This chapter describes Complex Hopfield Neural Network (CHNN), a complex-variable version of the Hopfield neural network, which can exist in both fixed point and oscillatory modes. Memories can be stored by a complex version of Hebbs rule. In the fixed-point mode, CHNN is similar to a continuous-time Hopfield network. In the oscillatory mode, when multiple patterns are stored, the network wanders chaotically among patterns. Presence of chaos in this mode is verified by appropriate time series analysis. It is shown that adaptive connections can be used to control chaos and increase memory capacity. Electronic realization of the network in oscillatory dynamics, with fixed and adaptive connections shows an interesting tradeoff between energy expenditure and retrieval performance. It is shown how the intrinsic chaos in CHNN can be used as a mechanism for annealing when the network is used for solving quadratic optimization problems. The networks applicability to chaotic synchronization is described.
It has been shown that by extending Hopfield’s real-valued model to complex –variable domain, it is possible to preserve the symmetric Hebbian synapses, while permitting the network to have oscillatory states (Chakravarthy & Ghosh, 1996). Pioneering work on complex-valued versions of Hopfield network was done by Hirose (1992). Other studies in the area of complex neural networks include complex backpropagation algorithm for training complex feedforward networks (Leung & Haykin, 1991; Nitta, 1997) and a similar extension for complex-valued recurrent neural networks (Mandic & Goh, 2004). For a comprehensive review of complex neural models the reader may consult (Hirose, 2003).