Wandering motion and cooperative phenomena in a chaotic neural network
M. Makishima and T. Shimizu
Graduate School of Engineering, Kokushikan University, Tokyo 154, Japan
As a model of associative memory, Hopfield proposed the spin-glass model. The model has the advantage that the total energy decreases monotonically . At the same time, however, the model has the difficulty that the system cannot escape from the spurious minimum if the system falls into it. To remove the difficulty several methods are proposed, for example, the stochastic method or the chaotic method.
In this paper we propose a new model for associative memory with discrete time, in which the network itself can search for minima of the energy successively by using the wandering motion and cooperative phenomena.
In this model the state of neuron i 
of the
consitituents of the network is specified by both the core variable
and the memory variable 
at step n. The core
variable is generated by using a map f(x,a) :
, where 
is a bifurcation
parameter at step n. The memory variable is the quantity
which describes the past history for the input information of neuron i.
Then the output 
of neuron i is defined by
. The memory variable is
determined by
, where
I is the external input, 
is the threshold, 
denotes the coupling constant between
neuron i and neuron j and 
is the decay constant. The
bifurcation parameter of the map f(x,a) is defined in terms
of by 
, where A, B
and C are constants. If the system once falls into some minimum of the
energy, the output remains constant and so will
increase or decrease monotonically. Therefore we introduce the threshold
with respect to 
to escape from the minimum. If
, the magnitude and sign of at step
n is updated according to 
,
where D is a constant.
By using this model we can clearly discuss the mechanism or the dynamics
of the wandering motion and cooperative phenomena in a chaotic neural
network to search for minima of the energy. We have applied the above
network to the problem of associative memory, where the system has
neurons and 3 patterns are stored. The network could retrieve all of
stored patterns successively. The ratio of correct retrieval was more
than 
. The neural network was also applied to TSP with 
neurons. The network could find the shortest route and the other very
quickly.