A statistical mechanics of an oscillator associative memory with scattered natural frequencies

We analyze an oscillator associative memory with scattered natural frequencies in memory retrieval states by a statistical mechanical method based on the SCSNA and the Sakaguchi-Kuramoto theory. The system with infinite stored patterns has a frustration on the synaptic weights. In addition, it is numerically shown that almost all oscillators synchronize under memory retrieval, but desynchronize under spurious memory retrieval, when setting optimal parameters. Thus, it is possible to determine whether the recalling process is successful or not using information about the synchrony/asynchrony. The solvable toy model presented in this paper may be a good candidate for showing the validity of the synchronized population coding in the brain proposed by Phillips and Singer. PACS numbers: 87.10.+e, 05.90.+m, 89.70.+c Typeset using REVTEX 1 Recently, oscillator neural networks have been attracting the attention of a growing number of researchers. Oscillator neural network models have the ability of flexible information processing compared to conventional models based on rate coding. This is because in oscillator models information is represented by many degrees of freedoms. Thus, a number of neural network models based on oscillatory activities are proposed in an engineering context. On the other hand, oscillator networks also have been attracting the attention of physicists for analogies of spin systems. In general, when the coupling among oscillators is sufficiently weak, the high-dimensional dynamics of a coupled oscillator system can be reduced to a phase description. There is a close analogy between the phase description of coupled oscillators and XY-spin models of magnetic materials. Therefore, we can discuss this system in the context of a statistical mechanism. There are three models of oscillator networks that are closely related to our investigation. Sakaguchi and Kuramoto [5] theoretically analyzed the mutual entrainment of uniformly coupled oscillators with scattered natural frequencies. Their model corresponds to ferromagnetic models of magnetic materials. Daido [6] numerically analyzed quasi-entrainment of randomly coupled oscillators with scattered natural frequencies. His model corresponds to Sherrigtion-Kirkpatrick model of spin glasses [3]. On the other hand, Cook’s model [4] is an associative memory with oscillatory elements with uniform natural frequencies, which is a class of expansions of the Hopfield model. Cook derived the model’s memory capacity by using the Replica theory. The latter two models have a frustration owing to a randomness of the coupling among oscillators. In this paper, we analyze an oscillator associative memory with scattered natural frequencies in memory retrieval states by a statistical mechanical method based on the SCSNA [10] and the Sakaguchi-Kuramoto theory [5]. This model is a non-equilibrium system, because asynchronous oscillators exist in the large population of oscillators. In this case, we can not define the Lyapunov function with “bottoms”. Instead, we discuss this system in the framework of the SCSNA and the Sakaguchi-Kuramoto theory, which can be applied to a non-equilibrium system. In the finite loading case, our theory coincides with the Sakaguchi-Kuramoto theory. When all oscillators have uniform natural frequencies, our theory is reduced to the previously proposed theories based on the SCSNA or the Replica method. There are three important properties of the Hopfield model [1,2]: storage capacity, the basin of attraction, and the existence of spurious memories. A serious problem in attractor type networks is avoiding being trapped in spurious equilibrium states in a relaxation process. The Lyapunov function can not be defined in the oscillator associative memory with scattered natural frequencies. When the variance of natural frequencies is sufficiently small, we can expect that all oscillators mutually synchronize in memory retrieval states based on the analogy of a behavior of the ferromagnetic model (uniformly coupled oscillators with scattered natural frequencies) [5]. In this case, an effective Lyapunov function exists in the system when the memory recalling process is successful. However, we can expect that oscillators desynchronize in spurious states from the analogy of the glassy oscillator model’s behavior (randomly coupled oscillators with scattered natural frequencies) [6]. In this case, an effective Lyapunov function does not exist when the memory recalling process is unsuccessful. Therefore, the existence of the effective Lyapunov function only when memory recalling process is successful is analogous with that of the associative memory model with 2 the nonmonotonic neurons [7]. In this paper, we numerically show the possibility of determining if the recalling process is successful or not using information about the synchrony/asynchrony. It is biologically plausible to assume that the natural frequencies of oscillatory activities are randomly distributed over the whole population, because each neuron has “individuality” in the real brain. In general, when the coupling is sufficiently weak, the high-dimensional dynamics of a coupled oscillator system can be reduced to the phase equation [8]. Let us consider the following simplified model, dφi dt = ωi + N