Methods for constructing limit cycles in oscillatory neural networks

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(1)Expressing a two-dimensional state by a complex variable z=x+iy and describing a dynamical system asdz(t)/dt=f(z(t))the problem of constructing a given set of limit cycles in the dynamical system was studied. However it turns out that this problem seems much more complicated than expected and satisfactory results have been obtained. However various difficulties and important observations on the problem have been clarified and they would be useful for future study on this problem.(2)Let us consider a neural network consisting of many oscillatory neurons with a single stable limit point. It has been pointed out that if all the frequencies of limit cycles are set to equal then an associative memory can be implemented by storing information in their phase differences. The present investigators studied various associative memories using discrete type neural networks in some detail, and analogous to these it is examined by various numerical experiments that most of associative memories using discrete type neural networks can be realized. However it was seen that since each phase can take continuous values a neural network consisting of oscillatory neurons could store much more information than discrete type neurons.(3)We also studied neural networks consisting of many oscillatory neurons with different frequencies of limit cycles. Although it has been also pointed out that the network is eventually decomposed into various groups according to their frequencies so that each group has the common frequency, its detailed behavior has not investigated but computer simulations show various interesting facts.