Boundary conditions and phase transitions in neural networks. Theoretical results

Neural Networks

Volumn 21, Issue 7, 2008, Pages 971-979

  Demongeot, Jacques ^a   Jézéquel, Christelle ^a   Sené, Sylvain ^a,b  

^a LABORATOIRE TIMC IMAG   (France)

^b ENS Lyon   (France)

Author keywords

Hopfield model extension; Phase transitions; Stochastic neural networks

Indexed keywords

ASYMPTOTIC ANALYSIS; BOUNDARY CONDITIONS; ITERATIVE METHODS; NUMERICAL METHODS; PHASE TRANSITIONS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS;

ASYMPTOTIC BEHAVIOUR; ASYMPTOTIC DYNAMICS; BOUNDARY INFLUENCES; FIXED BOUNDARY CONDITIONS; HOPFIELD MODELS; MATHEMATICAL APPROACH; STOCHASTIC NEURAL NETWORK; THEORETICAL STUDY;

HOPFIELD NEURAL NETWORKS;

ARTICLE; ARTIFICIAL NEURAL NETWORK; BEHAVIOR; DYNAMICS; HYPOTHESIS; MATHEMATICAL ANALYSIS; PHASE TRANSITION; PRIORITY JOURNAL; THEORETICAL MODEL;

EID: 51049084861     PISSN: 08936080     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.neunet.2008.04.003     Document Type: Article

References (47)

