A simple model of coupled dynamics of fast neurons and slow interactions, modelling self-organization in recurrent neural networks, leads naturally to an effective statistical mechanics characterized by a partition function which is an average over a replicated system. This is reminiscent of the replica trick used to study spin-glasses, but with the difference that the number of replicas has a physical meaning as the ratio of two temperatures and can be varied throughout the whole range of real values. The model has interesting phase consequences as a function of varying this ratio and external stimuli, and can be extended to a range of other models. As the basic archetypal model we consider a system of Ising spin neurons (J'i E {-I, I}, i E {I,..., N}, interacting via continuous-valued symmetric interactions, Iij, which themselves evolve in response to the states of the neurons. JijO"iO"j (2) i j and the subscript {Jij} indicates that the {Jij} are to be considered as quenched variables.

A.C.C. Coolen R.W. Penney D. Sherrington Dept. of Physics - Theoretical Physics University of Oxford 1 Keble Road, Oxford OXI 3NP, U.K. Abstract A simple model of coupled dynamics of fast neurons and slow interactions, modellingself-organization in recurrent neural networks, leads naturally to an effective statistical mechanics characterized by a partition function which is an average over a replicated system. This is reminiscent of the replica trick used to study spin-glasses, but with the difference that the number of replicas has a physical meaningas the ratio of two temperatures and can be varied throughout the whole range of real values. The model has interesting phaseconsequences as a function of varying this ratio and external stimuli, and can be extended to a range of other models. 1 A SIMPLE MODEL WITH FAST DYNAMIC NEURONS AND SLOW DYNAMIC INTERACTIONS As the basic archetypal model we consider a system of Ising spin neurons (J'i E {-I, I}, i E {I, ..., N}, interacting via continuous-valued symmetric interactions, Iij, which themselves evolve in response to the states of the neurons. The neurons are taken to have a stochastic field-alignment dynamics which is fast compared with the evolution rate of the interactions hj, such that on the timescale of Iii-dynamics the neurons are effectively in equilibrium according to a Boltzmann distribution, (1) 447 448 Cooien, Penney, and Sherrington where HVoj}({O"d) JijO"iO"j (2) i j and the subscript {Jij} indicates that the {Jij} are to be considered as quenched variables. In practice, several specific types of dynamics which obey detailed balance lead to the equilibrium distribution (1), such as a Markov process with single-spin flip Glauber dynamics [1]. The second term acts to limit the magnitude of hj; f3 is the characteristic inverse temperature of the interaction system. VNTJij(t) (4) where the effective Hamiltonian 11. ({ hj}) is given by 1 1 We now recognise (4) as having the form of a Langevin equation, so that the equilibrium distributionof the interaction system is given by a Boltzmann form. Z{3 (6) Coupled Dynamics of Fast Neurons and Slow Interactions 449 where n _ /j3. We may use Z as a generating functional to produce thermodynamic averagesof state variables I ( {O"d; {Jij}) in the combined system by adding suitable infinitesimal source terms to the neuron Hamiltonian (2): HP.j}({O"d) In fact, any real n is possible by tuning the ratio between the two {3's. In the formulation presented in this paper n is always nonnegative, but negative values are possible if the Hebbian rule of (3) is replaced by an anti-Hebbian form with (UiO"j) replaced by - (O"iO"j) (the case of negative n is being studied by Mezard and coworkers [7]).

A simple model of coupled dynamics of fast neurons and slow interactions, modelling self-organization in recurrent neural networks, leads naturally to an effective statistical mechanics characterized by a partition function which is an average over a replicated system. This is reminiscent of the replica trick used to study spin-glasses, but with the difference that the number of replicas has a physical meaning as the ratio of two temperatures and can be varied throughout the whole range of real values. The model has interesting phase consequences as a function of varying this ratio and external stimuli, and can be extended to a range of other models. As the basic archetypal model we consider a system of Ising spin neurons (J'i E {-I, I}, i E {I,..., N}, interacting via continuous-valued symmetric interactions, Iij, which themselves evolve in response to the states of the neurons. JijO"iO"j (2) i j and the subscript {Jij} indicates that the {Jij} are to be considered as quenched variables.