Neural Computation, Vol 10, 1527-1546, Copyright © 1998 by The MIT Press
Retrieval Dynamics in Oscillator Neural Networks
Toshio Aoyagi and Katsunori Kitano
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.
