, New York, 1994), Chap. 3 Google Scholar
  2. D. Amit, H. Gutfreund, H. Sompolinsky, Phys. Rev. A 32, 1007 (1985); D. Amit, H. Gutfreund, H. Sompolinsky, Phys. Rev. Lett. 55, 1530 (1985) CrossRefADSMathSciNetGoogle Scholar
  3. L.F. Abbott, S.B. Nelson, Nat. Neurosci. 3, 1178 (2000) CrossRefGoogle Scholar
  4. L.F. Abbott, T.B. Kepler, Model neurons: from Hodgkin–Huxley to Hopfield, in Statistical Mechanics of Neural Networks, edited by L. Garrido (Springer, Berlin, 1990) Google Scholar
  5. L.F. Abbott, C. van Vreeswijk, Phys. Rev. E 48, 1483 (1993) CrossRefADSGoogle Scholar
  6. E.D. Adrian, J. Physiol. (Lond.) 61, 49 (1926) Google Scholar
  7. E.D. Adrian, The Basis of Sensation (W.W. Norton, New York, 1928) Google Scholar
  8. S. Amari, IEEE Trans. Syst. Man. Cyb. SMC-2, 5 (1972) Google Scholar
  9. S. Amari, A Method of Statistical Neurodynamics (Kybernetik, 1974) Google Scholar
  10. S. Amari, K. Yoshida, K. Kanatani, SIAM J. Appl. Math. 33, 95 (1977) MATHCrossRefMathSciNetGoogle Scholar
  11. D.J. Amit, Modelling Brain Functions: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, 1989) Google Scholar
  12. V. Arnold, Équations Différentielles Ordinaires (Éditions Mir, Moscou) Google Scholar
  13. V. Arnold, Chapitre Supplémentaire de la Théorie des Équations Différentielles Ordinaires (Éditions Mir, Moscou) Google Scholar
  14. V. Arnold, A. Avez, Problèmes Ergodiques de la Mécanique Classique (Gauthier-Vilars, 1967) Google Scholar
  15. A. Babloyantz, C. Nicolis, J.M. Salazar, Phys. Lett. A 111, 152 (1985) CrossRefADSGoogle Scholar
  16. A. Babloyantz, A. Destexhe, Proc. Natl. Acad. Sci. USA 83, 3513 (1986) CrossRefADSGoogle Scholar
  17. A. Babloyantz, A. Destexhe, edited by M. Candill, C. Butler, Proc. IEEE. First Int. Conf. Neural Networks 4, 31 (1987) Google Scholar
  18. A. Babloyantz, A. Destexhe, edited by M. Markus, S. Muller, G. Nicolis, Springer Ser. Synerg. 39, 307 (1988) MathSciNetGoogle Scholar
  19. P. Bak, How Nature Works: The Science of Self-organized Criticality (Springer-Verlag, 1996; Oxford University Press, 1997) Google Scholar
  20. Ph. Blanchard, B. Cessac, T. Krüger, J. Stat. Phys. 88, 307 (1997) MATHCrossRefMathSciNetGoogle Scholar
  21. Ph. Blanchard, B. Cessac, T. Krüger, J. Stat. Phys. 98, 375 (2000) MATHCrossRefMathSciNetGoogle Scholar
  22. D.H. Chialvo, P. Bak, Neuroscience 90, 1137 (1999) CrossRefGoogle Scholar
  23. H. Berry, M. Quoy, Structure and Dynamics of Random Recurrent Neural Networks, 2005 (submitted) Google Scholar
  24. G.Q. Bi, M.M. Poo, J. Neurosci. 18, 10464 (1998) Google Scholar
  25. G. Basti, A. Perrone, IEEE I, 657 (1989) Google Scholar
  26. M. Benaim, Dynamiques d'activation et dynamiques d'apprentissage des réseaux de neurones, Thèse de doctorat, Toulouse (1992) Google Scholar
  27. W. Bialek, F. Rieke, R.R. de Ruyter van Stevenick, D. Warland, Science 252, 1854 (1991) CrossRefADSGoogle Scholar
  28. K. Binder, A. Young, Rev. Mod. Phys. 58, 801 (1986) CrossRefADSGoogle Scholar
