Neural Networks (1998) 11:495-508

Eugene M. Izhikevich

Systems Science Center, Box 7606,
Arizona State University,
Tempe, AZ 85287-7606.

Abstract. The cusp bifurcation provides one of the simplest routes leading to bistability and hysteresis in neuron dynamics. We show that weakly connected networks of neurons near cusp bifurcations that satisfy a certain adaptation condition have quite interesting and complicated dynamics. First we prove that any such network can be transformed into a canonical model by an appropriate continuous change of variables. Then we show that the canonical model can operate as a multiple attractor neural network or as a globally asymptotically stable neural network depending on the choice of parameters.

Keywords: Weakly connected neural networks, multiple cusp bifurcations, multiple pitchfork bifurcations, canonical models, Hebbian learning, bistability of perception

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