1. Albert R., and Barabási A.-L. Statistical mechanics of complex networks. Reviews of Modern Physics 74 1 (2002) 47-97
2. Amit D.J., Gutfreund H., and Sompolinsky H. Spin-glass models of neural networks. Physical Review A 32 2 (1985) 1007-1018
3. Aracena J., González M., Zuñiga A., Mendez M.A., and Cambiazo V. Regulatory network for cell shape changes during drosophila ventral furrow formation. Journal of Theoretical Biology 239 (2006) 49-62
4. Ayoubi R.A., Ziade H.A., and Bayoumi M.A. Fault tolerant Hopfield associative memory on torus. Proceedings of the 18th IEEE international symposium on defect and fault tolerance in VLSI systems (2003), IEEE Computer Society 369 (8 pages)
5. Bahi J.M., and Contassot-Vivier S. Stability of fully asynchronous discrete-time discrete-state dynamic networks. IEEE Transactions on Neural Networks 13 6 (2002) 1353-1363
6. Barabási A.-L., and Oltvai Z.N. Network biology: Understanding the cell's functional organization. Nature Reviews Genetics 5 (2004) 101-113
7. Cangiano, L. (2004). Mechanisms of rhythm generation in the Lamprey locomotor network. Ph.D. thesis. Karolinska Institute, Stockholm
8. Crombez G. Finding common fixed points of a class of paracontractions. Acta Mathematica Hungarica 103 3 (2004) 233-241
9. 2. Proceedings of the symposium on methods in the study of critical phenomena. Springer series in synergetics Vol. 9 (1981), Springer 254-267
10. Demongeot J. Coupling of Markov processes and Holley's inequalities for Gibbs measures. Transactions of the IXth Prague conference on information theory, statistical decision functions and random processes. Transactions of the Prague conference on information theory Vol. 9A (1983), Springer 183-189
11. Demongeot J., Benaouda D., and Jézéquel C. "Dynamical confinement" in neural networks and cell cycle. Chaos 5 1 (1995) 167-173
12. Demongeot J., Benaouda D., Nérot O., and Jézéquel C. Random simulation and confiners: Their application to neural networks. Acta Biotheoretica 42 2-3 (1994) 203-213
13. Demongeot, J., & Sené, S. (2008). Boundary conditions and phase transitions in neural networks. Simulation results. Neural Networks. doi:10.1016/j.neunet.2008.04.005
14. Dobrushin R.L. The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Functional Analysis and Its Applications 2 4 (1968) 302-312
15. Dobrushin R.L., Kolafa J., and Shlosman S.B. Phase diagram of the two-dimensional Ising antiferromagnet (computer-assisted proof). Communications in Mathematical Physics 102 1 (1985) 89-103
16. Fatès N., and Morvan M. An experimental study of robustness to asynchronism for elementary cellular automata. Complex systems 16 1 (2005) 1-27
17. Forest L., Glade N., and Demongeot J. Liénard systems and potential-Hamiltonian decomposition - applications in biology. Comptes Rendus de l'Académie des Sciences Biologies 330 3 (2007) 1-10
18. François O. Ergodicité des Processus Neuronaux. Comptes Rendus de l'Académie des Sciences Paris, série 1 310 6 (1990) 435-440
19. François O., Demongeot J., and Hervé T. Convergence of a self-organizing stochastic neural network. Neural Networks 5 2 (1992) 277-282
20. Gertsik V.M., and Dobrushin R.L. Gibbs states in lattice models with two-step interaction. Functional Analysis and its Applications 8 3 (1974) 201-211
21. Gjuvsland A.B., Hayes B.J., Stig W., Omholt S.W., and Carlborg O. Statistical epistasis is a generic feature of gene regulatory networks. Genetics 175 (2007) 411-420
22. Grillner S. The motor infrastructure: From ion channels to neuronal networks. Nature Reviews Neuroscience 4 (2003) 573-586
23. Hopfield J.J. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences of the United States of America 79 8 (1982) 2554-2558
24. Huberman B.A., and Hogg T. Phase transitions in artificial intelligence systems. Artificial Intelligence 33 2 (1987) 155-171
25. Huepe C., and Aldana-González M. Dynamical phase transition in a neural network model with noise: An exact solution. Journal of Statistical Physics 108 3-4 (2002) 527-540
26. Ising E. Beitrag zur Theorie des Ferromagnetismus. Zeitschrift fur Physics 31 (1925) 253-258
27. Jézéquel C., Nérot O., and Demongeot J. "Dynamical confinement" in neural networks. Journal of Biological Systems 3 4 (1995) 1157-1165
28. Kaneko K., and Akutsu Y. Phase transitions in two-dimensional stochastic cellular automata. Journal of Physics A: Mathematical and General 19 2 (1986) 69-75
29. Kinzel W. Phase transitions of cellular automata. Zeitschrift für Physik B Condensed Matter 58 3 (1985) 229-244
30. Kryukov, V. I., & Kirillov, A. B. (1989). Phase transitions and metastability in neural nets. In Proceedings of the IEEE international joint conference on neural networks (pp. 761-766). Vol. 1
31. Kürten, K. E. (1987). Phase transitions in quasirandom neural networks. In Proceedings of the IEEE international conference on neural networks (pp. 197-204). Vol. 2
32. Luque B., and Solé R.V. Phase transitions in random networks: Simple analytic determination of critical points. Physical Review E 55 1 (1997) 257-260
33. Martinelli F. On the two-dimensional dynamical ising model in the phase coexistence region. Journal of Statistical Physics 76 5-6 (1994) 1179-1246
34. McFadden F.E., Peng Y., and Reggia J.A. Local conditions for phase transitions in neural networks with variable connection strengths. Neural Networks 6 5 (1993) 667-676
35. Müller T.H., Swandulla D., and Zeilhofer H.U. Synaptic connectivity in cultured hypothalamic neuronal networks. Journal of Neurophysiology 77 6 (1997) 3218-3225
36. Peretto P. Collective properties of neural networks: A statistical physics approach. Biological Cybernetics 50 1 (1984) 51-62
37. Picco P. Artificial neural networks. A review from physical and mathematical points of view. Annales de l'institut Henri Poincaré (A) Physique théorique 64 3 (1996) 289-307
38. Preston C.J. Gibbs states on countable sets (1974), Cambridge University Press, Cambridge
39. Reggia J.A., and Edwards M. Phase transitions in connectionist models having rapidly varying connection strengths. Neural Computation 2 4 (1990) 523-535
40. Regnault D., Schabanel N., and Thierry E. Progresses in the analysis of stochastic 2D cellular automata: A study of asynchronous 2D minority. Proceedings of the international symposium on mathematical foundations of computer science. Lecture notes in computer science Vol. 4708 (2007), Springer 320-332
41. Ruelle D. Statistical mechanics: Rigourous results (1969), W. A. Benjamin
42. Ruelle D. Thermodynamic formalism: The mathematical structure of equilibrium statistical mechanics (1984), Cambridge Mathematical Library
43. Shrager J., Hogg T., and Huberman B.A. Observation of phase transitions in spreading activation networks. Science 236 4805 (1987) 1092-1094
44. Solé, R. V., Luque, B., & Kauffman, S. (2000). Phase transitions in random networks with multiple states. Technical Report 00-02-011. Santa Fe Institute
45. van Hemmen J.L. Spin-glass models of neural networks. Physical Review A 34 4 (1986) 3435-3445
46. Weisbuch G., and Stauffer D. Phase transtition in cellular random Boolean nets. Journal de Physique 48 1 (1987) 11-18
47. Xu X., and Tsai W.T. Constructing associative memories using neural networks. Neural Networks 3 3 (1990) 301-309