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Collective properties of neural networks: A statistical physics approach. (English) Zbl 0561.92020
The collective properties of neural networks are studied by the application of general methods of statistical mechanics. In a situation like neural network, the construction of Hamiltonians is fraught with difficulties. One way is to simulate the equilibrium of large systems; the other consists in modelling the network as a stochastic process evolving in a Markov manner naturally the evolution equation where its general equilibrium solution offers a way of identifying the Hamiltonian. Several models in vogue in physics including the Ising models are reviewed and adapted to the case of assembly of neurons.
Reviewer: S.K.Srinivasan

92F05 Other natural sciences (mathematical treatment)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
92Cxx Physiological, cellular and medical topics
Full Text: DOI
[1] Amari, S.I.: Neural theory of association and concept formation. Biol. Cybern. 26, 175–185 (1977) · Zbl 0364.92016 · doi:10.1007/BF00365229
[2] Anninos, P.A., Beek, B., Csermely, T.J., Harth, E.H., Pertile, G.: Dynamics of neural structures. J. Theor. Biol. 26, 121–148 (1970) · doi:10.1016/S0022-5193(70)80036-4
[3] Binder, K.: In: Fundamental problems in statistical mechanics. Cohen, E.O.M. (ed.). Amsterdam: North-Holland 1981
[4] Braitenberg, V.: Cell assemblies in the cerebral cortex. In: Theoretical approaches to complex systems. Heim, R., Palm, G. (eds.), p. 171. Berlin, Heidelberg, New York: Springer 1978
[5] Caianiello, E.R., de Luca, A., Ricciardi, L.M.: Reverberations and control of neural network. Kybernetik 4, 10–18 (1967) · doi:10.1007/BF00288821
[6] Choi, M.Y., Huberman, B.A.: Digital dynamics and the simulation of magnetic systems. Phys. Rev. B28, 2547–2554 (1983) · doi:10.1103/PhysRevB.28.2547
[7] Cooper, L.N.: A possible organization of animal memory and learning. In: Collective properties of physical systems. Nobel Symp. 24, 252–264 (1973)
[8] Feldman, J.L., Cowan, J.D.: Large scale activity in neural theory with application to motoneuron pool responses. Biol. Cybern. 17, 29–38 (1975) · Zbl 0291.92015 · doi:10.1007/BF00326707
[9] Fukushima, K.: A model of associative memory in the brain. Kybernetik 12, 58–63 (1973) · doi:10.1007/BF00272461
[10] Glauber, R.J.: Time-dependent statistics of the Ising model. Phys. Rev. 4, 294–307 (1963) · Zbl 0145.24003
[11] Hebb, D.O.: The organization of behavior. New York: Wiley 1949
[12] Van Hemmen, J.L.: Classical spin-glass model. Phys. Rev. Lett. 49, 409–412 (1982) · doi:10.1103/PhysRevLett.49.409
[13] Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982) · Zbl 1369.92007 · doi:10.1073/pnas.79.8.2554
[14] Ill condensed matter: Les Houches summer school-session 1978. Balian, R., Maynard, R., Toulouse, G. (eds.) Amsterdam: North-Holland 1979
[15] Ingber, L.: Statistical mechanics of neocortical interactions. I. Basic formulation. Physica 5D, 83–107 (1982)
[16] Ingber, L.: Statistical mechanics of neocortical interactions. Dynamics of synaptic modifications. Phys. Rev. A8, 395–416 (1983) · doi:10.1103/PhysRevA.28.395
[17] Katz, B., Miledi, R.: A study of synaptic transmission in the absence of nerve impulses. J. Physiol. 192, 407–436 (1967)
[18] Kirkpatrick, S., Sherrington, D.: Infinite-ranged models of spinglasses. Phys. Rev. B17, 4384–4403 (1978) · doi:10.1103/PhysRevB.17.4384
[19] Little, W.A.: The existence of persistent states in the brain. Math. Biosci. 19, 101–120 (1974) · Zbl 0272.92011 · doi:10.1016/0025-5564(74)90031-5
[20] Little, W.A., Shaw, G.L.: Analytic study of the memory storage capacity of a neural network. Math. Biosci. 39, 281–290 (1978) · Zbl 0395.92005 · doi:10.1016/0025-5564(78)90058-5
[21] Mattis, P.C.: Solvable spin systems with random interactions. Phys. Lett. A56, 421–422 (1976) · doi:10.1016/0375-9601(76)90396-0
[22] Von Neuman, J.: The computer and the brain. New Haven, London: Yale University Press 1958 · Zbl 0085.14106
[23] Parasi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754–1756 (1979) · doi:10.1103/PhysRevLett.43.1754
[24] Pastur, L.A., Figotin, A.L.: Theory of disordered spin systems. Teor. Mat. Fiz. 35, 193–210 (1978) · Zbl 0433.35072
[25] Thompson, R.S., Gibson, W.G.: Neural model with probabilistic firing behavior. I. General considerations. Math. Biosci. 56, 239–253 (1981) · Zbl 0457.92003 · doi:10.1016/0025-5564(81)90056-0
[26] Thompson, R.S., Gibson, W.G.: Neural model with probabilistic firing behavior. II. One- and two-neuron networks. Math. Biosci. 56, 255–285 (1981) · Zbl 0479.92006 · doi:10.1016/0025-5564(81)90057-2
[27] Vannimenus, J., Maillard, J.P., De Seze, L.: Ground-state correlations in the two-dimensional Ising frustation model. J. Phys. C 12, 4523–4532 (1979) · doi:10.1088/0022-3719/12/21/019
[28] Willwacher, G.: Fahigkeiten eines assoziativen Speichersystems im Vergleich zu Gehirnfunktionen. Biol. Cybern. 24, 181–198 (1976) · Zbl 0335.92008 · doi:10.1007/BF00335979
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