Frustration, stability, and delay-induced oscillations in a neural network model. (English) Zbl 0840.92003

Bélair, Jacques; Campbell, Sue Ann; van den Driessche, P.

SIAM J. Appl. Math. 56, No. 1, 245-255 (1996).

Summary: The effect of time delays on the linear stability of equilibria in an artificial neural network of Hopfield type is analyzed. The possibility of delay-induced oscillations occurring is characterized in terms of properties of the (not necessarily symmetric) connection matrix of the network. Such oscillations are possible exactly when the network is frustrated, equivalently when the signed digraph of the matrix does not require the Perron property. Nonlinear analysis (centre manifold computation) of a three-unit frustrated network is presented, giving the nature of the bifurcations taking place. A supercritical Hopf bifurcation is shown to occur, and a codimension-two bifurcation is unfolded.

MSC:

92B20	Neural networks for/in biological studies, artificial life and related topics
34K11	Oscillation theory of functional-differential equations
34K18	Bifurcation theory of functional-differential equations
15B48	Positive matrices and their generalizations; cones of matrices
05C20	Directed graphs (digraphs), tournaments
68T05	Learning and adaptive systems in artificial intelligence

Keywords:

centre manifold computation; time delays; linear stability of equilibria; neural network of Hopfield type; delay-induced oscillations; connection matrix; Perron property; three-unit frustrated network; supercritical Hopf bifurcation; codimension-two bifurcation