A discrete time neural network model with spiking neurons

1. Bak, P.: How Nature Works: the Science of Seld-Organized Criticality. Springer (1996), Oxford University Press (1997)

2. Blanchard Ph., Cessac B. and Krüger T. (1997). A dynamical system approach to SOC models of Zhang’s type. J. Stat. Phys. 88: 307–318

   Article  MATH  Google Scholar 

3. Blanchard Ph., Cessac B. and Krüger T. (2000). What can one learn about self-organized criticality from dynamical system theory?. J. Stat. Phys. 98: 375–404

   Article  MATH  Google Scholar 

4. Cessac B., Blanchard Ph., Krüger T. and Meunier J.L. (2004). J. Stat. Phys. 115(516): 1283–1326

   Article  Google Scholar 

5. Brunel N. and Hakim V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11: 1621–1671

   Article  Google Scholar 

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

   Article  MATH  MathSciNet  Google Scholar 

7. Cessac B. (1994). Occurence of chaos and AT line in random neural networks. Europhys. Lett. 26(8): 577–582

   Article  Google Scholar 

8. Cessac B. (1995). Increase in complexity in random neural networks. J. de Physique I (France) 5: 409–432

   Article  Google Scholar 

9. Cessac, B., Mazet, O., Samuelides, M., Soula, H.: Mean field theory for random recurrent spiking neural networks. NOLTA’05 (Non Linear Theory and its Applications) October 18–21, 2005, Brugge, Belgium

10. Cessac, B., Samuelides, M.: From Neuron to Neural Networks dynamics. EPJ Special Topics Topics in Dynamical Neural Networksá: From Large Scale Neural Networks to Motor Control and Vision (2007)

11. Cessac B. and Sepulchre J.A. (2004). Stable resonances and signal propagation in a chaotic network of coupled units. Phys. Rev. E 70: 056111

    Article  Google Scholar 

12. Cessac B. and Sepulchre J.A. (2006). Transmitting a signal by amplitude modulation in a chaotic network. Chaos 16: 013104

    Article  MathSciNet  Google Scholar 

13. Cessac, B., Sepulchre, J.A.: Linear Response in a class of simple systems far from equilibrium. Physica D 225(1): 13–28

14. Cessac, B., Viéville, T.: Biological plausibility of discrete time spiking neural networks (submited, 2007)

15. Cessac, B., Touboul, J.: A discrete time neural network model with spiking neurons: the case of random synapses (in preparation)

16. Chernov, N., Markarian, R.: Chaotic Billiards. American Mathematical Society, Providence (2006)

17. Coutinho R., Fernandez B., Lima R. and Meyroneinc A. (2006). Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52: 524–570

    Article  MATH  MathSciNet  Google Scholar 

18. Dauce E., Quoy M., Cessac B., Doyon B. and Samuelides M. (1998). Self- Organization and Dynamics reduction in recurrent networks: stimulus presentation and learning. Neural Netw 11: 521–533

    Article  Google Scholar 

19. Derrida B. (1987). Dynamical phaste transition in non-symmetric spin glasses. J. Phys. A Math. Gen. 20: 721–725

    Article  MathSciNet  Google Scholar 

20. Derrida B. and Flyvbjerg H. (1986). Multivalley structure in Kauffman’s model: analogy with spin glasses. J. Phys. A19: L1003–L1008

    Google Scholar 

21. Derrida B. and Pomeau Y. (1986). Random networks of automata: a simple annealed approximation. Europhys. Lett. 1: 45–49

    Article  Google Scholar 

22. Eckmann J.P. and Ruelle D. (1985). Ergodic Theory of Strange attractors. Rev. Mod. Phys. 57: 617

    Article  MathSciNet  Google Scholar 

23. Farcot, E.: Etude d’une classe d’équations différentielles affines par morceaux modélisant des réseaux de régulation biologique. Thèse de Doctorat, Grenoble, France (2005)

24. Faugeras, O., Papadopoulo, T., Touboul, J., Bossy, M., Tanre, E., Talay, D.: The Statistics of Spikes Trains for Some Simple Types of Neuron Models. In: Proceedings of the NeuroComp 2006 Conference, Pont-à-Mousson, France (2006)