  29. L. Boltzmann, Lectures on Gas Theory (Dover, New York, 1995), Translation by S. Brush Google Scholar
  30. A.J. Bray, M.A. Moore, J. Phys. C 13, L469 (1980) Google Scholar
  31. A. Roxin, N. Brunel, D. Hansel, Phys. Rev. Lett., 2005 (in press) Google Scholar
  32. Carr, Applications of Center Manifold Theory (Springer-Verlag, New-York; Heidelberg, Berlin, 1981) Google Scholar
  33. See e.g. the web site http://elegans.swmed.edu/ and references therein Google Scholar
  34. B. Cessac, B. Doyon, M. Quoy, M. Samuelides, Physica D 74, 24 (1994) MATHCrossRefADSMathSciNetGoogle Scholar
  35. B. Cessac, J. Phys. A 27, L927 (1994) Google Scholar
  36. B. Cessac, Europhys. Lett. 26, 577 (1994) Google Scholar
  37. B. Cessac, Propriétés statistiques des dynamiques de réseaux neuromimétiques, Thèse Université Paul Sabatier, Toulouse, 1994 Google Scholar
  38. B. Cessac, J. Phys. I (France) 5, 409 (1995) CrossRefGoogle Scholar
  39. B. Cessac, Ph. Blanchard, T. Krüger, J.L. Meunier, J. Stat. Phys. 115, 1283 (2004) CrossRefMathSciNetGoogle Scholar
  40. B. Cessac, J.A. Sepulchre, Phys. Rev. E 70, 056111 (2004) CrossRefADSGoogle Scholar
  41. B. Cessac, J.A. Sepulchre, Chaos. 16, 013104 (2006) CrossRefADSMathSciNetGoogle Scholar
  42. B. Cessac, J.A. Sepulchre, Physica D, 2007 (to appear) Google Scholar
  43. B. Cessac, Dynamical and topological effects of hebbian learning in a simple neural network model (in preparation) Google Scholar
  44. B. Cessac, Some remarks about a discrete time neural network model with spiking neurons: Spontaneous dynamics (in preparation) Google Scholar
  45. B. Cessac, Some remarks about a discrete time neural network model with spiking neurons: synaptic plasticiy and thermodynamic formalism (in preparation) Google Scholar
  46. Carpenter (1977) Google Scholar
  47. J.P. Changeux, S. Dehaene, Cognition 33, 63 (1989) CrossRefGoogle Scholar
  48. N. Chernov, R. Markarian, Am. Math. Soc. (2006) Google Scholar
  49. D.R.J. Chillingworth, Differentiable Toplogy with a View to Applications (Pitman, London, 1976) Google Scholar
  50. M.A. Cohen, S. Grossberg, IEEE Trans. Syst., Man Cybernet. SMC-13 (1983) Google Scholar
  51. M. Cosnard, J. Demongeot, K. Lausberg, K. Lott, Attractors, Confiners, and Fractal Dimensions: Applications in Neuromodelling (Wuerz Publishing Ltd., 1993) in Math. Appl. Biol. Med. Google Scholar
  52. A. Crisanti, H. Sompolinsky, Phys. Rev. A 36, 4922 (1987) CrossRefADSMathSciNetGoogle Scholar
  53. A. Crisanti, H.J. Sommers, H. Sompolinsky, Chaos in Neural Networks: Chaotic solutions, 1990 (preprint) Google Scholar
  54. J. Cronin, Mathematical Aspects of Hodgkin–Huxley Theory (Cambridge University Press, Cambridge, 1987) Google Scholar
  55. O. David, K.J. Friston, NeuroImage 20, 1743 (2003) CrossRefGoogle Scholar
  56. E. Dauce, M. Quoy, B. Cessac, B. Doyon, M. Samuelides, Neural Netw. 11, 521 (1998) CrossRefGoogle Scholar
  57. E. Daucé, Adaptation dynamique et apprentissage dans des réseaux de neurones récurrents aléatoires, thèse troisième cycle (Toulouse, 2000) Google Scholar
