In this paper we give under an appropriate theoretical framework a cha
racterization about neural networks (evolving in a binary set of state
s) which admit an energy. We prove that a neural network, iterated seq
uentially, admits an energy if and only if the weight verifies two con
ditions: the diagonal elements are non-negative and the associated inc
idence graph does not admit non-quasi-symmetric circuits. In this situ
ation the dynamics are robust with respect to a class of small changes
of the weight matrix. Further, for the parallel update we prove that
a necessary and sufficient condition to admit an energy is that the in
cidence graph does not contain non-quasi-symmetric circuits. (C) 1997
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