25. Freidlin M.I. and Wentzell A.D. (1998). Random Perturbations of Dynamical Systems. Springer, New York

    MATH  Google Scholar 

26. Gerstner W. and Kistler W.M. (2002). Spiking Neuron Models. Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge

    MATH  Google Scholar 

27. Guckenheimer J. and Holmes Ph. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer, Heidelberg

    Google Scholar 

28. Guikhman, I., Skorhokod, A.: Introduction à la théorie des processus aléatoires. Editions Mir, Moscou (1980)

29. Hodgkin A.L. and Huxley A.F. (1952). Current carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol. (Lond.) 116: 449–472

    Google Scholar 

30. Hodgkin A.L. and Huxley A.F. (1952). A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. J. Physiol. (Lond.) 117: 500–544

    Google Scholar 

31. Hunt B.R., Sauer T. and Yorke J.A. (1992). Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces. Bull. Am. Math. Soc. 27: 217–238

    MATH  MathSciNet  Google Scholar 

32. Izhikevich E.M. (2003). Simple model of spiking neurons. IEEE Trans Neural Netw 14: 1569–572

    Article  Google Scholar 

33. Kauffman, S.A.: J. Theor. Biol. 22 (1969)

34. Katok A. and Hasselblatt B. (1998). Introduction to the Modern Theory of Dynamical Systems. Kluwer, Dordrecht

    Google Scholar 

35. Keller G. (1998). Equilibrium states in ergodic theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

36. Kruglikov, B., Rypdal, M.: Entropy via multiplicity. arXivmath.DS/0505019

37. Kruglikov, B., Rypdal, M.: A piece-wise affine contracting map with positive entropy. arXiv math.DS/0504187

38. Langton C.G. (1990). Computation at the edge of chaos. Physica D 42: 12–37

    Article  MathSciNet  Google Scholar 

39. Lima R. and Ugalde E. (2006). Dynamical complexity of discrete time regulatory networks. Nonlinearity 19: 237–259

    Article  MATH  MathSciNet  Google Scholar 

40. Maass W. and Bishop C. (1998). Pulsed Neural Networks. MIT Press, Cambridge

    MATH  Google Scholar 

41. Mézard M., Parisi G. and Virasoro M.A. (1987). Spin-glass theory and beyond. World scientific, Singapore

    MATH  Google Scholar 

42. Packard, N.: Adaptation towards the edge of chaos. In: Kelso, J.A.S, Mandell, A.J., Shlesinger, M.F. (eds.) Dynamic Patterns in Complex Systems. pp. 293–301. World Scientific, Singapore (1988)

43. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Collection Asterisque 187–188, Société Mathématique de France (1990)

44. Rieke, F., Warland, D., de Ruyter van Steveninck, R., Bialek, W.: Spikes: exploring the neural code. MIT Press (1999)

45. Roxin N.B. and Hansel D. (2005). The role of delays in shaping the spatio-temporal dynamics of neuronal activity in large networks. Phys. Rev. Lett. 94: 238103

    Article  Google Scholar 

46. Rudolph M. and Destexhe A. (2006). Analytical integrate and fire neuron models with conductance-based dynamics for event driven simulation strategies. Neural Comput. 18: 2146–2210

    Article  MATH  MathSciNet  Google Scholar 

47. Sinai Ya.G. (1972). Gibbs measures in ergodic theory. Russ. Math. Surv. 27(4): 21–69

    Article  MathSciNet  Google Scholar 

48. Ruelle D. (1978). Thermodynamic Formalism. Addison-Wesley, Reading

    MATH  Google Scholar 

49. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes. in Math., vol. 470. Springer, Berlin (1975)

50. Sherrington, D.: An introduction and overview is given of the theory of spin glasses and its application. cond-mat/9806289

51. Sherrington D. and Kirkpatrick S. (1975). Solvable model of spin glass. Phys. Rev. Lett. 35(26): 1792

    Article  Google Scholar 

52. Samuelides, M., Cessac, B.: Random recurrent neural Networks. review paper, to appear in, topics in neural networks. Eur. Phys. J. (topical issue), Springer, Heidelberg (2006)

53. Schneidman, E., Berry, M.J., Segev, R., Bialek, W.: Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 04701, vol. 440 (2006)

    Google Scholar 

54. Soula, H.: Dynamique et plasticité dans les réseaux de neurones à impulsions. thèse de doctorat, INSA Lyon (2005)

55. Soula H., Beslon G. and Mazet O. (2006). Spontaneous dynamics of assymmetric random recurrent spiking neural networks. Neural Comput. 18: 1

    Article  MathSciNet  Google Scholar 

Download references