  58. A. Guillot, E. Daucé (Éds), Approche Dynamique de la Cognition Artificielle (Lavoisier, Paris, 2002) Google Scholar
  59. A.M.O. De Almeida, D.J. Thouless, J. Phys. A 11, 983 (1978) CrossRefADSGoogle Scholar
  60. B. Doyon, B. Cessac, M. Quoy, M. Samuelides, Acta Biotheoretica. 42, 215 (1994) CrossRefGoogle Scholar
  61. S. Doi, S. Kumagai, Non linear dynamics of small scale biophysical neural networks, in Biophysical Neural Networks, edited by R.R. Poznanski (Mary Ann Liebert, Inc., Larchmont, NY, 2001), p. 261 Google Scholar
  62. Y. Dudai, The Neurobiology of Memory (Oxford University Press, Oxford, 1989) Google Scholar
  63. B. Doyon, B. Cessac, M. Quoy, M. Samuelides, Int. J. Bifurc. Chaos 3, 279 (1993) MATHCrossRefGoogle Scholar
  64. J.P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985) CrossRefADSMathSciNetGoogle Scholar
  65. R. Eckhorn, R. Bauer, W. Jordan, M. Brosch, W. Kruse, M. Munk, H.J. Reitboeck, Biol. Cybernet. 60, 121 (1988) CrossRefGoogle Scholar
  66. A. Edelman, The circular law and the probability that a random matrix has k real eigenvalues, 1 (1993) Google Scholar
  67. A. Edwards, J. Phys. A 11, 983 (1978) CrossRefGoogle Scholar
  68. G.B. Ermentrout, N. Kopell, SIAM J. Math. Anal. 15, 215 (1984) MATHCrossRefMathSciNetGoogle Scholar
  69. W.J. Freeman Biol. Cyber. 56 (1987) 139-150 Google Scholar
  70. W.J. Freeman, Y. Yao, B. Burke, Neural Netw. 1, 277-288 (1988) CrossRefGoogle Scholar
  71. R. FitzHugh, Biophys. J. 1, 445-466 (1961) Google Scholar
  72. R.M. Fitzsimonds, H.J. Song, M.M. Poo, Nature 31 (1997); 388 (6641), 427-8 CrossRefGoogle Scholar
  73. A. Bovier, V. Gayrard, J. Stat. Phys. 69, 597-627 (1993) CrossRefMathSciNetGoogle Scholar
  74. D. Gallez, A. Babloyantz, Biol. Cybern. 64, 381-392 (1991) CrossRefGoogle Scholar
  75. J.M. Gambaudo, C. Tresser, Transition vers le chaos pour des applications continues de degré un du cercle, in Le chaos, théorie et expériences, Collection CEA (1988) Google Scholar
  76. S. Geman, Ann. Prob. 8, 252-261 (1980) MATHMathSciNetGoogle Scholar
  77. The Genesis simulator, http://www.genesis-sim.org/GENESIS/genesis.html Google Scholar
  78. W. Gerstner, W.M. Kistler, Spiking Neuron Models. Single Neurons, Populations, Plasticity (Cambridge University Press, Cambridge, 2002) Google Scholar
  79. V.L. Girko, Theor. Prob. Appl. 29, 694-706 (1984) CrossRefMathSciNetGoogle Scholar
  80. J.-L. Gouzé, J. Biol. Syst. 6, 11–15 (1998) MATHCrossRefGoogle Scholar
  81. C.M. Gray, W. Singer, Neurosci. Suppl. 1301P (1987) Google Scholar
  82. C.M. Gray, P. Koenig, A.K. Engel, W. Singer, Nature 338, 334-337 (1989) CrossRefADSGoogle Scholar
  83. F. Grimbert, O. Faugeras, Neural Comput. (2006) to appear Google Scholar
  84. J. Guckenheimer, P. Holmes, Non Linear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (Springer-Verlag, Berlin, 1983) Google Scholar
  85. J. Guckenheimer, I.S. Labouriau, Bull. Math. Biol. 55, 937-952 (1993) MATHGoogle Scholar
