We present an analytical approach that allows us to treat the long-time behavior of the recalling process in an oscillator neural network. It is well known that in coupled oscillatory neuronal systems, under suitable conditions, the original dynamics can be reduced to a simpler phase dynamics. In this description, the phases of the oscillators can be regarded as the timings of the neuronal spikes. To attempt an analytical treatment of the recalling dynamics of such a system, we study a simplified model in which we discretize time and assume a synchronous updating rule. The theoretical results show that the retrieval dynamics is described by recursion equations for some macroscopic parameters, such as an overlap with the retrieval pattern. We then treat the noise components in the local field, which arise from the learning of the unretrieved patterns, as gaussian variables. However, we take account of the temporal correlation between these noise components at different times. In particular, we find that this correlation is essential for correctly predicting the behavior of the retrieval process in the case of autoassociative memory. From the derived equations, the maximal storage capacity and the basin of attraction are calculated and graphically displayed. We also consider the more general case that the network retrieves an ordered sequence of phase patterns. In both cases, the basin of attraction remains sufficiently wide to recall the memorized pattern from a noisy one, even near saturation. The validity of these theoretical results is supported by numerical simulations. We believe that this model serves as a convenient starting point for the theoretical study of retrieval dynamics in general oscillatory systems.
Optimization and Stability of Heat Engines: The Role of Entropy Evolution