  86. J. Guckenheimer, O. Ricardo, SIAM J. Appl. Dyn. Syst. 1, 105-114 (2002) MATHCrossRefMathSciNetGoogle Scholar
  87. J. Guckenheimer, P. Worfolk, Dynamical systems: Some computational problems, NATO ASI, Bifurcations and Periodic Orbits of Vector Fields, Conference Proceedings and http://arxiv.org/abs/chao-dyn/9304010' (1993) Google Scholar
  88. B. Hassard, J. Theoret. Biol. 71, 401-420 (1978) CrossRefMathSciNetGoogle Scholar
  89. D.O. Hebb, The Organization of Behaviour (John Wiley & Sons, New York, 1949) Google Scholar
  90. B. Hille, Ionic Channels of Excitable Membranes, 2nd edn. (Sinauer Associates, Sunderland, Mass, 1992) Google Scholar
  91. L.J. Graham, The Surf-Hippo Neuron Simulation System, http://www.neurophys.biomedicale.univ-paris5.fr/~graham/surf-hippo-files/Surf-Hippo.README.html Google Scholar
  92. M.W. Hirsch Neural Networks 2, 331-349 (1989) Google Scholar
  93. A.L. Hodgkin, A.F. Huxley, J. Physiol. (Lond.) 116, 449-472 (1952) Google Scholar
  94. A.L. Hodgkin, A.F. Huxley, J. Physiol. (Lond.) 117, 500-544 (1952) Google Scholar
  95. J.J. Hopfield, Proc. Natl. Acad. Sci. USA 79, 2554-2558 (1981) CrossRefADSMathSciNetGoogle Scholar
  96. J.J. Hopfield, Nature 376, 33-36 (1995) CrossRefADSGoogle Scholar
  97. J.J. Hopfield, Tank, Biol. Cybern. 52, 141-152 (1985) MATHMathSciNetGoogle Scholar
  98. F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks (Springer-Verlag, New York, 1997) Google Scholar
  99. G. Iooss, A. Chenciner, ARMA 69, 109-198 (1979) MATHCrossRefADSMathSciNetGoogle Scholar
  100. E.M. Izhikevich, Bifurcations in brain dynamics, Ph.D. thesis, Department of Mathematics, Michigan State University (1996) Google Scholar
  101. J.J. Jack, D. Noble, R.W. Tsien, Electric current flow in excitable cells (Clarendon Press, Oxford 1975) Google Scholar
  102. B.H. Jansen, G. Rit, Biol. Cybern. 73, 357-366 (1995) MATHGoogle Scholar
  103. A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Kluwer, 1998) Google Scholar
  104. J. Keener, J. Sneyd, Mathematical Physiology, volume 8 of Interdisciplinary Applied Mathematics (Springer, New York, 1998) Google Scholar
  105. T.B. Kepler, L.F. Abbott, E. Marder, Biol. Cybern. 66, 381-387 (1992) MATHCrossRefGoogle Scholar
  106. S.R. Kelso, A.H. Ganong, T.H. Brown, Proc. Natl. Acad. Sci. USA 83, 5326-5330 (1986) CrossRefADSGoogle Scholar
  107. C. Koch, Biophysics of Computation (Oxford University Press, New York, 1999) Google Scholar
  108. C. Koch, O. Bernander, R.J. Douglas, J. Comput. Neurosci. 2, 63-82 (1995) MATHCrossRefGoogle Scholar
  109. J.P. Lasalle, J. Diff. Eq. 4, 57-65 (1968) MATHCrossRefMathSciNetGoogle Scholar
  110. I.S. Labouriau, SIAM J., Math. Anal. 20, 1-12 (1989) MATHCrossRefMathSciNetGoogle Scholar
  111. F.H. Lopes da Silva, A. van Rotterdam, P Barts, E. van Heusden, W. Burr, Model of neuronal populations, the basic mechanism of rhythmicity, in Progress in brain research, edited by M.A. Corner, D.F. Swaab (Elsevier, Amsterdam) 45, 281-308 (1976) Google Scholar
  112. W.S. Mac Cullogh, W. Pitts, Bull. Math. Biophys. 5, 115-133 (1943) CrossRefMathSciNetGoogle Scholar
  113. R.S. MacKay, C. Tresser, Physica D 19, 206-237 (1986) MATHCrossRefADSMathSciNetGoogle Scholar
  114. C.M. Marcus, R.M. Westrevelt, Phys. Rev. A 40, 501-504 (1989) CrossRefADSGoogle Scholar
  115. J.E. Marsden, M. Mac Craken, in The Hopf Bifurcation and Its Applications (Springer-Verlag, New York; Heidelberg, Berlin, 1976) Google Scholar
  116. M.V. Mascagni, A.S. Sherman, Numerical methods for neuronal modeling, in edited by Methods in Neuronal Modeling Christof Koch Idan Segev (MIT Press, Cambridge, MA, 1998) Google Scholar
  117. M. Mézard, G. Parisi, M.A. Virasoro, Spin-glass Theory and Beyond (Singapore World Scientific, 1987) Google Scholar
  118. J. Milnor, Com. Math. Phys. 99, 177 (1985) MATHCrossRefADSMathSciNetGoogle Scholar
  119. L. Molgedey, J. Schuchardt, H.G. Schuster, Phys. Rev. Let. 69, 3717-3719 (1992) CrossRefADSGoogle Scholar
  120. C. Morris, H. Lecar, Biophys. J. 35, 193-213 (1981) Google Scholar
  121. E.F. Mishchenko, N.Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Translated from Russian by F.M.C. Goodspeed (Plenum, New York, 1980) Google Scholar
  122. J.S. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061-2070 (1962) CrossRefGoogle Scholar
  123. M. Nelson, Rinzel, J. The Hodgkin-Huxley model, in The Book of Genesis, edited by J.M. Bower, Beeman, Chap. 4 (Springer, New York, 1995), pp. 27-51 Google Scholar
  124. G. Parisi, J. Phys. A. 19, L675-680 (1988) Google Scholar
  125. H. Poincaré, Oeuvres complètes, Jacques Gabay Google Scholar
  126. M.W. Oram, M.C. Wiener, R. Lestienne, B.J. Richmond, J. Neurophysiol. 81, 3021-3033 (1999) Google Scholar
  127. M. Pollicott, Invent. Math. 81, 413-426 (1985); D. Ruelle, J. Diff. Geom. 25, 99-116 (1987) MATHCrossRefADSMathSciNetGoogle Scholar
  128. F. Rieke, D. Warland, R. de Ruyter van Steveninck, W. Bialek, Spikes – Exploring the Neural Code (MIT Press, Cambridge, MA, 1996) Google Scholar
  129. J. Rinzel, Excitation dynamics: insights from simplified membrane models. Federation Proc. 44, 2944-2946 (1985) Google Scholar
  130. J. Rinzel, G.B. Ermentrout, Analysis of neuronal excitability and oscillations, Methods in Neuronal Modeling, edited by C. Koch, I. Segev (MIT Press, Cambridge, MA, 1989) Google Scholar
  131. J. Rinzel, R. Miller, Math. Biosci. 49, 22-59 (1980) CrossRefMathSciNetGoogle Scholar
  132. D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory (Academic Press, 1989) Google Scholar
  133. D. Ruelle, F. Takens, Commun. Math. Phys. 20, 167-192 (1971) MATHCrossRefADSMathSciNetGoogle Scholar
  134. D. Ruelle, J. Stat. Phys. 95, 393-468 (1999) MATHCrossRefMathSciNetGoogle Scholar
  135. M. Samuelides, B. Doyon, B. Cessac, M. Quoy, Math. of Neural Networks, 312-317 (1996) Google Scholar
  136. O. Moynot, M. Samuelides, Probab. Theory Relat. Fields 123, 41-75 (2002) MATHCrossRefMathSciNetGoogle Scholar
  137. M. Quoy, Apprentissage dans les réseaux neuromimétiques à dynamique de base chaotique, Thèse ENSAE, Toulouse, 1994 Google Scholar
  138. M. Quoy, B. Doyon, M. Samuelides, Hebbian Learning in Discrete Time Chaotic Neural Networks (WCNN, Washington DC, 1995) Google Scholar
  139. M. Quoy, E. Daucé, Visual and motor learning using a chaotic recurrent neural network: application to the control of a mobile robot, in Neural Computation (Berlin, 2000) Google Scholar
  140. E. Daucé, M. Quoy, Random Recurrent Neural Networks for Autonomous System Design (SAB Paris, France, 2000) Google Scholar
  141. E. Daucé, M. Quoy, B. Doyon, Biol. Cybern. 87, 185-198 (2002) MATHCrossRefGoogle Scholar
  142. C.A. Skarda, W.J. Freeman, Behav. Brain Sci. 10, 161-195 (1987) Google Scholar
  143. M.N. Shadlen, W.T. Newsome, Curr. Opin. Neurobiol. 4, 569-579 (1994) CrossRefGoogle Scholar
  144. D. Sherrington, An introduction and overview is given of the theory of spin glasses and its application, cond-mat/9806289 (1998) Google Scholar
  145. C. Soulé, ComPlexUs, 1, 123-133 (2003) Google Scholar
  146. Ya.G. Sinai, Russ. Math. Surveys, 27, 21-69 (1972); D. Ruelle, Thermodynamic formalism (Addison-Wesley Reading, Massachusetts, 1978); R. Bowen, Lect. Notes. Math. 470 (Springer-Verlag, Berlin, 1975) Google Scholar
  147. D. Sherrington, S. Kirkpatrick, Phys. Rev. Let. 35, 1792 (1975) CrossRefADSGoogle Scholar
  148. S. Smale J. Math. Biol. 3, 5-7 (1976) Google Scholar
  149. H.L. Smith, SIAM Rev. 30, 87-113 (1988) MATHCrossRefMathSciNetGoogle Scholar
  150. W.R. Softky, Curr. Opin. Neurobiol. 5, 239-247 (1995) CrossRefGoogle Scholar
  151. H. Sompolinsky, A. Crisanti, H.J. Sommers, Phys. Rev. Lett. 61, 259-262 (1988) CrossRefADSMathSciNetGoogle Scholar
  152. R. Thomas, On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations, in Numerical Methods in the Study of Critical Phenomena, of Springer-Verlag in Synergetics 9, 180-193 (1981) Google Scholar
  153. D.J. Thouless, P.W. Anderson, R.J. Palmer, Philos. Mag. 35, 593-601 (1977) Google Scholar
  154. S. Thorpe, D. Fize, C. Marlot, Nature 381, 520-522 (1996) CrossRefADSGoogle Scholar
  155. M. Viana, Stochastic dynamics of deterministic systems Google Scholar
  156. G. Boffeta, G. Lacorata, S. Musacchio, A. Vulpiani, Chaos, 13, 806 (2003) and references therein Google Scholar
  157. R.F. Williams, Publ. Math. IHES 43, 169 (1974) Google Scholar
  158. M. Yoshioka, Chaos synchronization in gap-junction-coupled neurons, ArXiv nlin.CD/0505054 (2005) Google Scholar
  159. M. Samuelides, B. Cessac, Eur. Phys. J. Special Topics 142, 89-122 (2007) Google Scholar
  160. L. Perrinet, Eur. Phys. J. Special Topics 142, 163-225 (2007) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  • B. Cessac
    • 1
    • 2
    • 3
  • M. Samuelides
    • 4
  1. 1.INRIA, 2004 route des LuciolesSophia-AntipolisFrance
  2. 2.INLN, 1361 route des LuciolesValbonneFrance
  3. 3.Université de NiceNiceFrance
  4. 4.École Nationale Supérieure de l'Aéronautique et de l'espace and ONERA/DTIM, 2 Av. E. BelinToulouseFrance

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